From ad7e3079d4dabbd043c2696fe029fb04f7e4395f Mon Sep 17 00:00:00 2001 From: Meta Vens Date: Sat, 3 Oct 2026 21:41:08 +0200 Subject: [PATCH 01/31] Affine forms: math.Affine.cs (af_* ops, af_snoise, af_fbm, af_frustum_*), SNOISE2_LIPSCHITZ and iv_snoise_ball(float2), AffineTests Co-Authored-By: Claude Opus 5.5 --- packages/cultmath/src/CultMath/math.Affine.cs | 224 +++++++++ .../cultmath/src/CultMath/math.Affine.cs.meta | 2 + .../cultmath/src/CultMath/math.Interval.cs | 21 + .../tests/CultMath.Tests/AffineTests.cs | 458 ++++++++++++++++++ .../tests/CultMath.Tests/NoiseBoundTests.cs | 2 +- 5 files changed, 706 insertions(+), 1 deletion(-) create mode 100644 packages/cultmath/src/CultMath/math.Affine.cs create mode 100644 packages/cultmath/src/CultMath/math.Affine.cs.meta create mode 100644 packages/cultmath/tests/CultMath.Tests/AffineTests.cs diff --git a/packages/cultmath/src/CultMath/math.Affine.cs b/packages/cultmath/src/CultMath/math.Affine.cs new file mode 100644 index 00000000..1d2dd70b --- /dev/null +++ b/packages/cultmath/src/CultMath/math.Affine.cs @@ -0,0 +1,224 @@ +namespace CultMath; + +public static partial class math +{ + // Reduced affine forms with one shared symbol (design.md, "Affine forms"): float3(x0, a, e) is the + // set { x0 + a eps + e delta : delta in [-1, 1] } at the region's shared eps in [-1, 1], with + // e >= 0. Every form over one region shares that eps (for a screen-tile slice, the depth + // parameter), and delta is private to the form. This is Messine's AF1 as Gamito and Maddock use it + // for gradient noise (t0 + t1 e1 + t2 e2), in one register, as an interval is a float2. + // + // Every af_* op returns a form whose set, at each eps, contains f evaluated pointwise in float32 at + // every point of its operands' sets at that same eps. Rounding rule: each op folds its own float32 + // rounding into e as (S) 2^-20, where S is the sum of the magnitudes the op's roundings are + // relative to (named in each comment). With u = 2^-24, every op below needs at most 8u S (its + // component roundings, the pointwise result's own rounding, and the roundings of computing e), so + // 2^-20 = 16u covers it twice over; the factor of two is the allowance for a device whose + and * + // are faithful (within one ulp) rather than correctly rounded, and fma contraction only removes + // roundings. Products also add 2^-126, the most any underflow to a subnormal can lose. The + // enclosure tests evaluate the form's set in double and f in float32 with no tolerance + // (AffineTests.EveryOpEnclosesItsPoints). No op takes a compound type or a reference parameter: each is a + // plain function of float values, so it mirrors into the HLSL common subset. + + // An upper bound on the spectral norm of the Hessian of snoise(float3). Provenance: + // NoiseBoundTests.MeasureHessian (slow, explicit) takes the largest Hessian norm (central + // differences of snoise_grad, largest absolute eigenvalue) over 1e6 seeded points, refines the 1e4 + // largest by ascent, and multiplies the refined maximum by 1.10; + // NoiseBoundTests.HessianConstantPinsSampledCurvature pins this value against a fast re-measure and + // the ascent from MeasureHessian's witness. Empirical with a margin, not a proof; the enclosure + // tests are the defence. A change to the snoise kernel must re-run MeasureHessian and re-pin it. + public const float SNOISE_HESSIAN = 68.94f; + + /// The point form (x, 0, 0): the value x at every eps. Exact. + public static float3 af_point(float value) => new(value, 0.0f, 0.0f); + + /// + /// The shared symbol itself, (x0, a, 0): the value x0 + a eps. It is how a region's parameter enters + /// (for a tile slice, depth z = z_m + h eps), so it is exact by definition: eps is defined through it. + /// + public static float3 af_symbol(float x0, float a) => new(x0, a, 0.0f); + + /// + /// Every value of the interval [lo, hi], uncorrelated with eps: ((lo + hi) / 2, 0, (hi - lo) / 2). + /// Rounding: x0 and e each round once, and a value of the interval is within e of x0 only up to + /// those; S = |x0| + e. + /// + public static float3 af_from_iv(float2 interval) + { + var x0 = (interval.x + interval.y) * 0.5f; + var e = (interval.y - interval.x) * 0.5f; + return new float3(x0, 0.0f, e + (abs(x0) + e) * 9.5367431640625e-7f); + } + + /// + /// The interval of every value of the form over every eps: x0 -+ (|a| + e), the bridge to iv_*. The + /// endpoints round inward by at most u (|x0| + |a| + e) each, so each is moved outward by + /// (|x0| + |a| + e) 2^-20. + /// + public static float2 af_range(float3 x) + { + var r = abs(x.y) + x.z; + var w = (abs(x.x) + r) * 9.5367431640625e-7f; + return new float2(x.x - r - w, x.x + r + w); + } + + /// + /// Encloses x + y: (x0 + y0, a_x + a_y, e_x + e_y). The shared symbol keeps the two forms + /// correlated, so af_sub(x, x) has a = 0. Rounding: S = |x0| + |a| + e of the result. + /// + public static float3 af_add(float3 x, float3 y) + { + var x0 = x.x + y.x; + var a = x.y + y.y; + var e = x.z + y.z; + return new float3(x0, a, e + (abs(x0) + abs(a) + e) * 9.5367431640625e-7f); + } + + /// Encloses x - y: (x0 - y0, a_x - a_y, e_x + e_y). Rounding as af_add. + public static float3 af_sub(float3 x, float3 y) + { + var x0 = x.x - y.x; + var a = x.y - y.y; + var e = x.z + y.z; + return new float3(x0, a, e + (abs(x0) + abs(a) + e) * 9.5367431640625e-7f); + } + + /// Encloses -x: (-x0, -a, e). Exact. + public static float3 af_neg(float3 x) => new(-x.x, -x.y, x.z); + + /// + /// Encloses x * s: (x0 s, a s, e |s|). Rounding: S = |x0| + |a| + e of the result, plus 2^-126 + /// for underflow. + /// + public static float3 af_scale(float3 x, float s) + { + var x0 = x.x * s; + var a = x.y * s; + var e = x.z * abs(s); + return new float3(x0, a, e + (abs(x0) + abs(a) + e) * 9.5367431640625e-7f + 1.17549435e-38f); + } + + /// Encloses x + v for v in the interval, uncorrelated with eps: af_add(x, af_from_iv(interval)). + public static float3 af_add_iv(float3 x, float2 interval) => af_add(x, af_from_iv(interval)); + + /// + /// Encloses x * y, AF1's product: (x0 y0, x0 a_y + y0 a_x, |x0| e_y + |y0| e_x + (|a_x| + e_x)(|a_y| + e_y)). + /// The last term condenses the eps^2 term a_x a_y and every cross term with an error symbol into e, + /// so a form multiplied by itself is wider than its square: there is no af_sqr until a consumer + /// names one. Rounding: a can cancel, so S = |x0| + |x0 a_y| + |y0 a_x| + e, plus 2^-126 for + /// underflow. + /// + public static float3 af_mul(float3 x, float3 y) + { + var x0 = x.x * y.x; + var p = x.x * y.y; + var q = y.x * x.y; + var a = p + q; + var e = abs(x.x) * y.z + abs(y.x) * x.z + (abs(x.y) + x.z) * (abs(y.y) + y.z); + return new float3(x0, a, e + (abs(x0) + abs(p) + abs(q) + e) * 9.5367431640625e-7f + 1.17549435e-38f); + } + + /// + /// Encloses snoise(p) for every p = centre + axis eps + w with |w| <= radius, from one snoise_grad at + /// the centre (value n, gradient g). With R = |axis| + radius it returns the narrower of two sound + /// forms: the Lipschitz form (n, 0, L R), L = SNOISE_LIPSCHITZ, and the centred form + /// (n, g . axis, |g| radius + M R^2 / 2), M = SNOISE_HESSIAN. The centred form is Taylor's theorem: + /// snoise(c + d) = n + g . d + r with |r| <= M |d|^2 / 2, d = axis eps + w, g . w <= |g| radius. + /// radius must be at least 0. + /// + /// Both forms carry an allowance for the float32 evaluation of snoise, which is not the exact + /// function: 2^-14 + 2^-12 R + L 2^-20 (|centre|_1 + R). The skew and the corner offsets round the + /// input point by at most about 2u |p|_1 before any kernel term is formed (the hashing is exact on + /// integers below 2^24), so a float32 snoise of p is the exact one at a point within 2u |p|_1, which + /// moves the value by at most L 2u |p|_1; for both p and the centre, with the caller's scaling of the + /// point by a frequency (one rounding) and a factor of two for a GPU, 16u = 2^-20. The kernel's + /// absolute rounding is at most 105 x four corners x terms of magnitude below 0.5^4 x 1.8 over about + /// 30 operations, under 2^-15, doubled to 2^-14; the gradient's error times |d| <= R is under 2^-13 R, + /// doubled to 2^-12 R. AffineTests.SnoiseFormEnclosesBall checks the result with no tolerance on the + /// CPU, and the fixture's enclosure check measures it on a GPU. Rounding of the form itself: S = + /// |n| + |a| + e. The Lipschitz form is chosen when its half-width L R is at most the centred form's + /// |a| + |g| radius + M R^2 / 2, so the total width is never more than the Lipschitz form's. + /// + public static float3 af_snoise(float3 centre, float3 axis, float radius) + { + var g = snoise_grad(centre); + var gradient = new float3(g.x, g.y, g.z); + var reach = length(axis) + radius; + var lipschitz = SNOISE_LIPSCHITZ * reach; + var a = dot(gradient, axis); + var remainder = length(gradient) * radius + 0.5f * SNOISE_HESSIAN * reach * reach; + var allowance = 6.103515625e-5f + 2.44140625e-4f * reach + + SNOISE_LIPSCHITZ * 9.5367431640625e-7f * (abs(centre.x) + abs(centre.y) + abs(centre.z) + reach); + var centred = abs(a) + remainder < lipschitz; + var slope = centred ? a : 0.0f; + var e = (centred ? remainder : lipschitz) + allowance; + return new float3(g.w, slope, e + (abs(g.w) + abs(slope) + e) * 9.5367431640625e-7f); + } + + /// + /// Encloses the fBm value, sum of a_i snoise(p f_i) over the octaves, for every p = centre + axis eps + /// + w with |w| <= radius: the octave sum of af_snoise(centre f_i, axis f_i, radius |f_i|) scaled by + /// a_i, with frequency f_i and amplitude a_i compounded exactly as fbm_grad compounds them (its .w is + /// what AffineTests.SnoiseFormEnclosesBall encloses). The rounding of p f_i, inside fbm_grad, and of + /// the scaled centre and axis here, moves each point by at most 3u (|centre f_i|_1 + R f_i), inside + /// af_snoise's allowance for its input point. octaves is clamped to [0, 16] as there. + /// + public static float3 af_fbm(float3 centre, float3 axis, float radius, int octaves, float lacunarity, float gain) + { + octaves = clamp(octaves, 0, 16); + var amplitude = 1.0f; + var frequency = 1.0f; + var sum = new float3(0.0f, 0.0f, 0.0f); + for (var i = 0; i < octaves; i++) + { + sum = af_add(sum, af_scale(af_snoise(centre * frequency, axis * frequency, radius * abs(frequency)), amplitude)); + frequency *= lacunarity; + amplitude *= gain; + } + + return sum; + } + + /// + /// The axis of a screen tile's slice over the depth segment [z0, z1], for af_frustum_ball's centre: + /// (m_c, 1) h with h = (z1 - z0) / 2, in the camera frame of iv_frustum_ball (a pinhole at the + /// origin looking down +z, the ray of slope m the points (m z, z)). Its z component is h itself, the + /// symbol's coefficient for depth: z = z_m + h eps. Its rounding is af_frustum_ball's. + /// + public static float3 af_frustum_axis(float2 centreSlope, float z0, float z1) + { + var h = (z1 - z0) * 0.5f; + return new float3(centreSlope.x * h, centreSlope.y * h, h); + } + + /// + /// The affine slice of a screen tile over [z0, z1]: a centre (xyz) and radius (w) such that every point + /// of every ray of the tile at depth z in [z0, z1] is centre + axis eps(z) + w', |w'| <= radius, with + /// axis = af_frustum_axis(centreSlope, z0, z1), z_m = (z0 + z1) / 2, h = (z1 - z0) / 2 (both float32, + /// as computed here) and eps(z) = clamp((z - z_m) / h, -1, 1) (0 where h is 0). The centre is + /// (m_c z_m, z_m) and the radius z1 footprintPerDepth + warpVariation, where footprintPerDepth is + /// iv_frustum_ball's and warpVariation is what the caller's warp can add once its value at the centre + /// is applied to the centre (design.md, "Affine forms", the centred warp). Derivation: a point of + /// slope m_c + e (|e| <= footprintPerDepth) at depth z is (m_c z_m, z_m) + (m_c, 1)(z - z_m) + + /// (e z, 0), and (m_c, 1)(z - z_m) = axis eps(z) + (m_c, 1) r(z), where the clamp residual + /// |r(z)| = |z - z_m - h eps(z)| is at most u (|z_m| + h), one rounding each of z_m and h. A caller + /// taking depth as af_symbol(z_m, h) adds the same residual, 2^-23 (|z_m| + h) covers it, to that + /// form's e. Rounding: the centre's products and the axis's products round once each (u |c|_1 and + /// u |axis|_1 against |eps| <= 1), the residual is at most u (|c|_1 + |axis|_1) since + /// |(m_c, 1)| <= |m_c|_1 + 1, and the radius rounds at most four times; 2^-20 (|c|_1 + |axis|_1 + + /// radius) covers it twice over (AffineTests.FrustumSliceEnclosesEveryRayPoint, extreme families + /// included). The caller rotates and translates the centre into world space, rotates the axis, and + /// scales all three by its noise frequency; the rounding of that transform is the caller's. + /// + public static float4 af_frustum_ball(float2 centreSlope, float z0, float z1, float footprintPerDepth, float warpVariation) + { + var zm = (z0 + z1) * 0.5f; + var h = (z1 - z0) * 0.5f; + var cx = centreSlope.x * zm; + var cy = centreSlope.y * zm; + var spread = (abs(centreSlope.x) + abs(centreSlope.y) + 1.0f) * h; + var radius = z1 * footprintPerDepth + warpVariation; + radius += (abs(cx) + abs(cy) + abs(zm) + spread + radius) * 9.5367431640625e-7f; + return new float4(cx, cy, zm, radius); + } +} diff --git a/packages/cultmath/src/CultMath/math.Affine.cs.meta b/packages/cultmath/src/CultMath/math.Affine.cs.meta new file mode 100644 index 00000000..34ba0047 --- /dev/null +++ b/packages/cultmath/src/CultMath/math.Affine.cs.meta @@ -0,0 +1,2 @@ +fileFormatVersion: 2 +guid: 86b87ffb6c8a4d72b067a64f9f8a06ce diff --git a/packages/cultmath/src/CultMath/math.Interval.cs b/packages/cultmath/src/CultMath/math.Interval.cs index 91e99f87..deedea32 100644 --- a/packages/cultmath/src/CultMath/math.Interval.cs +++ b/packages/cultmath/src/CultMath/math.Interval.cs @@ -20,6 +20,14 @@ public static partial class math // snoise kernel must re-run MeasureLipschitz and re-pin this constant. public const float SNOISE_LIPSCHITZ = 10.099261f; + // Lipschitz constant of snoise(float2) in its input, with SNOISE_LIPSCHITZ's provenance rule: + // NoiseBoundTests.MeasureLipschitz2 (slow, explicit) takes the largest gradient length (central + // differences over exact float steps) over 1e6 seeded points, refines the 1e4 largest by ascent and + // multiplies by 1.10; NoiseBoundTests.Lipschitz2ConstantPins pins it against a fast re-measure and + // the ascent from MeasureLipschitz2's witness. A change to the 2D snoise kernel must re-run + // MeasureLipschitz2 and re-pin this constant. + public const float SNOISE2_LIPSCHITZ = 9.19f; + /// The point interval [x, x]. public static float2 iv_point(float value) => new(value, value); @@ -142,6 +150,19 @@ public static partial class math return new(max(n - e, -1.0f), min(n + e, 1.0f)); } + /// + /// Encloses snoise(x) for every 2D x within distance radius of centre: [n - L2 r, n + L2 r] intersected + /// with [-1, 1], n = snoise(centre), L2 = SNOISE2_LIPSCHITZ. radius must be at least 0. One snoise + /// evaluation, value only. A consumer's 2D flow built from snoise(float2) varies over a ball by at + /// most the width this gives (NoiseBoundTests.CentredWarpStaysEnclosed). + /// + public static float2 iv_snoise_ball(float2 centre, float radius) + { + var n = snoise(centre); + var e = SNOISE2_LIPSCHITZ * radius; + return new(max(n - e, -1.0f), min(n + e, 1.0f)); + } + /// /// Encloses the fBm value, sum of a_i * snoise(x * f_i) over the octaves, for every x within distance /// radius of centre: the octave sum of iv_snoise_ball(centre * f_i, radius * |f_i|) scaled by a_i, diff --git a/packages/cultmath/tests/CultMath.Tests/AffineTests.cs b/packages/cultmath/tests/CultMath.Tests/AffineTests.cs new file mode 100644 index 00000000..38ccd071 --- /dev/null +++ b/packages/cultmath/tests/CultMath.Tests/AffineTests.cs @@ -0,0 +1,458 @@ +using CultMath; +using Xunit; +using static CultMath.math; + +namespace CultMath.Tests; + +/// +/// The affine forms of math.Affine.cs (design.md, "Affine forms"). A form (x0, a, e) is the set +/// x0 + a eps + e delta at the region's shared eps; every check evaluates that set in double and the +/// pointwise function in float32, with no tolerance. Operand values are float32 values drawn inside +/// their operands' sets at one shared eps, endpoints (eps and delta at -1 and 1) included. +/// +public sealed class AffineTests +{ + private readonly ITestOutputHelper output; + + public AffineTests(ITestOutputHelper output) => this.output = output; + + private static float Uniform(System.Random random, float lo, float hi) => lo + (hi - lo) * random.NextSingle(); + + private static float LogUniform(System.Random random, float lo, float hi) => + MathF.Exp(Uniform(random, MathF.Log(lo), MathF.Log(hi))); + + private static float Signed(System.Random random, float magnitude) => random.Next(2) == 0 ? -magnitude : magnitude; + + // Whether value lies in the form's set at eps, in double. + private static bool InForm(float3 form, float eps, double value) => + Math.Abs(value - (form.x + (double)form.y * eps)) <= form.z; + + // A form with its centre in [-20, 20] and |a| and e log-uniform over [1e-5, 20]; one in eight has + // a = 0 and one in eight e = 0. + private static float3 DrawForm(System.Random random) + { + var x0 = Uniform(random, -20.0f, 20.0f); + var a = random.Next(8) == 0 ? 0.0f : Signed(random, LogUniform(random, 1.0e-5f, 20.0f)); + var e = random.Next(8) == 0 ? 0.0f : LogUniform(random, 1.0e-5f, 20.0f); + return new float3(x0, a, e); + } + + private static float2 DrawInterval(System.Random random) + { + var lo = Uniform(random, -20.0f, 20.0f); + return new float2(lo, lo + (random.Next(8) == 0 ? 0.0f : LogUniform(random, 1.0e-5f, 40.0f))); + } + + // A float32 value in the form's set at eps, delta at an endpoint when asked; NaN when rounding leaves + // no float of the set near the drawn point (a set narrower than an ulp). + private static float Member(System.Random random, float3 form, float eps, bool endpoint) + { + var centre = form.x + (double)form.y * eps; + var delta = endpoint ? (random.Next(2) == 0 ? -1.0 : 1.0) : Uniform(random, -1.0f, 1.0f); + var value = (float)(centre + form.z * delta); + for (var k = 0; k < 4 && !InForm(form, eps, value); k++) + value = value > centre ? MathF.BitDecrement(value) : MathF.BitIncrement(value); + return InForm(form, eps, value) ? value : float.NaN; + } + + private static float MemberOf(System.Random random, float2 interval, bool endpoint) => + endpoint ? (random.Next(2) == 0 ? interval.x : interval.y) : Math.Clamp(Uniform(random, interval.x, interval.y), interval.x, interval.y); + + private readonly record struct Operands(float3 X, float3 Y, float2 Iv, float S); + + private readonly record struct Values(float X, float Y, float V, float S); + + // One op under the harness: Build gives the result's membership test (eps, value) from the operand + // forms, F the pointwise float32 function of operand values. + private sealed record Op(string Name, Func> Build, Func F); + + private static Func Form(float3 form) => (eps, value) => InForm(form, eps, value); + + private static Func Range(float2 range) => (_, value) => value >= range.x && value <= range.y; + + private static readonly Op[] Ops = + { + new("af_point", o => Form(af_point(o.X.x)), v => v.S), + new("af_symbol", o => Form(af_symbol(o.X.x, o.X.y)), v => v.S), + new("af_from_iv", o => Form(af_from_iv(o.Iv)), v => v.V), + new("af_range", o => Range(af_range(o.X)), v => v.X), + new("af_add", o => Form(af_add(o.X, o.Y)), v => v.X + v.Y), + new("af_sub", o => Form(af_sub(o.X, o.Y)), v => v.X - v.Y), + new("af_neg", o => Form(af_neg(o.X)), v => -v.X), + new("af_scale", o => Form(af_scale(o.X, o.S)), v => v.X * v.S), + new("af_add_iv", o => Form(af_add_iv(o.X, o.Iv)), v => v.X + v.V), + new("af_mul", o => Form(af_mul(o.X, o.Y)), v => v.X * v.Y), + }; + + // Runs one op over `forms` seeded operand draws x 16 (eps, delta) draws; returns the misses and up to + // four witnesses. The first four draws per form put eps at an endpoint, the first eight delta. + // af_point and af_symbol are checked on their defining value: the point x0 itself, and x0 + a eps in + // double (the symbol defines eps, so it has no float32 evaluation to enclose). + private static (int Misses, List Witness) Harness(Op op, int seed, int forms = 10_000, Func? draw = null) + { + draw ??= DrawForm; + var random = new System.Random(seed); + var misses = 0; + var witness = new List(); + for (var t = 0; t < forms; t++) + { + var operands = new Operands(draw(random), draw(random), DrawInterval(random), random.Next(8) == 0 ? 0.0f : Uniform(random, -4.0f, 4.0f)); + var contains = op.Build(operands); + for (var d = 0; d < 16; d++) + { + var eps = d < 4 ? (d % 2 == 0 ? -1.0f : 1.0f) : Uniform(random, -1.0f, 1.0f); + var endpoint = d < 8; + var x = Member(random, operands.X, eps, endpoint); + var y = Member(random, operands.Y, eps, endpoint); + if (float.IsNaN(x) || float.IsNaN(y)) + continue; + var v = MemberOf(random, operands.Iv, endpoint); + double f = op.Name switch + { + "af_point" => operands.X.x, + "af_symbol" => operands.X.x + (double)operands.X.y * eps, + _ => op.F(new Values(x, y, v, operands.S)), + }; + if (!contains(eps, f)) + { + misses++; + if (witness.Count < 4) + witness.Add($"{op.Name}: X {operands.X}, Y {operands.Y}, iv {operands.Iv}, s {operands.S:R}; eps {eps:R}, values ({x:R}, {y:R}, {v:R}) give {f:R}"); + } + } + } + + return (misses, witness); + } + + [Fact] + public void EveryOpEnclosesItsPoints() + { + var failures = new List(); + foreach (var op in Ops) + { + var (misses, witness) = Harness(op, 0xAF00); + if (misses > 0) + failures.Add($"{op.Name}: {misses} misses" + Environment.NewLine + string.Join(Environment.NewLine, witness)); + } + + Assert.True(failures.Count == 0, string.Join(Environment.NewLine, failures)); + } + + private static float3 Shrink(float3 form) => new(form.x, form.y, form.z > 0.0f ? MathF.BitDecrement(form.z) : 0.0f); + + // Each op with its rounding absorption removed: the propagated error alone, computed in float32. + private static readonly Op[] Unabsorbed = + { + new("af_from_iv", o => Form(new float3((o.Iv.x + o.Iv.y) * 0.5f, 0.0f, (o.Iv.y - o.Iv.x) * 0.5f)), v => v.V), + new("af_range", o => Range(new float2(o.X.x - (MathF.Abs(o.X.y) + o.X.z), o.X.x + (MathF.Abs(o.X.y) + o.X.z))), v => v.X), + new("af_add", o => Form(new float3(o.X.x + o.Y.x, o.X.y + o.Y.y, o.X.z + o.Y.z)), v => v.X + v.Y), + new("af_sub", o => Form(new float3(o.X.x - o.Y.x, o.X.y - o.Y.y, o.X.z + o.Y.z)), v => v.X - v.Y), + new("af_scale", o => Form(new float3(o.X.x * o.S, o.X.y * o.S, o.X.z * MathF.Abs(o.S))), v => v.X * v.S), + new("af_add_iv", o => Form(new float3(o.X.x + (o.Iv.x + o.Iv.y) * 0.5f, o.X.y, o.X.z + (o.Iv.y - o.Iv.x) * 0.5f)), v => v.X + v.V), + new("af_mul", o => Form(new float3(o.X.x * o.Y.x, o.X.x * o.Y.y + o.Y.x * o.X.y, + MathF.Abs(o.X.x) * o.Y.z + MathF.Abs(o.Y.x) * o.X.z + (MathF.Abs(o.X.y) + o.X.z) * (MathF.Abs(o.Y.y) + o.Y.z))), v => v.X * v.Y), + }; + + // One-ulp mutants that survive, each with the reason. A rounding absorption bounds the worst case + // over every operand value, and 2^-20 S is 16u S, at least eight ulps of the larger of |x0|, |a| and + // e; one ulp less of e leaves the rest of that margin, which no sampled rounding can cross. + private static readonly Dictionary OneUlpSurvivors = new() + { + ["af_from_iv"] = "the absorption (|x0| + e) 2^-20 is at least eight ulps of e over the one rounding of x0 and e it covers", + ["af_range"] = "each endpoint moves outward by (|x0| + |a| + e) 2^-20, eight ulps past the one rounding it covers", + ["af_add"] = "the absorption 2^-20 S is eight ulps of S's largest term past the three roundings it covers", + ["af_sub"] = "as af_add", + ["af_scale"] = "as af_add", + ["af_add_iv"] = "two absorptions, af_from_iv's and af_add's", + ["af_mul"] = "the absorption 2^-20 (|x0| + |x0 a_y| + |y0 a_x| + e) is eight ulps past the products it covers", + }; + + /// + /// The rounding absorptions are load-bearing: with each op's absorption removed (Unabsorbed), the + /// harness finds a miss for every op. Each op's e shrunk by one ulp is run too; it survives for every + /// op, because the absorption is a worst-case bound many ulps wide, and each survivor is named with + /// its reason in OneUlpSurvivors. A one-ulp mutant that starts failing, or one that is not named, + /// fails this test, so the list cannot go stale. + /// + [Fact] + public void EveryRoundingTermIsCaught() + { + var problems = new List(); + foreach (var mutant in Unabsorbed) + { + var (misses, _) = Harness(mutant, 0xAF01); + output.WriteLine($"AF-REPORT {mutant.Name} without absorption: {misses} misses"); + if (misses == 0) + problems.Add($"{mutant.Name}: removing the rounding absorption is not caught"); + } + + foreach (var op in Ops.Where(o => o.Name is not ("af_point" or "af_symbol" or "af_neg"))) + { + var shrunk = op.Name == "af_range" + ? op with { Build = o => { var r = af_range(o.X); return Range(new float2(MathF.BitIncrement(r.x), MathF.BitDecrement(r.y))); } } + : op with { Build = o => Form(Shrink(BuildForm(op.Name, o))) }; + var (misses, _) = Harness(shrunk, 0xAF02); + output.WriteLine($"AF-REPORT {op.Name} with e one ulp short: {misses} misses"); + var named = OneUlpSurvivors.ContainsKey(op.Name); + if (misses == 0 && !named) + problems.Add($"{op.Name}: e one ulp short survives and is not named"); + if (misses > 0 && named) + problems.Add($"{op.Name}: e one ulp short is now caught; remove it from OneUlpSurvivors"); + } + + Assert.True(problems.Count == 0, string.Join(Environment.NewLine, problems)); + } + + private static float3 BuildForm(string name, Operands o) => name switch + { + "af_from_iv" => af_from_iv(o.Iv), + "af_add" => af_add(o.X, o.Y), + "af_sub" => af_sub(o.X, o.Y), + "af_scale" => af_scale(o.X, o.S), + "af_add_iv" => af_add_iv(o.X, o.Iv), + "af_mul" => af_mul(o.X, o.Y), + _ => throw new ArgumentException(name), + }; + + /// + /// The shared symbol correlates: for x = af_symbol(x0, a), af_sub(x, x) and af_add(x, af_neg(x)) have + /// a = 0 and e at most their own rounding, and af_sub(af_scale(x, 2), af_add(x, x)) has a = 0. A + /// mutant of af_add, af_sub or af_scale that condensed a into e would leave e of order |a|. + /// + [Fact] + public void SharedSymbolCorrelates() + { + var random = new System.Random(0xAF03); + for (var t = 0; t < 10_000; t++) + { + var x = af_symbol(Uniform(random, -20.0f, 20.0f), Signed(random, LogUniform(random, 1.0e-5f, 20.0f))); + foreach (var (name, form) in new[] + { + ("af_sub(x, x)", af_sub(x, x)), + ("af_add(x, af_neg(x))", af_add(x, af_neg(x))), + ("af_sub(af_scale(x, 2), af_add(x, x))", af_sub(af_scale(x, 2.0f), af_add(x, x))), + }) + { + var rounding = (MathF.Abs(x.x) + MathF.Abs(x.y)) * 1.0e-5f; + Assert.True(form.y == 0.0f && form.z <= rounding, $"{name} for x = {x} is {form}: a must be 0 and e at most {rounding:R}"); + } + } + } + + // Forms whose range straddles zero (|x0| below |a| + e) or is one-signed (|x0| above it). + private static float3 DrawSigned(System.Random random, bool straddle) + { + var a = Signed(random, LogUniform(random, 1.0e-3f, 10.0f)); + var e = LogUniform(random, 1.0e-3f, 10.0f); + var reach = MathF.Abs(a) + e; + return new float3(Signed(random, straddle ? Uniform(random, 0.0f, 0.9f * reach) : reach * Uniform(random, 1.1f, 4.0f)), a, e); + } + + /// + /// af_mul encloses the product over forms that straddle zero and forms that are one-signed, every + /// sign combination. The eps^2 term a_x a_y is what the condensed e carries for eps at +-1: a mutant + /// dropping |a_x a_y| from e misses there. + /// + [Fact] + public void MulEnclosesEverySignCase() + { + var mul = Ops.Single(o => o.Name == "af_mul"); + var dropped = mul with + { + Build = o => + { + var r = af_mul(o.X, o.Y); + return Form(new float3(r.x, r.y, r.z - MathF.Abs(o.X.y * o.Y.y))); + }, + }; + foreach (var (sx, sy) in new[] { (true, true), (true, false), (false, true), (false, false) }) + { + var seed = 0xAF04 + (sx ? 1 : 0) + (sy ? 2 : 0); + var flip = 0; + float3 Draw(System.Random r) => DrawSigned(r, (flip++ % 2 == 0) ? sx : sy); + var (misses, witness) = Harness(mul, seed, 5_000, Draw); + Assert.True(misses == 0, $"af_mul, straddling ({sx}, {sy}): {misses} misses" + Environment.NewLine + string.Join(Environment.NewLine, witness)); + flip = 0; + var (caught, _) = Harness(dropped, seed, 5_000, Draw); + Assert.True(caught > 0, $"af_mul without |a_x a_y|, straddling ({sx}, {sy}), is not caught"); + } + } + + // af_snoise's float32 allowance, as its doc comment states it, for the width comparison below. + private static double Allowance(float3 centre, double reach) => + Math.ScaleB(1.0, -14) + Math.ScaleB(reach, -12) + + SNOISE_LIPSCHITZ * Math.ScaleB(Math.Abs(centre.x) + Math.Abs(centre.y) + Math.Abs(centre.z) + reach, -20); + + private static float3 UnitVector(System.Random random) + { + float3 v; + do + { + v = new float3(Uniform(random, -1.0f, 1.0f), Uniform(random, -1.0f, 1.0f), Uniform(random, -1.0f, 1.0f)); + } + while (dot(v, v) > 1.0f || dot(v, v) < 1.0e-4f); + + return normalize(v); + } + + // A float32 point of the set centre + axis eps + w, |w| <= radius, checked in double; the first eight + // w are on the surface (pulled in by 2^-20 so their rounding stays inside). NaN-free: a point whose + // rounding leaves the set is reported as null and skipped. + private static float3? PointOf(System.Random random, float3 centre, float3 axis, float radius, float eps, int index) + { + var w = UnitVector(random) * (index < 8 ? radius * (1.0f - 9.5367431640625e-7f) : radius * MathF.Cbrt(random.NextSingle())); + var p = centre + axis * eps + w; + var (dx, dy, dz) = (p.x - (centre.x + (double)axis.x * eps), p.y - (centre.y + (double)axis.y * eps), p.z - (centre.z + (double)axis.z * eps)); + return Math.Sqrt(dx * dx + dy * dy + dz * dz) <= radius ? p : null; + } + + /// + /// af_snoise and af_fbm over 2,000 (centre, axis, radius) each: centres in [-20, 20]^3, radius + /// log-uniform in [1e-4, 2], |axis| uniform in [0, 2] (an eighth exactly 0). 64 points + /// centre + axis eps + w per case, eps at -1 and 1 for the first eight; snoise (fbm_grad(...).w) in + /// float32 lies in the form at eps, no tolerance. Dropping the M R^2 / 2 remainder from a centred + /// form misses. af_snoise's total width is never more than the Lipschitz form's, L R plus the + /// allowance. + /// + [Fact] + public void SnoiseFormEnclosesBall() + { + var random = new System.Random(0xAF10); + var (misses, remainderMisses, skipped, centredCases) = (0, 0, 0, 0); + var witness = new List(); + for (var b = 0; b < 4000; b++) + { + var fbm = b >= 2000; + var centre = new float3(Uniform(random, -20.0f, 20.0f), Uniform(random, -20.0f, 20.0f), Uniform(random, -20.0f, 20.0f)); + var axis = random.Next(8) == 0 ? new float3(0.0f, 0.0f, 0.0f) : UnitVector(random) * Uniform(random, 0.0f, 2.0f); + var radius = LogUniform(random, 1.0e-4f, 2.0f); + var (octaves, lacunarity, gain) = (random.Next(0, 7), Uniform(random, 1.5f, 2.5f), Uniform(random, 0.3f, 0.7f)); + var form = fbm ? af_fbm(centre, axis, radius, octaves, lacunarity, gain) : af_snoise(centre, axis, radius); + var reach = (double)length(axis) + radius; + var remainder = 0.5 * SNOISE_HESSIAN * reach * reach; + var mutant = new float3(form.x, form.y, (float)(form.z - remainder)); + if (!fbm) + { + var lipschitz = (SNOISE_LIPSCHITZ * reach + Allowance(centre, reach) + Math.ScaleB(Math.Abs(form.x), -20)) * (1.0 + Math.ScaleB(1.0, -18)); + Assert.True(Math.Abs(form.y) + form.z <= lipschitz, $"af_snoise({centre}, {axis}, {radius:R}) = {form} is wider than the Lipschitz form's half-width {lipschitz:R}"); + centredCases += form.y != 0.0f ? 1 : 0; + } + + for (var i = 0; i < 64; i++) + { + var eps = i < 8 ? (i % 2 == 0 ? -1.0f : 1.0f) : Uniform(random, -1.0f, 1.0f); + var p = PointOf(random, centre, axis, radius, eps, i); + if (p is not { } point) + { + skipped++; + continue; + } + + var value = fbm ? fbm_grad(point, octaves, lacunarity, gain).w : snoise(point); + if (!InForm(form, eps, value)) + { + misses++; + if (witness.Count < 4) + witness.Add($"{(fbm ? $"af_fbm(..., {octaves}, {lacunarity:R}, {gain:R})" : "af_snoise")}({centre}, {axis}, {radius:R}) = {form} misses {value:R} at eps {eps:R}, {point}"); + } + + if (!fbm && form.y != 0.0f && !InForm(mutant, eps, value)) + remainderMisses++; + } + } + + output.WriteLine($"AF-REPORT af_snoise/af_fbm enclosure: {misses} misses, {skipped} points skipped (rounded out of the set), {centredCases} of 2000 af_snoise cases centred, {remainderMisses} misses with the remainder dropped"); + Assert.True(misses == 0, string.Join(Environment.NewLine, witness)); + Assert.True(skipped < 64 * 4000 / 100, $"{skipped} points rounded out of their sets"); + Assert.True(remainderMisses > 0, "dropping M R^2 / 2 from the centred form is not caught"); + } + + /// + /// At radius 1e-3 and |axis| = 1e-3, af_snoise takes the centred form in every one of 2,000 seeded + /// cases (a is nonzero), and its mean total width is under half of the Lipschitz form's 2 L R. A + /// mutant that always returns the Lipschitz form has a = 0. + /// + [Fact] + public void SnoiseChoosesTheCentredFormWhereNarrower() + { + var random = new System.Random(0xAF11); + var sum = 0.0; + for (var b = 0; b < 2000; b++) + { + var centre = new float3(Uniform(random, -20.0f, 20.0f), Uniform(random, -20.0f, 20.0f), Uniform(random, -20.0f, 20.0f)); + var axis = UnitVector(random) * 1.0e-3f; + var form = af_snoise(centre, axis, 1.0e-3f); + var lipschitz = 2.0 * SNOISE_LIPSCHITZ * 2.0e-3; + var width = 2.0 * (Math.Abs(form.y) + form.z); + Assert.True(form.y != 0.0f && width < lipschitz, $"af_snoise({centre}, {axis}, 1e-3) = {form}: not the centred form, or no narrower than 2 L R = {lipschitz:R}"); + sum += width / lipschitz; + } + + output.WriteLine($"AF-REPORT af_snoise at radius 1e-3: mean width over 2 L R {sum / 2000:F3}"); + Assert.True(sum / 2000 < 0.5, $"mean width over 2 L R is {sum / 2000:F3}, not under half"); + } + + /// + /// af_frustum_ball and af_frustum_axis over iv_frustum_ball's families (NoiseBoundTests.DrawBallCase: + /// the r2 domain and the degenerate, thin, wide-slope, far, from-the-camera, tiny-focal and huge-warp + /// extremes), 2,000 tiles each, warp passed as the warp variation: every exact ray point (m z, z), + /// formed in double from the float32 slope and depth at the footprint's corners at both depth ends + /// (and 56 more for the r2 domain), moved by a full-length outward flow of length warp, is within + /// radius of centre + axis eps(z), eps(z) = clamp((z - z_m) / h, -1, 1) with z_m and h the centre's z + /// and the axis's z (0 when h is 0). The radius is pinned to the derivation both ways: + /// r <= w <= r + 2^-19 (r + |c|_1 + |axis|_1), r = z1 footprintPerDepth + warp in double. + /// + [Fact] + public void FrustumSliceEnclosesEveryRayPoint() + { + var random = new System.Random(0xAF12); + string[] families = { "r2 domain", "degenerate", "thin", "wide slope", "far", "from the camera", "tiny focal", "huge warp" }; + var failures = new Dictionary(); + var witness = new List(); + foreach (var family in families) + { + failures[family] = 0; + for (var t = 0; t < 2000; t++) + { + var (n, focal, slope, z0, z1, warp) = NoiseBoundTests.DrawBallCase(random, family); + var half = n / (2.0f * focal); + var footprint = n / (MathF.Sqrt(2.0f) * focal); + var axis = af_frustum_axis(slope, z0, z1); + var ball = af_frustum_ball(slope, z0, z1, footprint, warp); + var (cx, cy, cz, w) = ((double)ball.x, (double)ball.y, (double)ball.z, (double)ball.w); + var r = z1 * (double)footprint + warp; + var c1 = Math.Abs(cx) + Math.Abs(cy) + Math.Abs(cz); + var a1 = Math.Abs((double)axis.x) + Math.Abs((double)axis.y) + Math.Abs((double)axis.z); + if (!(r <= w && w <= r + Math.ScaleB(r + c1 + a1, -19))) + { + failures[family]++; + if (witness.Count < 8) + witness.Add($"{family} pin: af_frustum_ball({slope}, {z0:R}, {z1:R}, {footprint:R}, {warp:R}) = {ball}, derivation {r:R}"); + } + + for (var i = 0; i < (family == "r2 domain" ? 64 : 8); i++) + { + var offset = i < 8 + ? new float2((i & 1) == 0 ? -half : half, (i & 2) == 0 ? -half : half) + : new float2(Uniform(random, -half, half), Uniform(random, -half, half)); + var z = i < 8 ? ((i & 4) == 0 ? z0 : z1) : Uniform(random, z0, z1); + var m = slope + offset; + var eps = axis.z == 0.0f ? 0.0 : Math.Clamp((z - cz) / axis.z, -1.0, 1.0); + var (qx, qy, qz) = (cx + axis.x * eps, cy + axis.y * eps, cz + axis.z * eps); + var (dx, dy, dz) = (m.x * (double)z - qx, m.y * (double)z - qy, z - qz); + var away = Math.Sqrt(dx * dx + dy * dy + dz * dz); + var distance = away + warp; + if (!(distance <= w)) + { + failures[family]++; + if (witness.Count < 8) + witness.Add($"{family}: af_frustum_ball({slope}, {z0:R}, {z1:R}, {footprint:R}, {warp:R}) = {ball}, axis {axis} misses slope {m}, depth {z:R} by {(distance - w) / w:R} r"); + } + } + } + } + + Assert.True(failures.Values.Sum() == 0, string.Join(", ", families.Select(f => $"{f} {failures[f]}")) + Environment.NewLine + string.Join(Environment.NewLine, witness)); + } +} diff --git a/packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs b/packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs index 2b00ca88..fc0ba0ac 100644 --- a/packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs +++ b/packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs @@ -121,7 +121,7 @@ public sealed class NoiseBoundTests // One tile case of TileBallEnclosesEveryRaySegment: the r2 domain, or one of the extreme families // where float32 rounding of the centre is largest against the radius (Soul's verdict s1). - private static (int N, float Focal, float2 Slope, float Z0, float Z1, float Warp) DrawBallCase(System.Random random, string family) + internal static (int N, float Focal, float2 Slope, float Z0, float Z1, float Warp) DrawBallCase(System.Random random, string family) { int[] sizes = { 1, 4, 8, 16 }; float[] depths = { 12.0f, 1000.0f, 2400.0f }; From 4137285e58efc2b5c7d4242d45929a57de08cc03 Mon Sep 17 00:00:00 2001 From: Meta Vens Date: Sat, 3 Oct 2026 21:54:36 +0200 Subject: [PATCH 02/31] Deep well (c), centred warp, affine FogField bound, range ceiling; SNOISE_HESSIAN and SNOISE2_LIPSCHITZ measured and pinned; cull map (c) is FogField.DeepWell (ruling wells-scale-order-of-magnitude), measured under the centred-warp interval bound and the affine bound on the same seeds; the shallow wells are the printed (c-band) row. The warp-enlarged ball and the warp-carrying detail ball in FogField.Bound are gone. (c) asserts affine cheaper than the centred interval at both warps, with the 0.5 cull floor under both (ruling affine-wells-contract-cheaper-than-interval). Co-Authored-By: Claude Opus 5.5 --- packages/cultmath/src/CultMath/math.Affine.cs | 2 +- .../cultmath/src/CultMath/math.Interval.cs | 2 +- .../tests/CultMath.Tests/AffineTests.cs | 21 +- .../tests/CultMath.Tests/NoiseBoundTests.cs | 889 ++++++++++++++++-- 4 files changed, 833 insertions(+), 81 deletions(-) diff --git a/packages/cultmath/src/CultMath/math.Affine.cs b/packages/cultmath/src/CultMath/math.Affine.cs index 1d2dd70b..4a116c7d 100644 --- a/packages/cultmath/src/CultMath/math.Affine.cs +++ b/packages/cultmath/src/CultMath/math.Affine.cs @@ -27,7 +27,7 @@ public static partial class math // NoiseBoundTests.HessianConstantPinsSampledCurvature pins this value against a fast re-measure and // the ascent from MeasureHessian's witness. Empirical with a margin, not a proof; the enclosure // tests are the defence. A change to the snoise kernel must re-run MeasureHessian and re-pin it. - public const float SNOISE_HESSIAN = 68.94f; + public const float SNOISE_HESSIAN = 56.050385f; /// The point form (x, 0, 0): the value x at every eps. Exact. public static float3 af_point(float value) => new(value, 0.0f, 0.0f); diff --git a/packages/cultmath/src/CultMath/math.Interval.cs b/packages/cultmath/src/CultMath/math.Interval.cs index deedea32..192abf2a 100644 --- a/packages/cultmath/src/CultMath/math.Interval.cs +++ b/packages/cultmath/src/CultMath/math.Interval.cs @@ -26,7 +26,7 @@ public static partial class math // multiplies by 1.10; NoiseBoundTests.Lipschitz2ConstantPins pins it against a fast re-measure and // the ascent from MeasureLipschitz2's witness. A change to the 2D snoise kernel must re-run // MeasureLipschitz2 and re-pin this constant. - public const float SNOISE2_LIPSCHITZ = 9.19f; + public const float SNOISE2_LIPSCHITZ = 8.117002f; /// The point interval [x, x]. public static float2 iv_point(float value) => new(value, value); diff --git a/packages/cultmath/tests/CultMath.Tests/AffineTests.cs b/packages/cultmath/tests/CultMath.Tests/AffineTests.cs index 38ccd071..b70de3f4 100644 --- a/packages/cultmath/tests/CultMath.Tests/AffineTests.cs +++ b/packages/cultmath/tests/CultMath.Tests/AffineTests.cs @@ -297,14 +297,23 @@ public sealed class AffineTests } // A float32 point of the set centre + axis eps + w, |w| <= radius, checked in double; the first eight - // w are on the surface (pulled in by 2^-20 so their rounding stays inside). NaN-free: a point whose - // rounding leaves the set is reported as null and skipped. + // w are on the surface. Where the float32 sum rounds a point out of the set, w is pulled in by the + // overshoot and 2^-21 (|centre|_1 + |axis|_1) and the sum retried; a point that still rounds out is + // reported as null and skipped. private static float3? PointOf(System.Random random, float3 centre, float3 axis, float radius, float eps, int index) { - var w = UnitVector(random) * (index < 8 ? radius * (1.0f - 9.5367431640625e-7f) : radius * MathF.Cbrt(random.NextSingle())); - var p = centre + axis * eps + w; - var (dx, dy, dz) = (p.x - (centre.x + (double)axis.x * eps), p.y - (centre.y + (double)axis.y * eps), p.z - (centre.z + (double)axis.z * eps)); - return Math.Sqrt(dx * dx + dy * dy + dz * dz) <= radius ? p : null; + var w = UnitVector(random) * (index < 8 ? radius : radius * MathF.Cbrt(random.NextSingle())); + for (var k = 0; k < 6; k++) + { + var p = centre + axis * eps + w; + var (dx, dy, dz) = (p.x - (centre.x + (double)axis.x * eps), p.y - (centre.y + (double)axis.y * eps), p.z - (centre.z + (double)axis.z * eps)); + var distance = Math.Sqrt(dx * dx + dy * dy + dz * dz); + if (distance <= radius) + return p; + w *= (float)(Math.Max(0.0, 2.0 * radius - distance - Math.ScaleB(Math.Abs(centre.x) + Math.Abs(centre.y) + Math.Abs(centre.z) + Math.Abs(axis.x) + Math.Abs(axis.y) + Math.Abs(axis.z), -21)) / radius); + } + + return null; } /// diff --git a/packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs b/packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs index fc0ba0ac..2e6f23b8 100644 --- a/packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs +++ b/packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs @@ -5,9 +5,9 @@ using static CultMath.math; namespace CultMath.Tests; /// -/// The noise bounds (design.md, "Intervals"): iv_snoise_ball and iv_fbm_ball enclose their functions over -/// a ball, a domain warp bounded by D is enclosed by enlarging the radius by D, SNOISE_LIPSCHITZ has -/// measured provenance, iv_frustum_ball encloses a screen tile's rays over a depth segment, the bound +/// The noise bounds (design.md, "Intervals" and "Affine forms"): iv_snoise_ball and iv_fbm_ball enclose +/// their functions over a ball, a domain warp bounded by D is enclosed by enlarging the radius by D, or +/// by the centred warp, SNOISE_LIPSCHITZ, SNOISE2_LIPSCHITZ and SNOISE_HESSIAN have measured provenance, iv_frustum_ball encloses a screen tile's rays over a depth segment, the bound /// composed from an analytic envelope and the noise ball encloses the density, and a tile march over /// fields with authored empty space is measured against a dense one for the same transmittance. Lines /// starting IV-REPORT carry the numbers the cut report quotes. @@ -251,10 +251,11 @@ public sealed class NoiseBoundTests /// /// intervals-enclose for the composition the site and Aetheria use: over 2,000 seeded tile slices - /// (Aetheria's wells, inside the fog, the void at every grid point, the shipped void and LOD-far; - /// N in {1, 4, 8, 16}; a camera drawn per tile) the bound composed from the envelope over the slice's - /// box, its fade, and the noise ball enlarged by the warp encloses the density at 64 points of the - /// slice, each sampled through the field's own warp, with no tolerance. The void's footprint-truncated density (octave weights) is checked + /// (the deep well, the shallow wells and inside the fog, each under the centred-warp interval bound + /// and the affine bound; the void at every grid point, the shipped void and LOD-far; N in + /// {1, 4, 8, 16}; a camera drawn per tile) the bound composed from the envelope over the slice's box, + /// its fade, and the noise enclosure encloses the density at 64 points of the slice, each sampled + /// through the field's own warp, with no tolerance. The void's footprint-truncated density (octave weights) is checked /// against the same bound; a quarter of the slices are LOD-far's, where the weights are strictly /// between 0 and 1 and at 0, and the test asserts it sampled both. /// @@ -263,7 +264,15 @@ public sealed class NoiseBoundTests { var random = new System.Random(0xE7B0); int[] sizes = { 1, 4, 8, 16 }; - var fields = new List { FogField.Wells(FogField.AetheriaWarp), FogField.InsideFog(FogField.AetheriaWarp) }; + var fog = new List(); + foreach (var mode in new[] { BoundMode.Interval, BoundMode.Affine }) + { + fog.Add(FogField.DeepWell(FogField.AetheriaWarp, mode)); + fog.Add(FogField.Wells(FogField.AetheriaWarp, mode)); + fog.Add(FogField.InsideFog(FogField.AetheriaWarp, mode)); + } + + var fields = new List(); foreach (var rh in VoidField.HollowRadii) foreach (var offset in VoidField.CameraOffsets) foreach (var ramp in VoidField.Ramps) @@ -274,7 +283,7 @@ public sealed class NoiseBoundTests var (partial, zero) = (0, 0); for (var t = 0; t < 2000; t++) { - var field = t % 2 == 0 ? fields[(t / 2) % 2] : t % 4 == 1 ? fields[^1] : fields[2 + random.Next(fields.Count - 2)]; + var field = t % 2 == 0 ? fog[(t / 2) % fog.Count] : t % 4 == 1 ? fields[^1] : fields[random.Next(fields.Count)]; var tile = field.DrawTile(random, sizes[random.Next(sizes.Length)]); var near = field.Grid[0]; var far = field.Grid[^1]; @@ -474,6 +483,482 @@ public sealed class NoiseBoundTests Assert.True(MathF.Abs(ratio - 1.0f) <= 0.005f, $"SNOISE_LIPSCHITZ is {ratio:R} times 1.10 x the re-measured {refined:R}; re-run MeasureLipschitz and re-pin"); } + // ---- The curvature and 2D Lipschitz constants (math.Affine.cs, math.Interval.cs) ---- + + // Central differences over exact float32 steps: the step actually taken is (p + h) - p in float32, so + // the quotient is not skewed by the rounding of p + h far from the origin. + private static double Step(float x, float h, out double back) { back = x - (double)(x - h); return (x + h) - (double)x; } + + // The spectral norm (largest absolute eigenvalue) of the symmetrised Hessian of snoise(float3) at p, + // from central differences of snoise_grad at h = 1e-3. + private static double HessianNorm(float3 p) + { + const float h = 1.0e-3f; + var m = new double[3, 3]; + for (var a = 0; a < 3; a++) + { + var e = new float3(a == 0 ? h : 0.0f, a == 1 ? h : 0.0f, a == 2 ? h : 0.0f); + var forward = Step(a == 0 ? p.x : a == 1 ? p.y : p.z, h, out var back); + var gp = snoise_grad(p + e); + var gm = snoise_grad(p - e); + m[0, a] = (gp.x - (double)gm.x) / (forward + back); + m[1, a] = (gp.y - (double)gm.y) / (forward + back); + m[2, a] = (gp.z - (double)gm.z) / (forward + back); + } + + var (a00, a11, a22) = (m[0, 0], m[1, 1], m[2, 2]); + var (a01, a02, a12) = ((m[0, 1] + m[1, 0]) / 2, (m[0, 2] + m[2, 0]) / 2, (m[1, 2] + m[2, 1]) / 2); + var p1 = a01 * a01 + a02 * a02 + a12 * a12; + if (p1 == 0.0) + return Math.Max(Math.Abs(a00), Math.Max(Math.Abs(a11), Math.Abs(a22))); + var q = (a00 + a11 + a22) / 3; + var p2 = (a00 - q) * (a00 - q) + (a11 - q) * (a11 - q) + (a22 - q) * (a22 - q) + 2 * p1; + var s = Math.Sqrt(p2 / 6); + var (b00, b11, b22, b01, b02, b12) = ((a00 - q) / s, (a11 - q) / s, (a22 - q) / s, a01 / s, a02 / s, a12 / s); + var r = Math.Clamp((b00 * (b11 * b22 - b12 * b12) - b01 * (b01 * b22 - b12 * b02) + b02 * (b01 * b12 - b11 * b02)) / 2, -1.0, 1.0); + var phi = Math.Acos(r) / 3; + var largest = q + 2 * s * Math.Cos(phi); + var smallest = q + 2 * s * Math.Cos(phi + 2 * Math.PI / 3); + return Math.Max(Math.Abs(largest), Math.Abs(smallest)); + } + + // The gradient length of snoise(float2) at p, from central differences at h = 1e-3. + private static double Gradient2Norm(float2 p) + { + const float h = 1.0e-3f; + var fx = Step(p.x, h, out var bx); + var fy = Step(p.y, h, out var by); + var gx = (snoise(p + new float2(h, 0.0f)) - (double)snoise(p - new float2(h, 0.0f))) / (fx + bx); + var gy = (snoise(p + new float2(0.0f, h)) - (double)snoise(p - new float2(0.0f, h))) / (fy + by); + return Math.Sqrt(gx * gx + gy * gy); + } + + // Hill-climbs f from p as AscendGradientNorm does: the ascent direction is f's central difference, + // the step grows on success and halves on failure. Returns the largest value reached. + private static double Ascend(Func f, float3 p) + { + const float h = 1.0e-3f; + var step = 1.0e-2f; + var best = f(p); + for (var k = 0; k < 200 && step > 1.0e-6f; k++) + { + var d = new float3( + (float)(f(p + new float3(h, 0.0f, 0.0f)) - f(p - new float3(h, 0.0f, 0.0f))), + (float)(f(p + new float3(0.0f, h, 0.0f)) - f(p - new float3(0.0f, h, 0.0f))), + (float)(f(p + new float3(0.0f, 0.0f, h)) - f(p - new float3(0.0f, 0.0f, h)))); + if (dot(d, d) == 0.0f) + break; + var q = p + normalize(d) * step; + var value = f(q); + if (value > best) + { + p = q; + best = value; + step *= 1.5f; + } + else + { + step *= 0.5f; + } + } + + return best; + } + + private static double Ascend2(Func f, float2 p) => Ascend(q => f(new float2(q.x, q.y)), new float3(p.x, p.y, 0.0f)); + + // The largest f over `samples` seeded points of [-256, 256]^dims, the largest after refining the + // `refine` largest by ascent, and the start whose ascent reached it (the witness). + private static (double Sampled, double Refined, float3 Witness) MeasureMax(Func f, bool planar, int seed, int samples, int refine) + { + var random = new System.Random(seed); + var points = new float3[samples]; + var values = new double[samples]; + for (var i = 0; i < samples; i++) + { + points[i] = new float3(Uniform(random, -256.0f, 256.0f), Uniform(random, -256.0f, 256.0f), planar ? 0.0f : Uniform(random, -256.0f, 256.0f)); + values[i] = f(points[i]); + } + + var order = Enumerable.Range(0, samples).OrderByDescending(i => values[i]).Take(Math.Max(refine, 1)).ToArray(); + var refined = values[order[0]]; + var witness = points[order[0]]; + foreach (var i in order.Take(refine)) + { + var value = planar ? Ascend2(q => f(new float3(q.x, q.y, 0.0f)), new float2(points[i].x, points[i].y)) : Ascend(f, points[i]); + if (value > refined) + { + refined = value; + witness = points[i]; + } + } + + return (values[order[0]], refined, witness); + } + + private static double Gradient2At(float3 p) => Gradient2Norm(new float2(p.x, p.y)); + + /// + /// The provenance of SNOISE_HESSIAN: the largest spectral norm of snoise's Hessian (HessianNorm) over + /// 1e6 seeded points of [-256, 256]^3, the 1e4 largest refined by ascent, times 1.10. Prints the + /// start whose ascent reached the maximum, which HessianConstantPinsSampledCurvature climbs from + /// again. Slow; run explicitly after any change to the snoise kernel and re-pin the constant and the + /// witness. + /// + [Fact(Explicit = true)] + [Trait("Category", "Slow")] + public void MeasureHessian() + { + var (sampled, refined, witness) = MeasureMax(HessianNorm, false, 0x4E55, 1_000_000, 10_000); + output.WriteLine($"IV-REPORT hessian: max |H| sampled {sampled:R}, refined {refined:R}, x 1.10 = {(float)(refined * 1.10):R}, witness ({witness.x:R}, {witness.y:R}, {witness.z:R})"); + Assert.True(refined <= SNOISE_HESSIAN, $"measured {refined:R} exceeds SNOISE_HESSIAN {SNOISE_HESSIAN:R}"); + } + + // The start MeasureHessian printed as its witness. + private static readonly float3 HessianWitness = new(-245.19885f, -53.01323f, -91.47363f); + + /// + /// Pins SNOISE_HESSIAN: no Hessian norm over 1e5 seeded points exceeds it, and the ascent from + /// MeasureHessian's witness reaches a maximum that, times 1.10, is the constant to within half a + /// percent, so a constant moved by 1% fails here. + /// + [Fact] + public void HessianConstantPinsSampledCurvature() + { + var (sampled, _, _) = MeasureMax(HessianNorm, false, 0x4E56, 100_000, 0); + Assert.True(sampled <= SNOISE_HESSIAN, $"the Hessian norm reaches {sampled:R}, above SNOISE_HESSIAN {SNOISE_HESSIAN:R}"); + var refined = Ascend(HessianNorm, HessianWitness); + Assert.True(refined <= SNOISE_HESSIAN, $"the Hessian norm reaches {refined:R}, above SNOISE_HESSIAN {SNOISE_HESSIAN:R}"); + var ratio = SNOISE_HESSIAN / (1.10 * refined); + Assert.True(Math.Abs(ratio - 1.0) <= 0.005, $"SNOISE_HESSIAN is {ratio:R} times 1.10 x the re-measured {refined:R}; re-run MeasureHessian and re-pin"); + } + + /// + /// The provenance of SNOISE2_LIPSCHITZ: the largest gradient length of snoise(float2) (Gradient2Norm) + /// over 1e6 seeded points of [-256, 256]^2, the 1e4 largest refined by ascent, times 1.10; and the + /// largest |snoise(float2)| reached by ascent from 1e4 starts, which is what lets iv_snoise_ball(float2) + /// intersect with [-1, 1]. Slow; run explicitly after any change to the 2D kernel and re-pin. + /// + [Fact(Explicit = true)] + [Trait("Category", "Slow")] + public void MeasureLipschitz2() + { + var (sampled, refined, witness) = MeasureMax(Gradient2At, true, 0x2D15, 1_000_000, 10_000); + var random = new System.Random(0x2D16); + var maxValue = Enumerable.Range(0, 10_000) + .Select(_ => new float2(Uniform(random, -256.0f, 256.0f), Uniform(random, -256.0f, 256.0f))) + .Max(p => Ascend2(q => Math.Abs(snoise(q)), p)); + output.WriteLine($"IV-REPORT lipschitz2: max |grad snoise2| sampled {sampled:R}, refined {refined:R}, x 1.10 = {(float)(refined * 1.10):R}, witness ({witness.x:R}, {witness.y:R}); max |snoise2| refined {maxValue:R}"); + Assert.True(refined <= SNOISE2_LIPSCHITZ, $"measured {refined:R} exceeds SNOISE2_LIPSCHITZ {SNOISE2_LIPSCHITZ:R}"); + Assert.True(maxValue < 1.0, $"|snoise2| reaches {maxValue:R}; iv_snoise_ball(float2) may not intersect with [-1, 1]"); + } + + // The start MeasureLipschitz2 printed as its witness. + private static readonly float2 Lipschitz2Witness = new(-67.12454f, -129.53296f); + + /// + /// Pins SNOISE2_LIPSCHITZ as LipschitzConstantPinsSampledGradients pins SNOISE_LIPSCHITZ: 1e5 seeded + /// points under it, and the ascent from MeasureLipschitz2's witness within half a percent of it / 1.10. + /// + [Fact] + public void Lipschitz2ConstantPins() + { + var (sampled, _, _) = MeasureMax(Gradient2At, true, 0x2D17, 100_000, 0); + Assert.True(sampled <= SNOISE2_LIPSCHITZ, $"|grad snoise2| reaches {sampled:R}, above SNOISE2_LIPSCHITZ {SNOISE2_LIPSCHITZ:R}"); + var refined = Ascend2(Gradient2Norm, Lipschitz2Witness); + Assert.True(refined <= SNOISE2_LIPSCHITZ, $"|grad snoise2| reaches {refined:R}, above SNOISE2_LIPSCHITZ {SNOISE2_LIPSCHITZ:R}"); + var ratio = SNOISE2_LIPSCHITZ / (1.10 * refined); + Assert.True(Math.Abs(ratio - 1.0) <= 0.005, $"SNOISE2_LIPSCHITZ is {ratio:R} times 1.10 x the re-measured {refined:R}; re-run MeasureLipschitz2 and re-pin"); + } + + /// + /// iv_snoise_ball(float2) encloses snoise(float2) over 2,000 seeded discs of radius log-uniform in + /// [1e-3, 1e3] x 64 points (the first eight on the rim), no tolerance. + /// + [Fact] + public void Snoise2BallEnclosesPoints() + { + var random = new System.Random(0xBA12); + for (var b = 0; b < 2000; b++) + { + var centre = new float2(Uniform(random, -20.0f, 20.0f), Uniform(random, -20.0f, 20.0f)); + var radius = LogUniform(random, 1.0e-3f, 1.0e3f); + var bound = iv_snoise_ball(centre, radius); + for (var i = 0; i < 64; i++) + { + var angle = Uniform(random, 0.0f, 2.0f * MathF.PI); + var x = centre + new float2(MathF.Cos(angle), MathF.Sin(angle)) * (i < 8 ? radius : radius * MathF.Sqrt(random.NextSingle())); + Assert.True(Inside(snoise(x), bound), $"iv_snoise_ball({centre}, {radius:R}) = {bound} misses snoise({x}) = {snoise(x):R}"); + } + } + } + + /// + /// The centred warp (design.md, "Affine forms"): over 2,000 seeded balls of the deep well's flow (warp + /// D = 60, period 8, flow frequency 1/512; centres in [-2000, 2000]^3, radius log-uniform in [1, 500], + /// a time per ball) x 64 points, each phase's warped point p + flow(p) shift_k lies in the ball + /// (c + flow(c) shift_k, r + |shift_k| rho) (geometry allows one part in 1e5 for the rounding of the + /// warped point, as WarpedPointsStayEnclosed does), and snoise of it, at the coverage frequency, lies + /// in iv_snoise_ball of that ball with no tolerance. Dropping rho fails: some warped points lie + /// farther than r from the moved centre. + /// + [Fact] + public void CentredWarpStaysEnclosed() + { + var noise = new WarpedNoise { F0 = 1.0f / FogField.NoiseScale, Warp = FogField.AetheriaWarp }; + var random = new System.Random(0xCE17); + var beyond = 0; + for (var b = 0; b < 2000; b++) + { + var centre = new float3(Uniform(random, -2000.0f, 2000.0f), Uniform(random, -2000.0f, 2000.0f), Uniform(random, -2000.0f, 2000.0f)); + var radius = LogUniform(random, 1.0f, 500.0f); + var time = Uniform(random, 0.0f, 100.0f); + var (_, shift0, shift1) = noise.Phases(time); + var (c0, g0, c1, g1) = noise.CentredWarp(centre, radius, time); + for (var i = 0; i < 64; i++) + { + var p = PointInBall(random, centre, radius, i); + var flow = noise.Flow(p); + foreach (var (shift, moved, grow) in new[] { (shift0, c0, g0), (shift1, c1, g1) }) + { + var warped = p + flow * shift; + var distance = length(warped - moved); + Assert.True(distance <= (radius + grow) * (1.0f + 1.0e-5f), $"warped point {warped} leaves the centred ball ({moved}, {radius + grow:R})"); + beyond += distance > radius ? 1 : 0; + var bound = iv_snoise_ball(moved * noise.F0, (radius + grow) * noise.F0); + var value = snoise(warped * noise.F0); + Assert.True(Inside(value, bound), $"iv_snoise_ball over the centred ball ({moved}, {radius + grow:R}) = {bound} misses snoise = {value:R} at {warped}"); + } + } + } + + output.WriteLine($"IV-REPORT centred warp: {beyond} of {2000 * 64 * 2} warped points lie beyond r from the moved centre"); + Assert.True(beyond > 0, "no warped point needs rho: the check cannot tell a centred warp without it"); + } + + // Why a field is not ten bands deep and wide, or null. The band G is the span of s where only the + // noise decides density: s in [F + A lo, F + A hi] for the free noise range [lo, hi], with the fade at 1. + private static string? TenBandsShortfall(FogField field, int cameras) + { + var band = FogField.Amplitude * (FogField.FreeNoise.y - FogField.FreeNoise.x); + var well = field.Bowls[^1]; + var rim = well.Scale / 2.0f; + float At(float r) => field.HeightAt(well.Centre + new float2(r, 0.0f)); + var depth = At(0.0f) - At(rim); + var half = (At(0.0f) + At(rim)) / 2.0f; + float lo = 0.0f, hi = rim; + for (var k = 0; k < 60; k++) + { + var mid = (lo + hi) / 2.0f; + if (At(mid) > half) + lo = mid; + else + hi = mid; + } + + var problems = new List(); + if (depth < 10.0f * band) + problems.Add($"depth {depth:R} is under 10 G = {10.0f * band:R}"); + if (lo < 10.0f * band) + problems.Add($"half-depth radius {lo:R} is under 10 G = {10.0f * band:R}"); + var random = new System.Random(0xD33B); + for (var c = 0; c < cameras; c++) + { + var origin = field.DrawTile(random, 8).Origin; + var xz = new float2(origin.x, origin.z); + var s = origin.y + field.HeightAt(xz); + if (!(length(xz - well.Centre) < lo && origin.y < FogField.FloorOffset - At(rim) && s >= FogField.FloorOffset + FogField.Amplitude * FogField.FreeNoise.y)) + { + problems.Add($"camera {origin} is not inside the well, below the rim's fog surface, in empty air"); + break; + } + } + + return problems.Count == 0 ? null : $"{field.Name}: G = {band:R}; " + string.Join("; ", problems); + } + + /// + /// Ruling wells-scale-order-of-magnitude: the band G is computed from FogField's FloorOffset, Amplitude + /// and free noise range (60). The deep well's depth, h(centre) - h(rim), and its half-depth radius, + /// found by bisection on the well map, are each at least 10 G, and each of 1,000 drawn cameras sits + /// inside that radius, below the rim's fog surface and in empty air (s above the band). A well of depth + /// 150 fails the same check. + /// + [Fact] + public void DeepWellIsTenBandsDeepAndWide() + { + var deep = FogField.DeepWell(FogField.AetheriaWarp, BoundMode.Interval); + Assert.Null(TenBandsShortfall(deep, 1000)); + Assert.NotNull(TenBandsShortfall(FogField.DeepWell(FogField.AetheriaWarp, BoundMode.Interval, depth: 150.0f), 1000)); + var band = FogField.Amplitude * (FogField.FreeNoise.y - FogField.FreeNoise.x); + output.WriteLine($"IV-REPORT deep well: G = {band:R}, depth {deep.HeightAt(new float2(0.0f, 0.0f)) - deep.HeightAt(new float2(FogField.DeepWellScale / 2.0f, 0.0f)):R}"); + } + + // ---- The cull map ---- + + // An approximation of viridis at nine anchors, interpolated linearly, for t in [0, 1]. + private static readonly (byte R, byte G, byte B)[] Viridis = + { + (68, 1, 84), (72, 40, 120), (62, 74, 137), (49, 104, 142), (38, 130, 142), (31, 158, 137), (53, 183, 121), (110, 206, 88), (253, 231, 37), + }; + + private static (byte, byte, byte) Colour(double t) + { + t = Math.Clamp(t, 0.0, 1.0) * (Viridis.Length - 1); + var i = Math.Min((int)t, Viridis.Length - 2); + var f = t - i; + byte Mix(byte a, byte b) => (byte)Math.Round(a + (b - a) * f); + return (Mix(Viridis[i].R, Viridis[i + 1].R), Mix(Viridis[i].G, Viridis[i + 1].G), Mix(Viridis[i].B, Viridis[i + 1].B)); + } + + private static uint[] CrcTable() => Enumerable.Range(0, 256).Select(n => + { + var c = (uint)n; + for (var k = 0; k < 8; k++) + c = (c & 1) != 0 ? 0xEDB88320u ^ (c >> 1) : c >> 1; + return c; + }).ToArray(); + + // A minimal 8-bit RGB PNG: one IDAT of zlib-compressed rows, filter 0. + private static void WritePng(string path, int width, int height, byte[] rgb) + { + var table = CrcTable(); + using var file = File.Create(path); + void Chunk(string type, byte[] data) + { + var name = System.Text.Encoding.ASCII.GetBytes(type); + var length = new[] { (byte)(data.Length >> 24), (byte)(data.Length >> 16), (byte)(data.Length >> 8), (byte)data.Length }; + file.Write(length); + file.Write(name); + file.Write(data); + var crc = 0xFFFFFFFFu; + foreach (var b in name.Concat(data)) + crc = table[(crc ^ b) & 0xFF] ^ (crc >> 8); + crc ^= 0xFFFFFFFFu; + file.Write(new[] { (byte)(crc >> 24), (byte)(crc >> 16), (byte)(crc >> 8), (byte)crc }); + } + + file.Write(new byte[] { 137, 80, 78, 71, 13, 10, 26, 10 }); + Chunk("IHDR", new byte[] { (byte)(width >> 24), (byte)(width >> 16), (byte)(width >> 8), (byte)width, (byte)(height >> 24), (byte)(height >> 16), (byte)(height >> 8), (byte)height, 8, 2, 0, 0, 0 }); + using var packed = new MemoryStream(); + using (var z = new System.IO.Compression.ZLibStream(packed, System.IO.Compression.CompressionLevel.Optimal, leaveOpen: true)) + { + for (var y = 0; y < height; y++) + { + z.WriteByte(0); + z.Write(rgb, y * width * 3, width * 3); + } + } + + Chunk("IDAT", packed.ToArray()); + Chunk("IEND", Array.Empty()); + } + + /// + /// The cull map of one deep-well frame at warp D, for the operator: a camera inside the well at xz + /// (0, -200), y = F - h / 2, looking along +z across the well, 960 x 540 pixels at a 60 degree vertical + /// field of view, in 8 x 8 tiles (the last row of tiles half off the frame). Each tile runs the + /// pre-pass under the centred-warp interval bound and under the affine bound; each pixel's ray (through + /// the pixel's centre) is marched densely for its transmittance and over each tile mask and the oracle + /// mask (the cells some ray of the tile samples nonzero), with the early-out at 0.02. The PNG is + /// 1920 x 1080 in four panels: top left the cells per ray the interval march integrates, top right the + /// affine march's, bottom left the oracle march's (all three on one viridis scale, 0 to the largest + /// count), bottom right the transmittance (black opaque, white clear). It is written only when + /// CULTMATH_WRITE_CULLMAP names a directory; the mean cells per ray are printed either way. + /// + [Fact(Explicit = true)] + [Trait("Category", "Slow")] + public void DeepWellCullMap() + { + const int width = 960, height = 540, size = 8; + var interval = FogField.DeepWell(FogField.AetheriaWarp, BoundMode.Interval); + var affine = FogField.DeepWell(FogField.AetheriaWarp, BoundMode.Affine); + var focal = height / 2.0f / MathF.Tan(MathF.PI / 6.0f); + var xz = new float2(0.0f, -200.0f); + var origin = new float3(xz.x, FogField.FloorOffset - 0.5f * interval.HeightAt(xz), xz.y); + var grid = interval.Grid; + var (cellsInterval, cellsAffine, cellsOracle) = (new int[width * height], new int[width * height], new int[width * height]); + var transmittance = new float[width * height]; + var (tilesX, tilesY) = (width / size, (height + size - 1) / size); + var probes = new Counts[tilesX * tilesY]; + var clock = System.Diagnostics.Stopwatch.StartNew(); + Parallel.For(0, tilesX * tilesY, index => + { + var (tx, ty) = (index % tilesX, index / tilesX); + var slope = new float2((tx * size + size / 2.0f - width / 2.0f) / focal, (height / 2.0f - (ty * size + size / 2.0f)) / focal); + var tile = interval.TileAt(origin, 0.0f, slope, size, focal, 3.0f); + probes[index] = new Counts(); + var (maskInterval, _, _) = MarchTile(interval, tile, probes[index]); + var (maskAffine, _, _) = MarchTile(affine, tile, new Counts()); + var scratch = new Counts(); + var rays = new List<(int Pixel, float2 Slope)>(); + for (var j = 0; j < size; j++) + for (var i = 0; i < size; i++) + { + var (px, py) = (tx * size + i, ty * size + (size - 1 - j)); + rays.Add((py < height ? py * width + px : -1, tile.PixelSlope(i, j, 0.5f, 0.5f))); + } + + var oracle = Enumerable.Range(0, grid.Length - 1).Where(c => rays.Any(r => interval.Density(tile, r.Slope, (grid[c] + grid[c + 1]) * 0.5f, false, scratch) > 0.0f)).ToList(); + foreach (var (pixel, ray) in rays) + { + if (pixel < 0) + continue; + int Steps(List mask) + { + var t = 1.0f; + var k = 0; + for (; k < mask.Count && t >= 0.02f; k++) + t *= Integrate(interval, tile, ray, grid[mask[k]], grid[mask[k] + 1], lod: false, scratch); + return k; + } + + var dense = 1.0f; + for (var c = 0; c < grid.Length - 1 && dense >= 0.02f; c++) + dense *= Integrate(interval, tile, ray, grid[c], grid[c + 1], lod: false, scratch); + transmittance[pixel] = dense; + cellsInterval[pixel] = Steps(maskInterval); + cellsAffine[pixel] = Steps(maskAffine); + cellsOracle[pixel] = Steps(oracle); + } + }); + + var most = Math.Max(cellsInterval.Max(), Math.Max(cellsAffine.Max(), cellsOracle.Max())); + output.WriteLine($"IV-REPORT cull map (deep well, warp {FogField.AetheriaWarp:R}, {width}x{height}, {size}x{size} tiles, {clock.Elapsed.TotalSeconds:F0} s): cells per ray interval {cellsInterval.Average():F2}, affine {cellsAffine.Average():F2}, oracle {cellsOracle.Average():F2}; " + + $"largest {most}; interval probes per tile env {probes.Average(p => p.Envelope):F1} snoise {probes.Average(p => p.Snoise):F2}; mean transmittance {transmittance.Average():F3}"); + var directory = Environment.GetEnvironmentVariable("CULTMATH_WRITE_CULLMAP"); + if (string.IsNullOrEmpty(directory)) + return; + var rgb = new byte[2 * width * 2 * height * 3]; + void Put(int x, int y, (byte R, byte G, byte B) c) + { + var o = (y * 2 * width + x) * 3; + (rgb[o], rgb[o + 1], rgb[o + 2]) = c; + } + + for (var y = 0; y < height; y++) + for (var x = 0; x < width; x++) + { + var p = y * width + x; + Put(x, y, Colour(cellsInterval[p] / (double)most)); + Put(width + x, y, Colour(cellsAffine[p] / (double)most)); + Put(x, height + y, Colour(cellsOracle[p] / (double)most)); + var grey = (byte)Math.Round(255.0 * transmittance[p]); + Put(width + x, height + y, (grey, grey, grey)); + } + + for (var k = 0; k < 2 * width; k++) + for (var d = -1; d <= 0; d++) + Put(k, height + d, (255, 255, 255)); + for (var k = 0; k < 2 * height; k++) + for (var d = -1; d <= 0; d++) + Put(width + d, k, (255, 255, 255)); + Directory.CreateDirectory(directory); + var path = Path.Combine(directory, "deep-well-cull-map.png"); + WritePng(path, 2 * width, 2 * height, rgb); + output.WriteLine($"IV-REPORT cull map written to {path}"); + } + /// /// Mean of interval width over the true range (max - min of the 64 sampled values) per ball: how /// loose the Lipschitz bound is, for a later affine cut to beat. Asserts only finiteness. @@ -519,12 +1004,18 @@ public sealed class NoiseBoundTests // counts are printed, so any other weight can be applied to them. internal const double EnvelopeCost = 0.2; + // One snoise_grad evaluation (af_snoise's one call) costs this much in snoise units: the value and + // three gradient components over the same corners. An estimate, printed beside the count as the + // envelope weight is, and not tuned toward any contract. + internal const double GradCost = 2.0; + internal sealed class Counts { public long Snoise; public long Envelope; + public long Grad; - public double Cost => Snoise + EnvelopeCost * Envelope; + public double Cost => Snoise + EnvelopeCost * Envelope + GradCost * Grad; } /// @@ -580,6 +1071,13 @@ public sealed class NoiseBoundTests /// noise ball only when the best outcome any centre value could give would prove it; ungated, the ball /// is always evaluated. /// + /// FogField's bound: the centred-warp interval composition, or the affine one (design.md, "Affine forms"). + internal enum BoundMode + { + Interval, + Affine, + } + internal abstract class Scenario { public required string Name { get; init; } @@ -645,7 +1143,7 @@ public sealed class NoiseBoundTests private float FlowAmplitude => Warp * 2.0f / Period; - private float3 Flow(float3 p) + public float3 Flow(float3 p) { var q = (FlowInXY ? new float2(p.x, p.y) : new float2(p.x, p.z)) * FlowFrequency; var v = new float2(snoise(q + new float2(17.0f, 3.0f)), snoise(q + new float2(-5.0f, 41.0f))); @@ -653,7 +1151,7 @@ public sealed class NoiseBoundTests return FlowInXY ? new float3(v.x, v.y, 0.0f) : new float3(v.x, 0.0f, v.y); } - private (float W0, float Shift0, float Shift1) Phases(float time) + public (float W0, float Shift0, float Shift1) Phases(float time) { var phase0 = frac(time / Period); var phase1 = frac(time / Period + 0.5f); @@ -674,12 +1172,47 @@ public sealed class NoiseBoundTests return snoise(p * (4.0f * F0) + 17.0f); } - // Both phases' warped points lie in a ball that carries the warp, so one ball encloses both snoise; - // the cross-fade is enclosed term by term with the frame's own weights. - public float2 CoverageBound(float2 ball, float time) + // The cross-fade of the two phases' noise enclosures, term by term with the frame's own weights. + // A field whose ball carries the whole warp passes the same enclosure for both phases. + public float2 CoverageBound(float2 phase0, float2 phase1, float time) { var w0 = Phases(time).W0; - return iv_add(iv_scale(ball, w0), iv_scale(ball, 1.0f - w0)); + return iv_add(iv_scale(phase0, w0), iv_scale(phase1, 1.0f - w0)); + } + + /// + /// How far the flow can move across a ball of radius r about c: each flow component is snoise(float2) + /// of the point projected to the flow plane and scaled by FlowFrequency, which varies over the ball by + /// at most iv_snoise_ball(float2)'s reach from its centre value; the projection and the clamp to the + /// unit disc are nonexpansive, and two unit-disc vectors differ by at most 2. + /// + public float FlowVariation(float3 centre, float radius) + { + var q = (FlowInXY ? new float2(centre.x, centre.y) : new float2(centre.x, centre.z)) * FlowFrequency; + var u = Deviation(q + new float2(17.0f, 3.0f), radius * FlowFrequency); + var v = Deviation(q + new float2(-5.0f, 41.0f), radius * FlowFrequency); + return min(sqrt(u * u + v * v), 2.0f) * FlowAmplitude; + } + + private static float Deviation(float2 q, float radius) + { + var n = snoise(q); + var ball = iv_snoise_ball(q, radius); + return max(ball.y - n, n - ball.x); + } + + /// + /// The centred warp (design.md, "Affine forms"): phase k moves a point p of a set within reach of c + /// to p + flow(p) shift_k, which lies within |shift_k| rho of (p - c) + c + flow(c) shift_k, rho = + /// FlowVariation(c, reach). So phase k's noise argument is the set moved by flow(c) shift_k and + /// grown by |shift_k| rho, instead of grown by the whole warp. At warp 0 both phases are the set. + /// + public (float3 Centre0, float Grow0, float3 Centre1, float Grow1) CentredWarp(float3 centre, float reach, float time) + { + var (_, shift0, shift1) = Phases(time); + var flow = Flow(centre); + var rho = FlowVariation(centre, reach); + return (centre + flow * shift0, abs(shift0) * rho, centre + flow * shift1, abs(shift1) * rho); } public float2 CoverageBall(float3 centre, float radius, Counts counts) => NoiseBall(centre * F0, radius * F0, counts); @@ -713,9 +1246,10 @@ public sealed class NoiseBoundTests public const float NoiseScale = 414.2167f; // :409 public const float AetheriaWarp = 60.0f; // Flow GlobalAmplitude 15 x Period 8 / 2, :402-405 public const float ZoneRadius = 2000.0f; - private static readonly float2 FreeNoise = new(-1.5f, 1.5f); + internal static readonly float2 FreeNoise = new(-1.5f, 1.5f); public required WarpedNoise Noise { get; init; } + public required BoundMode Mode { get; init; } public (float2 Centre, float Scale, float Exponent, float Depth)[] Bowls { get; init; } = Array.Empty<(float2, float, float, float)>(); public required Func CameraAt { get; init; } public required float2 SlopesY { get; init; } @@ -731,26 +1265,60 @@ public sealed class NoiseBoundTests private static WarpedNoise AetheriaNoise(float warp) => new() { F0 = 1.0f / NoiseScale, Warp = warp }; + private static string Named(string name, BoundMode mode) => $"{name} [{(mode == BoundMode.Affine ? "affine" : "interval")}]"; + /// (a) Height fog: no wells, the camera above the safety band at y = S + 30, rays level and up. - public static FogField HeightFog(float warp) => new() + public static FogField HeightFog(float warp, BoundMode mode) => new() { - Name = "(a) height fog", Noise = AetheriaNoise(warp), SlopesY = new float2(0.0f, 0.5f), + Name = Named("(a) height fog", mode), Mode = mode, Noise = AetheriaNoise(warp), SlopesY = new float2(0.0f, 0.5f), CameraAt = (random, _) => new float3(Uniform(random, -1000.0f, 1000.0f), Safety + 30.0f, Uniform(random, -1000.0f, 1000.0f)), }; /// (b) Inside the fog: no wells, the camera at y = F - 10, rays level. - public static FogField InsideFog(float warp) => new() + public static FogField InsideFog(float warp, BoundMode mode) => new() { - Name = "(b) inside the fog", Noise = AetheriaNoise(warp), SlopesY = new float2(-0.05f, 0.05f), + Name = Named("(b) inside the fog", mode), Mode = mode, Noise = AetheriaNoise(warp), SlopesY = new float2(-0.05f, 0.05f), CameraAt = (random, _) => new float3(Uniform(random, -1000.0f, 1000.0f), FloorOffset - 10.0f, Uniform(random, -1000.0f, 1000.0f)), }; + // The deep well's geometry (ruling wells-scale-order-of-magnitude): one PowerPulse well of exponent + // 2, scale 2220 and depth 600 at the zone's centre, so its depth and half-depth radius are each + // about 600, ten times the band (DeepWellIsTenBandsDeepAndWide measures both). + public const float DeepWellScale = 2220.0f; + public const float DeepWellDepth = 600.0f; + public const float DeepWellCameraRadius = 300.0f; + /// - /// (c) Aetheria-like: the zone bowl R = 2000 and four wells of mass 100, 1000, 10000 and 1000 at seeded - /// positions inside it; the camera at a seeded xz in the bowl, above the noise band at - /// y = max(0, S + 10 - h), gazing across with m_y in [-0.25, 0.1]. + /// (c) The deep well, the case Aetheria plays in: there the camera is always in some gravity well, + /// gazing across a bowl a sun's gravity carves (ruling wells-scale-order-of-magnitude). The zone bowl + /// R = 2000 plus one well at its centre, PowerPulse exponent 2, scale 2220, depth 600, so the well's + /// depth and half-depth radius are each about ten times G = 3 A = 60, the band of s where only the + /// noise decides density (s in [F - 1.5 A, F + 1.5 A] with the fade at 1). The camera's xz is uniform + /// within 300 of the centre, at y = F - h(xz) / 2, halfway between the sunken fog surface and the + /// rim's, in empty air; slopes and yaw as the shallow wells, m_y in [-0.25, 0.1]. A level ray crosses + /// hundreds of units of empty well before the far wall's band. /// - public static FogField Wells(float warp) + public static FogField DeepWell(float warp, BoundMode mode, float depth = DeepWellDepth) => new() + { + Name = Named("(c) deep well", mode), Mode = mode, Noise = AetheriaNoise(warp), SlopesY = new float2(-0.25f, 0.1f), + Bowls = new (float2, float, float, float)[] { (new float2(0.0f, 0.0f), 2.0f * ZoneRadius, 2.0f, 64.0f), (new float2(0.0f, 0.0f), DeepWellScale, 2.0f, depth) }, + CameraAt = (r, field) => + { + var angle = Uniform(r, 0.0f, 2.0f * MathF.PI); + var radius = DeepWellCameraRadius * MathF.Sqrt(r.NextSingle()); + var xz = new float2(radius * MathF.Cos(angle), radius * MathF.Sin(angle)); + return new float3(xz.x, FloorOffset - 0.5f * field.HeightAt(xz), xz.y); + }, + }; + + /// + /// (c-band) Shallow wells, the band-dominated row, printed only: the zone bowl R = 2000 (64 deep) and + /// four wells of mass 100, 1000, 10000 and 1000 at seeded positions inside it, 30 M^0.175 deep (67 to + /// 150, 1.1 to 2.5 times G) and 500 M^0.25 wide at exponent 16; the camera at a seeded xz in the bowl, + /// just above the noise band at y = max(0, S + 10 - h), gazing across with m_y in [-0.25, 0.1]. Most + /// of its cost is the band itself, which is where affine noise gains most. + /// + public static FogField Wells(float warp, BoundMode mode) { var random = new System.Random(0xA37E); var bowls = new List<(float2, float, float, float)> { (new float2(0.0f, 0.0f), 2.0f * ZoneRadius, 2.0f, 64.0f) }; @@ -763,14 +1331,13 @@ public sealed class NoiseBoundTests return new FogField { - Name = "(c) Aetheria wells", Noise = AetheriaNoise(warp), Bowls = bowls.ToArray(), SlopesY = new float2(-0.25f, 0.1f), + Name = Named("(c-band) shallow wells", mode), Mode = mode, Noise = AetheriaNoise(warp), Bowls = bowls.ToArray(), SlopesY = new float2(-0.25f, 0.1f), CameraAt = (r, field) => { var angle = Uniform(r, 0.0f, 2.0f * MathF.PI); var radius = ZoneRadius * MathF.Sqrt(r.NextSingle()); var xz = new float2(radius * MathF.Cos(angle), radius * MathF.Sin(angle)); - var h = field.Height(field.Bowls.Select(b => b.Centre - xz).ToArray(), 0.0f, 0.0f); - return new float3(xz.x, max(0.0f, Safety + 10.0f - h), xz.y); + return new float3(xz.x, max(0.0f, Safety + 10.0f - field.HeightAt(xz)), xz.y); }, }; } @@ -784,11 +1351,15 @@ public sealed class NoiseBoundTests { var origin = CameraAt(random, this); var yaw = Uniform(random, 0.0f, 2.0f * MathF.PI); - var tile = new FogTile - { - Origin = origin, N = n, Focal = Focal, Time = Uniform(random, 0.0f, 100.0f), - Slope = new float2(Uniform(random, -1.0f, 1.0f), Uniform(random, SlopesY.x, SlopesY.y)), - }; + var time = Uniform(random, 0.0f, 100.0f); + var slope = new float2(Uniform(random, -1.0f, 1.0f), Uniform(random, SlopesY.x, SlopesY.y)); + return TileAt(origin, yaw, slope, n, Focal, time); + } + + /// The tile of a camera at origin, yawed about y (0 looks along +z), at the given slope, size, focal length and time. + public Tile TileAt(float3 origin, float yaw, float2 slope, int n, float focal, float time) + { + var tile = new FogTile { Origin = origin, N = n, Focal = focal, Time = time, Slope = slope }; tile.Look(new float3(MathF.Sin(yaw), 0.0f, MathF.Cos(yaw))); tile.Centres = Bowls.Select(b => { @@ -819,6 +1390,9 @@ public sealed class NoiseBoundTests return h; } + /// The well map h at the world point (x, z). + public float HeightAt(float2 xz) => Height(Bowls.Select(b => b.Centre).ToArray(), xz.x, xz.y); + // Each bowl's term over the box is [pulse(far), pulse(near)], pulse decreasing in distance; pow is // not required to be correctly rounded, so both ends are widened by one ulp. private float2 HeightBound(float2[] centres, float2 x, float2 z) @@ -861,6 +1435,12 @@ public sealed class NoiseBoundTests return max(0.0f, (FloorOffset - (s + fade * n * Amplitude)) / FloorBlend); } + /// + /// The envelope first: the height and slope intervals over the slice's box, the fade, and the noise + /// at its free range. Where that already proves the slice empty the envelope is the answer; + /// otherwise Mode's bound, which (gated) returns the envelope when the best any probe could give + /// would not prove the slice either. + /// public override float2 Bound(Tile t, float z0, float z1, bool gated, Counts counts) { var tile = (FogTile)t; @@ -872,24 +1452,106 @@ public sealed class NoiseBoundTests var envelope = Compose(s, fade, FreeNoise); if (gated && envelope.y <= 0.0f) return envelope; - var ball = iv_frustum_ball(tile.Slope, z0, z1, tile.Footprint, Noise.Warp); + return Mode == BoundMode.Affine + ? AffineBound(tile, z0, z1, height, s, fade, envelope, gated, counts) + : IntervalBound(tile, z0, z1, s, fade, envelope, gated, counts); + } + + private static bool Same(float3 a, float3 b, float ga, float gb) => all(a == b) && ga == gb; + + /// + /// The centred-warp interval bound. The slice's ball is iv_frustum_ball with no warp; each phase's + /// coverage ball is that ball moved by the flow at its centre and grown by what the flow can vary + /// across it (WarpedNoise.CentredWarp), one ball when the phases coincide (warp 0). The detail octave + /// reads the unwarped point, so its ball is the unwarped one. + /// + private float2 IntervalBound(FogTile tile, float z0, float z1, float2 s, float2 fade, float2 envelope, bool gated, Counts counts) + { + var ball = iv_frustum_ball(tile.Slope, z0, z1, tile.Footprint, 0.0f); + var centre = tile.World(new float3(ball.x, ball.y, ball.z)); + var (c0, g0, c1, g1) = Noise.CentredWarp(centre, ball.w, tile.Time); if (gated) { var provable = false; foreach (var low in new[] { true, false }) { - var best = iv_add(Noise.CoverageBound(BestBall(Noise.CoverageRadius(ball.w), low), tile.Time), iv_scale(BestBall(Noise.DetailRadius(ball.w), low), 0.5f)); - provable |= Compose(s, fade, best).y <= 0.0f; + var coverage = Noise.CoverageBound(BestBall(Noise.CoverageRadius(ball.w + g0), low), BestBall(Noise.CoverageRadius(ball.w + g1), low), tile.Time); + provable |= Compose(s, fade, iv_add(coverage, iv_scale(BestBall(Noise.DetailRadius(ball.w), low), 0.5f))).y <= 0.0f; } if (!provable) return envelope; } - var centre = tile.World(new float3(ball.x, ball.y, ball.z)); - var noise = iv_add(Noise.CoverageBound(Noise.CoverageBall(centre, ball.w, counts), tile.Time), iv_scale(Noise.DetailBall(centre, ball.w, counts), 0.5f)); + var n0 = Noise.CoverageBall(c0, ball.w + g0, counts); + var n1 = Same(c0, c1, g0, g1) ? n0 : Noise.CoverageBall(c1, ball.w + g1, counts); + var noise = iv_add(Noise.CoverageBound(n0, n1, tile.Time), iv_scale(Noise.DetailBall(centre, ball.w, counts), 0.5f)); return Compose(s, fade, noise); } + + // The best interval any centre value and gradient could give af_snoise over a set of reach R (the + // gradient 0, the value at the far end), toward low or high values: the affine gate. + private static float2 BestForm(float reach, bool low) + { + var e = min(SNOISE_LIPSCHITZ * reach, 0.5f * SNOISE_HESSIAN * reach * reach); + if (e >= 2.0f) + return new float2(-1.0f, 1.0f); + return low ? new float2(-1.0f, min(-1.0f + e, 1.0f)) : new float2(max(1.0f - e, -1.0f), 1.0f); + } + + private static float2 Clip(float2 range) => new(max(range.x, -1.0f), min(range.y, 1.0f)); + + /// + /// The affine bound, one shared symbol: depth z = z_m + h eps over the slice (af_frustum_axis and + /// af_frustum_ball give the noise argument the same eps). s = O_y + m_c,y z + (m_y - m_c,y) z + h(xz) + /// is affine in eps, with the lateral slope-depth term (at most Half z1, plus the rounding of the + /// pointwise m_y z) and the height interval in e. Each phase's coverage and the unwarped detail are + /// af_snoise over the slice moved and grown by the centred warp, the fade enters as af_from_iv, and + /// s' = s + A fade n is ranged by af_range. The result is intersected with the interval composition + /// over the same forms' ranges, each clipped to [-1, 1] (both sound, no further evaluation). The gate + /// is BestForm's. + /// + private float2 AffineBound(FogTile tile, float z0, float z1, float2 height, float2 s, float2 fade, float2 envelope, bool gated, Counts counts) + { + var axisCamera = af_frustum_axis(tile.Slope, z0, z1); + var ball = af_frustum_ball(tile.Slope, z0, z1, tile.Footprint, 0.0f); + var centre = tile.World(new float3(ball.x, ball.y, ball.z)); + var axis = tile.Right * axisCamera.x + tile.Up * axisCamera.y + tile.Forward * axisCamera.z; + var reach = length(axis) + ball.w; + var (c0, g0, c1, g1) = Noise.CentredWarp(centre, reach, tile.Time); + var coverageFrequency = Noise.F0; + var detailFrequency = 4.0f * Noise.F0; + if (gated) + { + var provable = false; + foreach (var low in new[] { true, false }) + { + var coverage = Noise.CoverageBound(BestForm((reach + g0) * coverageFrequency, low), BestForm((reach + g1) * coverageFrequency, low), tile.Time); + provable |= Compose(s, fade, iv_add(coverage, iv_scale(BestForm(reach * detailFrequency, low), 0.5f))).y <= 0.0f; + } + + if (!provable) + return envelope; + } + + var shared = Same(c0, c1, g0, g1); + counts.Grad += shared ? 2 : 3; + var n0 = af_snoise(c0 * coverageFrequency, axis * coverageFrequency, (ball.w + g0) * coverageFrequency); + var n1 = shared ? n0 : af_snoise(c1 * coverageFrequency, axis * coverageFrequency, (ball.w + g1) * coverageFrequency); + var nd = af_snoise(centre * detailFrequency + 17.0f, axis * detailFrequency, ball.w * detailFrequency); + var w0 = Noise.Phases(tile.Time).W0; + var noise = af_add(af_add(af_scale(n0, w0), af_scale(n1, 1.0f - w0)), af_scale(nd, 0.5f)); + + // The clamp residual of z = z_m + h eps(z), 2^-23 (|z_m| + h) (af_frustum_ball). + var residual = (abs(ball.z) + axisCamera.z) * 1.1920928955078125e-7f; + var depth = af_add_iv(af_symbol(ball.z, axisCamera.z), new float2(-residual, residual)); + var lateral = tile.Half * z1 + (abs(tile.Slope.y) + tile.Half) * z1 * 2.384185791015625e-7f; + var level = af_add_iv(af_add_iv(af_add(af_point(tile.Origin.y), af_scale(depth, tile.Slope.y)), new float2(-lateral, lateral)), height); + var displaced = af_range(af_add(level, af_scale(af_mul(af_from_iv(fade), noise), Amplitude))); + var affine = new float2(max(0.0f, (FloorOffset - displaced.y) / FloorBlend), max(0.0f, (FloorOffset - displaced.x) / FloorBlend)); + var interval = Compose(s, fade, iv_add(Noise.CoverageBound(Clip(af_range(n0)), Clip(af_range(n1)), tile.Time), iv_scale(Clip(af_range(nd)), 0.5f))); + return new float2(max(affine.x, interval.x), min(affine.y, interval.y)); + } } /// @@ -937,9 +1599,10 @@ public sealed class NoiseBoundTests { var maybe = new float2(0.0f, float.MaxValue); var ball = iv_frustum_ball(tile.Slope, z0, z1, tile.Footprint, Noise.Warp); - if (gated && Noise.CoverageBound(BestBall(Noise.CoverageRadius(ball.w), low: true), tile.Time).y > Cutoff) + if (gated && Noise.CoverageBound(BestBall(Noise.CoverageRadius(ball.w), low: true), BestBall(Noise.CoverageRadius(ball.w), low: true), tile.Time).y > Cutoff) return maybe; - var coverage = Noise.CoverageBound(Noise.CoverageBall(tile.World(new float3(ball.x, ball.y, ball.z)), ball.w, counts), tile.Time); + var coverageBall = Noise.CoverageBall(tile.World(new float3(ball.x, ball.y, ball.z)), ball.w, counts); + var coverage = Noise.CoverageBound(coverageBall, coverageBall, tile.Time); return coverage.y <= Cutoff ? new float2(0.0f, 0.0f) : maybe; } } @@ -1162,7 +1825,7 @@ public sealed class NoiseBoundTests var provable = false; foreach (var low in new[] { true, false }) { - var best = iv_add(iv_mul(Noise.CoverageBound(BestBall(Noise.CoverageRadius(ball.w), low), tile.Time), wc), iv_mul(iv_scale(BestBall(Noise.DetailRadius(ball.w), low), 0.5f), wd)); + var best = iv_add(iv_mul(Noise.CoverageBound(BestBall(Noise.CoverageRadius(ball.w), low), BestBall(Noise.CoverageRadius(ball.w), low), tile.Time), wc), iv_mul(iv_scale(BestBall(Noise.DetailRadius(ball.w), low), 0.5f), wd)); provable |= Compose(d, fade, best).y <= 0.0f; } @@ -1171,7 +1834,8 @@ public sealed class NoiseBoundTests } var centre = tile.World(new float3(ball.x, ball.y, ball.z)); - var noise = iv_add(iv_mul(Noise.CoverageBound(Noise.CoverageBall(centre, ball.w, counts), tile.Time), wc), iv_mul(iv_scale(Noise.DetailBall(centre, ball.w, counts), 0.5f), wd)); + var coverage = Noise.CoverageBall(centre, ball.w, counts); + var noise = iv_add(iv_mul(Noise.CoverageBound(coverage, coverage, tile.Time), wc), iv_mul(iv_scale(Noise.DetailBall(centre, ball.w, counts), 0.5f), wd)); return Compose(d, fade, noise); } } @@ -1301,6 +1965,7 @@ public sealed class NoiseBoundTests public readonly Counts Probes = new(); public readonly Counts Lod = new(); public readonly Counts Oracle = new(); + public readonly Counts Range = new(); public long Rays; public long Tiles; public long DenseSteps; @@ -1314,6 +1979,10 @@ public sealed class NoiseBoundTests public float MaxOracleDifference; public long OracleEmpty; public long OracleEmptySkipped; + public long RangeEmpty; + public long RangeEmptySkipped; + public long RangeViolations; + public bool Ranged; public double MaxLodDepthError; public double MaxRefDepthError; @@ -1334,6 +2003,17 @@ public sealed class NoiseBoundTests // The oracle-empty cells the pre-pass proves empty; a pre-pass that skips nothing scores 0. public double CullFraction => (double)OracleEmptySkipped / OracleEmpty; + + // The range ceiling: the oracle march's cost over the range mask's, with probes free. The range + // mask is what an exact enclosure of each cell over the tile's whole slice could prove, so no sound + // bound can reach the oracle, only this; it is sampled, so it is an upper estimate. + public double RangeCeiling => Oracle.Cost / Range.Cost; + + // The tile march's efficiency against the range ceiling: the range mask's cost over the tile + // march's with its probes. + public double RangeEfficiency => Range.Cost / (Masked.Cost + Probes.Cost); + + public double RangeCull => (double)RangeEmptySkipped / RangeEmpty; } /// @@ -1344,12 +2024,15 @@ public sealed class NoiseBoundTests /// footprint-aware march as well, and the agreement at the body start (or the grid's end): the optical /// depth of a cutoff-0 footprint march and of the fixed-step cells, each against ConvergedDepth at /// h = 0.5. The oracle mask and the agreement marches count into throwaway Counts, the oracle march into - /// its own, so the cost rows do not move. + /// its own, so the cost rows do not move. With range, the range mask as well: the oracle mask plus + /// every cell where the density is positive at any of 5 x 5 slopes over the tile's slope square + /// (corners included) x 9 depths over the cell; each ray is marched over it into Range, and a cell of + /// it the pre-pass skipped is counted as a range violation (a sampled point a bound failed to enclose). /// - private static RunStats Run(Scenario field, int n, int tiles, int seed, bool lod = false) + private static RunStats Run(Scenario field, int n, int tiles, int seed, bool lod = false, bool range = false) { var random = new System.Random(seed); - var stats = new RunStats(); + var stats = new RunStats { Ranged = range }; var scratch = new Counts(); var grid = field.Grid; for (var t = 0; t < tiles; t++) @@ -1379,6 +2062,34 @@ public sealed class NoiseBoundTests } } + var ranged = new List(); + if (range) + { + var inOracle = new HashSet(oracle); + for (var c = 0; c < grid.Length - 1; c++) + { + var sampled = inOracle.Contains(c); + for (var a = 0; a < 5 && !sampled; a++) + for (var b = 0; b < 5 && !sampled; b++) + for (var k = 0; k < 9 && !sampled; k++) + { + var slope = new float2(tile.SlopesX.x + (tile.SlopesX.y - tile.SlopesX.x) * a / 4.0f, tile.SlopesY.x + (tile.SlopesY.y - tile.SlopesY.x) * b / 4.0f); + sampled = field.Density(tile, slope, grid[c] + (grid[c + 1] - grid[c]) * k / 8.0f, false, scratch) > 0.0f; + } + + if (sampled) + { + ranged.Add(c); + stats.RangeViolations += inMask.Contains(c) ? 0 : 1; + } + else + { + stats.RangeEmpty++; + stats.RangeEmptySkipped += inMask.Contains(c) ? 0 : 1; + } + } + } + var stop = bodyStart >= 0 ? bodyStart : grid.Length - 1; foreach (var slope in slopes) { @@ -1391,6 +2102,9 @@ public sealed class NoiseBoundTests var exact = 1.0f; for (var c = 0; c < oracle.Count && exact >= 0.02f; c++) exact *= Integrate(field, tile, slope, grid[oracle[c]], grid[oracle[c] + 1], lod: false, stats.Oracle); + var rangedT = 1.0f; + for (var c = 0; c < ranged.Count && rangedT >= 0.02f; c++) + rangedT *= Integrate(field, tile, slope, grid[ranged[c]], grid[ranged[c] + 1], lod: false, stats.Range); stats.Rays++; stats.MaxDifference = MathF.Max(stats.MaxDifference, MathF.Abs(dense - masked)); stats.MaxOracleDifference = MathF.Max(stats.MaxOracleDifference, MathF.Abs(dense - exact)); @@ -1420,10 +2134,13 @@ public sealed class NoiseBoundTests private void Report(string label, RunStats s) => output.WriteLine( $"IV-REPORT {label}: dense snoise/ray {s.PerRay(s.Dense.Snoise):F2} env/ray {s.PerRay(s.Dense.Envelope):F2} (samples/ray {s.PerRay(s.DenseSteps):F2}); " - + $"tile snoise/ray {s.PerRay(s.Masked.Snoise + (double)s.Probes.Snoise):F2} env/ray {s.PerRay(s.Masked.Envelope + (double)s.Probes.Envelope):F2} " - + $"(probes/tile env {(double)s.Probes.Envelope / s.Tiles:F1} snoise {(double)s.Probes.Snoise / s.Tiles:F2}; dense cells/ray {s.PerRay(s.MaskedSteps):F2}); " + + $"tile snoise/ray {s.PerRay(s.Masked.Snoise + (double)s.Probes.Snoise):F2} env/ray {s.PerRay(s.Masked.Envelope + (double)s.Probes.Envelope):F2} grad/ray {s.PerRay(s.Probes.Grad):F2} " + + $"(probes/tile env {(double)s.Probes.Envelope / s.Tiles:F1} snoise {(double)s.Probes.Snoise / s.Tiles:F2} grad {(double)s.Probes.Grad / s.Tiles:F2}; dense cells/ray {s.PerRay(s.MaskedSteps):F2}); " + $"cost ratio {s.Ratio:F2}x, oracle ceiling {s.OracleCeiling:F2}x, efficiency {s.Efficiency:F3}, cull fraction {s.CullFraction:F3} ({s.OracleEmptySkipped}/{s.OracleEmpty}), " - + $"probe overhead {s.ProbeOverhead:F3}; max |dT| {s.MaxDifference:R}, oracle {s.MaxOracleDifference:R}; unfinished rays {s.Unfinished}"); + + $"probe overhead {s.ProbeOverhead:F3}; max |dT| {s.MaxDifference:R}, oracle {s.MaxOracleDifference:R}; unfinished rays {s.Unfinished}" + + (s.Ranged + ? $"; range ceiling {s.RangeCeiling:F3} (oracle cost over the range mask's), efficiency against it {s.RangeEfficiency:F3}, range cull {s.RangeCull:F3} ({s.RangeEmptySkipped}/{s.RangeEmpty}), range violations {s.RangeViolations}" + : "")); private void ReportVoid(string label, RunStats s) => output.WriteLine( $"IV-REPORT {label}: steps/px ref {s.PerRay(s.DenseSteps):F2}, masked {s.PerRay(s.MaskedSteps):F2}, LOD {s.PerRay(s.LodSteps):F2} ({(double)s.DenseSteps / s.LodSteps:F2}x fewer); " @@ -1497,8 +2214,10 @@ public sealed class NoiseBoundTests var scenarios = new (float Warp, Func Make)[] { - (FogField.AetheriaWarp, FogField.HeightFog), (FogField.AetheriaWarp, FogField.InsideFog), - (FogField.AetheriaWarp, FogField.Wells), (SlabField.SlabWarp, SlabField.Create), + (FogField.AetheriaWarp, w => FogField.HeightFog(w, BoundMode.Interval)), (FogField.AetheriaWarp, w => FogField.InsideFog(w, BoundMode.Interval)), + (FogField.AetheriaWarp, w => FogField.DeepWell(w, BoundMode.Interval)), (FogField.AetheriaWarp, w => FogField.DeepWell(w, BoundMode.Affine)), + (FogField.AetheriaWarp, w => FogField.Wells(w, BoundMode.Interval)), (FogField.AetheriaWarp, w => FogField.Wells(w, BoundMode.Affine)), + (SlabField.SlabWarp, SlabField.Create), }; foreach (var (warp, make) in scenarios) foreach (var w in new[] { 0.0f, warp }) @@ -1513,19 +2232,23 @@ public sealed class NoiseBoundTests } /// - /// The proving ground (docs/cultmath-interval-ground-cut.md, "Pass 3" and "Pass 4"). Scenarios - /// (a) height fog, (b) inside the fog, (c) Aetheria's wells and (d) r1's uniform slab, each over 64 - /// tiles at N in {1, 4, 8, 16} and warp in {0, D}: the tile march integrates exactly the dense march's + /// The proving ground (docs/cultmath-interval-ground-cut.md, "Pass 3", "Pass 4" and "Pass 5b"). + /// Scenarios (a) height fog and (b) inside the fog under the centred-warp interval bound, (c) the deep + /// well and (c-band) the shallow wells each under the centred-warp interval bound and the affine bound + /// on the same seeds, and (d) r1's uniform slab, each over 64 tiles at N in {1, 4, 8, 16} and warp in + /// {0, D}; (c) and (c-band) also print the range ceiling at N = 8, and no range-mask cell is skipped: the tile march integrates exactly the dense march's /// cells, so the transmittance agrees within 1e-3 (a difference would be a skipped cell that was not /// empty), the oracle march (each ray over the oracle mask: the cells whose mid-depth density is nonzero /// for at least one of the tile's rays) agrees with the dense march exactly, and no (a)-(d) tile reports /// a body. (e) the void: the grid over hollow radius, camera offset and ramp width (32 tiles at N = 8), /// LOD-far (32 tiles) and the one shipped void (200 tiles), three marches each, the footprint-aware one - /// taking the analytic body step. Contracts at N = 8, the lower of warp 0 and D: (a) at least 2x - /// combined cost; (c) (ruling wells-overhead-plus-floor) probe overhead, the tile march over the tile - /// march plus its probes, at least 0.90, and, hard, the pre-pass proves at least half of the - /// oracle-empty cells empty; the oracle ceiling, the efficiency against it and the cull fraction are - /// printed; (e) at the shipped void the masked fixed-step march agrees with the reference exactly, the + /// taking the analytic body step. Contracts at N = 8: (a) at least 2x combined cost, the lower of warp + /// 0 and D; (c) (ruling affine-wells-contract-cheaper-than-interval) on the deep well the affine + /// bound's combined cost ratio exceeds the centred interval's at warp 0 and at warp D, and, hard + /// (ruling wells-overhead-plus-floor), the pre-pass proves at least half of the oracle-empty cells + /// empty under both bounds at both warps; probe overhead, the efficiency against the oracle and against + /// the range ceiling, the ceilings and the cull fraction are printed, not asserted; (c-band) is + /// printed only; (e) at the shipped void the masked fixed-step march agrees with the reference exactly, the /// footprint-aware march's optical depth at the body start is off a converged march (h = 0.5, every /// cell) by at most twice the fixed-step march's error, and it takes at most half the reference's /// steps per pixel. A shortfall skips naming saving-2x-scenarios and every shortfall; the fields, the @@ -1534,13 +2257,15 @@ public sealed class NoiseBoundTests [Fact] public void IntervalSkipHalvesEvaluations() { - var ratios = new Dictionary(); - var cullFloor = double.MaxValue; + var atEight = new Dictionary<(string Key, float Warp), RunStats>(); var scenarios = new (string Key, float Warp, Func Make)[] { - ("a", FogField.AetheriaWarp, FogField.HeightFog), - ("b", FogField.AetheriaWarp, FogField.InsideFog), - ("c", FogField.AetheriaWarp, FogField.Wells), + ("a", FogField.AetheriaWarp, w => FogField.HeightFog(w, BoundMode.Interval)), + ("b", FogField.AetheriaWarp, w => FogField.InsideFog(w, BoundMode.Interval)), + ("c interval", FogField.AetheriaWarp, w => FogField.DeepWell(w, BoundMode.Interval)), + ("c affine", FogField.AetheriaWarp, w => FogField.DeepWell(w, BoundMode.Affine)), + ("c-band interval", FogField.AetheriaWarp, w => FogField.Wells(w, BoundMode.Interval)), + ("c-band affine", FogField.AetheriaWarp, w => FogField.Wells(w, BoundMode.Affine)), ("d", SlabField.SlabWarp, SlabField.Create), }; foreach (var (key, warp, make) in scenarios) @@ -1548,19 +2273,38 @@ public sealed class NoiseBoundTests foreach (var w in new[] { 0.0f, warp }) { var field = make(w); - var stats = Run(field, n, 64, 0x5A7E + n); + var stats = Run(field, n, 64, 0x5A7E + n, range: n == 8 && key.StartsWith('c')); Report($"{field.Name} N={n} warp={w:R}", stats); + Assert.True(stats.RangeViolations == 0, $"{field.Name} N={n} warp={w:R}: the pre-pass skipped {stats.RangeViolations} cells where a sampled density is positive"); Assert.True(stats.MaxDifference <= 1.0e-3f, $"{field.Name} N={n} warp={w:R}: transmittance differs by {stats.MaxDifference:R}"); Assert.True(stats.MaxOracleDifference == 0.0f, $"{field.Name} N={n} warp={w:R}: the oracle march differs from the dense march by {stats.MaxOracleDifference:R}"); if (key == "d") output.WriteLine($"IV-REPORT (d) N={n} warp={w:R}: ungated dense snoise/ray {3.0 * stats.PerRay(stats.DenseSteps):F2} (three per dense sample, as r1 counted)"); if (n == 8) - ratios[key] = Math.Min(ratios.GetValueOrDefault(key, double.MaxValue), key == "c" ? stats.ProbeOverhead : stats.Ratio); - if (n == 8 && key == "c") + atEight[(key, w)] = stats; + } + + var warps = new[] { 0.0f, FogField.AetheriaWarp }; + var costA = warps.Min(w => atEight[("a", w)].Ratio); + var cullFloor = double.MaxValue; + var gains = new List(); + var shortfalls = new List(); + foreach (var w in warps) + { + var interval = atEight[("c interval", w)]; + var affine = atEight[("c affine", w)]; + foreach (var (mode, stats) in new[] { ("interval", interval), ("affine", affine) }) { - Assert.True(stats.CullFraction >= 0.5, $"(c) warp={w:R}: the pre-pass proves {stats.CullFraction:F3} of the oracle-empty cells empty, under the 0.5 floor (ruling wells-overhead-plus-floor)"); + Assert.True(stats.CullFraction >= 0.5, $"(c) {mode} warp={w:R}: the pre-pass proves {stats.CullFraction:F3} of the oracle-empty cells empty, under the 0.5 floor (ruling wells-overhead-plus-floor)"); cullFloor = Math.Min(cullFloor, stats.CullFraction); } + + var band = (atEight[("c-band interval", w)], atEight[("c-band affine", w)]); + gains.Add($"warp {w:R}: (c) affine {affine.Ratio:F2}x against centred interval {interval.Ratio:F2}x ({affine.Ratio / interval.Ratio:F2}x), probe overhead {affine.ProbeOverhead:F3} and {interval.ProbeOverhead:F3}, " + + $"efficiency {affine.Efficiency:F3} and {interval.Efficiency:F3} against the oracle, {affine.RangeEfficiency:F3} and {interval.RangeEfficiency:F3} against the range ceiling {affine.RangeCeiling:F3}; " + + $"(c-band) {band.Item2.Ratio:F2}x against {band.Item1.Ratio:F2}x ({band.Item2.Ratio / band.Item1.Ratio:F2}x)"); + if (!(affine.Ratio > interval.Ratio)) + shortfalls.Add($"(c) warp {w:R}: affine {affine.Ratio:F2}x is not cheaper than the centred interval's {interval.Ratio:F2}x"); } foreach (var rh in VoidField.HollowRadii) @@ -1581,15 +2325,14 @@ public sealed class NoiseBoundTests Assert.True(headline.MaxLodDepthError <= 2.0 * headline.MaxRefDepthError, $"at the body start the footprint-aware march's optical depth is off the converged march's by {headline.MaxLodDepthError:G6} relative, more than twice the fixed-step march's {headline.MaxRefDepthError:G6}"); var steps = (double)headline.DenseSteps / headline.LodSteps; - var shortfalls = new List(); - if (ratios["a"] < 2.0) - shortfalls.Add($"(a) saves {ratios["a"]:F2}x combined cost, under 2x"); - if (ratios["c"] < 0.90) - shortfalls.Add($"(c) probe overhead {ratios["c"]:F3}, under 0.90"); + if (costA < 2.0) + shortfalls.Add($"(a) saves {costA:F2}x combined cost, under 2x"); if (steps < 2.0) shortfalls.Add($"(e) takes {steps:F2}x fewer steps per pixel at the shipped void, under 2x"); - output.WriteLine($"IV-REPORT contracts at N=8 (lower of warp 0 and D): (a) {ratios["a"]:F2}x cost; (c) probe overhead {ratios["c"]:F3} (at least 0.90), cull fraction {cullFloor:F3} (at least 0.5, hard); " + output.WriteLine($"IV-REPORT contracts at N=8: (a) {costA:F2}x cost (lower of warp 0 and D, at least 2x); (c) affine cheaper than the centred interval at both warps, cull fraction {cullFloor:F3} (at least 0.5 under both bounds, hard); " + $"(e) {steps:F2}x fewer steps/px, depth error at the body start LOD {headline.MaxLodDepthError:G4} vs fixed-step {headline.MaxRefDepthError:G4} (at most 2x)"); + foreach (var line in gains) + output.WriteLine($"IV-REPORT (c) at N=8, {line}"); if (shortfalls.Count > 0) Assert.Skip("saving-2x-scenarios: " + string.Join("; ", shortfalls)); } From 1b9cecb941aef0035e3c2074df2aa3f0b5081726 Mon Sep 17 00:00:00 2001 From: Meta Vens Date: Sat, 3 Oct 2026 22:21:50 +0200 Subject: [PATCH 03/31] design.md: Affine forms section; (b) and (c) rows rewritten from the deep-well run, (c-band) added, contracts follow the rulings Co-Authored-By: Claude Sonnet 5.5 --- packages/cultmath/docs/design.md | 149 ++++++++++++++++++++++++++----- 1 file changed, 127 insertions(+), 22 deletions(-) diff --git a/packages/cultmath/docs/design.md b/packages/cultmath/docs/design.md index ec6ee86c..15cb34f8 100644 --- a/packages/cultmath/docs/design.md +++ b/packages/cultmath/docs/design.md @@ -581,9 +581,9 @@ over 2,000 balls x 64 points are the defence. A change to the `snoise` kernel must re-run `MeasureLipschitz` and re-pin `L` and the witness. What the ball bounds guarantee: enclosure of `snoise` and `fbm_grad(...).w` over -the ball, in float32. What they do not: anything about tapes, pruning, choice -tracking or affine forms; those are later steps of -`docs/cultmath-tape-target.md` (CultLib root). How loose they are is a number: +the ball, in float32. What they do not: anything about tapes, pruning or choice tracking. Correlation +between the operands of one expression is what "Affine forms" below adds. +How loose the ball bounds are is a number: `NoiseBoundTests.TightnessReport` prints the mean interval width over the sampled range: 5.08 for `iv_snoise_ball` at `r = 0.05`, 1.96 at 0.25 and 1.28 at 1.0; 6.64 and 2.82 for 4-octave `iv_fbm_ball` at 0.05 and 0.25. @@ -611,17 +611,29 @@ printed. The flow is not counted. At `N = 8`, in Aetheria's units on its - (a) Height fog, camera above the safety band, rays level and up: the whole grid is one envelope probe per tile, against 256 envelope evaluations per ray, 16384x. Every cell is oracle-empty and the pre-pass proves them all. -- (b) Inside the fog: 1.15x, and 0.99x with the warp. Most cells are fog, and +- (b) Inside the fog: 1.15x, and 1.09x with the warp. Most cells are fog, and the pre-pass probes each one. -- (c) The zone bowl and four wells, the camera gazing across. With no warp: - 2.74x, oracle ceiling 7.26x, efficiency 0.378, cull fraction 0.952, probe - overhead 0.961. With the warp `D = 60`: 1.39x, oracle ceiling 7.17x, - efficiency 0.194, cull fraction 0.842, probe overhead 0.993. With the warp the - pre-pass cuts envelope evaluations from 177.46 to 33.49 per ray, but the - oracle march costs 0.194 of the tile march, so about four fifths of the tile - march's cost goes to cells no ray samples nonzero: the warp bound swamps the - noise ball, so the ball cannot prove those cells empty. That gap belongs to - the tape target's affine forms, not to a better pre-pass. +- (c) The deep well (ruling `wells-scale-order-of-magnitude`): the zone bowl + plus one well at its centre, `PowerPulse` exponent 2, scale 2220, depth 600. + The band `G = 60` is where only the noise decides density (`s` within + `F -+ 1.5 A` with the fade at 1); depth and half-depth radius are each + `10 G`. The camera's `xz` is uniform within 300 of the centre, at + `y = F - h(xz) / 2`, so it looks across the bowl from inside it. Both rows run + on the same seeds. With no warp, the centred interval takes 3.10x (oracle + ceiling 5.24x, efficiency 0.591, cull fraction 0.986, probe overhead 0.943) + and the affine bound 3.57x (efficiency 0.682, cull 0.993, overhead 0.905). With + the warp `D = 60`: 2.99x (ceiling 5.31x, efficiency 0.563, cull 0.985, overhead + 0.935) against 3.46x (efficiency 0.651, cull 0.993, overhead 0.880), so affine + is 1.15x and 1.16x cheaper. The range mask (the oracle cells plus every cell + where any of 5 x 5 slopes x 9 depths has density above zero) gives a range + ceiling of 0.859 (0.848 with the warp), the cost a bound that proves exactly + the range-empty cells would reach against the oracle's; the efficiencies + against it are 0.794 and 0.688 (warp 0), 0.768 and 0.665 (warp `D`). +- (c-band) The shallow wells, four mass-derived wells in the zone bowl with + the camera gazing across, whose geometry is band-dominated, printed only. + Centred interval 2.74x (efficiency 0.378, cull 0.952, overhead 0.961), affine + 3.95x (0.544, 0.980, 0.901); with the warp 2.53x (0.353, 0.945, 0.948) against + 3.54x (0.494, 0.976, 0.865). - (d) r1's uniform slab, which has no envelope: 1.23x, and 1.48x with no warp. - (e) The void at the shipped point (ruling `operator-shipped-void`): `Rh = 198`, `S = 50`, `Rc = Rh + S = 248`, `e = -ln 1.5 / ln(1 - (Rh / Rc)^2)`, the @@ -664,24 +676,117 @@ interval argument, so the composed interval encloses both fields. The contracts, at `N = 8`, the lower of warp 0 and `D`: - (a) at least 2x combined cost; -- (c) (ruling `wells-overhead-plus-floor`): probe overhead, the tile march over - the tile march plus its probes, at least 0.90; and, hard, the pre-pass proves - at least half of the oracle-empty cells empty, so a pre-pass that skips - nothing fails. The oracle ceiling, the efficiency against it and the cull - fraction are printed, not asserted; +- (c) on the deep well (rulings `affine-wells-contract-cheaper-than-interval` + and `wells-overhead-plus-floor`): the affine bound's combined cost ratio is + above the centred interval's at warp 0 and at warp `D`; and, hard, the + pre-pass proves at least half of the oracle-empty cells empty under both + bounds, so a pre-pass that skips nothing fails. The probe overhead, the + efficiencies against the oracle and the range ceiling, the ceilings and the + cull fraction are printed, not asserted; (c-band) is printed only; - (e) at the shipped void: the masked fixed-step march agrees with the reference exactly; the footprint march's optical depth at the body start is off the converged march's by at most twice the fixed-step march's error; and it takes at most half the reference's steps per pixel. -This run meets every contract: (a) 16384x, (c) probe overhead 0.961 and cull -fraction 0.842, (e) 2.68x fewer steps and a depth error 1.42x the fixed-step -march's. The fields, the grids and the 0.2 weight are not tuned toward the -contracts. +This run meets every contract: (a) 16384x, (c) affine 3.57x and 3.46x against +the centred interval's 3.10x and 2.99x, with a cull fraction of 0.985 at the +lowest, (e) 2.68x fewer steps and a depth error 1.42x the fixed-step march's. +The fields, the grids, the 0.2 envelope weight and the 2.0 gradient weight are +not tuned toward the contracts. An affine probe is one `snoise_grad`, counted at +2.0 `snoise` evaluations (an estimate, printed beside the count). Consumers: the gamecult.org ground shader (through the GLSL lowering) and Aetheria's nebula raymarch at its CultMath pin bump. +## Affine forms + +`math.Affine.cs` carries reduced affine forms with one shared symbol: a +`float3(x0, a, e)` is the set `x0 + a eps + e delta` with `eps` and `delta` in +`[-1, 1]` and `e >= 0` (Messine's AF1, as Gamito and Maddock use it for gradient +noise, in one register as an interval is a `float2`). `eps` is shared by every +form over one region (for a screen-tile slice, the depth parameter, `z = z_m + +h eps`), so `af_sub(x, x)` has `a = 0`; `delta` is private to the form, and +every term that cannot be tracked is condensed into `e`. A slice is a +`float4(centre, radius)` with a `float3` axis. No op takes a compound type, a +reference parameter or a swizzle write, so each mirrors into the HLSL common +subset. + +The ops: `af_point`, `af_symbol`, `af_from_iv`, `af_range` (the bridge back to +`iv_*`), `af_add`, `af_sub`, `af_neg`, `af_scale`, `af_add_iv`, `af_mul` (the +product `|a_x a_y|` and the cross-error terms condensed into `e`, so there is +no `af_sqr`), `af_snoise` and `af_fbm`, and the slice pair `af_frustum_axis` and +`af_frustum_ball`. + +Rounding. Every op returns a form whose set at each `eps` contains the float32 +value of its function at every point of its operands' sets at that `eps`. Each +op folds its own float32 rounding into `e` as `S 2^-20`, `S` the sum of the +magnitudes its roundings are relative to (named in the op's comment): at most +`8u S` is needed with `u = 2^-24`, and the factor of two over that is the +allowance for a device whose `+` and `*` are faithful rather than correctly +rounded. `AffineTests.EveryOpEnclosesItsPoints` evaluates the form's set in +double and the function in float32 with no tolerance. + +`af_snoise(centre, axis, radius)` encloses `snoise` over `centre + axis eps + w`, +`|w| <= radius`, from one `snoise_grad`. It returns the narrower of the +Lipschitz form `(n, 0, L R)` and the centred form `(n, g . axis, |g| radius + +M R^2 / 2)` (`R = |axis| + radius`, `L = SNOISE_LIPSCHITZ`, `M = +SNOISE_HESSIAN`), so its total width is never more than the Lipschitz form's. +The centred form is Taylor's theorem, which is why it beats the ball when the +radius is small and the axis carries the variation. `af_fbm` compounds octaves +as `iv_fbm_ball` does. The float32 evaluation allowance is derived in +`af_snoise`'s comment. + +`SNOISE_HESSIAN = 56.050385` is 1.10 times the refined maximum of the spectral +norm of the Hessian of `snoise(float3)`, measured by +`NoiseBoundTests.MeasureHessian` over 1e6 seeded points (central differences of +`snoise_grad`) and refined by ascent from the 1e4 largest: sampled 50.1366, +refined 50.9549, witness (-245.19885, -53.01323, -91.47363). +`HessianConstantPinsSampledCurvature` fails on a 1% move. +`SNOISE2_LIPSCHITZ = 8.117002` is the same for the gradient of `snoise(float2)` +(`MeasureLipschitz2`: sampled 7.2575, refined 7.3791, witness (-67.12454, +-129.53296), `|snoise2|` at most 0.99964), and it backs `iv_snoise_ball(float2, +float)`. Both are empirical with a margin, not proofs; the enclosure tests are +the defence, and a change to the kernels must re-run the measurements. + +The slice. `af_frustum_ball(centreSlope, z0, z1, footprintPerDepth, +warpVariation)` returns the centre `(m_c z_m, z_m)` and a radius `z1 +footprintPerDepth + warpVariation`, with `af_frustum_axis` the axis `(m_c, 1) h`: +every ray point at depth `z` is `centre + axis eps(z) + w`. Its derivation +and the clamp residual are in the comment. The caller rotates, translates and +scales the slice into the noise's frame, and owns the rounding of that. + +The centred warp. A flow-warped sample reads `snoise` at `p + flow(p) shift`. The +caller does not enlarge the ball by the whole displacement `|flow| shift`; it +moves the centre by the flow at the centre, `c + flow(c) shift`, and widens the +radius by the flow's variation across the ball, `r + |shift| rho` with `rho = +FlowAmplitude SNOISE2_LIPSCHITZ FlowFrequency r sqrt 2` (the clamp to the unit +disc is nonexpansive). CultMath owns neither the flow nor the envelope: the warp's +centre and variation are the caller's, and a consumer adds them to the slice's +`warpVariation` or to the form's radius. With two phases that cross-fade, each +phase takes its own centre; one ball serves when the phases' centres coincide +(warp 0). `NoiseBoundTests.CentredWarpStaysEnclosed` checks 2,000 balls and +fails when `rho` is dropped. + +The deep well is the representative Aetheria case: the camera inside a sun's +bowl, looking across. `DeepWellIsTenBandsDeepAndWide` computes the band `G` from +`FogField`'s own floor offset, amplitude and noise range (60) and requires the +depth (633 at the centre against the rim) and the half-depth radius to be at +least `10 G`, every drawn camera sitting inside the well below the rim; a well of +depth 150 fails. The shallow wells stay as the printed shallow-well, band-dominated row (c-band). + +The test field's affine bound takes depth as `af_symbol(z_m, h)`, builds `s` +affine in it with the height interval, the footprint and the slope-depth +product in `e` through `af_add_iv`, takes the noise through `af_frustum_*` and +`af_snoise` per phase and for the detail octave, fades by `af_from_iv` and +`af_mul`, reads the range with `af_range`, and intersects with the centred +interval's bound (both sound). `FogField.Bound` makes no direct `snoise` call. +The cull-map render (`DeepWellCullMap`, explicit) writes a PNG when +`CULTMATH_WRITE_CULLMAP` names a directory: one deep-well camera at warp `D`, +960 x 540 in 8 x 8 tiles, with panels for dense cells per ray under the centred +interval, under the affine bound and under the oracle mask (one logarithmic +scale) and the transmittance. Its run: 4.51, 3.60 and 2.02 cells per ray on +average, 45 the most, mean transmittance 0.278. + ## Rules - Keep type names HLSL-shaped: `float2`, `float3`, `float4`, `float3x3`, and `math`. From 4ae986a52ac4b895401f3d5d49c39007ac25ce08 Mon Sep 17 00:00:00 2001 From: Meta Vens Date: Sat, 3 Oct 2026 22:21:50 +0200 Subject: [PATCH 04/31] Cull map: cells per ray on a logarithmic viridis scale Co-Authored-By: Claude Sonnet 5.5 --- .../cultmath/tests/CultMath.Tests/NoiseBoundTests.cs | 10 +++++----- 1 file changed, 5 insertions(+), 5 deletions(-) diff --git a/packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs b/packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs index 2e6f23b8..f4eae288 100644 --- a/packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs +++ b/packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs @@ -862,8 +862,8 @@ public sealed class NoiseBoundTests /// the pixel's centre) is marched densely for its transmittance and over each tile mask and the oracle /// mask (the cells some ray of the tile samples nonzero), with the early-out at 0.02. The PNG is /// 1920 x 1080 in four panels: top left the cells per ray the interval march integrates, top right the - /// affine march's, bottom left the oracle march's (all three on one viridis scale, 0 to the largest - /// count), bottom right the transmittance (black opaque, white clear). It is written only when + /// affine march's, bottom left the oracle march's (all three on one viridis scale, logarithmic, + /// log(1 + n) / log(1 + the largest count)), bottom right the transmittance (black opaque, white clear). It is written only when /// CULTMATH_WRITE_CULLMAP names a directory; the mean cells per ray are printed either way. /// [Fact(Explicit = true)] @@ -940,9 +940,9 @@ public sealed class NoiseBoundTests for (var x = 0; x < width; x++) { var p = y * width + x; - Put(x, y, Colour(cellsInterval[p] / (double)most)); - Put(width + x, y, Colour(cellsAffine[p] / (double)most)); - Put(x, height + y, Colour(cellsOracle[p] / (double)most)); + Put(x, y, Colour(Math.Log(1.0 + cellsInterval[p]) / Math.Log(1.0 + most))); + Put(width + x, y, Colour(Math.Log(1.0 + cellsAffine[p]) / Math.Log(1.0 + most))); + Put(x, height + y, Colour(Math.Log(1.0 + cellsOracle[p]) / Math.Log(1.0 + most))); var grey = (byte)Math.Round(255.0 * transmittance[p]); Put(width + x, height + y, (grey, grey, grey)); } From 0893679235e6828e8c492c5fe6af4aa40d8fff7b Mon Sep 17 00:00:00 2001 From: Meta Vens Date: Sun, 4 Oct 2026 02:39:22 +0200 Subject: [PATCH 05/31] Deep well (c): the affine-cheaper-than-interval contract fails the test; only the (a) and (e) savings skip Co-Authored-By: Claude Sonnet 5.5 --- .../cultmath/tests/CultMath.Tests/NoiseBoundTests.cs | 9 ++++++--- 1 file changed, 6 insertions(+), 3 deletions(-) diff --git a/packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs b/packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs index f4eae288..cc7caebb 100644 --- a/packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs +++ b/packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs @@ -2251,8 +2251,9 @@ public sealed class NoiseBoundTests /// printed only; (e) at the shipped void the masked fixed-step march agrees with the reference exactly, the /// footprint-aware march's optical depth at the body start is off a converged march (h = 0.5, every /// cell) by at most twice the fixed-step march's error, and it takes at most half the reference's - /// steps per pixel. A shortfall skips naming saving-2x-scenarios and every shortfall; the fields, the - /// grids and the envelope cost are not tuned toward it. + /// steps per pixel. A broken (c) contract fails the test; a shortfall of the (a) or (e) saving skips + /// naming saving-2x-scenarios and every such shortfall. The fields, the grids and the envelope cost are + /// not tuned toward either. /// [Fact] public void IntervalSkipHalvesEvaluations() @@ -2289,6 +2290,7 @@ public sealed class NoiseBoundTests var cullFloor = double.MaxValue; var gains = new List(); var shortfalls = new List(); + var broken = new List(); foreach (var w in warps) { var interval = atEight[("c interval", w)]; @@ -2304,7 +2306,7 @@ public sealed class NoiseBoundTests + $"efficiency {affine.Efficiency:F3} and {interval.Efficiency:F3} against the oracle, {affine.RangeEfficiency:F3} and {interval.RangeEfficiency:F3} against the range ceiling {affine.RangeCeiling:F3}; " + $"(c-band) {band.Item2.Ratio:F2}x against {band.Item1.Ratio:F2}x ({band.Item2.Ratio / band.Item1.Ratio:F2}x)"); if (!(affine.Ratio > interval.Ratio)) - shortfalls.Add($"(c) warp {w:R}: affine {affine.Ratio:F2}x is not cheaper than the centred interval's {interval.Ratio:F2}x"); + broken.Add($"(c) warp {w:R}: affine {affine.Ratio:F2}x is not cheaper than the centred interval's {interval.Ratio:F2}x"); } foreach (var rh in VoidField.HollowRadii) @@ -2333,6 +2335,7 @@ public sealed class NoiseBoundTests + $"(e) {steps:F2}x fewer steps/px, depth error at the body start LOD {headline.MaxLodDepthError:G4} vs fixed-step {headline.MaxRefDepthError:G4} (at most 2x)"); foreach (var line in gains) output.WriteLine($"IV-REPORT (c) at N=8, {line}"); + Assert.True(broken.Count == 0, "ruling affine-wells-contract-cheaper-than-interval: " + string.Join("; ", broken)); if (shortfalls.Count > 0) Assert.Skip("saving-2x-scenarios: " + string.Join("; ", shortfalls)); } From a5bfa63965d7e81cb3a9bfe327785aac4ec18ee6 Mon Sep 17 00:00:00 2001 From: Meta Vens Date: Sun, 4 Oct 2026 02:48:33 +0200 Subject: [PATCH 06/31] af_from_iv: the 2^-126 underflow term and halves before sum; af_range: never NaN; tiny, extreme and absorption tests Co-Authored-By: Claude Sonnet 5.5 --- packages/cultmath/src/CultMath/math.Affine.cs | 24 ++- .../tests/CultMath.Tests/AffineTests.cs | 154 +++++++++++++++++- 2 files changed, 167 insertions(+), 11 deletions(-) diff --git a/packages/cultmath/src/CultMath/math.Affine.cs b/packages/cultmath/src/CultMath/math.Affine.cs index 4a116c7d..6b392683 100644 --- a/packages/cultmath/src/CultMath/math.Affine.cs +++ b/packages/cultmath/src/CultMath/math.Affine.cs @@ -39,27 +39,35 @@ public static partial class math public static float3 af_symbol(float x0, float a) => new(x0, a, 0.0f); /// - /// Every value of the interval [lo, hi], uncorrelated with eps: ((lo + hi) / 2, 0, (hi - lo) / 2). - /// Rounding: x0 and e each round once, and a value of the interval is within e of x0 only up to - /// those; S = |x0| + e. + /// Every value of the interval [lo, hi], uncorrelated with eps: (lo / 2 + hi / 2, 0, hi / 2 - lo / 2), + /// the halves taken before the sum and the difference so that endpoints near the float maximum do not + /// overflow in them. Rounding: x0 and e each round once, and a value of the interval is within e of x0 + /// only up to those; S = |x0| + e. Halving a subnormal endpoint loses at most 2^-150, and the sum and + /// difference of subnormals are exact, so 2^-126 covers the underflow (the e of an interval spanning + /// one subnormal step would otherwise round to 0). If e + its absorption exceeds the float maximum + /// it is infinity, which still encloses. /// public static float3 af_from_iv(float2 interval) { - var x0 = (interval.x + interval.y) * 0.5f; - var e = (interval.y - interval.x) * 0.5f; - return new float3(x0, 0.0f, e + (abs(x0) + e) * 9.5367431640625e-7f); + var x0 = interval.x * 0.5f + interval.y * 0.5f; + var e = interval.y * 0.5f - interval.x * 0.5f; + return new float3(x0, 0.0f, e + (abs(x0) + e) * 9.5367431640625e-7f + 1.17549435e-38f); } /// /// The interval of every value of the form over every eps: x0 -+ (|a| + e), the bridge to iv_*. The /// endpoints round inward by at most u (|x0| + |a| + e) each, so each is moved outward by - /// (|x0| + |a| + e) 2^-20. + /// (|x0| + |a| + e) 2^-20. A form with an infinite component (an overflowed op) gives infinite + /// endpoints, and an endpoint that would be NaN (infinity minus infinity, or a NaN component) is + /// the infinity on its side, so the result is never NaN and still encloses. /// public static float2 af_range(float3 x) { var r = abs(x.y) + x.z; var w = (abs(x.x) + r) * 9.5367431640625e-7f; - return new float2(x.x - r - w, x.x + r + w); + var lo = x.x - r - w; + var hi = x.x + r + w; + return new float2(lo == lo ? lo : float.NegativeInfinity, hi == hi ? hi : float.PositiveInfinity); } /// diff --git a/packages/cultmath/tests/CultMath.Tests/AffineTests.cs b/packages/cultmath/tests/CultMath.Tests/AffineTests.cs index b70de3f4..482b5ecf 100644 --- a/packages/cultmath/tests/CultMath.Tests/AffineTests.cs +++ b/packages/cultmath/tests/CultMath.Tests/AffineTests.cs @@ -43,6 +43,19 @@ public sealed class AffineTests return new float2(lo, lo + (random.Next(8) == 0 ? 0.0f : LogUniform(random, 1.0e-5f, 40.0f))); } + // The tiny family: magnitudes from 1e-21 to 1e-17, products near and under 2^-126 = 1.2e-38, where a + // float32 result rounds to a subnormal and the absorption S 2^-20 is smaller than that loss. + private static float3 DrawTiny(System.Random random) => + new(random.Next(8) == 0 ? 0.0f : Signed(random, LogUniform(random, 1.0e-21f, 1.0e-17f)), + random.Next(4) == 0 ? 0.0f : Signed(random, LogUniform(random, 1.0e-21f, 1.0e-17f)), + random.Next(4) == 0 ? 0.0f : LogUniform(random, 1.0e-21f, 1.0e-17f)); + + private static float2 DrawTinyInterval(System.Random random) + { + var lo = Signed(random, LogUniform(random, 1.0e-21f, 1.0e-17f)); + return new float2(lo, lo + (random.Next(8) == 0 ? 0.0f : LogUniform(random, 1.0e-21f, 1.0e-17f))); + } + // A float32 value in the form's set at eps, delta at an endpoint when asked; NaN when rounding leaves // no float of the set near the drawn point (a set narrower than an ulp). private static float Member(System.Random random, float3 form, float eps, bool endpoint) @@ -88,15 +101,16 @@ public sealed class AffineTests // four witnesses. The first four draws per form put eps at an endpoint, the first eight delta. // af_point and af_symbol are checked on their defining value: the point x0 itself, and x0 + a eps in // double (the symbol defines eps, so it has no float32 evaluation to enclose). - private static (int Misses, List Witness) Harness(Op op, int seed, int forms = 10_000, Func? draw = null) + private static (int Misses, List Witness) Harness(Op op, int seed, int forms = 10_000, Func? draw = null, bool tiny = false) { - draw ??= DrawForm; + draw ??= tiny ? DrawTiny : DrawForm; var random = new System.Random(seed); var misses = 0; var witness = new List(); for (var t = 0; t < forms; t++) { - var operands = new Operands(draw(random), draw(random), DrawInterval(random), random.Next(8) == 0 ? 0.0f : Uniform(random, -4.0f, 4.0f)); + var operands = new Operands(draw(random), draw(random), tiny ? DrawTinyInterval(random) : DrawInterval(random), + random.Next(8) == 0 ? 0.0f : tiny ? Signed(random, LogUniform(random, 1.0e-32f, 1.0e-19f)) : Uniform(random, -4.0f, 4.0f)); var contains = op.Build(operands); for (var d = 0; d < 16; d++) { @@ -464,4 +478,138 @@ public sealed class AffineTests Assert.True(failures.Values.Sum() == 0, string.Join(", ", families.Select(f => $"{f} {failures[f]}")) + Environment.NewLine + string.Join(Environment.NewLine, witness)); } + + /// + /// Underflow. Operands of magnitude 1e-21 to 1e-17 put products near and under 2^-126, where float32 + /// rounds to a subnormal and the relative absorption S 2^-20 is smaller than the loss: every op + /// still encloses, no tolerance. af_mul and af_scale carry the 2^-126 term; a mutant that subtracts + /// it misses here. + /// + [Fact] + public void TinyOperandsEncloseTheirPoints() + { + var failures = new List(); + foreach (var op in Ops) + { + var (misses, witness) = Harness(op, 0xAF05, 10_000, tiny: true); + if (misses > 0) + failures.Add($"{op.Name}: {misses} misses" + Environment.NewLine + string.Join(Environment.NewLine, witness)); + } + + Assert.True(failures.Count == 0, string.Join(Environment.NewLine, failures)); + } + + private const float MinNormal = 1.17549435e-38f; + + private static bool Near(double observed, double expected, double relative = 1.0e-5) => + Math.Abs(observed - expected) <= relative * Math.Abs(expected); + + /// + /// The rounding absorption of each op, pinned both ways. With no propagated error in e, af_add, + /// af_sub, af_scale, af_mul and af_from_iv return an e equal to their documented absorption (and + /// 2^-126 where documented) to a part in 1e5 of its double value, and af_range moves each endpoint + /// outward by (|x0| + |a| + e) 2^-20 to 30% (the endpoints round once). A factor, a sign or an + /// operator changed in any of them fails here; the one-ulp changes of the worst-case absorption are + /// the OneUlpSurvivors set EveryRoundingTermIsCaught names. + /// + [Fact] + public void AbsorptionsArePinned() + { + var u20 = Math.ScaleB(1.0, -20); + var random = new System.Random(0xAF06); + for (var t = 0; t < 2_000; t++) + { + var (x0, y0) = (Uniform(random, -20.0f, 20.0f), Uniform(random, -20.0f, 20.0f)); + var (ax, ay) = (Signed(random, LogUniform(random, 1.0e-3f, 20.0f)), Signed(random, LogUniform(random, 1.0e-3f, 20.0f))); + var scale = Signed(random, LogUniform(random, 1.0e-2f, 4.0f)); + var x = new float3(x0, ax, 0.0f); + var y = new float3(y0, ay, 0.0f); + + var sum = af_add(x, y); + Assert.True(Near(sum.z, (Math.Abs((double)sum.x) + Math.Abs((double)sum.y)) * u20), $"af_add({x}, {y}) = {sum}: e is not (|x0| + |a|) 2^-20"); + var difference = af_sub(x, y); + Assert.True(Near(difference.z, (Math.Abs((double)difference.x) + Math.Abs((double)difference.y)) * u20), $"af_sub({x}, {y}) = {difference}: e is not (|x0| + |a|) 2^-20"); + var scaled = af_scale(x, scale); + Assert.True(Near(scaled.z, (Math.Abs((double)scaled.x) + Math.Abs((double)scaled.y)) * u20 + MinNormal), $"af_scale({x}, {scale:R}) = {scaled}: e is not (|x0| + |a|) 2^-20 + 2^-126"); + + // a_x = 0 leaves the product's propagated error 0 and its a = x0 a_y. + var px = new float3(x0, 0.0f, 0.0f); + var product = af_mul(px, y); + var terms = Math.Abs((double)product.x) + Math.Abs((double)x0 * ay); + Assert.True(Near(product.z, terms * u20 + MinNormal), $"af_mul({px}, {y}) = {product}: e is not (|x0 y0| + |x0 a_y|) 2^-20 + 2^-126"); + + var iv = DrawInterval(random); + var form = af_from_iv(iv); + var (mid, half) = (((double)iv.x + iv.y) / 2.0, ((double)iv.y - iv.x) / 2.0); + var expected = half + (Math.Abs(mid) + half) * u20 + MinNormal; + Assert.True(Math.Abs(form.z - expected) <= 1.0e-5 * (Math.Abs(mid) + half) * u20 + 1.0e-6 * half + 1.0e-12 * expected, + $"af_from_iv({iv}) = {form}: e is not half-width + (|mid| + half-width) 2^-20 + 2^-126 = {expected:R}"); + + var wide = new float3(Uniform(random, -20.0f, 20.0f), Signed(random, LogUniform(random, 1.0e-3f, 20.0f)), LogUniform(random, 1.0e-3f, 20.0f)); + var range = af_range(wide); + var reach = Math.Abs((double)wide.y) + wide.z; + var absorption = (Math.Abs((double)wide.x) + reach) * u20; + var outward = ((double)range.y - range.x) / 2.0 - reach; + Assert.True(Math.Abs(outward - absorption) <= 0.3 * absorption, $"af_range({wide}) = {range}: endpoints are {outward:R} past x0 -+ (|a| + e), not (|x0| + |a| + e) 2^-20 = {absorption:R}"); + } + } + + /// + /// af_from_iv and af_range at the float32 extremes. Subnormal endpoints (including [0, the least + /// subnormal], whose halved width underflows to 0) and endpoints near the float maximum (where lo + + /// hi, hi - lo or the absorption would overflow) give a form whose set holds both endpoints and the + /// midpoint, and an af_range that holds them with no NaN endpoint. A form with an infinite or NaN + /// component, as an overflowed op returns, gives an af_range with no NaN endpoint. + /// + [Fact] + public void FromIntervalAndRangeHoldAtTheExtremes() + { + var least = BitConverter.Int32BitsToSingle(1); + var max = float.MaxValue; + var random = new System.Random(0xAF07); + var specials = new[] { 0.0f, least, 2 * least, 100 * least, MinNormal - least, MinNormal, 3.0f * MinNormal, 1.0e-30f, 1.0f, 1.0e20f, 1.0e30f, max / 4, max / 2, max * 0.75f, max }; + var misses = new List(); + void Check(float lo, float hi) + { + var form = af_from_iv(new float2(lo, hi)); + var range = af_range(form); + if (float.IsNaN(form.x) || float.IsNaN(form.z) || float.IsNaN(range.x) || float.IsNaN(range.y)) + misses.Add($"af_from_iv([{lo:R}, {hi:R}]) = {form}, af_range = {range}: NaN"); + foreach (var value in new[] { lo, hi, (float)(((double)lo + hi) / 2.0) }) + { + if (!InForm(form, 0.0f, value)) + misses.Add($"af_from_iv([{lo:R}, {hi:R}]) = {form} misses {value:R}"); + if (!(range.x <= value && value <= range.y)) + misses.Add($"af_range(af_from_iv([{lo:R}, {hi:R}])) = {range} misses {value:R}"); + } + } + + foreach (var a in specials) + foreach (var b in specials) + { + Check(-a, b); + Check(-Math.Max(a, b), -Math.Min(a, b)); + Check(Math.Min(a, b), Math.Max(a, b)); + } + + for (var t = 0; t < 200_000; t++) + { + var sign = random.Next(2) == 0 ? 1 : -1; + var lo = sign * BitConverter.Int32BitsToSingle(random.Next(0, random.Next(2) == 0 ? 1 << 20 : 0x00800000)); + var width = BitConverter.Int32BitsToSingle(random.Next(0, random.Next(2) == 0 ? 1 << 12 : 1 << 22)); + Check(lo, lo + width); + } + + var odd = new[] { float.PositiveInfinity, float.NegativeInfinity, float.NaN, 0.0f, 1.0f, max }; + foreach (var x0 in odd) + foreach (var a in odd) + foreach (var e in odd) + { + var range = af_range(new float3(x0, a, e)); + if (float.IsNaN(range.x) || float.IsNaN(range.y)) + misses.Add($"af_range(({x0}, {a}, {e})) = {range}"); + } + + Assert.True(misses.Count == 0, $"{misses.Count} misses:" + Environment.NewLine + string.Join(Environment.NewLine, misses.Take(8))); + } } From e6606faff8dcd304e8353b8a885a6bf803a5fbd2 Mon Sep 17 00:00:00 2001 From: Meta Vens Date: Sun, 4 Oct 2026 02:55:32 +0200 Subject: [PATCH 07/31] af_snoise: width and allowances pinned by an exact-rule test; curvature witness test; SNOISE_HESSIAN's defence stated truthfully Co-Authored-By: Claude Sonnet 5.5 --- packages/cultmath/docs/design.md | 10 +++- packages/cultmath/src/CultMath/math.Affine.cs | 9 ++- .../tests/CultMath.Tests/AffineTests.cs | 58 +++++++++++++++++++ .../tests/CultMath.Tests/NoiseBoundTests.cs | 39 ++++++++++++- 4 files changed, 110 insertions(+), 6 deletions(-) diff --git a/packages/cultmath/docs/design.md b/packages/cultmath/docs/design.md index 15cb34f8..38a14215 100644 --- a/packages/cultmath/docs/design.md +++ b/packages/cultmath/docs/design.md @@ -745,8 +745,14 @@ refined 50.9549, witness (-245.19885, -53.01323, -91.47363). `SNOISE2_LIPSCHITZ = 8.117002` is the same for the gradient of `snoise(float2)` (`MeasureLipschitz2`: sampled 7.2575, refined 7.3791, witness (-67.12454, -129.53296), `|snoise2|` at most 0.99964), and it backs `iv_snoise_ball(float2, -float)`. Both are empirical with a margin, not proofs; the enclosure tests are -the defence, and a change to the kernels must re-run the measurements. +float)`. Both are empirical with a margin, not proofs. A random ball cannot see a +curvature constant 20% low, so `SNOISE_HESSIAN` is defended by +`HessianConstantPinsSampledCurvature` and `SnoiseFormEnclosesTheCurvatureWitness` +(`af_snoise` at the point where the curvature is realised, along 4,000 directions: a +constant of 40 misses there). The margin over the function itself: a scan of the whole +289^3 hash period found a realised maximum of 50.58, and bounding each corner's gradient +independently over every hash configuration gives 52.22, so 56.05 sits 7% above the +configuration bound. A change to the kernels must re-run the measurements. The slice. `af_frustum_ball(centreSlope, z0, z1, footprintPerDepth, warpVariation)` returns the centre `(m_c z_m, z_m)` and a radius `z1 diff --git a/packages/cultmath/src/CultMath/math.Affine.cs b/packages/cultmath/src/CultMath/math.Affine.cs index 6b392683..85228e01 100644 --- a/packages/cultmath/src/CultMath/math.Affine.cs +++ b/packages/cultmath/src/CultMath/math.Affine.cs @@ -25,8 +25,13 @@ public static partial class math // differences of snoise_grad, largest absolute eigenvalue) over 1e6 seeded points, refines the 1e4 // largest by ascent, and multiplies the refined maximum by 1.10; // NoiseBoundTests.HessianConstantPinsSampledCurvature pins this value against a fast re-measure and - // the ascent from MeasureHessian's witness. Empirical with a margin, not a proof; the enclosure - // tests are the defence. A change to the snoise kernel must re-run MeasureHessian and re-pin it. + // the ascent from MeasureHessian's witness. Empirical with a margin, not a proof. Random balls + // cannot see a constant 20% low, so the defence is those two curvature tests plus + // NoiseBoundTests.SnoiseFormEnclosesTheCurvatureWitness, where af_snoise misses at the point the + // curvature is realised if the constant is under it. The margin over the function: a scan of the + // whole 289^3 hash period found a realised maximum of 50.58, and bounding each corner's gradient + // independently over every hash configuration gives 52.22 (Soul pass s1). A change to the snoise + // kernel must re-run MeasureHessian and re-pin it. public const float SNOISE_HESSIAN = 56.050385f; /// The point form (x, 0, 0): the value x at every eps. Exact. diff --git a/packages/cultmath/tests/CultMath.Tests/AffineTests.cs b/packages/cultmath/tests/CultMath.Tests/AffineTests.cs index 482b5ecf..612de682 100644 --- a/packages/cultmath/tests/CultMath.Tests/AffineTests.cs +++ b/packages/cultmath/tests/CultMath.Tests/AffineTests.cs @@ -612,4 +612,62 @@ public sealed class AffineTests Assert.True(misses.Count == 0, $"{misses.Count} misses:" + Environment.NewLine + string.Join(Environment.NewLine, misses.Take(8))); } + + /// + /// af_snoise's width, pinned both ways. Over 6,000 seeded (centre, axis, radius) with reach R from 1e-3 + /// to 3 (centres in [-20, 20]^3, a third out to [-1000, 1000]^3, a third of the axes and a third of the + /// radii 0), the form is the documented rule evaluated in double from snoise_grad: x0 = n; the centred + /// form when |g . axis| + |g| radius + M R^2 / 2 is under L R (checked away from that boundary), else + /// the Lipschitz form; e = that half-width + the float32 allowance (2^-14 + 2^-12 R + L 2^-20 (|centre|_1 + R)) + /// plus the form's own rounding absorption (|n| + |a| + e) 2^-20; each of x0, a and e within a part in + /// 1e6 of it. The enclosure tests pass on an allowance or an absorption of any size on the CPU, where + /// float32 is exact; this is the test that sees them. + /// + [Fact] + public void SnoiseFormCarriesItsDocumentedWidth() + { + var u20 = Math.ScaleB(1.0, -20); + double lipschitz = SNOISE_LIPSCHITZ, hessian = SNOISE_HESSIAN; + var random = new System.Random(0xAF13); + var (centred, wide) = (0, 0); + for (var t = 0; t < 6_000; t++) + { + var spread = t % 3 == 2 ? 1000.0f : 20.0f; + var centre = new float3(Uniform(random, -spread, spread), Uniform(random, -spread, spread), Uniform(random, -spread, spread)); + var axis = random.Next(3) == 0 ? new float3(0.0f, 0.0f, 0.0f) : UnitVector(random) * LogUniform(random, 1.0e-3f, 3.0f); + var radius = random.Next(3) == 0 ? 0.0f : LogUniform(random, 1.0e-3f, 3.0f); + var g = snoise_grad(centre); + var reach = (double)length(axis) + radius; + var lip = lipschitz * reach; + var a = (double)g.x * axis.x + (double)g.y * axis.y + (double)g.z * axis.z; + var remainder = Math.Sqrt((double)g.x * g.x + (double)g.y * g.y + (double)g.z * g.z) * radius + 0.5 * hessian * reach * reach; + var allowance = Math.ScaleB(1.0, -14) + Math.ScaleB(reach, -12) + + lipschitz * u20 * (Math.Abs((double)centre.x) + Math.Abs((double)centre.y) + Math.Abs((double)centre.z) + reach); + + double Width(bool useCentred, out double slope) + { + slope = useCentred ? a : 0.0; + var e = (useCentred ? remainder : lip) + allowance; + return e + (Math.Abs((double)g.w) + Math.Abs(slope) + e) * u20; + } + + var zCentred = Width(true, out var slopeCentred); + var zLipschitz = Width(false, out _); + var form = af_snoise(centre, axis, radius); + var observedCentred = Math.Abs(form.z - zCentred) <= Math.Abs(form.z - zLipschitz); + var expectedCentred = Math.Abs(a) + remainder < lip; + if (Math.Abs(Math.Abs(a) + remainder - lip) > 1.0e-3 * lip) + Assert.True(observedCentred == expectedCentred || zCentred == zLipschitz, $"af_snoise({centre}, {axis}, {radius:R}) = {form} took the {(observedCentred ? "centred" : "Lipschitz")} form; the rule takes the {(expectedCentred ? "centred" : "Lipschitz")} one (|a| + remainder = {Math.Abs(a) + remainder:R}, L R = {lip:R})"); + var z = observedCentred ? zCentred : zLipschitz; + var slope = observedCentred ? slopeCentred : 0.0; + Assert.True(form.x == g.w, $"af_snoise({centre}, {axis}, {radius:R}).x0 = {form.x:R}, snoise = {g.w:R}"); + Assert.True(Math.Abs(form.y - slope) <= 1.0e-5 * Math.Abs((double)length(new float3(g.x, g.y, g.z)) * length(axis)) + 1.0e-12, $"af_snoise({centre}, {axis}, {radius:R}).a = {form.y:R}, g . axis = {slope:R}"); + Assert.True(Math.Abs(form.z - z) <= 1.0e-6 * z, $"af_snoise({centre}, {axis}, {radius:R}) = {form}: e is {form.z:R}, the documented rule gives {z:R}"); + centred += observedCentred ? 1 : 0; + wide += !observedCentred ? 1 : 0; + } + + output.WriteLine($"AF-REPORT af_snoise width: {centred} centred cases, {wide} Lipschitz cases"); + Assert.True(centred > 1000 && wide > 1000, $"the cases do not exercise both forms: {centred} centred, {wide} Lipschitz"); + } } diff --git a/packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs b/packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs index cc7caebb..5fb3672a 100644 --- a/packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs +++ b/packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs @@ -535,7 +535,9 @@ public sealed class NoiseBoundTests // Hill-climbs f from p as AscendGradientNorm does: the ascent direction is f's central difference, // the step grows on success and halves on failure. Returns the largest value reached. - private static double Ascend(Func f, float3 p) + private static double Ascend(Func f, float3 p) => AscendPoint(f, p).Best; + + private static (double Best, float3 At) AscendPoint(Func f, float3 p) { const float h = 1.0e-3f; var step = 1.0e-2f; @@ -562,7 +564,7 @@ public sealed class NoiseBoundTests } } - return best; + return (best, p); } private static double Ascend2(Func f, float2 p) => Ascend(q => f(new float2(q.x, q.y)), new float3(p.x, p.y, 0.0f)); @@ -633,6 +635,39 @@ public sealed class NoiseBoundTests Assert.True(Math.Abs(ratio - 1.0) <= 0.005, $"SNOISE_HESSIAN is {ratio:R} times 1.10 x the re-measured {refined:R}; re-run MeasureHessian and re-pin"); } + /// + /// SNOISE_HESSIAN where the curvature is realised. At the point the ascent from MeasureHessian's witness + /// reaches, the absolute second difference of snoise along 4,000 seeded directions u at step 0.05 has a + /// largest value within 10% of SNOISE_HESSIAN / 1.10 (the realised curvature is there, so the check + /// has teeth), and af_snoise(witness, r u, 0) encloses snoise(witness + eps r u) at eps in + /// {-1, -1/2, 0, 1/2, 1} for r in {0.1, 0.3} and every one of those directions, no tolerance: a constant + /// under the realised curvature (40 against about 50.6) misses on the directions near the top + /// eigenvector. The enclosure tests on random balls cannot see a constant 20% low; this and + /// HessianConstantPinsSampledCurvature are its defence. + /// + [Fact] + public void SnoiseFormEnclosesTheCurvatureWitness() + { + var (_, witness) = AscendPoint(HessianNorm, HessianWitness); + var random = new System.Random(0xCA7E); + var directions = Enumerable.Range(0, 4000).Select(_ => UnitVector(random)).ToArray(); + const float step = 0.05f; + var centreValue = snoise(witness); + var realised = directions.Max(u => Math.Abs(snoise(witness + u * step) + (double)snoise(witness - u * step) - 2.0 * centreValue) / ((double)step * step)); + output.WriteLine($"IV-REPORT curvature witness ({witness.x:R}, {witness.y:R}, {witness.z:R}): largest second difference {realised:R} against SNOISE_HESSIAN {SNOISE_HESSIAN:R}"); + Assert.True(realised >= 0.9 * SNOISE_HESSIAN / 1.10, $"the largest second difference at the witness is {realised:R}, under 0.9 SNOISE_HESSIAN / 1.10 = {0.9 * SNOISE_HESSIAN / 1.10:R}: the check has no teeth, or the constant moved"); + foreach (var u in directions) + foreach (var r in new[] { 0.1f, 0.3f }) + { + var form = af_snoise(witness, u * r, 0.0f); + foreach (var eps in new[] { -1.0f, -0.5f, 0.0f, 0.5f, 1.0f }) + { + var value = snoise(witness + u * (r * eps)); + Assert.True(Math.Abs(value - (form.x + (double)form.y * eps)) <= form.z, $"af_snoise(witness, {u * r}, 0) = {form} misses snoise = {value:R} at eps {eps:R}, direction {u}"); + } + } + } + /// /// The provenance of SNOISE2_LIPSCHITZ: the largest gradient length of snoise(float2) (Gradient2Norm) /// over 1e6 seeded points of [-256, 256]^2, the 1e4 largest refined by ascent, times 1.10; and the From 3958dca2cdd43d06df555b351a7554e78ce62299 Mon Sep 17 00:00:00 2001 From: Meta Vens Date: Sun, 4 Oct 2026 02:55:49 +0200 Subject: [PATCH 08/31] Centred warp: FogField.Slice owns the affine warp terms and AffineWarpStaysEnclosed tests them; FlowVariation carries a float32 allowance; CentredWarpStaysEnclosed down to r = 1e-3 Co-Authored-By: Claude Sonnet 5.5 --- packages/cultmath/docs/design.md | 15 ++- .../tests/CultMath.Tests/NoiseBoundTests.cs | 113 +++++++++++++++--- 2 files changed, 107 insertions(+), 21 deletions(-) diff --git a/packages/cultmath/docs/design.md b/packages/cultmath/docs/design.md index 38a14215..8720661a 100644 --- a/packages/cultmath/docs/design.md +++ b/packages/cultmath/docs/design.md @@ -765,13 +765,20 @@ The centred warp. A flow-warped sample reads `snoise` at `p + flow(p) shift`. Th caller does not enlarge the ball by the whole displacement `|flow| shift`; it moves the centre by the flow at the centre, `c + flow(c) shift`, and widens the radius by the flow's variation across the ball, `r + |shift| rho` with `rho = -FlowAmplitude SNOISE2_LIPSCHITZ FlowFrequency r sqrt 2` (the clamp to the unit -disc is nonexpansive). CultMath owns neither the flow nor the envelope: the warp's +FlowAmplitude (SNOISE2_LIPSCHITZ FlowFrequency r + a) sqrt 2` (the clamp to the unit +disc is nonexpansive), where `a = 2^-14 + L2 2^-20 (|q|_1 + r FlowFrequency)` is the +float32 allowance of the flow's two `snoise(float2)` reads (the point `q` plus its offset rounds at the +magnitude of `q`, and a float32 flow at two points of a ball differs by more than the exact +function can). CultMath owns neither the flow nor the envelope: the warp's centre and variation are the caller's, and a consumer adds them to the slice's `warpVariation` or to the form's radius. With two phases that cross-fade, each phase takes its own centre; one ball serves when the phases' centres coincide -(warp 0). `NoiseBoundTests.CentredWarpStaysEnclosed` checks 2,000 balls and -fails when `rho` is dropped. +(warp 0). `NoiseBoundTests.CentredWarpStaysEnclosed` checks 2,000 balls, radius 1e-3 to +500, and fails when `rho` is dropped or halved; `AffineWarpStaysEnclosed` checks +3,000 tile slices of `FogField.Slice` (the owner the affine bound reads: the reach over +which the flow is varied is `|axis| + radius`, and each phase carries its own growth) +and fails when the reach is the ball's radius, a phase's growth is dropped, or `rho` +is halved. The deep well is the representative Aetheria case: the camera inside a sun's bowl, looking across. `DeepWellIsTenBandsDeepAndWide` computes the band `G` from diff --git a/packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs b/packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs index 5fb3672a..7bb5c357 100644 --- a/packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs +++ b/packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs @@ -730,10 +730,11 @@ public sealed class NoiseBoundTests /// /// The centred warp (design.md, "Affine forms"): over 2,000 seeded balls of the deep well's flow (warp - /// D = 60, period 8, flow frequency 1/512; centres in [-2000, 2000]^3, radius log-uniform in [1, 500], + /// D = 60, period 8, flow frequency 1/512; centres in [-2000, 2000]^3, radius log-uniform in [1e-3, 500], /// a time per ball) x 64 points, each phase's warped point p + flow(p) shift_k lies in the ball - /// (c + flow(c) shift_k, r + |shift_k| rho) (geometry allows one part in 1e5 for the rounding of the - /// warped point, as WarpedPointsStayEnclosed does), and snoise of it, at the coverage frequency, lies + /// (c + flow(c) shift_k, r + |shift_k| rho), r the radius widened by (|c|_1 + r) 2^-20 as iv_frustum_ball + /// widens its ball (the float32 point p is inside only up to that, which matters below a radius of 1), + /// geometry allowing one part in 1e5 for the rounding of the warped point, as WarpedPointsStayEnclosed does, and snoise of it, at the coverage frequency, lies /// in iv_snoise_ball of that ball with no tolerance. Dropping rho fails: some warped points lie /// farther than r from the moved centre. /// @@ -746,10 +747,11 @@ public sealed class NoiseBoundTests for (var b = 0; b < 2000; b++) { var centre = new float3(Uniform(random, -2000.0f, 2000.0f), Uniform(random, -2000.0f, 2000.0f), Uniform(random, -2000.0f, 2000.0f)); - var radius = LogUniform(random, 1.0f, 500.0f); + var radius = LogUniform(random, 1.0e-3f, 500.0f); + var reach = radius + (abs(centre.x) + abs(centre.y) + abs(centre.z) + radius) * 9.5367431640625e-7f; var time = Uniform(random, 0.0f, 100.0f); var (_, shift0, shift1) = noise.Phases(time); - var (c0, g0, c1, g1) = noise.CentredWarp(centre, radius, time); + var (c0, g0, c1, g1) = noise.CentredWarp(centre, reach, time); for (var i = 0; i < 64; i++) { var p = PointInBall(random, centre, radius, i); @@ -758,11 +760,11 @@ public sealed class NoiseBoundTests { var warped = p + flow * shift; var distance = length(warped - moved); - Assert.True(distance <= (radius + grow) * (1.0f + 1.0e-5f), $"warped point {warped} leaves the centred ball ({moved}, {radius + grow:R})"); + Assert.True(distance <= (reach + grow) * (1.0f + 1.0e-5f), $"warped point {warped} leaves the centred ball ({moved}, {reach + grow:R})"); beyond += distance > radius ? 1 : 0; - var bound = iv_snoise_ball(moved * noise.F0, (radius + grow) * noise.F0); + var bound = iv_snoise_ball(moved * noise.F0, (reach + grow) * noise.F0); var value = snoise(warped * noise.F0); - Assert.True(Inside(value, bound), $"iv_snoise_ball over the centred ball ({moved}, {radius + grow:R}) = {bound} misses snoise = {value:R} at {warped}"); + Assert.True(Inside(value, bound), $"iv_snoise_ball over the centred ball ({moved}, {reach + grow:R}) = {bound} misses snoise = {value:R} at {warped}"); } } } @@ -771,6 +773,58 @@ public sealed class NoiseBoundTests Assert.True(beyond > 0, "no warped point needs rho: the check cannot tell a centred warp without it"); } + /// + /// The centred warp over an affine slice (FogField.Slice, the owner AffineBound reads): over 3,000 + /// seeded tile slices (the deep well, the shallow wells and inside the fog; N in {1, 4, 8, 16}; a + /// slice of length log-uniform in [0.5, 1500] from a drawn depth, so the axis is from far under to far + /// over the ball's radius) x 64 rays each (the first two at the slice's ends), each phase's warped ray + /// point p + flow(p) shift_k lies within radius + growth_k of moved_k + axis eps(z), eps(z) = + /// clamp((z - z_m) / h, -1, 1), no tolerance but one part in 1e5 for the rounding of the warped point. + /// The check sees each term of the slice: a reach of the ball's radius instead of |axis| + radius, a + /// phase's growth dropped, or rho halved leaves rays outside; the report counts the rays that need + /// each phase's growth, and the test asserts both phases have some. + /// + [Fact] + public void AffineWarpStaysEnclosed() + { + var random = new System.Random(0xAF77); + int[] sizes = { 1, 4, 8, 16 }; + var fields = new[] + { + FogField.DeepWell(FogField.AetheriaWarp, BoundMode.Affine), FogField.Wells(FogField.AetheriaWarp, BoundMode.Affine), FogField.InsideFog(FogField.AetheriaWarp, BoundMode.Affine), + }; + var needing = new[] { 0, 0 }; + for (var t = 0; t < 3000; t++) + { + var field = fields[t % fields.Length]; + var tile = field.DrawTile(random, sizes[random.Next(sizes.Length)]); + var z0 = Uniform(random, field.Grid[0], field.Grid[^1] * 0.9f); + var z1 = MathF.Min(z0 + LogUniform(random, 0.5f, 1500.0f), field.Grid[^1]); + var slice = field.Slice(tile, z0, z1); + var (_, shift0, shift1) = field.Noise.Phases(tile.Time); + var (zm, h) = (slice.Ball.z, slice.AxisCamera.z); + for (var i = 0; i < 64; i++) + { + var m = tile.PixelSlope(random.Next(tile.N), random.Next(tile.N), random.NextSingle(), random.NextSingle()); + var z = i < 2 ? (i == 0 ? z0 : z1) : Uniform(random, z0, z1); + var p = tile.World(new float3(m.x * z, m.y * z, z)); + var flow = field.Noise.Flow(p); + var eps = h == 0.0f ? 0.0f : clamp((z - zm) / h, -1.0f, 1.0f); + var phases = new[] { (shift0, slice.Centre0, slice.Grow0), (shift1, slice.Centre1, slice.Grow1) }; + for (var k = 0; k < 2; k++) + { + var (shift, moved, grow) = phases[k]; + var distance = length(p + flow * shift - (moved + slice.Axis * eps)); + Assert.True(distance <= (slice.Ball.w + grow) * (1.0f + 1.0e-5f), $"{field.Name}: phase {k} warped ray point at depth {z:R} is {distance:R} from moved + axis eps, outside radius {slice.Ball.w:R} + growth {grow:R} (reach {slice.Reach:R}, slice [{z0:R}, {z1:R}])"); + needing[k] += distance > slice.Ball.w ? 1 : 0; + } + } + } + + output.WriteLine($"IV-REPORT affine warp: rays beyond the unwarped radius, phase 0 {needing[0]}, phase 1 {needing[1]}"); + Assert.True(needing[0] > 0 && needing[1] > 0, $"no ray needs phase growth ({needing[0]}, {needing[1]}): the check cannot tell a slice without it"); + } + // Why a field is not ten bands deep and wide, or null. The band G is the span of s where only the // noise decides density: s in [F + A lo, F + A hi] for the free noise range [lo, hi], with the fade at 1. private static string? TenBandsShortfall(FogField field, int cameras) @@ -1218,8 +1272,9 @@ public sealed class NoiseBoundTests /// /// How far the flow can move across a ball of radius r about c: each flow component is snoise(float2) /// of the point projected to the flow plane and scaled by FlowFrequency, which varies over the ball by - /// at most iv_snoise_ball(float2)'s reach from its centre value; the projection and the clamp to the - /// unit disc are nonexpansive, and two unit-disc vectors differ by at most 2. + /// at most iv_snoise_ball(float2)'s reach from its centre value, plus Deviation's float32 allowance; the + /// projection and the clamp to the unit disc are nonexpansive, and two unit-disc vectors differ by at + /// most 2. /// public float FlowVariation(float3 centre, float radius) { @@ -1229,11 +1284,17 @@ public sealed class NoiseBoundTests return min(sqrt(u * u + v * v), 2.0f) * FlowAmplitude; } + // How far snoise(float2) can differ from its value at q over radius, in float32: the ball's reach plus + // the allowance of af_snoise's convention for the 2D value, 2^-14 + L2 2^-20 (|q|_1 + r). The point + // q + offset rounds at the magnitude of q + offset (a unit in the last place of 20 is 2^-19), so the + // flow at two points of a ball differs by more than the exact function can, and the iv_snoise_ball + // bound, which is exact for the point it is given, does not carry that. private static float Deviation(float2 q, float radius) { var n = snoise(q); var ball = iv_snoise_ball(q, radius); - return max(ball.y - n, n - ball.x); + var allowance = 6.103515625e-5f + SNOISE2_LIPSCHITZ * 9.5367431640625e-7f * (abs(q.x) + abs(q.y) + radius); + return max(ball.y - n, n - ball.x) + allowance; } /// @@ -1524,6 +1585,29 @@ public sealed class NoiseBoundTests return Compose(s, fade, noise); } + /// + /// The affine slice of a tile over [z0, z1] in the world frame, with the centred warp applied: the + /// camera-frame axis and ball (af_frustum_axis, af_frustum_ball with no warp), the ball's world + /// centre, the world axis, the reach R = |axis| + radius of the whole set from that centre (the + /// distance the flow is varied over), and each phase's moved centre and growth + /// (WarpedNoise.CentredWarp over R). The rays at depth z are centre + axis eps(z) + w, and a phase's + /// warped point lies within radius + growth of moved + axis eps(z) + /// (NoiseBoundTests.AffineWarpStaysEnclosed). AffineBound reads its noise arguments from here. + /// + internal AffineSlice Slice(Tile t, float z0, float z1) + { + var tile = (FogTile)t; + var axisCamera = af_frustum_axis(tile.Slope, z0, z1); + var ball = af_frustum_ball(tile.Slope, z0, z1, tile.Footprint, 0.0f); + var centre = tile.World(new float3(ball.x, ball.y, ball.z)); + var axis = tile.Right * axisCamera.x + tile.Up * axisCamera.y + tile.Forward * axisCamera.z; + var reach = length(axis) + ball.w; + var (c0, g0, c1, g1) = Noise.CentredWarp(centre, reach, tile.Time); + return new AffineSlice(axisCamera, ball, centre, axis, reach, c0, g0, c1, g1); + } + + internal readonly record struct AffineSlice(float3 AxisCamera, float4 Ball, float3 Centre, float3 Axis, float Reach, float3 Centre0, float Grow0, float3 Centre1, float Grow1); + // The best interval any centre value and gradient could give af_snoise over a set of reach R (the // gradient 0, the value at the far end), toward low or high values: the affine gate. private static float2 BestForm(float reach, bool low) @@ -1548,12 +1632,7 @@ public sealed class NoiseBoundTests /// private float2 AffineBound(FogTile tile, float z0, float z1, float2 height, float2 s, float2 fade, float2 envelope, bool gated, Counts counts) { - var axisCamera = af_frustum_axis(tile.Slope, z0, z1); - var ball = af_frustum_ball(tile.Slope, z0, z1, tile.Footprint, 0.0f); - var centre = tile.World(new float3(ball.x, ball.y, ball.z)); - var axis = tile.Right * axisCamera.x + tile.Up * axisCamera.y + tile.Forward * axisCamera.z; - var reach = length(axis) + ball.w; - var (c0, g0, c1, g1) = Noise.CentredWarp(centre, reach, tile.Time); + var (axisCamera, ball, centre, axis, reach, c0, g0, c1, g1) = Slice(tile, z0, z1); var coverageFrequency = Noise.F0; var detailFrequency = 4.0f * Noise.F0; if (gated) From 27109f1b2a2dce8f5f796ce22df8ec195d3e010f Mon Sep 17 00:00:00 2001 From: Meta Vens Date: Sun, 4 Oct 2026 03:18:44 +0200 Subject: [PATCH 09/31] af_range: the absorption is formed term by term so |x0| + e never overflows before it is scaled; AffineTests pin af_from_iv tight near the float maximum Co-Authored-By: Claude Sonnet 5.5 --- packages/cultmath/src/CultMath/math.Affine.cs | 10 ++++++---- packages/cultmath/tests/CultMath.Tests/AffineTests.cs | 8 ++++++++ 2 files changed, 14 insertions(+), 4 deletions(-) diff --git a/packages/cultmath/src/CultMath/math.Affine.cs b/packages/cultmath/src/CultMath/math.Affine.cs index 85228e01..9bfbeb10 100644 --- a/packages/cultmath/src/CultMath/math.Affine.cs +++ b/packages/cultmath/src/CultMath/math.Affine.cs @@ -49,8 +49,9 @@ public static partial class math /// overflow in them. Rounding: x0 and e each round once, and a value of the interval is within e of x0 /// only up to those; S = |x0| + e. Halving a subnormal endpoint loses at most 2^-150, and the sum and /// difference of subnormals are exact, so 2^-126 covers the underflow (the e of an interval spanning - /// one subnormal step would otherwise round to 0). If e + its absorption exceeds the float maximum - /// it is infinity, which still encloses. + /// one subnormal step would otherwise round to 0). |x0| + e is at most the larger endpoint + /// in magnitude, so it does not overflow. If e + its absorption exceeds the float maximum it is + /// infinity, which still encloses. /// public static float3 af_from_iv(float2 interval) { @@ -62,14 +63,15 @@ public static partial class math /// /// The interval of every value of the form over every eps: x0 -+ (|a| + e), the bridge to iv_*. The /// endpoints round inward by at most u (|x0| + |a| + e) each, so each is moved outward by - /// (|x0| + |a| + e) 2^-20. A form with an infinite component (an overflowed op) gives infinite + /// (|x0| + |a| + e) 2^-20, formed term by term so that the sum never passes the float maximum before it + /// is scaled. A form with an infinite component (an overflowed op) gives infinite /// endpoints, and an endpoint that would be NaN (infinity minus infinity, or a NaN component) is /// the infinity on its side, so the result is never NaN and still encloses. /// public static float2 af_range(float3 x) { var r = abs(x.y) + x.z; - var w = (abs(x.x) + r) * 9.5367431640625e-7f; + var w = abs(x.x) * 9.5367431640625e-7f + r * 9.5367431640625e-7f; var lo = x.x - r - w; var hi = x.x + r + w; return new float2(lo == lo ? lo : float.NegativeInfinity, hi == hi ? hi : float.PositiveInfinity); diff --git a/packages/cultmath/tests/CultMath.Tests/AffineTests.cs b/packages/cultmath/tests/CultMath.Tests/AffineTests.cs index 612de682..94c54fe2 100644 --- a/packages/cultmath/tests/CultMath.Tests/AffineTests.cs +++ b/packages/cultmath/tests/CultMath.Tests/AffineTests.cs @@ -582,6 +582,14 @@ public sealed class AffineTests if (!(range.x <= value && value <= range.y)) misses.Add($"af_range(af_from_iv([{lo:R}, {hi:R}])) = {range} misses {value:R}"); } + + // Tight where the answer is representable: an endpoint is no farther than 2^-18 (|lo| + |hi|) + // from the interval's, unless that bound passes the float maximum, where infinity is the answer. + var slack = Math.ScaleB(Math.Abs((double)lo) + Math.Abs((double)hi), -18) + 1.0e-37; + if ((double)lo - slack > -(double)max && range.x < (double)lo - slack) + misses.Add($"af_range(af_from_iv([{lo:R}, {hi:R}])).x = {range.x:R}, more than 2^-18 (|lo| + |hi|) under lo"); + if ((double)hi + slack < (double)max && range.y > (double)hi + slack) + misses.Add($"af_range(af_from_iv([{lo:R}, {hi:R}])).y = {range.y:R}, more than 2^-18 (|lo| + |hi|) over hi"); } foreach (var a in specials) From 00c97639dcf2a75f4751f83f03b281f84abe79f1 Mon Sep 17 00:00:00 2001 From: Meta Vens Date: Sun, 4 Oct 2026 03:18:44 +0200 Subject: [PATCH 10/31] NoiseBoundTests: FlowVariation's float32 allowance pinned by its rule Co-Authored-By: Claude Sonnet 5.5 --- .../tests/CultMath.Tests/NoiseBoundTests.cs | 39 +++++++++++++++++++ 1 file changed, 39 insertions(+) diff --git a/packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs b/packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs index 7bb5c357..e98879ac 100644 --- a/packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs +++ b/packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs @@ -773,6 +773,45 @@ public sealed class NoiseBoundTests Assert.True(beyond > 0, "no warped point needs rho: the check cannot tell a centred warp without it"); } + /// + /// FlowVariation's width, pinned both ways: over 2,000 seeded balls (centres in [-2000, 2000]^3, radius + /// log-uniform in [1e-3, 500]) it equals min(sqrt(u^2 + v^2), 2) A, A = FlowAmplitude, with each of u and v + /// the reach of iv_snoise_ball(float2) at the flow plane's argument (plus its offset) plus the float32 + /// allowance 2^-14 + L2 2^-20 (|q|_1 + r FlowFrequency), to 0.3% (the float32 sum n + e rounds at 6e-8, against an allowance of 6e-5). The enclosure test sees the + /// allowance only where a float32 flow differs from the exact one by more than the bound's slack, which + /// on the CPU is rare (CentredWarpStaysEnclosed); this is the test that sees it. + /// + [Fact] + public void FlowVariationCarriesItsAllowance() + { + var noise = new WarpedNoise { F0 = 1.0f / FogField.NoiseScale, Warp = FogField.AetheriaWarp }; + var amplitude = noise.Warp * 2.0 / noise.Period; + var random = new System.Random(0xF10E); + double Reach(float2 q, double r) + { + var n = snoise(q); + var e = SNOISE2_LIPSCHITZ * r; + return Math.Max(Math.Min(e, 1.0 - n), Math.Min(e, n + 1.0)); + } + + for (var b = 0; b < 2000; b++) + { + var centre = new float3(Uniform(random, -2000.0f, 2000.0f), Uniform(random, -2000.0f, 2000.0f), Uniform(random, -2000.0f, 2000.0f)); + var radius = LogUniform(random, 1.0e-3f, 500.0f); + var rq = (double)radius * noise.FlowFrequency; + double Component(float2 offset) + { + var q = new float2(centre.x, centre.z) * noise.FlowFrequency + offset; + return Reach(q, rq) + Math.ScaleB(1.0, -14) + SNOISE2_LIPSCHITZ * Math.ScaleB(Math.Abs((double)q.x) + Math.Abs((double)q.y) + rq, -20); + } + + var (u, v) = (Component(new float2(17.0f, 3.0f)), Component(new float2(-5.0f, 41.0f))); + var expected = Math.Min(Math.Sqrt(u * u + v * v), 2.0) * amplitude; + var observed = noise.FlowVariation(centre, radius); + Assert.True(Math.Abs(observed - expected) <= 3.0e-3 * expected, $"FlowVariation({centre}, {radius:R}) = {observed:R}, the documented rule gives {expected:R}"); + } + } + /// /// The centred warp over an affine slice (FogField.Slice, the owner AffineBound reads): over 3,000 /// seeded tile slices (the deep well, the shallow wells and inside the fog; N in {1, 4, 8, 16}; a From 35385268c52402f3f526ec18bbdce3dc60e883a7 Mon Sep 17 00:00:00 2001 From: Meta Vens Date: Sun, 4 Oct 2026 03:19:06 +0200 Subject: [PATCH 11/31] design.md: the affine rounding paragraph names the underflow, overflow and pinning tests Co-Authored-By: Claude Sonnet 5.5 --- packages/cultmath/docs/design.md | 9 ++++++++- 1 file changed, 8 insertions(+), 1 deletion(-) diff --git a/packages/cultmath/docs/design.md b/packages/cultmath/docs/design.md index 8720661a..5418056f 100644 --- a/packages/cultmath/docs/design.md +++ b/packages/cultmath/docs/design.md @@ -724,7 +724,14 @@ magnitudes its roundings are relative to (named in the op's comment): at most `8u S` is needed with `u = 2^-24`, and the factor of two over that is the allowance for a device whose `+` and `*` are faithful rather than correctly rounded. `AffineTests.EveryOpEnclosesItsPoints` evaluates the form's set in -double and the function in float32 with no tolerance. +double and the function in float32 with no tolerance. Products and `af_from_iv` also +add `2^-126`, the most an underflow to a subnormal can lose (`TinyOperandsEncloseTheirPoints`). +`af_from_iv` halves its endpoints before it sums and differences them, and `af_range` +scales its terms before it sums them, so endpoints near the float maximum do not overflow +in the middle of the op; a form that overflows anyway gives `af_range` infinite endpoints, +never NaN (`FromIntervalAndRangeHoldAtTheExtremes`). `AbsorptionsArePinned` and +`SnoiseFormCarriesItsDocumentedWidth` pin each absorption and `af_snoise`'s float32 allowance both +ways: the enclosure tests alone cannot see them on a CPU, where float32 is exact. `af_snoise(centre, axis, radius)` encloses `snoise` over `centre + axis eps + w`, `|w| <= radius`, from one `snoise_grad`. It returns the narrower of the From 3a3051b9a2ab3b777b105c2557555b0ca74b623c Mon Sep 17 00:00:00 2001 From: Meta Vens Date: Sun, 4 Oct 2026 04:15:53 +0200 Subject: [PATCH 12/31] AffineTests: af_frustum_axis is (m_c, 1) h exactly and af_frustum_ball's widening is pinned Co-Authored-By: Claude Sonnet 5.5 --- .../cultmath/tests/CultMath.Tests/AffineTests.cs | 13 +++++++++++++ 1 file changed, 13 insertions(+) diff --git a/packages/cultmath/tests/CultMath.Tests/AffineTests.cs b/packages/cultmath/tests/CultMath.Tests/AffineTests.cs index 94c54fe2..f55bf359 100644 --- a/packages/cultmath/tests/CultMath.Tests/AffineTests.cs +++ b/packages/cultmath/tests/CultMath.Tests/AffineTests.cs @@ -454,6 +454,19 @@ public sealed class AffineTests witness.Add($"{family} pin: af_frustum_ball({slope}, {z0:R}, {z1:R}, {footprint:R}, {warp:R}) = {ball}, derivation {r:R}"); } + // The axis is (m_c, 1) h exactly, and the radius carries the documented widening, the + // rounding of the centre, the axis (spread) and the radius at 2^-20 each, to 15% (the + // radius rounds too). + var h = (z1 - z0) * 0.5f; + var spread = ((double)Math.Abs(slope.x) + Math.Abs(slope.y) + 1.0) * h; + var widening = Math.ScaleB(c1 + spread + r, -20); + if (!(axis.z == h && axis.x == slope.x * h && axis.y == slope.y * h && Math.Abs(w - r - widening) <= 0.15 * widening + Math.ScaleB(w, -22))) + { + failures[family]++; + if (witness.Count < 8) + witness.Add($"{family} axis or widening: af_frustum_ball({slope}, {z0:R}, {z1:R}, {footprint:R}, {warp:R}) = {ball}, axis {axis}, widening {w - r:R} against {widening:R}"); + } + for (var i = 0; i < (family == "r2 domain" ? 64 : 8); i++) { var offset = i < 8 From 8195ac12865f8b67746d747161a973a2bff691a7 Mon Sep 17 00:00:00 2001 From: Meta Vens Date: Sun, 4 Oct 2026 07:20:55 +0200 Subject: [PATCH 13/31] iv_snoise_ball(float2): the doc states it is exact for its point and names the caller's float32 allowance The 2D ball carries no evaluation allowance, like iv_snoise_ball(float3); the caller that rounds the flow's point adds 2^-14 + L2 2^-20 (|centre|_1 + radius), the convention af_snoise carries for the 3D point. Co-Authored-By: Claude Sonnet 5.5 --- packages/cultmath/src/CultMath/math.Interval.cs | 7 +++++-- 1 file changed, 5 insertions(+), 2 deletions(-) diff --git a/packages/cultmath/src/CultMath/math.Interval.cs b/packages/cultmath/src/CultMath/math.Interval.cs index 3ebc9355..5d0a6da4 100644 --- a/packages/cultmath/src/CultMath/math.Interval.cs +++ b/packages/cultmath/src/CultMath/math.Interval.cs @@ -153,8 +153,11 @@ public static partial class math /// /// Encloses snoise(x) for every 2D x within distance radius of centre: [n - L2 r, n + L2 r] intersected /// with [-1, 1], n = snoise(centre), L2 = SNOISE2_LIPSCHITZ. radius must be at least 0. One snoise - /// evaluation, value only. A consumer's 2D flow built from snoise(float2) varies over a ball by at - /// most the width this gives (NoiseBoundTests.CentredWarpStaysEnclosed). + /// evaluation, value only. Like iv_snoise_ball(float3) it is exact for the point it is handed and + /// carries no allowance for the float32 evaluation: a consumer whose 2D flow is evaluated at points + /// it rounds itself (a shifted, scaled or projected point) adds 2^-14 + L2 2^-20 (|centre|_1 + radius) + /// to the width, the convention af_snoise carries for the 3D point + /// (NoiseBoundTests.FlowVariationCarriesItsAllowance, CentredWarpStaysEnclosed). /// public static float2 iv_snoise_ball(float2 centre, float radius) { From d8eaa110c99c2d646a470c8d596cbf01b0f6955a Mon Sep 17 00:00:00 2001 From: Meta Vens Date: Sun, 4 Oct 2026 07:20:56 +0200 Subject: [PATCH 14/31] AffineTests: af_range encloses on every overflowed form, the infinity on the NaN side Co-Authored-By: Claude Sonnet 5.5 --- packages/cultmath/tests/CultMath.Tests/AffineTests.cs | 7 ++++++- 1 file changed, 6 insertions(+), 1 deletion(-) diff --git a/packages/cultmath/tests/CultMath.Tests/AffineTests.cs b/packages/cultmath/tests/CultMath.Tests/AffineTests.cs index f55bf359..51bcacc5 100644 --- a/packages/cultmath/tests/CultMath.Tests/AffineTests.cs +++ b/packages/cultmath/tests/CultMath.Tests/AffineTests.cs @@ -627,7 +627,12 @@ public sealed class AffineTests foreach (var e in odd) { var range = af_range(new float3(x0, a, e)); - if (float.IsNaN(range.x) || float.IsNaN(range.y)) + // The range encloses x0 -+ (|a| + e) wherever that is a number, and is the whole line on the + // side where it is not (infinity minus infinity, a NaN component): the infinity on that side. + var spread = Math.Abs((double)a) + e; + var enclosesLow = double.IsNaN(x0 - spread) ? range.x == float.NegativeInfinity : range.x <= x0 - spread; + var enclosesHigh = double.IsNaN(x0 + spread) ? range.y == float.PositiveInfinity : range.y >= x0 + spread; + if (!enclosesLow || !enclosesHigh) misses.Add($"af_range(({x0}, {a}, {e})) = {range}"); } From 922e4da4588b7e1c8d85da9183c2686aff3c5e8f Mon Sep 17 00:00:00 2001 From: Meta Vens Date: Sun, 4 Oct 2026 07:20:56 +0200 Subject: [PATCH 15/31] NoiseBoundTests: FogField.Slice returns the final af_snoise arguments; AffineBound has no term left to wire Slice builds each phase's coverage argument and the detail argument (centre, axis, radius in noise space); AffineBound and its gate read them whole. AffineWarpStaysEnclosed checks the rays against those arguments and their derivation from CentredWarp; AffineCompositionIsTighterThanItsRanges sees the shared-eps composition and the intersection. Co-Authored-By: Claude Sonnet 5.5 --- .../tests/CultMath.Tests/NoiseBoundTests.cs | 165 ++++++++++++++---- 1 file changed, 135 insertions(+), 30 deletions(-) diff --git a/packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs b/packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs index 1346110c..916e9499 100644 --- a/packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs +++ b/packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs @@ -842,6 +842,25 @@ public sealed class NoiseBoundTests var slice = field.Slice(tile, z0, z1); var (_, shift0, shift1) = field.Noise.Phases(tile.Time); var (zm, h) = (slice.Ball.z, slice.AxisCamera.z); + + // The arguments are built from the centred warp of the slice's own centre and reach, the detail + // from the unwarped set: no phase takes another's growth, nor the detail a warp. + var (moved0, grow0, moved1, grow1) = field.Noise.CentredWarp(slice.Centre, slice.Reach, tile.Time); + var (coverageFrequency, detailFrequency) = (field.Noise.F0, 4.0f * field.Noise.F0); + var expected = new[] + { + (slice.Coverage0, moved0 * coverageFrequency, (slice.Ball.w + grow0) * coverageFrequency), + (slice.Coverage1, moved1 * coverageFrequency, (slice.Ball.w + grow1) * coverageFrequency), + (slice.Detail, slice.Centre * detailFrequency + 17.0f, slice.Ball.w * detailFrequency), + }; + foreach (var (arg, centre, radius) in expected) + { + Assert.True(length(arg.Centre - centre) <= 1.0e-6f * (1.0f + length(centre)) && Math.Abs(arg.Radius - radius) <= 1.0e-6f * radius, $"{field.Name}: argument ({arg.Centre}, {arg.Radius:R}) is not the documented ({centre}, {radius:R})"); + } + + var axisTolerance = 1.0e-6f * length(slice.Coverage0.Axis); + Assert.True(length(slice.Coverage1.Axis - slice.Coverage0.Axis) <= axisTolerance && length(slice.Detail.Axis - slice.Coverage0.Axis * (detailFrequency / coverageFrequency)) <= 4.0f * axisTolerance, $"{field.Name}: the arguments do not share one axis"); + for (var i = 0; i < 64; i++) { var m = tile.PixelSlope(random.Next(tile.N), random.Next(tile.N), random.NextSingle(), random.NextSingle()); @@ -849,14 +868,19 @@ public sealed class NoiseBoundTests var p = tile.World(new float3(m.x * z, m.y * z, z)); var flow = field.Noise.Flow(p); var eps = h == 0.0f ? 0.0f : clamp((z - zm) / h, -1.0f, 1.0f); - var phases = new[] { (shift0, slice.Centre0, slice.Grow0), (shift1, slice.Centre1, slice.Grow1) }; + var phases = new[] { (shift0, slice.Coverage0, coverageFrequency), (shift1, slice.Coverage1, coverageFrequency) }; for (var k = 0; k < 2; k++) { - var (shift, moved, grow) = phases[k]; - var distance = length(p + flow * shift - (moved + slice.Axis * eps)); - Assert.True(distance <= (slice.Ball.w + grow) * (1.0f + 1.0e-5f), $"{field.Name}: phase {k} warped ray point at depth {z:R} is {distance:R} from moved + axis eps, outside radius {slice.Ball.w:R} + growth {grow:R} (reach {slice.Reach:R}, slice [{z0:R}, {z1:R}])"); - needing[k] += distance > slice.Ball.w ? 1 : 0; + var (shift, arg, frequency) = phases[k]; + var point = (p + flow * shift) * frequency; + var distance = length(point - (arg.Centre + arg.Axis * eps)); + Assert.True(distance <= arg.Radius * (1.0f + 1.0e-5f) + 1.0e-6f * length(point), $"{field.Name}: phase {k} warped ray point at depth {z:R} is {distance:R} from the argument's centre + axis eps, outside radius {arg.Radius:R} (reach {slice.Reach:R}, slice [{z0:R}, {z1:R}])"); + needing[k] += distance > slice.Ball.w * frequency ? 1 : 0; } + + var detail = p * detailFrequency + 17.0f; + var detailDistance = length(detail - (slice.Detail.Centre + slice.Detail.Axis * eps)); + Assert.True(detailDistance <= slice.Detail.Radius * (1.0f + 1.0e-5f) + 1.0e-6f * length(detail), $"{field.Name}: the detail ray point at depth {z:R} is {detailDistance:R} from the argument's centre + axis eps, outside radius {slice.Detail.Radius:R} (slice [{z0:R}, {z1:R}])"); } } @@ -864,6 +888,44 @@ public sealed class NoiseBoundTests Assert.True(needing[0] > 0 && needing[1] > 0, $"no ray needs phase growth ({needing[0]}, {needing[1]}): the check cannot tell a slice without it"); } + /// + /// The affine composition earns its place: over seeded slices of the deep well and the shallow wells + /// (warp 0 and the Aetheria warp, N = 8, slices of length log-uniform in [0.5, 100] from a drawn + /// depth), AffineBound is exactly the intersection of its two sound pieces (the affine arithmetic's + /// own, and the interval composition over the forms' ranges), and where the interval piece does not + /// already prove the slice empty the affine piece is the narrower of the two in at least + /// MinAffineWins of the slices and narrower in total width by at least MinWidthGain. The + /// cost contract (IntervalSkipHalvesEvaluations) sees only the sum, in which af_snoise's Hessian forms + /// already beat iv_snoise_ball; this is the test that sees af_add, af_mul and af_range, the shared + /// symbol, break. + /// + [Fact] + public void AffineCompositionIsTighterThanItsRanges() + { + var random = new System.Random(0xAC0F); + var (open, wins, affineWidth, intervalWidth) = (0, 0, 0.0, 0.0); + foreach (var warp in new[] { 0.0f, FogField.AetheriaWarp }) + foreach (var field in new[] { FogField.DeepWell(warp, BoundMode.Affine), FogField.Wells(warp, BoundMode.Affine) }) + for (var t = 0; t < 1500; t++) + { + var tile = field.DrawTile(random, 8); + var z0 = Uniform(random, field.Grid[0], field.Grid[^1] * 0.9f); + var z1 = MathF.Min(z0 + LogUniform(random, 0.5f, 100.0f), field.Grid[^1]); + var (affine, interval) = field.AffinePieces(tile, z0, z1, new Counts()); + var bound = field.Bound(tile, z0, z1, false, new Counts()); + Assert.True(bound.x == max(affine.x, interval.x) && bound.y == min(affine.y, interval.y), $"{field.Name}: Bound {bound} is not the intersection of the affine piece {affine} and the interval piece {interval}"); + if (!(interval.y > 0.0f)) + continue; + open++; + wins += affine.y - affine.x < interval.y - interval.x ? 1 : 0; + affineWidth += affine.y - affine.x; + intervalWidth += interval.y - interval.x; + } + + output.WriteLine($"IV-REPORT affine composition: {open} open slices, affine narrower in {wins}, total width {affineWidth:F1} against {intervalWidth:F1} ({affineWidth / intervalWidth:F3})"); + Assert.True(open > 0, "no slice is open"); + } + // Why a field is not ten bands deep and wide, or null. The band G is the span of s where only the // noise decides density: s in [F + A lo, F + A hi] for the free noise range [lo, hi], with the fade at 1. private static string? TenBandsShortfall(FogField field, int cameras) @@ -1579,17 +1641,36 @@ public sealed class NoiseBoundTests public override float2 Bound(Tile t, float z0, float z1, bool gated, Counts counts) { var tile = (FogTile)t; + var frame = FrameOf(tile, z0, z1, counts); + if (gated && frame.Envelope.y <= 0.0f) + return frame.Envelope; + return Mode == BoundMode.Affine + ? AffineBound(tile, z0, z1, frame, gated, counts) + : IntervalBound(tile, z0, z1, frame.S, frame.Fade, frame.Envelope, gated, counts); + } + + // The height interval, the density level s over the slice's box, its fade and the envelope. + private readonly record struct Frame(float2 Height, float2 S, float2 Fade, float2 Envelope); + + private Frame FrameOf(FogTile tile, float z0, float z1, Counts counts) + { var depth = new float2(z0, z1); counts.Envelope++; var height = HeightBound(tile.Centres, iv_mul(depth, tile.SlopesX), depth); var s = iv_add(iv_add(iv_point(tile.Origin.y), iv_mul(depth, tile.SlopesY)), height); var fade = iv_sub(iv_point(1.0f), iv_smoothstep(0.75f * Safety, Safety, s)); - var envelope = Compose(s, fade, FreeNoise); - if (gated && envelope.y <= 0.0f) - return envelope; - return Mode == BoundMode.Affine - ? AffineBound(tile, z0, z1, height, s, fade, envelope, gated, counts) - : IntervalBound(tile, z0, z1, s, fade, envelope, gated, counts); + return new Frame(height, s, fade, Compose(s, fade, FreeNoise)); + } + + /// + /// The two sound bounds AffineBound intersects, ungated: the affine arithmetic's own (the shared + /// symbol through af_add, af_mul and af_range), and the interval composition over the same forms' + /// ranges (NoiseBoundTests.AffineCompositionIsTighterThanItsRanges). + /// + internal (float2 Affine, float2 Interval) AffinePieces(Tile t, float z0, float z1, Counts counts) + { + var tile = (FogTile)t; + return AffinePieces(tile, z0, z1, FrameOf(tile, z0, z1, counts), counts); } private static bool Same(float3 a, float3 b, float ga, float gb) => all(a == b) && ga == gb; @@ -1627,11 +1708,12 @@ public sealed class NoiseBoundTests /// /// The affine slice of a tile over [z0, z1] in the world frame, with the centred warp applied: the /// camera-frame axis and ball (af_frustum_axis, af_frustum_ball with no warp), the ball's world - /// centre, the world axis, the reach R = |axis| + radius of the whole set from that centre (the - /// distance the flow is varied over), and each phase's moved centre and growth - /// (WarpedNoise.CentredWarp over R). The rays at depth z are centre + axis eps(z) + w, and a phase's - /// warped point lies within radius + growth of moved + axis eps(z) - /// (NoiseBoundTests.AffineWarpStaysEnclosed). AffineBound reads its noise arguments from here. + /// centre, the world axis, and the reach R = |axis| + radius of the whole set from that centre (the + /// distance the flow is varied over). It returns the final af_snoise arguments, so a consumer has no + /// term left to wire: each phase's coverage argument is the set moved by the flow at its centre + /// (WarpedNoise.CentredWarp over R) and grown by what the flow can vary across it, and the detail + /// argument is the unwarped set. The rays at depth z lie within an argument's radius of its centre + /// + axis eps(z), eps(z) = clamp((z - z_m) / h, -1, 1) (NoiseBoundTests.AffineWarpStaysEnclosed). /// internal AffineSlice Slice(Tile t, float z0, float z1) { @@ -1642,10 +1724,25 @@ public sealed class NoiseBoundTests var axis = tile.Right * axisCamera.x + tile.Up * axisCamera.y + tile.Forward * axisCamera.z; var reach = length(axis) + ball.w; var (c0, g0, c1, g1) = Noise.CentredWarp(centre, reach, tile.Time); - return new AffineSlice(axisCamera, ball, centre, axis, reach, c0, g0, c1, g1); + var coverageFrequency = Noise.F0; + var detailFrequency = 4.0f * Noise.F0; + return new AffineSlice( + axisCamera, ball, centre, reach, + new NoiseArg(c0 * coverageFrequency, axis * coverageFrequency, (ball.w + g0) * coverageFrequency), + new NoiseArg(c1 * coverageFrequency, axis * coverageFrequency, (ball.w + g1) * coverageFrequency), + new NoiseArg(centre * detailFrequency + 17.0f, axis * detailFrequency, ball.w * detailFrequency)); } - internal readonly record struct AffineSlice(float3 AxisCamera, float4 Ball, float3 Centre, float3 Axis, float Reach, float3 Centre0, float Grow0, float3 Centre1, float Grow1); + /// One af_snoise argument in noise space: the centre, the axis along eps, and the radius. + internal readonly record struct NoiseArg(float3 Centre, float3 Axis, float Radius) + { + /// The distance of the whole set from its centre: the gate's reach. + public float Reach => length(Axis) + Radius; + + public float3 Snoise() => af_snoise(Centre, Axis, Radius); + } + + internal readonly record struct AffineSlice(float3 AxisCamera, float4 Ball, float3 Centre, float Reach, NoiseArg Coverage0, NoiseArg Coverage1, NoiseArg Detail); // The best interval any centre value and gradient could give af_snoise over a set of reach R (the // gradient 0, the value at the far end), toward low or high values: the affine gate. @@ -1669,29 +1766,37 @@ public sealed class NoiseBoundTests /// over the same forms' ranges, each clipped to [-1, 1] (both sound, no further evaluation). The gate /// is BestForm's. /// - private float2 AffineBound(FogTile tile, float z0, float z1, float2 height, float2 s, float2 fade, float2 envelope, bool gated, Counts counts) + private float2 AffineBound(FogTile tile, float z0, float z1, Frame frame, bool gated, Counts counts) { - var (axisCamera, ball, centre, axis, reach, c0, g0, c1, g1) = Slice(tile, z0, z1); - var coverageFrequency = Noise.F0; - var detailFrequency = 4.0f * Noise.F0; if (gated) { + var slice = Slice(tile, z0, z1); + var (s, fade) = (frame.S, frame.Fade); var provable = false; foreach (var low in new[] { true, false }) { - var coverage = Noise.CoverageBound(BestForm((reach + g0) * coverageFrequency, low), BestForm((reach + g1) * coverageFrequency, low), tile.Time); - provable |= Compose(s, fade, iv_add(coverage, iv_scale(BestForm(reach * detailFrequency, low), 0.5f))).y <= 0.0f; + var coverage = Noise.CoverageBound(BestForm(slice.Coverage0.Reach, low), BestForm(slice.Coverage1.Reach, low), tile.Time); + provable |= Compose(s, fade, iv_add(coverage, iv_scale(BestForm(slice.Detail.Reach, low), 0.5f))).y <= 0.0f; } if (!provable) - return envelope; + return frame.Envelope; } - var shared = Same(c0, c1, g0, g1); + var (affine, interval) = AffinePieces(tile, z0, z1, frame, counts); + return new float2(max(affine.x, interval.x), min(affine.y, interval.y)); + } + + private (float2 Affine, float2 Interval) AffinePieces(FogTile tile, float z0, float z1, Frame frame, Counts counts) + { + var slice = Slice(tile, z0, z1); + var (ball, axisCamera) = (slice.Ball, slice.AxisCamera); + var (height, s, fade) = (frame.Height, frame.S, frame.Fade); + var shared = slice.Coverage0 == slice.Coverage1; counts.Grad += shared ? 2 : 3; - var n0 = af_snoise(c0 * coverageFrequency, axis * coverageFrequency, (ball.w + g0) * coverageFrequency); - var n1 = shared ? n0 : af_snoise(c1 * coverageFrequency, axis * coverageFrequency, (ball.w + g1) * coverageFrequency); - var nd = af_snoise(centre * detailFrequency + 17.0f, axis * detailFrequency, ball.w * detailFrequency); + var n0 = slice.Coverage0.Snoise(); + var n1 = shared ? n0 : slice.Coverage1.Snoise(); + var nd = slice.Detail.Snoise(); var w0 = Noise.Phases(tile.Time).W0; var noise = af_add(af_add(af_scale(n0, w0), af_scale(n1, 1.0f - w0)), af_scale(nd, 0.5f)); @@ -1703,7 +1808,7 @@ public sealed class NoiseBoundTests var displaced = af_range(af_add(level, af_scale(af_mul(af_from_iv(fade), noise), Amplitude))); var affine = new float2(max(0.0f, (FloorOffset - displaced.y) / FloorBlend), max(0.0f, (FloorOffset - displaced.x) / FloorBlend)); var interval = Compose(s, fade, iv_add(Noise.CoverageBound(Clip(af_range(n0)), Clip(af_range(n1)), tile.Time), iv_scale(Clip(af_range(nd)), 0.5f))); - return new float2(max(affine.x, interval.x), min(affine.y, interval.y)); + return (affine, interval); } } From 52bf488b511a8d621f120189a38b584ae3a7fb23 Mon Sep 17 00:00:00 2001 From: Meta Vens Date: Sun, 4 Oct 2026 07:26:34 +0200 Subject: [PATCH 16/31] NoiseBoundTests: the affine piece alone is wider in total; measure the intersection's gain Co-Authored-By: Claude Sonnet 5.5 --- .../tests/CultMath.Tests/NoiseBoundTests.cs | 22 ++++++++++--------- 1 file changed, 12 insertions(+), 10 deletions(-) diff --git a/packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs b/packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs index 916e9499..395ca389 100644 --- a/packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs +++ b/packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs @@ -893,17 +893,19 @@ public sealed class NoiseBoundTests /// (warp 0 and the Aetheria warp, N = 8, slices of length log-uniform in [0.5, 100] from a drawn /// depth), AffineBound is exactly the intersection of its two sound pieces (the affine arithmetic's /// own, and the interval composition over the forms' ranges), and where the interval piece does not - /// already prove the slice empty the affine piece is the narrower of the two in at least - /// MinAffineWins of the slices and narrower in total width by at least MinWidthGain. The - /// cost contract (IntervalSkipHalvesEvaluations) sees only the sum, in which af_snoise's Hessian forms - /// already beat iv_snoise_ball; this is the test that sees af_add, af_mul and af_range, the shared - /// symbol, break. + /// already prove the slice empty the affine piece is the narrower of the two in at least MinWins of + /// the slices and the intersection is narrower than the interval piece in total width by at least + /// MinGain. The affine piece alone is wider in total (the shared symbol pays where the forms are + /// correlated and loses to the clipped ranges where they are not), so it is the intersection's gain + /// that measures it. The cost contract (IntervalSkipHalvesEvaluations) sees only the sum, in which + /// af_snoise's Hessian forms already beat iv_snoise_ball; this is the test that sees af_add, af_mul and + /// af_range, the shared symbol, break. /// [Fact] - public void AffineCompositionIsTighterThanItsRanges() + public void AffineCompositionTightensItsRanges() { var random = new System.Random(0xAC0F); - var (open, wins, affineWidth, intervalWidth) = (0, 0, 0.0, 0.0); + var (open, wins, intersectionWidth, intervalWidth) = (0, 0, 0.0, 0.0); foreach (var warp in new[] { 0.0f, FogField.AetheriaWarp }) foreach (var field in new[] { FogField.DeepWell(warp, BoundMode.Affine), FogField.Wells(warp, BoundMode.Affine) }) for (var t = 0; t < 1500; t++) @@ -918,11 +920,11 @@ public sealed class NoiseBoundTests continue; open++; wins += affine.y - affine.x < interval.y - interval.x ? 1 : 0; - affineWidth += affine.y - affine.x; + intersectionWidth += bound.y - bound.x; intervalWidth += interval.y - interval.x; } - output.WriteLine($"IV-REPORT affine composition: {open} open slices, affine narrower in {wins}, total width {affineWidth:F1} against {intervalWidth:F1} ({affineWidth / intervalWidth:F3})"); + output.WriteLine($"IV-REPORT affine composition: {open} open slices, affine narrower in {wins} ({(double)wins / open:F3}), intersection width {intersectionWidth:F1} against the interval piece's {intervalWidth:F1} ({intersectionWidth / intervalWidth:F3})"); Assert.True(open > 0, "no slice is open"); } @@ -1665,7 +1667,7 @@ public sealed class NoiseBoundTests /// /// The two sound bounds AffineBound intersects, ungated: the affine arithmetic's own (the shared /// symbol through af_add, af_mul and af_range), and the interval composition over the same forms' - /// ranges (NoiseBoundTests.AffineCompositionIsTighterThanItsRanges). + /// ranges (NoiseBoundTests.AffineCompositionTightensItsRanges). /// internal (float2 Affine, float2 Interval) AffinePieces(Tile t, float z0, float z1, Counts counts) { From bdafdf3dd15645a0ef7a6cc617b81b164a971441 Mon Sep 17 00:00:00 2001 From: Meta Vens Date: Sun, 4 Oct 2026 07:28:03 +0200 Subject: [PATCH 17/31] NoiseBoundTests: the affine composition must win in 35% of open slices and narrow the interval piece Co-Authored-By: Claude Sonnet 5.5 --- packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs | 8 +++++--- 1 file changed, 5 insertions(+), 3 deletions(-) diff --git a/packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs b/packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs index 395ca389..110a8e24 100644 --- a/packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs +++ b/packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs @@ -893,9 +893,9 @@ public sealed class NoiseBoundTests /// (warp 0 and the Aetheria warp, N = 8, slices of length log-uniform in [0.5, 100] from a drawn /// depth), AffineBound is exactly the intersection of its two sound pieces (the affine arithmetic's /// own, and the interval composition over the forms' ranges), and where the interval piece does not - /// already prove the slice empty the affine piece is the narrower of the two in at least MinWins of - /// the slices and the intersection is narrower than the interval piece in total width by at least - /// MinGain. The affine piece alone is wider in total (the shared symbol pays where the forms are + /// already prove the slice empty the affine piece is the narrower of the two in at least 35% of + /// them and the intersection is narrower than the interval piece in total width by at least 0.5% + /// (measured: 43% and 1.1%). The affine piece alone is wider in total (the shared symbol pays where the forms are /// correlated and loses to the clipped ranges where they are not), so it is the intersection's gain /// that measures it. The cost contract (IntervalSkipHalvesEvaluations) sees only the sum, in which /// af_snoise's Hessian forms already beat iv_snoise_ball; this is the test that sees af_add, af_mul and @@ -926,6 +926,8 @@ public sealed class NoiseBoundTests output.WriteLine($"IV-REPORT affine composition: {open} open slices, affine narrower in {wins} ({(double)wins / open:F3}), intersection width {intersectionWidth:F1} against the interval piece's {intervalWidth:F1} ({intersectionWidth / intervalWidth:F3})"); Assert.True(open > 0, "no slice is open"); + Assert.True(wins >= 0.35 * open, $"the affine piece is narrower than the interval piece in {wins} of {open} open slices, under 35%"); + Assert.True(intersectionWidth <= 0.995 * intervalWidth, $"the intersection's width {intersectionWidth:F1} is not under 0.995 of the interval piece's {intervalWidth:F1}"); } // Why a field is not ten bands deep and wide, or null. The band G is the span of s where only the From 25e0aa3fe4ce948e598f9ce04466ce0951233986 Mon Sep 17 00:00:00 2001 From: Meta Vens Date: Sun, 4 Oct 2026 07:34:22 +0200 Subject: [PATCH 18/31] NoiseBoundTests: the slice's reach is |axis| + radius Co-Authored-By: Claude Sonnet 5.5 --- packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs | 2 ++ 1 file changed, 2 insertions(+) diff --git a/packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs b/packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs index 110a8e24..5eb13fc6 100644 --- a/packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs +++ b/packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs @@ -858,6 +858,8 @@ public sealed class NoiseBoundTests Assert.True(length(arg.Centre - centre) <= 1.0e-6f * (1.0f + length(centre)) && Math.Abs(arg.Radius - radius) <= 1.0e-6f * radius, $"{field.Name}: argument ({arg.Centre}, {arg.Radius:R}) is not the documented ({centre}, {radius:R})"); } + var reachRule = length(slice.Coverage0.Axis) / coverageFrequency + slice.Ball.w; + Assert.True(Math.Abs(slice.Reach - reachRule) <= 1.0e-5f * reachRule, $"{field.Name}: the reach {slice.Reach:R} is not |axis| + radius = {reachRule:R}"); var axisTolerance = 1.0e-6f * length(slice.Coverage0.Axis); Assert.True(length(slice.Coverage1.Axis - slice.Coverage0.Axis) <= axisTolerance && length(slice.Detail.Axis - slice.Coverage0.Axis * (detailFrequency / coverageFrequency)) <= 4.0f * axisTolerance, $"{field.Name}: the arguments do not share one axis"); From aad21e4b36ea6e896616675f6f120deb8b04f12b Mon Sep 17 00:00:00 2001 From: Meta Vens Date: Sun, 4 Oct 2026 07:37:25 +0200 Subject: [PATCH 19/31] design.md: the (c) rows re-measured on the snoise tie fix (the centred interval 3.20x and 3.12x, affine unchanged) Co-Authored-By: Claude Sonnet 5.5 --- packages/cultmath/docs/design.md | 18 +++++++++--------- 1 file changed, 9 insertions(+), 9 deletions(-) diff --git a/packages/cultmath/docs/design.md b/packages/cultmath/docs/design.md index d41c2605..fa9f0542 100644 --- a/packages/cultmath/docs/design.md +++ b/packages/cultmath/docs/design.md @@ -624,20 +624,20 @@ printed. The flow is not counted. At `N = 8`, in Aetheria's units on its `F -+ 1.5 A` with the fade at 1); depth and half-depth radius are each `10 G`. The camera's `xz` is uniform within 300 of the centre, at `y = F - h(xz) / 2`, so it looks across the bowl from inside it. Both rows run - on the same seeds. With no warp, the centred interval takes 3.10x (oracle - ceiling 5.24x, efficiency 0.591, cull fraction 0.986, probe overhead 0.943) + on the same seeds. With no warp, the centred interval takes 3.20x (oracle + ceiling 5.24x, efficiency 0.611, cull fraction 0.988, probe overhead 0.940) and the affine bound 3.57x (efficiency 0.682, cull 0.993, overhead 0.905). With - the warp `D = 60`: 2.99x (ceiling 5.31x, efficiency 0.563, cull 0.985, overhead - 0.935) against 3.46x (efficiency 0.651, cull 0.993, overhead 0.880), so affine - is 1.15x and 1.16x cheaper. The range mask (the oracle cells plus every cell + the warp `D = 60`: 3.12x (ceiling 5.31x, efficiency 0.588, cull 0.987, overhead + 0.929) against 3.46x (efficiency 0.651, cull 0.993, overhead 0.880), so affine + is 1.12x and 1.11x cheaper. The range mask (the oracle cells plus every cell where any of 5 x 5 slopes x 9 depths has density above zero) gives a range ceiling of 0.859 (0.848 with the warp), the cost a bound that proves exactly the range-empty cells would reach against the oracle's; the efficiencies - against it are 0.794 and 0.688 (warp 0), 0.768 and 0.665 (warp `D`). + against it are 0.794 and 0.712 (warp 0), 0.768 and 0.694 (warp `D`). - (c-band) The shallow wells, four mass-derived wells in the zone bowl with the camera gazing across, whose geometry is band-dominated, printed only. - Centred interval 2.74x (efficiency 0.378, cull 0.952, overhead 0.961), affine - 3.95x (0.544, 0.980, 0.901); with the warp 2.53x (0.353, 0.945, 0.948) against + Centred interval 3.01x (efficiency 0.414, cull 0.960, overhead 0.958), affine + 3.95x (0.544, 0.980, 0.901); with the warp 2.75x (0.383, 0.954, 0.944) against 3.54x (0.494, 0.976, 0.865). - (d) r1's uniform slab, which has no envelope: 1.23x, and 1.48x with no warp. - (e) The void at the shipped point (ruling `operator-shipped-void`): `Rh = 198`, @@ -694,7 +694,7 @@ The contracts, at `N = 8`, the lower of warp 0 and `D`: at most half the reference's steps per pixel. This run meets every contract: (a) 16384x, (c) affine 3.57x and 3.46x against -the centred interval's 3.10x and 2.99x, with a cull fraction of 0.985 at the +the centred interval's 3.20x and 3.12x, with a cull fraction of 0.987 at the lowest, (e) 2.68x fewer steps and a depth error 1.42x the fixed-step march's. The fields, the grids, the 0.2 envelope weight and the 2.0 gradient weight are not tuned toward the contracts. An affine probe is one `snoise_grad`, counted at From 118e85fd3a9bb809fbb3413a978a2a81dbab1edf Mon Sep 17 00:00:00 2001 From: Meta Vens Date: Sun, 4 Oct 2026 07:37:35 +0200 Subject: [PATCH 20/31] design.md: the slice returns the final af_snoise arguments; the composition test Co-Authored-By: Claude Sonnet 5.5 --- packages/cultmath/docs/design.md | 14 ++++++++++---- 1 file changed, 10 insertions(+), 4 deletions(-) diff --git a/packages/cultmath/docs/design.md b/packages/cultmath/docs/design.md index fa9f0542..e2e7df14 100644 --- a/packages/cultmath/docs/design.md +++ b/packages/cultmath/docs/design.md @@ -787,10 +787,16 @@ centre and variation are the caller's, and a consumer adds them to the slice's phase takes its own centre; one ball serves when the phases' centres coincide (warp 0). `NoiseBoundTests.CentredWarpStaysEnclosed` checks 2,000 balls, radius 1e-3 to 500, and fails when `rho` is dropped or halved; `AffineWarpStaysEnclosed` checks -3,000 tile slices of `FogField.Slice` (the owner the affine bound reads: the reach over -which the flow is varied is `|axis| + radius`, and each phase carries its own growth) -and fails when the reach is the ball's radius, a phase's growth is dropped, or `rho` -is halved. +3,000 tile slices of `FogField.Slice`, which returns the final `af_snoise` arguments +(each phase's moved centre and grown radius, the detail's unwarped set, the shared axis), +so the affine bound wires nothing itself: the reach over which the flow is varied is +`|axis| + radius`, each phase carries its own growth, and the gate reads each argument's +reach. It fails when the reach is the ball's radius or the larger of the two, a phase's +growth is dropped or taken from the other phase, the axis or the detail radius is scaled, +or `rho` is halved. `AffineCompositionTightensItsRanges` sees the shared-symbol +composition itself: the bound is exactly the intersection of the affine arithmetic's piece +and the interval composition over the forms' ranges, and the affine piece is the narrower +of the two in at least 35% of the open slices (the piece alone is wider in total). The deep well is the representative Aetheria case: the camera inside a sun's bowl, looking across. `DeepWellIsTenBandsDeepAndWide` computes the band `G` from From 03359a23652425adb6ea36b461d03c70206a0e65 Mon Sep 17 00:00:00 2001 From: Meta Vens Date: Sun, 4 Oct 2026 09:24:27 +0200 Subject: [PATCH 21/31] NoiseBoundTests: the gate never gives up a cull the ungated bound proves (pins the loosening direction) Co-Authored-By: Claude Sonnet 5.5 --- .../tests/CultMath.Tests/NoiseBoundTests.cs | 36 +++++++++++++++++++ 1 file changed, 36 insertions(+) diff --git a/packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs b/packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs index 5eb13fc6..d8b579f5 100644 --- a/packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs +++ b/packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs @@ -932,6 +932,42 @@ public sealed class NoiseBoundTests Assert.True(intersectionWidth <= 0.995 * intervalWidth, $"the intersection's width {intersectionWidth:F1} is not under 0.995 of the interval piece's {intervalWidth:F1}"); } + /// + /// The gate only ever gives up what the probes could not have delivered. Gated, a slice the best + /// conceivable probe cannot prove empty returns the envelope; so a slice the ungated bound proves + /// empty (its upper density 0) must be proved empty gated too. A gate that errs toward giving up + /// (the reach inflated, a gradient or Hessian term overstated) loosens the bound exactly there and + /// costs only culling, which no enclosure test sees; a gate that errs the other way only costs probes, + /// which the cost contract (IntervalSkipHalvesEvaluations) sees. Over seeded slices of both wells at + /// warp 0 and the Aetheria warp, N = 8, in both modes, no slice the ungated bound proves empty is + /// left open by the gated one, and at least 200 are proved empty. + /// + [Fact] + public void GateNeverGivesUpAProvableCull() + { + var random = new System.Random(0x6A7E); + var (proved, lost) = (0, new List()); + foreach (var mode in new[] { BoundMode.Interval, BoundMode.Affine }) + foreach (var warp in new[] { 0.0f, FogField.AetheriaWarp }) + foreach (var field in new[] { FogField.DeepWell(warp, mode), FogField.Wells(warp, mode) }) + for (var t = 0; t < 1500; t++) + { + var tile = field.DrawTile(random, 8); + var z0 = Uniform(random, field.Grid[0], field.Grid[^1] * 0.9f); + var z1 = MathF.Min(z0 + LogUniform(random, 0.5f, 100.0f), field.Grid[^1]); + if (field.Bound(tile, z0, z1, false, new Counts()).y > 0.0f) + continue; + proved++; + var gated = field.Bound(tile, z0, z1, true, new Counts()); + if (gated.y > 0.0f) + lost.Add($"{field.Name} {mode} warp {warp}: slice [{z0:R}, {z1:R}] is proved empty ungated but the gate leaves it at {gated}"); + } + + output.WriteLine($"IV-REPORT gate: {proved} slices proved empty ungated, {lost.Count} left open by the gate"); + Assert.True(proved >= 200, $"only {proved} slices are proved empty ungated: the check cannot tell a gate that loses them"); + Assert.True(lost.Count == 0, $"{lost.Count} provable culls given up by the gate, first: {lost.FirstOrDefault()}"); + } + // Why a field is not ten bands deep and wide, or null. The band G is the span of s where only the // noise decides density: s in [F + A lo, F + A hi] for the free noise range [lo, hi], with the fade at 1. private static string? TenBandsShortfall(FogField field, int cameras) From 962aa45d473c144697648dc1ae4c7bcdbbe25551 Mon Sep 17 00:00:00 2001 From: Meta Vens Date: Sun, 4 Oct 2026 09:33:28 +0200 Subject: [PATCH 22/31] design.md: rows (b), (d), (e) and the closing paragraph re-measured on the snoise tie fix (one tree) Co-Authored-By: Claude Sonnet 5.5 --- packages/cultmath/docs/design.md | 15 ++++++++------- 1 file changed, 8 insertions(+), 7 deletions(-) diff --git a/packages/cultmath/docs/design.md b/packages/cultmath/docs/design.md index e2e7df14..a46ca91a 100644 --- a/packages/cultmath/docs/design.md +++ b/packages/cultmath/docs/design.md @@ -616,7 +616,7 @@ printed. The flow is not counted. At `N = 8`, in Aetheria's units on its - (a) Height fog, camera above the safety band, rays level and up: the whole grid is one envelope probe per tile, against 256 envelope evaluations per ray, 16384x. Every cell is oracle-empty and the pre-pass proves them all. -- (b) Inside the fog: 1.15x, and 1.09x with the warp. Most cells are fog, and +- (b) Inside the fog: 1.16x, and 1.11x with the warp. Most cells are fog, and the pre-pass probes each one. - (c) The deep well (ruling `wells-scale-order-of-magnitude`): the zone bowl plus one well at its centre, `PowerPulse` exponent 2, scale 2220, depth 600. @@ -639,19 +639,20 @@ printed. The flow is not counted. At `N = 8`, in Aetheria's units on its Centred interval 3.01x (efficiency 0.414, cull 0.960, overhead 0.958), affine 3.95x (0.544, 0.980, 0.901); with the warp 2.75x (0.383, 0.954, 0.944) against 3.54x (0.494, 0.976, 0.865). -- (d) r1's uniform slab, which has no envelope: 1.23x, and 1.48x with no warp. +- (d) r1's uniform slab, which has no envelope: 1.38x with the warp 0.05, and + 1.67x with no warp. - (e) The void at the shipped point (ruling `operator-shipped-void`): `Rh = 198`, `S = 50`, `Rc = Rh + S = 248`, `e = -ln 1.5 / ln(1 - (Rh / Rc)^2)`, the shipped camera, 200 tiles. The fixed-step reference takes 36.34 steps per - pixel. The same cells masked by the pre-pass take 20.13, with identical - transmittance. The footprint-aware march takes 13.54: 2.68x fewer. `snoise` - per pixel goes from 32.98 to 26.69, and combined cost from 40.25 to 29.52 with + pixel. The same cells masked by the pre-pass take 20.08, with identical + transmittance. The footprint-aware march takes 13.50: 2.69x fewer. `snoise` + per pixel goes from 32.98 to 26.70, and combined cost from 40.25 to 29.54 with the probes, 1.36x. Every one of the 200 tiles proves a body. Before the body both marches are measured against a converged one (midpoint quadrature at `h = 0.5` over every cell, trusting no pre-pass): the relative optical-depth error at the body start is at most 0.0608 for the fixed-step march and 0.08639 for the footprint march, 1.42x. Over the grid of hollow radius, camera offset - and ramp width the steps ratio runs from 1.57x (`Rh = 100`, low, `S = 150`) to + and ramp width the steps ratio runs from 1.59x (`Rh = 100`, low, `S = 150`) to 27.45x (`Rh = 1000`, centred, `S = 10`), the cost ratio from 1.26x to 3.89x, and the depth-error ratio from 0.85 to 4.72 (`Rh = 200`, centred, `S = 150`, where the errors are 0.01504 and 0.003188). LOD-far takes 14.19x fewer steps; @@ -695,7 +696,7 @@ The contracts, at `N = 8`, the lower of warp 0 and `D`: This run meets every contract: (a) 16384x, (c) affine 3.57x and 3.46x against the centred interval's 3.20x and 3.12x, with a cull fraction of 0.987 at the -lowest, (e) 2.68x fewer steps and a depth error 1.42x the fixed-step march's. +lowest, (e) 2.69x fewer steps and a depth error 1.42x the fixed-step march's. The fields, the grids, the 0.2 envelope weight and the 2.0 gradient weight are not tuned toward the contracts. An affine probe is one `snoise_grad`, counted at 2.0 `snoise` evaluations (an estimate, printed beside the count). From e16edc0b70801fe1b7bf0733f1f7b497957d70ee Mon Sep 17 00:00:00 2001 From: Meta Vens Date: Sun, 4 Oct 2026 09:47:49 +0200 Subject: [PATCH 23/31] Affine forms in HLSL: CultMath.Affine.hlsl mirrors math.Affine.cs bit for bit; iv_snoise_ball(float2); the GLSL lowering carries them Co-Authored-By: Claude Sonnet 5.5 --- packages/cultmath/README.md | 3 +- .../cultmath/scripts/build-unity-package.ps1 | 4 +- .../cultmath/shaders/CultMath.Affine.hlsl | 130 ++++++++++++++++ .../shaders/CultMath.Affine.hlsl.meta | 7 + .../cultmath/shaders/CultMath.Interval.hlsl | 11 ++ packages/cultmath/shaders/CultMath.glsl | 144 +++++++++++++++++- packages/cultmath/shaders/CultMath.hlsl | 1 + .../tests/CultMath.Tests/HlslMirrorTests.cs | 1 + .../HlslSourceCompatibilityTests.cs | 6 +- .../Shaders/CultMath.Affine.hlsl | 130 ++++++++++++++++ .../Shaders/CultMath.Affine.hlsl.meta | 7 + .../Shaders/CultMath.Interval.hlsl | 11 ++ .../Shaders/CultMath.hlsl | 1 + 13 files changed, 452 insertions(+), 4 deletions(-) create mode 100644 packages/cultmath/shaders/CultMath.Affine.hlsl create mode 100644 packages/cultmath/shaders/CultMath.Affine.hlsl.meta create mode 100644 packages/cultmath/unity/org.gamecult.cultmath/Shaders/CultMath.Affine.hlsl create mode 100644 packages/cultmath/unity/org.gamecult.cultmath/Shaders/CultMath.Affine.hlsl.meta diff --git a/packages/cultmath/README.md b/packages/cultmath/README.md index 0b12819e..e5df5f1e 100644 --- a/packages/cultmath/README.md +++ b/packages/cultmath/README.md @@ -34,7 +34,8 @@ shader semantics. parity. HLSL already owns `float2`, `float3`, and `float4`; the include exposes `cultmath_*` functions for shared semantics HLSL does not provide. It includes `shaders/CultMath.Phacelle.hlsl` (MPL-2.0) and - `shaders/CultMath.Interval.hlsl` from the same directory, so ship all three. + `shaders/CultMath.Interval.hlsl` and `shaders/CultMath.Affine.hlsl` from the same + directory, so ship all four. - interval arithmetic for culling empty space with a proof: `iv_*` functions over `float2(lo, hi)` (`iv_add`, `iv_mul`, `iv_exp`, `iv_smoothstep`, ...) and `iv_snoise_ball`/`iv_fbm_ball`, which bound `snoise` and `fbm_grad` over a diff --git a/packages/cultmath/scripts/build-unity-package.ps1 b/packages/cultmath/scripts/build-unity-package.ps1 index 7fb6f7a6..f232d6eb 100644 --- a/packages/cultmath/scripts/build-unity-package.ps1 +++ b/packages/cultmath/scripts/build-unity-package.ps1 @@ -84,7 +84,9 @@ $requiredTemplateFiles = @( "Shaders\CultMath.Phacelle.hlsl", "Shaders\CultMath.Phacelle.hlsl.meta", "Shaders\CultMath.Interval.hlsl", - "Shaders\CultMath.Interval.hlsl.meta" + "Shaders\CultMath.Interval.hlsl.meta", + "Shaders\CultMath.Affine.hlsl", + "Shaders\CultMath.Affine.hlsl.meta" ) foreach ($relativePath in $requiredTemplateFiles) { if (-not (Test-Path -LiteralPath (Join-Path $templateRoot $relativePath) -PathType Leaf)) { diff --git a/packages/cultmath/shaders/CultMath.Affine.hlsl b/packages/cultmath/shaders/CultMath.Affine.hlsl new file mode 100644 index 00000000..2c91a00d --- /dev/null +++ b/packages/cultmath/shaders/CultMath.Affine.hlsl @@ -0,0 +1,130 @@ +#ifndef CULTMATH_AFFINE_HLSL +#define CULTMATH_AFFINE_HLSL + +// Reduced affine forms with one shared symbol over float3(x0, a, e): the set { x0 + a eps + e delta : +// delta in [-1, 1] } at the region's shared eps in [-1, 1], e >= 0. Every cultmath_af_* function mirrors +// math.Affine.cs bit for bit; see that file and docs/design.md ("Affine forms") for each op's enclosure +// and the float32 rounding allowance it folds into e. Included by CultMath.hlsl after +// CultMath.Interval.hlsl, whose CULTMATH_SNOISE_LIPSCHITZ it shares, and after its own +// definition of cultmath_snoise_grad. + +// An upper bound on the spectral norm of the Hessian of cultmath_snoise(float3); provenance in +// math.Affine.cs (SNOISE_HESSIAN). +static const float CULTMATH_SNOISE_HESSIAN = 56.050385; + +float3 cultmath_af_point(float value) { return float3(value, 0.0, 0.0); } +float3 cultmath_af_symbol(float x0, float a) { return float3(x0, a, 0.0); } + +float3 cultmath_af_from_iv(float2 interval) +{ + float x0 = interval.x * 0.5 + interval.y * 0.5; + float e = interval.y * 0.5 - interval.x * 0.5; + return float3(x0, 0.0, e + (abs(x0) + e) * 9.5367431640625e-7 + 1.17549435e-38); +} + +// An endpoint that would be NaN is the infinity on its side. +float2 cultmath_af_range(float3 x) +{ + float r = abs(x.y) + x.z; + float w = abs(x.x) * 9.5367431640625e-7 + r * 9.5367431640625e-7; + float lo = x.x - r - w; + float hi = x.x + r + w; + return float2(lo == lo ? lo : asfloat(0xff800000u), hi == hi ? hi : asfloat(0x7f800000u)); +} + +float3 cultmath_af_add(float3 x, float3 y) +{ + float x0 = x.x + y.x; + float a = x.y + y.y; + float e = x.z + y.z; + return float3(x0, a, e + (abs(x0) + abs(a) + e) * 9.5367431640625e-7); +} + +float3 cultmath_af_sub(float3 x, float3 y) +{ + float x0 = x.x - y.x; + float a = x.y - y.y; + float e = x.z + y.z; + return float3(x0, a, e + (abs(x0) + abs(a) + e) * 9.5367431640625e-7); +} + +float3 cultmath_af_neg(float3 x) { return float3(-x.x, -x.y, x.z); } + +float3 cultmath_af_scale(float3 x, float s) +{ + float x0 = x.x * s; + float a = x.y * s; + float e = x.z * abs(s); + return float3(x0, a, e + (abs(x0) + abs(a) + e) * 9.5367431640625e-7 + 1.17549435e-38); +} + +float3 cultmath_af_add_iv(float3 x, float2 interval) { return cultmath_af_add(x, cultmath_af_from_iv(interval)); } + +float3 cultmath_af_mul(float3 x, float3 y) +{ + float x0 = x.x * y.x; + float p = x.x * y.y; + float q = y.x * x.y; + float a = p + q; + float e = abs(x.x) * y.z + abs(y.x) * x.z + (abs(x.y) + x.z) * (abs(y.y) + y.z); + return float3(x0, a, e + (abs(x0) + abs(p) + abs(q) + e) * 9.5367431640625e-7 + 1.17549435e-38); +} + +// The narrower of the Lipschitz form and the centred (Taylor) form over every p = centre + axis eps + w, +// |w| <= radius, from one cultmath_snoise_grad at the centre; the allowance for the float32 evaluation is +// in math.Affine.cs. +float3 cultmath_af_snoise(float3 centre, float3 axis, float radius) +{ + float4 g = cultmath_snoise_grad(centre); + float3 gradient = float3(g.x, g.y, g.z); + float reach = length(axis) + radius; + float lipschitz = CULTMATH_SNOISE_LIPSCHITZ * reach; + float a = dot(gradient, axis); + float remainder = length(gradient) * radius + 0.5 * CULTMATH_SNOISE_HESSIAN * reach * reach; + float allowance = 6.103515625e-5 + 2.44140625e-4 * reach + + CULTMATH_SNOISE_LIPSCHITZ * 9.5367431640625e-7 * (abs(centre.x) + abs(centre.y) + abs(centre.z) + reach); + bool centred = abs(a) + remainder < lipschitz; + float slope = centred ? a : 0.0; + float e = (centred ? remainder : lipschitz) + allowance; + return float3(g.w, slope, e + (abs(g.w) + abs(slope) + e) * 9.5367431640625e-7); +} + +// Octave sum of cultmath_af_snoise, compounding frequency and amplitude as the value-and-gradient fBm does. +float3 cultmath_af_fbm(float3 centre, float3 axis, float radius, int octaves, float lacunarity, float gain) +{ + octaves = clamp(octaves, 0, 16); + float amplitude = 1.0; + float frequency = 1.0; + float3 sum = float3(0.0, 0.0, 0.0); + for (int i = 0; i < octaves; i++) + { + sum = cultmath_af_add(sum, cultmath_af_scale(cultmath_af_snoise(centre * frequency, axis * frequency, radius * abs(frequency)), amplitude)); + frequency *= lacunarity; + amplitude *= gain; + } + + return sum; +} + +// The axis of a screen tile's slice over the depth segment [z0, z1]; camera frame of +// cultmath_iv_frustum_ball. +float3 cultmath_af_frustum_axis(float2 centreSlope, float z0, float z1) +{ + float h = (z1 - z0) * 0.5; + return float3(centreSlope.x * h, centreSlope.y * h, h); +} + +// The affine slice of a screen tile over [z0, z1]: centre xyz, radius w. Derivation in math.Affine.cs. +float4 cultmath_af_frustum_ball(float2 centreSlope, float z0, float z1, float footprintPerDepth, float warpVariation) +{ + float zm = (z0 + z1) * 0.5; + float h = (z1 - z0) * 0.5; + float cx = centreSlope.x * zm; + float cy = centreSlope.y * zm; + float spread = (abs(centreSlope.x) + abs(centreSlope.y) + 1.0) * h; + float radius = z1 * footprintPerDepth + warpVariation; + radius += (abs(cx) + abs(cy) + abs(zm) + spread + radius) * 9.5367431640625e-7; + return float4(cx, cy, zm, radius); +} + +#endif diff --git a/packages/cultmath/shaders/CultMath.Affine.hlsl.meta b/packages/cultmath/shaders/CultMath.Affine.hlsl.meta new file mode 100644 index 00000000..81cc7925 --- /dev/null +++ b/packages/cultmath/shaders/CultMath.Affine.hlsl.meta @@ -0,0 +1,7 @@ +fileFormatVersion: 2 +guid: 26f28fce38e14eed8b9b5dbd758e4ffc +ShaderIncludeImporter: + externalObjects: {} + userData: + assetBundleName: + assetBundleVariant: diff --git a/packages/cultmath/shaders/CultMath.Interval.hlsl b/packages/cultmath/shaders/CultMath.Interval.hlsl index 22c19221..bc084ace 100644 --- a/packages/cultmath/shaders/CultMath.Interval.hlsl +++ b/packages/cultmath/shaders/CultMath.Interval.hlsl @@ -9,6 +9,9 @@ // Lipschitz constant of cultmath_snoise(float3); provenance in math.Interval.cs (SNOISE_LIPSCHITZ). static const float CULTMATH_SNOISE_LIPSCHITZ = 7.9640074; +// Lipschitz constant of cultmath_snoise(float2); provenance in math.Interval.cs (SNOISE2_LIPSCHITZ). +static const float CULTMATH_SNOISE2_LIPSCHITZ = 8.117002; + float2 cultmath_iv_point(float value) { return float2(value, value); } float2 cultmath_iv_add(float2 a, float2 b) { return float2(a.x + b.x, a.y + b.y); } float2 cultmath_iv_sub(float2 a, float2 b) { return float2(a.x - b.y, a.y - b.x); } @@ -87,6 +90,14 @@ float2 cultmath_iv_snoise_ball(float3 centre, float radius) return float2(max(n - e, -1.0), min(n + e, 1.0)); } +// The 2D overload: [n - L2 r, n + L2 r] intersected with [-1, 1], n = cultmath_snoise(centre). One snoise, value only. +float2 cultmath_iv_snoise_ball(float2 centre, float radius) +{ + float n = cultmath_snoise(centre); + float e = CULTMATH_SNOISE2_LIPSCHITZ * radius; + return float2(max(n - e, -1.0), min(n + e, 1.0)); +} + // Octave sum of cultmath_iv_snoise_ball, compounding frequency and amplitude as the value-and-gradient fBm does. float2 cultmath_iv_fbm_ball(float3 centre, float radius, int octaves, float lacunarity, float gain) { diff --git a/packages/cultmath/shaders/CultMath.glsl b/packages/cultmath/shaders/CultMath.glsl index 50e8f340..99dbd4b8 100644 --- a/packages/cultmath/shaders/CultMath.glsl +++ b/packages/cultmath/shaders/CultMath.glsl @@ -1,4 +1,4 @@ -// Generated from shaders/CultMath.hlsl and CultMath.Interval.hlsl by GlslLowering (packages/cultmath/tests/CultMath.Tests/GlslLowering.cs). Do not edit; regenerate with CULTMATH_WRITE_GLSL=1 dotnet test --filter GlslMirrorTests. MIT. cultmath_phacelle is in CultMath.Phacelle.glsl, to be concatenated after this file by consumers that call it. +// Generated from shaders/CultMath.hlsl and CultMath.Interval.hlsl and CultMath.Affine.hlsl by GlslLowering (packages/cultmath/tests/CultMath.Tests/GlslLowering.cs). Do not edit; regenerate with CULTMATH_WRITE_GLSL=1 dotnet test --filter GlslMirrorTests. MIT. cultmath_phacelle is in CultMath.Phacelle.glsl, to be concatenated after this file by consumers that call it. #ifndef CULTMATH_GLSL #define CULTMATH_GLSL // CultMath HLSL mirror. @@ -518,6 +518,9 @@ CultCellular cultmath_cellular(vec3 p) // Lipschitz constant of cultmath_snoise(vec3); provenance in math.Interval.cs (SNOISE_LIPSCHITZ). const float CULTMATH_SNOISE_LIPSCHITZ = 7.9640074; +// Lipschitz constant of cultmath_snoise(vec2); provenance in math.Interval.cs (SNOISE2_LIPSCHITZ). +const float CULTMATH_SNOISE2_LIPSCHITZ = 8.117002; + vec2 cultmath_iv_point(float value) { return vec2(value, value); } vec2 cultmath_iv_add(vec2 a, vec2 b) { return vec2(a.x + b.x, a.y + b.y); } vec2 cultmath_iv_sub(vec2 a, vec2 b) { return vec2(a.x - b.y, a.y - b.x); } @@ -596,6 +599,14 @@ vec2 cultmath_iv_snoise_ball(vec3 centre, float radius) return vec2(max(n - e, -1.0), min(n + e, 1.0)); } +// The 2D overload: [n - L2 r, n + L2 r] intersected with [-1, 1], n = cultmath_snoise(centre). One snoise, value only. +vec2 cultmath_iv_snoise_ball(vec2 centre, float radius) +{ + float n = cultmath_snoise(centre); + float e = CULTMATH_SNOISE2_LIPSCHITZ * radius; + return vec2(max(n - e, -1.0), min(n + e, 1.0)); +} + // Octave sum of cultmath_iv_snoise_ball, compounding frequency and amplitude as the value-and-gradient fBm does. vec2 cultmath_iv_fbm_ball(vec3 centre, float radius, int octaves, float lacunarity, float gain) { @@ -631,6 +642,137 @@ vec4 cultmath_iv_frustum_ball(vec2 centreSlope, float z0, float z1, float footpr + + +// Reduced affine forms with one shared symbol over vec3(x0, a, e): the set { x0 + a eps + e delta : +// delta in [-1, 1] } at the region's shared eps in [-1, 1], e >= 0. Every cultmath_af_* function mirrors +// math.Affine.cs bit for bit; see that file and docs/design.md ("Affine forms") for each op's enclosure +// and the float32 rounding allowance it folds into e. Included by CultMath.hlsl after +// CultMath.Interval.hlsl, whose CULTMATH_SNOISE_LIPSCHITZ it shares, and after its own +// definition of cultmath_snoise_grad. + +// An upper bound on the spectral norm of the Hessian of cultmath_snoise(vec3); provenance in +// math.Affine.cs (SNOISE_HESSIAN). +const float CULTMATH_SNOISE_HESSIAN = 56.050385; + +vec3 cultmath_af_point(float value) { return vec3(value, 0.0, 0.0); } +vec3 cultmath_af_symbol(float x0, float a) { return vec3(x0, a, 0.0); } + +vec3 cultmath_af_from_iv(vec2 interval) +{ + float x0 = interval.x * 0.5 + interval.y * 0.5; + float e = interval.y * 0.5 - interval.x * 0.5; + return vec3(x0, 0.0, e + (abs(x0) + e) * 9.5367431640625e-7 + 1.17549435e-38); +} + +// An endpoint that would be NaN is the infinity on its side. +vec2 cultmath_af_range(vec3 x) +{ + float r = abs(x.y) + x.z; + float w = abs(x.x) * 9.5367431640625e-7 + r * 9.5367431640625e-7; + float lo = x.x - r - w; + float hi = x.x + r + w; + return vec2(lo == lo ? lo : uintBitsToFloat(0xff800000u), hi == hi ? hi : uintBitsToFloat(0x7f800000u)); +} + +vec3 cultmath_af_add(vec3 x, vec3 y) +{ + float x0 = x.x + y.x; + float a = x.y + y.y; + float e = x.z + y.z; + return vec3(x0, a, e + (abs(x0) + abs(a) + e) * 9.5367431640625e-7); +} + +vec3 cultmath_af_sub(vec3 x, vec3 y) +{ + float x0 = x.x - y.x; + float a = x.y - y.y; + float e = x.z + y.z; + return vec3(x0, a, e + (abs(x0) + abs(a) + e) * 9.5367431640625e-7); +} + +vec3 cultmath_af_neg(vec3 x) { return vec3(-x.x, -x.y, x.z); } + +vec3 cultmath_af_scale(vec3 x, float s) +{ + float x0 = x.x * s; + float a = x.y * s; + float e = x.z * abs(s); + return vec3(x0, a, e + (abs(x0) + abs(a) + e) * 9.5367431640625e-7 + 1.17549435e-38); +} + +vec3 cultmath_af_add_iv(vec3 x, vec2 interval) { return cultmath_af_add(x, cultmath_af_from_iv(interval)); } + +vec3 cultmath_af_mul(vec3 x, vec3 y) +{ + float x0 = x.x * y.x; + float p = x.x * y.y; + float q = y.x * x.y; + float a = p + q; + float e = abs(x.x) * y.z + abs(y.x) * x.z + (abs(x.y) + x.z) * (abs(y.y) + y.z); + return vec3(x0, a, e + (abs(x0) + abs(p) + abs(q) + e) * 9.5367431640625e-7 + 1.17549435e-38); +} + +// The narrower of the Lipschitz form and the centred (Taylor) form over every p = centre + axis eps + w, +// |w| <= radius, from one cultmath_snoise_grad at the centre; the allowance for the float32 evaluation is +// in math.Affine.cs. +vec3 cultmath_af_snoise(vec3 centre, vec3 axis, float radius) +{ + vec4 g = cultmath_snoise_grad(centre); + vec3 gradient = vec3(g.x, g.y, g.z); + float reach = length(axis) + radius; + float lipschitz = CULTMATH_SNOISE_LIPSCHITZ * reach; + float a = dot(gradient, axis); + float remainder = length(gradient) * radius + 0.5 * CULTMATH_SNOISE_HESSIAN * reach * reach; + float allowance = 6.103515625e-5 + 2.44140625e-4 * reach + + CULTMATH_SNOISE_LIPSCHITZ * 9.5367431640625e-7 * (abs(centre.x) + abs(centre.y) + abs(centre.z) + reach); + bool centred = abs(a) + remainder < lipschitz; + float slope = centred ? a : 0.0; + float e = (centred ? remainder : lipschitz) + allowance; + return vec3(g.w, slope, e + (abs(g.w) + abs(slope) + e) * 9.5367431640625e-7); +} + +// Octave sum of cultmath_af_snoise, compounding frequency and amplitude as the value-and-gradient fBm does. +vec3 cultmath_af_fbm(vec3 centre, vec3 axis, float radius, int octaves, float lacunarity, float gain) +{ + octaves = clamp(octaves, 0, 16); + float amplitude = 1.0; + float frequency = 1.0; + vec3 sum = vec3(0.0, 0.0, 0.0); + for (int i = 0; i < octaves; i++) + { + sum = cultmath_af_add(sum, cultmath_af_scale(cultmath_af_snoise(centre * frequency, axis * frequency, radius * abs(frequency)), amplitude)); + frequency *= lacunarity; + amplitude *= gain; + } + + return sum; +} + +// The axis of a screen tile's slice over the depth segment [z0, z1]; camera frame of +// cultmath_iv_frustum_ball. +vec3 cultmath_af_frustum_axis(vec2 centreSlope, float z0, float z1) +{ + float h = (z1 - z0) * 0.5; + return vec3(centreSlope.x * h, centreSlope.y * h, h); +} + +// The affine slice of a screen tile over [z0, z1]: centre xyz, radius w. Derivation in math.Affine.cs. +vec4 cultmath_af_frustum_ball(vec2 centreSlope, float z0, float z1, float footprintPerDepth, float warpVariation) +{ + float zm = (z0 + z1) * 0.5; + float h = (z1 - z0) * 0.5; + float cx = centreSlope.x * zm; + float cy = centreSlope.y * zm; + float spread = (abs(centreSlope.x) + abs(centreSlope.y) + 1.0) * h; + float radius = z1 * footprintPerDepth + warpVariation; + radius += (abs(cx) + abs(cy) + abs(zm) + spread + radius) * 9.5367431640625e-7; + return vec4(cx, cy, zm, radius); +} + + + + float cultmath_value_noise(vec2 position) { vec2 cell = floor(position); diff --git a/packages/cultmath/shaders/CultMath.hlsl b/packages/cultmath/shaders/CultMath.hlsl index e40cf0b6..6856a000 100644 --- a/packages/cultmath/shaders/CultMath.hlsl +++ b/packages/cultmath/shaders/CultMath.hlsl @@ -509,6 +509,7 @@ CultCellular cultmath_cellular(float3 p) #include "CultMath.Phacelle.hlsl" #include "CultMath.Interval.hlsl" +#include "CultMath.Affine.hlsl" float cultmath_value_noise(float2 position) { diff --git a/packages/cultmath/tests/CultMath.Tests/HlslMirrorTests.cs b/packages/cultmath/tests/CultMath.Tests/HlslMirrorTests.cs index 9e2dc9af..00a2e90a 100644 --- a/packages/cultmath/tests/CultMath.Tests/HlslMirrorTests.cs +++ b/packages/cultmath/tests/CultMath.Tests/HlslMirrorTests.cs @@ -66,6 +66,7 @@ public sealed class HlslMirrorTests [InlineData("CultMath.hlsl")] [InlineData("CultMath.Phacelle.hlsl")] [InlineData("CultMath.Interval.hlsl")] + [InlineData("CultMath.Affine.hlsl")] public void UnityPackageShaderIsIdenticalToTheCanonicalShader(string name) { var root = GetCultMathRoot(); diff --git a/packages/cultmath/tests/CultMath.Tests/HlslSourceCompatibilityTests.cs b/packages/cultmath/tests/CultMath.Tests/HlslSourceCompatibilityTests.cs index ce38dea4..b1e3d8f8 100644 --- a/packages/cultmath/tests/CultMath.Tests/HlslSourceCompatibilityTests.cs +++ b/packages/cultmath/tests/CultMath.Tests/HlslSourceCompatibilityTests.cs @@ -33,6 +33,10 @@ public sealed class HlslSourceCompatibilityTests Assert.NotNull(ShaderFunction(shader, "cultmath_phacelle", typeof(float3), typeof(float3), typeof(float), typeof(float))); Assert.NotNull(ShaderFunction(shader, "cultmath_iv_snoise_ball", typeof(float3), typeof(float))); Assert.NotNull(ShaderFunction(shader, "cultmath_iv_fbm_ball", typeof(float3), typeof(float), typeof(int), typeof(float), typeof(float))); + Assert.NotNull(ShaderFunction(shader, "cultmath_iv_snoise_ball", typeof(float2), typeof(float))); + Assert.NotNull(ShaderFunction(shader, "cultmath_af_snoise", typeof(float3), typeof(float3), typeof(float))); + Assert.NotNull(ShaderFunction(shader, "cultmath_af_fbm", typeof(float3), typeof(float3), typeof(float), typeof(int), typeof(float), typeof(float))); + Assert.NotNull(ShaderFunction(shader, "cultmath_af_frustum_ball", typeof(float2), typeof(float), typeof(float), typeof(float), typeof(float))); } // HLSL functions carry no access modifier, so they compile as private instance methods. @@ -133,7 +137,7 @@ public sealed class HlslSourceCompatibilityTests } Assert.True(mismatches.Count == 0, $"{mismatches.Count} mismatches:{Environment.NewLine}{string.Join(Environment.NewLine, mismatches.Take(12))}"); - Assert.Equal(53, compared.Count); + Assert.Equal(67, compared.Count); } /// diff --git a/packages/cultmath/unity/org.gamecult.cultmath/Shaders/CultMath.Affine.hlsl b/packages/cultmath/unity/org.gamecult.cultmath/Shaders/CultMath.Affine.hlsl new file mode 100644 index 00000000..2c91a00d --- /dev/null +++ b/packages/cultmath/unity/org.gamecult.cultmath/Shaders/CultMath.Affine.hlsl @@ -0,0 +1,130 @@ +#ifndef CULTMATH_AFFINE_HLSL +#define CULTMATH_AFFINE_HLSL + +// Reduced affine forms with one shared symbol over float3(x0, a, e): the set { x0 + a eps + e delta : +// delta in [-1, 1] } at the region's shared eps in [-1, 1], e >= 0. Every cultmath_af_* function mirrors +// math.Affine.cs bit for bit; see that file and docs/design.md ("Affine forms") for each op's enclosure +// and the float32 rounding allowance it folds into e. Included by CultMath.hlsl after +// CultMath.Interval.hlsl, whose CULTMATH_SNOISE_LIPSCHITZ it shares, and after its own +// definition of cultmath_snoise_grad. + +// An upper bound on the spectral norm of the Hessian of cultmath_snoise(float3); provenance in +// math.Affine.cs (SNOISE_HESSIAN). +static const float CULTMATH_SNOISE_HESSIAN = 56.050385; + +float3 cultmath_af_point(float value) { return float3(value, 0.0, 0.0); } +float3 cultmath_af_symbol(float x0, float a) { return float3(x0, a, 0.0); } + +float3 cultmath_af_from_iv(float2 interval) +{ + float x0 = interval.x * 0.5 + interval.y * 0.5; + float e = interval.y * 0.5 - interval.x * 0.5; + return float3(x0, 0.0, e + (abs(x0) + e) * 9.5367431640625e-7 + 1.17549435e-38); +} + +// An endpoint that would be NaN is the infinity on its side. +float2 cultmath_af_range(float3 x) +{ + float r = abs(x.y) + x.z; + float w = abs(x.x) * 9.5367431640625e-7 + r * 9.5367431640625e-7; + float lo = x.x - r - w; + float hi = x.x + r + w; + return float2(lo == lo ? lo : asfloat(0xff800000u), hi == hi ? hi : asfloat(0x7f800000u)); +} + +float3 cultmath_af_add(float3 x, float3 y) +{ + float x0 = x.x + y.x; + float a = x.y + y.y; + float e = x.z + y.z; + return float3(x0, a, e + (abs(x0) + abs(a) + e) * 9.5367431640625e-7); +} + +float3 cultmath_af_sub(float3 x, float3 y) +{ + float x0 = x.x - y.x; + float a = x.y - y.y; + float e = x.z + y.z; + return float3(x0, a, e + (abs(x0) + abs(a) + e) * 9.5367431640625e-7); +} + +float3 cultmath_af_neg(float3 x) { return float3(-x.x, -x.y, x.z); } + +float3 cultmath_af_scale(float3 x, float s) +{ + float x0 = x.x * s; + float a = x.y * s; + float e = x.z * abs(s); + return float3(x0, a, e + (abs(x0) + abs(a) + e) * 9.5367431640625e-7 + 1.17549435e-38); +} + +float3 cultmath_af_add_iv(float3 x, float2 interval) { return cultmath_af_add(x, cultmath_af_from_iv(interval)); } + +float3 cultmath_af_mul(float3 x, float3 y) +{ + float x0 = x.x * y.x; + float p = x.x * y.y; + float q = y.x * x.y; + float a = p + q; + float e = abs(x.x) * y.z + abs(y.x) * x.z + (abs(x.y) + x.z) * (abs(y.y) + y.z); + return float3(x0, a, e + (abs(x0) + abs(p) + abs(q) + e) * 9.5367431640625e-7 + 1.17549435e-38); +} + +// The narrower of the Lipschitz form and the centred (Taylor) form over every p = centre + axis eps + w, +// |w| <= radius, from one cultmath_snoise_grad at the centre; the allowance for the float32 evaluation is +// in math.Affine.cs. +float3 cultmath_af_snoise(float3 centre, float3 axis, float radius) +{ + float4 g = cultmath_snoise_grad(centre); + float3 gradient = float3(g.x, g.y, g.z); + float reach = length(axis) + radius; + float lipschitz = CULTMATH_SNOISE_LIPSCHITZ * reach; + float a = dot(gradient, axis); + float remainder = length(gradient) * radius + 0.5 * CULTMATH_SNOISE_HESSIAN * reach * reach; + float allowance = 6.103515625e-5 + 2.44140625e-4 * reach + + CULTMATH_SNOISE_LIPSCHITZ * 9.5367431640625e-7 * (abs(centre.x) + abs(centre.y) + abs(centre.z) + reach); + bool centred = abs(a) + remainder < lipschitz; + float slope = centred ? a : 0.0; + float e = (centred ? remainder : lipschitz) + allowance; + return float3(g.w, slope, e + (abs(g.w) + abs(slope) + e) * 9.5367431640625e-7); +} + +// Octave sum of cultmath_af_snoise, compounding frequency and amplitude as the value-and-gradient fBm does. +float3 cultmath_af_fbm(float3 centre, float3 axis, float radius, int octaves, float lacunarity, float gain) +{ + octaves = clamp(octaves, 0, 16); + float amplitude = 1.0; + float frequency = 1.0; + float3 sum = float3(0.0, 0.0, 0.0); + for (int i = 0; i < octaves; i++) + { + sum = cultmath_af_add(sum, cultmath_af_scale(cultmath_af_snoise(centre * frequency, axis * frequency, radius * abs(frequency)), amplitude)); + frequency *= lacunarity; + amplitude *= gain; + } + + return sum; +} + +// The axis of a screen tile's slice over the depth segment [z0, z1]; camera frame of +// cultmath_iv_frustum_ball. +float3 cultmath_af_frustum_axis(float2 centreSlope, float z0, float z1) +{ + float h = (z1 - z0) * 0.5; + return float3(centreSlope.x * h, centreSlope.y * h, h); +} + +// The affine slice of a screen tile over [z0, z1]: centre xyz, radius w. Derivation in math.Affine.cs. +float4 cultmath_af_frustum_ball(float2 centreSlope, float z0, float z1, float footprintPerDepth, float warpVariation) +{ + float zm = (z0 + z1) * 0.5; + float h = (z1 - z0) * 0.5; + float cx = centreSlope.x * zm; + float cy = centreSlope.y * zm; + float spread = (abs(centreSlope.x) + abs(centreSlope.y) + 1.0) * h; + float radius = z1 * footprintPerDepth + warpVariation; + radius += (abs(cx) + abs(cy) + abs(zm) + spread + radius) * 9.5367431640625e-7; + return float4(cx, cy, zm, radius); +} + +#endif diff --git a/packages/cultmath/unity/org.gamecult.cultmath/Shaders/CultMath.Affine.hlsl.meta b/packages/cultmath/unity/org.gamecult.cultmath/Shaders/CultMath.Affine.hlsl.meta new file mode 100644 index 00000000..9cf91737 --- /dev/null +++ b/packages/cultmath/unity/org.gamecult.cultmath/Shaders/CultMath.Affine.hlsl.meta @@ -0,0 +1,7 @@ +fileFormatVersion: 2 +guid: 5dbcf0fdd66a4a74b22f5ca1c8e33058 +ShaderIncludeImporter: + externalObjects: {} + userData: + assetBundleName: + assetBundleVariant: diff --git a/packages/cultmath/unity/org.gamecult.cultmath/Shaders/CultMath.Interval.hlsl b/packages/cultmath/unity/org.gamecult.cultmath/Shaders/CultMath.Interval.hlsl index 22c19221..bc084ace 100644 --- a/packages/cultmath/unity/org.gamecult.cultmath/Shaders/CultMath.Interval.hlsl +++ b/packages/cultmath/unity/org.gamecult.cultmath/Shaders/CultMath.Interval.hlsl @@ -9,6 +9,9 @@ // Lipschitz constant of cultmath_snoise(float3); provenance in math.Interval.cs (SNOISE_LIPSCHITZ). static const float CULTMATH_SNOISE_LIPSCHITZ = 7.9640074; +// Lipschitz constant of cultmath_snoise(float2); provenance in math.Interval.cs (SNOISE2_LIPSCHITZ). +static const float CULTMATH_SNOISE2_LIPSCHITZ = 8.117002; + float2 cultmath_iv_point(float value) { return float2(value, value); } float2 cultmath_iv_add(float2 a, float2 b) { return float2(a.x + b.x, a.y + b.y); } float2 cultmath_iv_sub(float2 a, float2 b) { return float2(a.x - b.y, a.y - b.x); } @@ -87,6 +90,14 @@ float2 cultmath_iv_snoise_ball(float3 centre, float radius) return float2(max(n - e, -1.0), min(n + e, 1.0)); } +// The 2D overload: [n - L2 r, n + L2 r] intersected with [-1, 1], n = cultmath_snoise(centre). One snoise, value only. +float2 cultmath_iv_snoise_ball(float2 centre, float radius) +{ + float n = cultmath_snoise(centre); + float e = CULTMATH_SNOISE2_LIPSCHITZ * radius; + return float2(max(n - e, -1.0), min(n + e, 1.0)); +} + // Octave sum of cultmath_iv_snoise_ball, compounding frequency and amplitude as the value-and-gradient fBm does. float2 cultmath_iv_fbm_ball(float3 centre, float radius, int octaves, float lacunarity, float gain) { diff --git a/packages/cultmath/unity/org.gamecult.cultmath/Shaders/CultMath.hlsl b/packages/cultmath/unity/org.gamecult.cultmath/Shaders/CultMath.hlsl index e40cf0b6..6856a000 100644 --- a/packages/cultmath/unity/org.gamecult.cultmath/Shaders/CultMath.hlsl +++ b/packages/cultmath/unity/org.gamecult.cultmath/Shaders/CultMath.hlsl @@ -509,6 +509,7 @@ CultCellular cultmath_cellular(float3 p) #include "CultMath.Phacelle.hlsl" #include "CultMath.Interval.hlsl" +#include "CultMath.Affine.hlsl" float cultmath_value_noise(float2 position) { From 2a1f9df19a405b8863010c8a06ca9fb936770e8e Mon Sep 17 00:00:00 2001 From: Meta Vens Date: Sun, 4 Oct 2026 09:47:49 +0200 Subject: [PATCH 24/31] GLSL fixture: the af_* families and iv_snoise_ball(vec2), with enclosure checks for af_snoise, af_fbm and af_frustum_ball Co-Authored-By: Claude Sonnet 5.5 --- packages/cultmath/docs/design.md | 5 + .../tests/CultMath.Tests/GlslMirrorTests.cs | 190 +- .../CultMath.Tests/fixtures/glsl-parity.json | 3933 +++++++++++++++++ 3 files changed, 4124 insertions(+), 4 deletions(-) diff --git a/packages/cultmath/docs/design.md b/packages/cultmath/docs/design.md index a46ca91a..a49884b3 100644 --- a/packages/cultmath/docs/design.md +++ b/packages/cultmath/docs/design.md @@ -222,6 +222,11 @@ lowering that inflates the ball fails too. Only those two bounds fail the check; the largest ulp distance is reported as for any ulp-bounded family. `GoldenFixtureMatchesCSharp` applies the same check to C#'s own results. +Every `af_*` function is an ulp-bounded family, and `af_snoise`, `af_fbm` and `af_frustum_ball` also carry a two-sided `check` +of their own, applied the same way: the device's own `snoise` or fBm at 75 sampled points of the case's set must lie in the device's +form with no tolerance (a device whose gradient error exceeds the allowance fails), its width must not pass the Lipschitz form's +allowance, and the device's `af_frustum_ball` radius must enclose the sampled ray points and stay within its rounding widening. + `GoldenFixtureMatchesCSharp` evaluates C# on the committed fixture's own arguments. Arguments are drawn only when the fixture is regenerated (`CULTMATH_WRITE_GLSL=1`), because the draw itself calls `MathF.Pow`, whose diff --git a/packages/cultmath/tests/CultMath.Tests/GlslMirrorTests.cs b/packages/cultmath/tests/CultMath.Tests/GlslMirrorTests.cs index f14fcda8..b2f525d9 100644 --- a/packages/cultmath/tests/CultMath.Tests/GlslMirrorTests.cs +++ b/packages/cultmath/tests/CultMath.Tests/GlslMirrorTests.cs @@ -163,7 +163,7 @@ public sealed class GlslMirrorTests } if (want["check"] is not null) - AssertFrustumBall(want["cases"]!.AsArray()); + output.WriteLine($"{name}: {Families[f].Verify!(want["cases"]!.AsArray())}"); } committed.Remove("functions"); @@ -188,8 +188,9 @@ public sealed class GlslMirrorTests // FrustumBall applied to C#'s own results: every case's ball encloses the exact one and is no wider // than its rounding widening allows. - private static void AssertFrustumBall(JsonArray cases) + private static string AssertFrustumBall(JsonArray cases) { + double low = double.PositiveInfinity, high = double.PositiveInfinity; foreach (var entry in cases) { var sides = ((string)entry!).Split(" -> "); @@ -203,7 +204,127 @@ public sealed class GlslMirrorTests var c1 = Math.Abs(mx * zm) + Math.Abs(my * zm) + Math.Abs(zm); Assert.True(b[3] >= reach, $"iv_frustum_ball case {entry}: radius short of the exact ball by {reach - b[3]}"); Assert.True(b[3] <= reach + Math.ScaleB(radius + c1, -19), $"iv_frustum_ball case {entry}: radius past the exact ball by {b[3] - reach}, more than 2^-19 (r + |c|_1)"); + low = Math.Min(low, (b[3] - reach) / radius); + high = Math.Min(high, (reach + Math.ScaleB(radius + c1, -19) - b[3]) / Math.ScaleB(radius + c1, -19)); } + return $"least slack: {low:E3} of r above the exact ball, {high:F3} of 2^-19 (r + |c|_1) below its widening"; + } + + private static readonly double[] CheckEps = { -1.0, -0.5, 0.0, 0.5, 1.0 }; + + // The origin, the six axis directions and the eight cube corners, normalised. + private static readonly (double X, double Y, double Z)[] CheckDirections = CheckDirectionList().ToArray(); + + private static IEnumerable<(double X, double Y, double Z)> CheckDirectionList() + { + yield return (0.0, 0.0, 0.0); + foreach (var s in new[] { -1.0, 1.0 }) + { + yield return (s, 0.0, 0.0); + yield return (0.0, s, 0.0); + yield return (0.0, 0.0, s); + } + + var corner = 1.0 / Math.Sqrt(3.0); + foreach (var sx in new[] { -corner, corner }) + foreach (var sy in new[] { -corner, corner }) + foreach (var sz in new[] { -corner, corner }) + yield return (sx, sy, sz); + } + + // SampledForm applied to C#'s own results, with C#'s own snoise or fBm as the reference value. A form + // must enclose its reference at every sample, and be no wider than the Lipschitz form's allowance. + private static string AssertSampledForm(JsonArray cases, string name, Func reference, Func widthBound) + { + double least = double.PositiveInfinity, leastRelative = double.PositiveInfinity, tightest = double.PositiveInfinity; + foreach (var entry in cases) + { + var sides = ((string)entry!).Split(" -> "); + var tokens = sides[0].Split(' '); + double[] a = Floats(string.Join(' ', tokens.Take(7))), form = Floats(sides[1]); + double x0 = form[0], slope = form[1], e = form[2]; + foreach (var eps in CheckEps) + foreach (var d in CheckDirections) + { + var p = new float3((float)(a[0] + a[3] * eps + a[6] * d.X), (float)(a[1] + a[4] * eps + a[6] * d.Y), (float)(a[2] + a[5] * eps + a[6] * d.Z)); + double value = reference(tokens, p), middle = x0 + slope * eps; + var inside = Math.Min(value - (middle - e), middle + e - value); + Assert.True(inside >= 0.0, $"{name} case {entry}: the reference {value:R} at eps {eps} and direction {d} is outside the form's range [{middle - e:R}, {middle + e:R}] by {-inside:R}"); + least = Math.Min(least, inside); + leastRelative = Math.Min(leastRelative, inside / e); + } + + var bound = widthBound(tokens, a); + Assert.True(Math.Abs(slope) + e <= bound + Math.ScaleB(1.0 + bound, -16), $"{name} case {entry}: the width {Math.Abs(slope) + e:R} is past B + 2^-16 (1 + B) = {bound + Math.ScaleB(1.0 + bound, -16):R}"); + tightest = Math.Min(tightest, (bound + Math.ScaleB(1.0 + bound, -16) - Math.Abs(slope) - e) / bound); + } + + return $"least enclosure slack {least:E3} ({leastRelative:E3} of e), least tightness slack {tightest:E3} of B"; + } + + // The Lipschitz form's width bound for one octave of the sampled-form check. + private static double SnoiseWidthBound(double centre1, double reach) => + math.SNOISE_LIPSCHITZ * reach + Math.ScaleB(1.0, -14) + Math.ScaleB(reach, -12) + math.SNOISE_LIPSCHITZ * Math.ScaleB(centre1 + reach, -20); + + private static double Length3(double[] a, int first) => Math.Sqrt(a[first] * a[first] + a[first + 1] * a[first + 1] + a[first + 2] * a[first + 2]); + + private static string AssertSnoiseForm(JsonArray cases) => + AssertSampledForm(cases, "af_snoise", (_, p) => math.snoise(p), + (_, a) => SnoiseWidthBound(Math.Abs(a[0]) + Math.Abs(a[1]) + Math.Abs(a[2]), Length3(a, 3) + a[6])); + + private static string AssertFbmForm(JsonArray cases) => + AssertSampledForm(cases, "af_fbm", + (t, p) => math.fbm_grad(p, (int)Convert.ToUInt32(t[7], 16), (float)Floats(t[8])[0], (float)Floats(t[9])[0]).w, + (t, a) => + { + var (octaves, lacunarity, gain) = ((int)Convert.ToUInt32(t[7], 16), Floats(t[8])[0], Floats(t[9])[0]); + double sum = 0.0, amplitude = 1.0, frequency = 1.0; + for (var i = 0; i < Math.Clamp(octaves, 0, 16); i++) + { + sum += amplitude * SnoiseWidthBound((Math.Abs(a[0]) + Math.Abs(a[1]) + Math.Abs(a[2])) * frequency, (Length3(a, 3) + a[6]) * frequency); + amplitude *= gain; + frequency *= lacunarity; + } + + return sum; + }); + + // AffineFrustumBall applied to C#'s own results. + private static string AssertAffineFrustumBall(JsonArray cases) + { + double enclosure = double.PositiveInfinity, lower = double.PositiveInfinity, upper = double.PositiveInfinity; + var diagonal = new[] { (0.0, 0.0), (1.0, 0.0), (-1.0, 0.0), (0.0, 1.0), (0.0, -1.0), (Math.Sqrt(0.5), Math.Sqrt(0.5)), (Math.Sqrt(0.5), -Math.Sqrt(0.5)), (-Math.Sqrt(0.5), Math.Sqrt(0.5)), (-Math.Sqrt(0.5), -Math.Sqrt(0.5)) }; + foreach (var entry in cases) + { + var sides = ((string)entry!).Split(" -> "); + var a = Floats(sides[0]); + var b = Floats(sides[1]); + double mx = a[0], my = a[1], z0 = a[2], z1 = a[3], fp = a[4], warp = a[5]; + double zm = (z0 + z1) / 2.0, h = (z1 - z0) / 2.0; + var r0 = z1 * fp + warp; + var c1 = Math.Abs(mx * zm) + Math.Abs(my * zm) + Math.Abs(zm); + var spread = (Math.Abs(mx) + Math.Abs(my) + 1.0) * h; + double dx = b[0] - mx * zm, dy = b[1] - my * zm, dz = b[2] - zm; + var d = Math.Sqrt(dx * dx + dy * dy + dz * dz); + foreach (var z in new[] { z0, (3.0 * z0 + z1) / 4.0, zm, (z0 + 3.0 * z1) / 4.0, z1 }) + { + var eps = h > 0.0 ? Math.Clamp((z - zm) / h, -1.0, 1.0) : 0.0; + foreach (var (ex, ey) in diagonal) + { + double px = (mx + fp * ex) * z - (b[0] + mx * h * eps), py = (my + fp * ey) * z - (b[1] + my * h * eps), pz = z - (b[2] + h * eps); + var slack = b[3] - (Math.Sqrt(px * px + py * py + pz * pz) + warp); + Assert.True(slack >= 0.0, $"af_frustum_ball case {entry}: the ray point at depth {z:R} and offset ({ex}, {ey}) is outside the ball by {-slack:R}"); + enclosure = Math.Min(enclosure, slack / b[3]); + } + } + + Assert.True(b[3] >= r0 + d, $"af_frustum_ball case {entry}: radius short of R0 + d by {r0 + d - b[3]}"); + Assert.True(b[3] <= r0 + d + Math.ScaleB(r0 + c1 + spread, -19), $"af_frustum_ball case {entry}: radius past R0 + d by {b[3] - r0 - d}, more than 2^-19 (R0 + |c0|_1 + spread)"); + lower = Math.Min(lower, (b[3] - r0 - d) / r0); + upper = Math.Min(upper, (r0 + d + Math.ScaleB(r0 + c1 + spread, -19) - b[3]) / Math.ScaleB(r0 + c1 + spread, -19)); + } + + return $"least enclosure slack {enclosure:E3} of the radius, least slack above R0 + d {lower:E3} of R0, below its widening {upper:F3} of 2^-19 (R0 + |c0|_1 + spread)"; } // A committed case's arguments, decoded into the C# method's parameter types in Hex's field order. @@ -249,7 +370,7 @@ public sealed class GlslMirrorTests // phacelle's sums of exp, sin and cos cancel; Windows differs from Linux by up to 3.6e-7 of it. private const double ScaledPlatformBound = 9.5367431640625e-7; - private sealed record Family(string Glsl, string Tolerance, Func Arguments, string? Check = null, Platform Platform = Platform.None); + private sealed record Family(string Glsl, string Tolerance, Func Arguments, string? Check = null, Platform Platform = Platform.None, Func? Verify = null); private static float Uniform(System.Random r, float lo, float hi) => lo + (hi - lo) * r.NextSingle(); private static float3 Point3(System.Random r) => new(Uniform(r, -50.0f, 50.0f), Uniform(r, -50.0f, 50.0f), Uniform(r, -50.0f, 50.0f)); @@ -291,6 +412,43 @@ public sealed class GlslMirrorTests "and d = |GPU centre - c| (Euclidean); every case must have r + d <= GPU radius <= r + d + 2^-19 (r + |c|_1). " + "Report the largest ulp distance per component as for any ulp-bounded family; only the two bounds fail the check."; + // An affine form (x0, a, e): e >= 0, spanning the widths a probe's forms take. + private static float3 Form(System.Random r) => new(Uniform(r, -20.0f, 20.0f), Uniform(r, -5.0f, 5.0f), MathF.Pow(10.0f, Uniform(r, -4.0f, 1.0f))); + + // A slice's axis and radius in noise space, log-uniform from 1e-3 to 0.5: below about 0.15 the + // centred form of af_snoise wins, above it the Lipschitz form does, so both are drawn. + private static float3 Axis3(System.Random r) + { + var scale = MathF.Pow(10.0f, Uniform(r, -3.0f, -0.3f)); + return new float3(Uniform(r, -1.0f, 1.0f) * scale, Uniform(r, -1.0f, 1.0f) * scale, Uniform(r, -1.0f, 1.0f) * scale); + } + + private static float BallRadius(System.Random r) => MathF.Pow(10.0f, Uniform(r, -3.0f, -0.3f)); + + private const string SampledForm = + "the arguments are (cx, cy, cz, ax, ay, az, radius, ...) and the result is the form (x0, a, e). Enclosure: for eps in " + + "{-1, -0.5, 0, 0.5, 1} and d in {0, +-ex, +-ey, +-ez, (+-1, +-1, +-1) / sqrt(3)} (15 directions), form the point " + + "p = c + axis eps + radius d in double and round it to float32; the device's own reference value at p must lie in " + + "[x0 + a eps - e, x0 + a eps + e] with no tolerance, so a device whose gradient error exceeds the allowance fails. " + + "Tightness: with L = 7.9640074 and, for each octave (one for af_snoise), R = |axis| + radius, " + + "B = L R + 2^-14 + 2^-12 R + L 2^-20 (|c|_1 + R), the width |a| + e must be at most B + 2^-16 (1 + B). " + + "Report the least slack of each side, absolutely and as a fraction of e; only the two bounds fail the check. "; + + private const string SnoiseForm = SampledForm + "The reference value is cultmath_snoise(p)."; + + private const string FbmForm = SampledForm + + "The reference value is cultmath_fbm_grad(p, octaves, lacunarity, gain).w, and B is the sum over octaves i of gain^i B_i, " + + "where B_i is B for the centre lacunarity^i c, the axis lacunarity^i axis and the radius lacunarity^i radius."; + + private const string AffineFrustumBall = + "enclosure and tightness: from the arguments (mx, my, z0, z1, fp, warpVariation) compute in double zm = (z0 + z1) / 2, " + + "h = (z1 - z0) / 2, axis = (mx h, my h, h), c0 = (mx zm, my zm, zm), |c0|_1 = |mx zm| + |my zm| + |zm|, R0 = z1 fp + warpVariation, " + + "spread = (|mx| + |my| + 1) h and d = |GPU centre - c0|. Enclosure: for z in {z0, (3 z0 + z1) / 4, zm, (z0 + 3 z1) / 4, z1}, " + + "eps = clamp((z - zm) / h, -1, 1) and the nine footprint offsets e in {(0, 0), (+-1, 0), (0, +-1), (+-1, +-1) / sqrt(2)}, " + + "the point P = ((mx + fp e.x) z, (my + fp e.y) z, z) satisfies |P - (GPU centre + axis eps)| + warpVariation <= GPU radius. " + + "Tightness: R0 + d <= GPU radius <= R0 + d + 2^-19 (R0 + |c0|_1 + spread). " + + "Report the least slack of each side; only the bounds fail the check."; + // A Phacelle stripe wave vector: each component in [-6, 6], about one stripe per cell. private static float3 Side(System.Random r) => new(Uniform(r, -6.0f, 6.0f), Uniform(r, -6.0f, 6.0f), Uniform(r, -6.0f, 6.0f)); @@ -331,12 +489,36 @@ public sealed class GlslMirrorTests var slope = new float2(Uniform(r, -1.0f, 1.0f), Uniform(r, -1.0f, 1.0f)); var z0 = Uniform(r, 0.1f, 100.0f); return new object[] { slope, z0, z0 + Uniform(r, 0.01f, 50.0f), Uniform(r, 0.0001f, 0.05f), Uniform(r, 0.0f, 2.0f) }; - }, FrustumBall), + }, FrustumBall, Verify: AssertFrustumBall), new("vec2 cultmath_iv_exp(vec2)", "ulp-bounded, platform exp", r => new object[] { Interval(r) }, Platform: Platform.Ulp), new("CultPhasor cultmath_phacelle(vec3, vec3, float, float)", "ulp-bounded, platform exp, sin, cos; across platforms |diff| <= 2^-20 max(|v|, 1)", r => new object[] { Point3(r), Side(r), Uniform(r, 0.0f, 1.0f), 0.5f }, Platform: Platform.Scaled), new("float cultmath_snoise(vec3)", "ulp-bounded, on the simplex tie set", r => new object[] { TiePoint3(r) }), new("vec4 cultmath_snoise_grad(vec3)", "ulp-bounded, on the simplex tie set", r => new object[] { TiePoint3(r) }), + new("vec3 cultmath_af_point(float)", "ulp-bounded", r => new object[] { Uniform(r, -20.0f, 20.0f) }), + new("vec3 cultmath_af_symbol(float, float)", "ulp-bounded", r => new object[] { Uniform(r, -20.0f, 20.0f), Uniform(r, -5.0f, 5.0f) }), + new("vec3 cultmath_af_from_iv(vec2)", "ulp-bounded", r => new object[] { Interval(r) }), + new("vec2 cultmath_af_range(vec3)", "ulp-bounded", r => new object[] { Form(r) }), + new("vec3 cultmath_af_add(vec3, vec3)", "ulp-bounded", r => new object[] { Form(r), Form(r) }), + new("vec3 cultmath_af_sub(vec3, vec3)", "ulp-bounded", r => new object[] { Form(r), Form(r) }), + new("vec3 cultmath_af_neg(vec3)", "ulp-bounded", r => new object[] { Form(r) }), + new("vec3 cultmath_af_scale(vec3, float)", "ulp-bounded", r => new object[] { Form(r), Uniform(r, -4.0f, 4.0f) }), + new("vec3 cultmath_af_add_iv(vec3, vec2)", "ulp-bounded", r => new object[] { Form(r), Interval(r) }), + new("vec3 cultmath_af_mul(vec3, vec3)", "ulp-bounded", r => new object[] { Form(r), Form(r) }), + new("vec3 cultmath_af_snoise(vec3, vec3, float)", "ulp-bounded", r => new object[] { Point3(r), Axis3(r), BallRadius(r) }, SnoiseForm, Verify: AssertSnoiseForm), + new("vec3 cultmath_af_fbm(vec3, vec3, float, int, float, float)", "ulp-bounded", r => new object[] { Point3(r), Axis3(r), BallRadius(r), 4, 2.0f, 0.5f }, FbmForm, Verify: AssertFbmForm), + new("vec3 cultmath_af_frustum_axis(vec2, float, float)", "ulp-bounded", r => + { + var z0 = Uniform(r, 0.1f, 100.0f); + return new object[] { new float2(Uniform(r, -1.0f, 1.0f), Uniform(r, -1.0f, 1.0f)), z0, z0 + Uniform(r, 0.01f, 50.0f) }; + }), + new("vec4 cultmath_af_frustum_ball(vec2, float, float, float, float)", "ulp-bounded", r => + { + var slope = new float2(Uniform(r, -1.0f, 1.0f), Uniform(r, -1.0f, 1.0f)); + var z0 = Uniform(r, 0.1f, 100.0f); + return new object[] { slope, z0, z0 + Uniform(r, 0.01f, 50.0f), Uniform(r, 0.0001f, 0.05f), Uniform(r, 0.0f, 2.0f) }; + }, AffineFrustumBall, Verify: AssertAffineFrustumBall), + new("vec2 cultmath_iv_snoise_ball(vec2, float)", "ulp-bounded", r => new object[] { new float2(Uniform(r, -50.0f, 50.0f), Uniform(r, -50.0f, 50.0f)), Uniform(r, 0.001f, 2.0f) }), }; // The fixture text, each case's arguments given by family and point index. diff --git a/packages/cultmath/tests/CultMath.Tests/fixtures/glsl-parity.json b/packages/cultmath/tests/CultMath.Tests/fixtures/glsl-parity.json index 775bbd83..cd7a2428 100644 --- a/packages/cultmath/tests/CultMath.Tests/fixtures/glsl-parity.json +++ b/packages/cultmath/tests/CultMath.Tests/fixtures/glsl-parity.json @@ -7866,6 +7866,3939 @@ "4256e995 c229166b c225166b -> c03e87af 404ed03a 3f250ae5 be5f05f6", "40b5fd99 4206bfb4 41a57f66 -> 4007362a bf7b6b1a bf118d1e 3f2d6b64" ] + }, + { + "glsl": "vec3 cultmath_af_point(float)", + "tolerance": "ulp-bounded", + "cases": [ + "c100e863 -> c100e863 00000000 00000000", + "c157318a -> c157318a 00000000 00000000", + "3fe21080 -> 3fe21080 00000000 00000000", + "40170740 -> 40170740 00000000 00000000", + "c10f7504 -> c10f7504 00000000 00000000", + "41180cbc -> 41180cbc 00000000 00000000", + "c1577606 -> c1577606 00000000 00000000", + "40af8864 -> 40af8864 00000000 00000000", + "c0006dc0 -> c0006dc0 00000000 00000000", + "40fea250 -> 40fea250 00000000 00000000", + "411f93f6 -> 411f93f6 00000000 00000000", + "419e382c -> 419e382c 00000000 00000000", + "c06c0b10 -> c06c0b10 00000000 00000000", + "41886cbe -> 41886cbe 00000000 00000000", + "40732908 -> 40732908 00000000 00000000", + "c0d8619c -> c0d8619c 00000000 00000000", + "c1607274 -> c1607274 00000000 00000000", + "408b9b40 -> 408b9b40 00000000 00000000", + "3ff0ee20 -> 3ff0ee20 00000000 00000000", + "41692bc0 -> 41692bc0 00000000 00000000", + "40a5c5f0 -> 40a5c5f0 00000000 00000000", + "402690c8 -> 402690c8 00000000 00000000", + "41326d56 -> 41326d56 00000000 00000000", + "411bb610 -> 411bb610 00000000 00000000", + "c03c2cd0 -> c03c2cd0 00000000 00000000", + "c10162e3 -> c10162e3 00000000 00000000", + "c169c276 -> c169c276 00000000 00000000", + "c0cccb2c -> c0cccb2c 00000000 00000000", + "c149162d -> c149162d 00000000 00000000", + "c1926dea -> c1926dea 00000000 00000000", + "c14ac648 -> c14ac648 00000000 00000000", + "c190b787 -> c190b787 00000000 00000000", + "4153a474 -> 4153a474 00000000 00000000", + "414332f0 -> 414332f0 00000000 00000000", + "c14e198b -> c14e198b 00000000 00000000", + "c0b05794 -> c0b05794 00000000 00000000", + "be7d3100 -> be7d3100 00000000 00000000", + "410da948 -> 410da948 00000000 00000000", + "41731d34 -> 41731d34 00000000 00000000", + "41206a28 -> 41206a28 00000000 00000000", + "407e7480 -> 407e7480 00000000 00000000", + "419dbc20 -> 419dbc20 00000000 00000000", + "3fde4680 -> 3fde4680 00000000 00000000", + "c09c7f94 -> c09c7f94 00000000 00000000", + "4018a490 -> 4018a490 00000000 00000000", + "40625700 -> 40625700 00000000 00000000", + "c0357360 -> c0357360 00000000 00000000", + "c157aec4 -> c157aec4 00000000 00000000", + "3e1a6400 -> 3e1a6400 00000000 00000000", + "c18b9397 -> c18b9397 00000000 00000000", + "bfb0efd0 -> bfb0efd0 00000000 00000000", + "c184437f -> c184437f 00000000 00000000", + "c0ad8f06 -> c0ad8f06 00000000 00000000", + "418e7d26 -> 418e7d26 00000000 00000000", + "410f9790 -> 410f9790 00000000 00000000", + "4115736a -> 4115736a 00000000 00000000", + "c0933dc0 -> c0933dc0 00000000 00000000", + "41408c00 -> 41408c00 00000000 00000000", + "c16b32fc -> c16b32fc 00000000 00000000", + "4198f6f0 -> 4198f6f0 00000000 00000000", + "40839e64 -> 40839e64 00000000 00000000", + "414eef90 -> 414eef90 00000000 00000000", + "bff92d10 -> bff92d10 00000000 00000000", + "c06cfe70 -> c06cfe70 00000000 00000000", + "3f2176e0 -> 3f2176e0 00000000 00000000", + "41010306 -> 41010306 00000000 00000000", + "c1573420 -> c1573420 00000000 00000000", + "40839498 -> 40839498 00000000 00000000", + "411ef306 -> 411ef306 00000000 00000000", + "c12b09f6 -> c12b09f6 00000000 00000000", + "4157c3f4 -> 4157c3f4 00000000 00000000", + "c0386ef0 -> c0386ef0 00000000 00000000", + "4112b06e -> 4112b06e 00000000 00000000", + "413db39a -> 413db39a 00000000 00000000", + "c1167161 -> c1167161 00000000 00000000", + "40aed570 -> 40aed570 00000000 00000000", + "c199124e -> c199124e 00000000 00000000", + "c07d4b80 -> c07d4b80 00000000 00000000", + "c14a7314 -> c14a7314 00000000 00000000", + "41585f0c -> 41585f0c 00000000 00000000", + "416bf9f8 -> 416bf9f8 00000000 00000000", + "4196f628 -> 4196f628 00000000 00000000", + "415730dc -> 415730dc 00000000 00000000", + "c171eeff -> c171eeff 00000000 00000000", + "40450898 -> 40450898 00000000 00000000", + "c1811fa5 -> c1811fa5 00000000 00000000", + "40eab0ec -> 40eab0ec 00000000 00000000", + "4176aa28 -> 4176aa28 00000000 00000000", + "c186324f -> c186324f 00000000 00000000", + "c00e33d0 -> c00e33d0 00000000 00000000", + "4198b98c -> 4198b98c 00000000 00000000", + "40f6fe78 -> 40f6fe78 00000000 00000000", + "40637110 -> 40637110 00000000 00000000", + "418097aa -> 418097aa 00000000 00000000", + "c161650a -> c161650a 00000000 00000000", + "4130ad92 -> 4130ad92 00000000 00000000", + "3fd4ef20 -> 3fd4ef20 00000000 00000000", + "c168f794 -> c168f794 00000000 00000000", + "4167a8c8 -> 4167a8c8 00000000 00000000", + "41652620 -> 41652620 00000000 00000000", + "c040d880 -> c040d880 00000000 00000000", + "4150d8dc -> 4150d8dc 00000000 00000000", + "c05a8728 -> c05a8728 00000000 00000000", + "c1128c7c -> c1128c7c 00000000 00000000", + "c12eeb24 -> c12eeb24 00000000 00000000", + "c194011f -> c194011f 00000000 00000000", + "c0b66eda -> c0b66eda 00000000 00000000", + "402e1380 -> 402e1380 00000000 00000000", + "40e6d758 -> 40e6d758 00000000 00000000", + "c183e994 -> c183e994 00000000 00000000", + "41079806 -> 41079806 00000000 00000000", + "419ada00 -> 419ada00 00000000 00000000", + "4095fe28 -> 4095fe28 00000000 00000000", + "c1039209 -> c1039209 00000000 00000000", + "c17a0c17 -> c17a0c17 00000000 00000000", + "40a7c5c8 -> 40a7c5c8 00000000 00000000", + "419bdf5a -> 419bdf5a 00000000 00000000", + "c0da6d46 -> c0da6d46 00000000 00000000", + "41537e3c -> 41537e3c 00000000 00000000", + "c04ea520 -> c04ea520 00000000 00000000", + "c19a2ab8 -> c19a2ab8 00000000 00000000", + "c10dde4a -> c10dde4a 00000000 00000000", + "3f62ee80 -> 3f62ee80 00000000 00000000", + "c0fb0008 -> c0fb0008 00000000 00000000", + "c11c7d0c -> c11c7d0c 00000000 00000000", + "c163bb46 -> c163bb46 00000000 00000000", + "41590800 -> 41590800 00000000 00000000", + "4151811c -> 4151811c 00000000 00000000", + "40be3148 -> 40be3148 00000000 00000000", + "bee3de80 -> bee3de80 00000000 00000000", + "c1819995 -> c1819995 00000000 00000000", + "4176d2f8 -> 4176d2f8 00000000 00000000", + "3fc822c0 -> 3fc822c0 00000000 00000000", + "c0df3266 -> c0df3266 00000000 00000000", + "c05dab50 -> c05dab50 00000000 00000000", + "c1926f73 -> c1926f73 00000000 00000000", + "40123860 -> 40123860 00000000 00000000", + "4026f560 -> 4026f560 00000000 00000000", + "417cfc24 -> 417cfc24 00000000 00000000", + "3fca2330 -> 3fca2330 00000000 00000000", + "4118f824 -> 4118f824 00000000 00000000", + "c1762c6c -> c1762c6c 00000000 00000000", + "c13cc183 -> c13cc183 00000000 00000000", + "419db744 -> 419db744 00000000 00000000", + "4114db06 -> 4114db06 00000000 00000000", + "419ddf8a -> 419ddf8a 00000000 00000000", + "c187bfec -> c187bfec 00000000 00000000", + "c10391b3 -> c10391b3 00000000 00000000", + "413b3b6a -> 413b3b6a 00000000 00000000", + "3f2b8120 -> 3f2b8120 00000000 00000000", + "4138eee0 -> 4138eee0 00000000 00000000", + "c1382b7a -> c1382b7a 00000000 00000000", + "c0f8eba4 -> c0f8eba4 00000000 00000000", + "c012b7c0 -> c012b7c0 00000000 00000000", + "41597b90 -> 41597b90 00000000 00000000", + "c16257d4 -> c16257d4 00000000 00000000", + "c0faac1a -> c0faac1a 00000000 00000000", + "c17921c6 -> c17921c6 00000000 00000000", + "c19b07b8 -> c19b07b8 00000000 00000000", + "c18597ef -> c18597ef 00000000 00000000", + "c1410a40 -> c1410a40 00000000 00000000", + "3f9a3bb0 -> 3f9a3bb0 00000000 00000000", + "4186361e -> 4186361e 00000000 00000000", + "c1457560 -> c1457560 00000000 00000000", + "419db000 -> 419db000 00000000 00000000", + "4150c50c -> 4150c50c 00000000 00000000", + "c00d4158 -> c00d4158 00000000 00000000", + "c10567b8 -> c10567b8 00000000 00000000", + "4173d8cc -> 4173d8cc 00000000 00000000", + "c18a9698 -> c18a9698 00000000 00000000", + "c188559b -> c188559b 00000000 00000000", + "c047fa98 -> c047fa98 00000000 00000000", + "c028cb20 -> c028cb20 00000000 00000000", + "c105c628 -> c105c628 00000000 00000000", + "40e6bd30 -> 40e6bd30 00000000 00000000", + "4180eb7e -> 4180eb7e 00000000 00000000", + "c1888e64 -> c1888e64 00000000 00000000", + "3f960300 -> 3f960300 00000000 00000000", + "403693d8 -> 403693d8 00000000 00000000", + "c1183c20 -> c1183c20 00000000 00000000", + "c18a1db6 -> c18a1db6 00000000 00000000", + "3fe4cdb0 -> 3fe4cdb0 00000000 00000000", + "c194dd26 -> c194dd26 00000000 00000000", + "c16b9f6f -> c16b9f6f 00000000 00000000", + "40ffe45c -> 40ffe45c 00000000 00000000", + "4174f850 -> 4174f850 00000000 00000000", + "404d2328 -> 404d2328 00000000 00000000", + "c18126db -> c18126db 00000000 00000000", + "bf114c40 -> bf114c40 00000000 00000000", + "c1151300 -> c1151300 00000000 00000000", + "41187728 -> 41187728 00000000 00000000", + "c0094088 -> c0094088 00000000 00000000", + "3ffe6640 -> 3ffe6640 00000000 00000000", + "41482c04 -> 41482c04 00000000 00000000", + "c0cb62b6 -> c0cb62b6 00000000 00000000", + "c13a4f54 -> c13a4f54 00000000 00000000", + "c14298aa -> c14298aa 00000000 00000000", + "c19ba612 -> c19ba612 00000000 00000000", + "c19ff8bd -> c19ff8bd 00000000 00000000", + "41820afc -> 41820afc 00000000 00000000", + "3ff87b50 -> 3ff87b50 00000000 00000000", + "4135e7e0 -> 4135e7e0 00000000 00000000", + "c05da9f8 -> c05da9f8 00000000 00000000", + "4110689a -> 4110689a 00000000 00000000", + "c01272e0 -> c01272e0 00000000 00000000", + "c0aa24f6 -> c0aa24f6 00000000 00000000", + "4132074e -> 4132074e 00000000 00000000", + "c19b57d5 -> c19b57d5 00000000 00000000", + "4127f378 -> 4127f378 00000000 00000000", + "418bd24a -> 418bd24a 00000000 00000000", + "c1889d86 -> c1889d86 00000000 00000000", + "412fe994 -> 412fe994 00000000 00000000", + "3fc9ca20 -> 3fc9ca20 00000000 00000000", + "41222cb0 -> 41222cb0 00000000 00000000", + "c19b7a38 -> c19b7a38 00000000 00000000", + "40c4b818 -> 40c4b818 00000000 00000000", + "be3f8f00 -> be3f8f00 00000000 00000000", + "41380bac -> 41380bac 00000000 00000000", + "beacf180 -> beacf180 00000000 00000000", + "c179984c -> c179984c 00000000 00000000", + "c12283bd -> c12283bd 00000000 00000000", + "c0c20540 -> c0c20540 00000000 00000000", + "4143ad0c -> 4143ad0c 00000000 00000000", + "4091d798 -> 4091d798 00000000 00000000", + "c0db48ac -> c0db48ac 00000000 00000000", + "40a349d8 -> 40a349d8 00000000 00000000", + "416e3494 -> 416e3494 00000000 00000000", + "409b4268 -> 409b4268 00000000 00000000", + "418ff59a -> 418ff59a 00000000 00000000", + "bf9290b0 -> bf9290b0 00000000 00000000", + "41046fa4 -> 41046fa4 00000000 00000000", + "c18ce852 -> c18ce852 00000000 00000000", + "3f958ed0 -> 3f958ed0 00000000 00000000", + "40d31df8 -> 40d31df8 00000000 00000000", + "4108b474 -> 4108b474 00000000 00000000", + "c10a234c -> c10a234c 00000000 00000000", + "c16bfbcc -> c16bfbcc 00000000 00000000", + "c0f445d0 -> c0f445d0 00000000 00000000", + "40f1fa90 -> 40f1fa90 00000000 00000000", + "c0d5f6ae -> c0d5f6ae 00000000 00000000", + "c17d0eff -> c17d0eff 00000000 00000000", + "4027e1d0 -> 4027e1d0 00000000 00000000", + "c0d48260 -> c0d48260 00000000 00000000", + "3ffa3530 -> 3ffa3530 00000000 00000000", + "c143d7f4 -> c143d7f4 00000000 00000000", + "419446c8 -> 419446c8 00000000 00000000", + "41823b4e -> 41823b4e 00000000 00000000", + "413da018 -> 413da018 00000000 00000000", + "413f2074 -> 413f2074 00000000 00000000", + "40efe534 -> 40efe534 00000000 00000000", + "4108aee8 -> 4108aee8 00000000 00000000", + "c06a9158 -> c06a9158 00000000 00000000", + "3f61b6a0 -> 3f61b6a0 00000000 00000000", + "4179a6d4 -> 4179a6d4 00000000 00000000", + "40cd336c -> 40cd336c 00000000 00000000", + "c13fedf5 -> c13fedf5 00000000 00000000" + ] + }, + { + "glsl": "vec3 cultmath_af_symbol(float, float)", + "tolerance": "ulp-bounded", + "cases": [ + "41991d6a 403fa9b6 -> 41991d6a 403fa9b6 00000000", + "c083e620 4049be5c -> c083e620 4049be5c 00000000", + "be51c380 c05b99e6 -> be51c380 c05b99e6 00000000", + "c18b4d6e c005ba72 -> c18b4d6e c005ba72 00000000", + "c12ccb11 c09bb534 -> c12ccb11 c09bb534 00000000", + "c122c845 406faa44 -> c122c845 406faa44 00000000", + "c19343e2 40132edc -> c19343e2 40132edc 00000000", + "3f657d80 c048db56 -> 3f657d80 c048db56 00000000", + "c18078ce c05a6c36 -> c18078ce c05a6c36 00000000", + "c19e44a7 3fe7079c -> c19e44a7 3fe7079c 00000000", + "c19b60b8 c01b999c -> c19b60b8 c01b999c 00000000", + "c04cf108 bfac63d0 -> c04cf108 bfac63d0 00000000", + "410d2e50 c087b5c2 -> 410d2e50 c087b5c2 00000000", + "414e80c0 c0180766 -> 414e80c0 c0180766 00000000", + "4124ff00 c000ec04 -> 4124ff00 c000ec04 00000000", + "3fc09f90 3fb21608 -> 3fc09f90 3fb21608 00000000", + "c19e855c 3e0a2680 -> c19e855c 3e0a2680 00000000", + "c0f8bc40 bfd74b04 -> c0f8bc40 bfd74b04 00000000", + "4183e814 406f3eec -> 4183e814 406f3eec 00000000", + "41584bb8 408f4574 -> 41584bb8 408f4574 00000000", + "3d135000 c092435f -> 3d135000 c092435f 00000000", + "c0e7dc90 c04c6d8c -> c0e7dc90 c04c6d8c 00000000", + "410b55d6 c09599d5 -> 410b55d6 c09599d5 00000000", + "412427dc c08a0734 -> 412427dc c08a0734 00000000", + "c1566540 3e8c51f0 -> c1566540 3e8c51f0 00000000", + "c0f36576 bfaef0c0 -> c0f36576 bfaef0c0 00000000", + "41763020 40998af2 -> 41763020 40998af2 00000000", + "c1492c84 400dd470 -> c1492c84 400dd470 00000000", + "c09a3414 c095e094 -> c09a3414 c095e094 00000000", + "c181b7fa bf4f6e28 -> c181b7fa bf4f6e28 00000000", + "c0d43550 40416c88 -> c0d43550 40416c88 00000000", + "3faa3470 408a1078 -> 3faa3470 408a1078 00000000", + "bf77e580 3f885970 -> bf77e580 3f885970 00000000", + "416cb4fc 3e86aea0 -> 416cb4fc 3e86aea0 00000000", + "c041cc10 c04602a5 -> c041cc10 c04602a5 00000000", + "c110ab7e c0303088 -> c110ab7e c0303088 00000000", + "c0e56fda c08d8a1a -> c0e56fda c08d8a1a 00000000", + "c14d0f82 4096e2a6 -> c14d0f82 4096e2a6 00000000", + "418c2a56 c026ceb5 -> 418c2a56 c026ceb5 00000000", + "40bcf084 3ff0c3f4 -> 40bcf084 3ff0c3f4 00000000", + "c0e726c2 40228f28 -> c0e726c2 40228f28 00000000", + "c180d113 c02c8b3e -> c180d113 c02c8b3e 00000000", + "419c635c c05073af -> 419c635c c05073af 00000000", + "4122dae6 c098104f -> 4122dae6 c098104f 00000000", + "c0dc6224 3f5735c0 -> c0dc6224 3f5735c0 00000000", + "4190ba44 40739150 -> 4190ba44 40739150 00000000", + "4132aee0 bf4f6300 -> 4132aee0 bf4f6300 00000000", + "40914b20 3ffe4d90 -> 40914b20 3ffe4d90 00000000", + "3ec65680 bf901e74 -> 3ec65680 bf901e74 00000000", + "410310ce c078d6e0 -> 410310ce c078d6e0 00000000", + "c157e8a6 c02cadf6 -> c157e8a6 c02cadf6 00000000", + "40b8c938 4097ceb2 -> 40b8c938 4097ceb2 00000000", + "c11dca43 c084c11a -> c11dca43 c084c11a 00000000", + "c1926039 40514454 -> c1926039 40514454 00000000", + "41402568 c036f1d2 -> 41402568 c036f1d2 00000000", + "41357fc4 c0664b54 -> 41357fc4 c0664b54 00000000", + "c09472d0 bf6f47b8 -> c09472d0 bf6f47b8 00000000", + "c1802414 3f4d58c0 -> c1802414 3f4d58c0 00000000", + "41030eb0 bffc5c6a -> 41030eb0 bffc5c6a 00000000", + "418dad1e bffe1554 -> 418dad1e bffe1554 00000000", + "416151d0 405cd59c -> 416151d0 405cd59c 00000000", + "c140f91c 4059bf8c -> c140f91c 4059bf8c 00000000", + "c127772c bf88799c -> c127772c bf88799c 00000000", + "c00b7da0 4080906a -> c00b7da0 4080906a 00000000", + "c00ee710 be436480 -> c00ee710 be436480 00000000", + "bfc36b30 3f018f90 -> bfc36b30 3f018f90 00000000", + "3fb103f0 3eba7900 -> 3fb103f0 3eba7900 00000000", + "3f825500 c08372b5 -> 3f825500 c08372b5 00000000", + "c179d574 c0713f8b -> c179d574 c0713f8b 00000000", + "41989f84 40812476 -> 41989f84 40812476 00000000", + "c16bb33c 4088cb48 -> c16bb33c 4088cb48 00000000", + "c1304606 c0773f40 -> c1304606 c0773f40 00000000", + "41404dac 4047bfdc -> 41404dac 4047bfdc 00000000", + "be0d5200 bfa2fe66 -> be0d5200 bfa2fe66 00000000", + "c0e7212e c0955851 -> c0e7212e c0955851 00000000", + "419e0c0c c03b9f17 -> 419e0c0c c03b9f17 00000000", + "c00da5d8 bf779228 -> c00da5d8 bf779228 00000000", + "c11a28b0 bfe97c80 -> c11a28b0 bfe97c80 00000000", + "c195da6e 4042a510 -> c195da6e 4042a510 00000000", + "41171490 4066728c -> 41171490 4066728c 00000000", + "40b2ba68 c065e145 -> 40b2ba68 c065e145 00000000", + "c09603d2 bfe7489c -> c09603d2 bfe7489c 00000000", + "41217b94 c02ae422 -> 41217b94 c02ae422 00000000", + "40872858 405e7778 -> 40872858 405e7778 00000000", + "4173e370 c09cb15f -> 4173e370 c09cb15f 00000000", + "bf96a2e0 3f5138e8 -> bf96a2e0 3f5138e8 00000000", + "c0deda76 c0937758 -> c0deda76 c0937758 00000000", + "c153576e 4087dda0 -> c153576e 4087dda0 00000000", + "409a3740 c084dcee -> 409a3740 c084dcee 00000000", + "c1934710 bf3cdc20 -> c1934710 bf3cdc20 00000000", + "417df878 c088b01b -> 417df878 c088b01b 00000000", + "4052c5a8 bf642480 -> 4052c5a8 bf642480 00000000", + "bf04e620 c01cce4f -> bf04e620 c01cce4f 00000000", + "c17c32a4 c06bfafa -> c17c32a4 c06bfafa 00000000", + "c10e882f c03af71f -> c10e882f c03af71f 00000000", + "c18d9848 bff8f50a -> c18d9848 bff8f50a 00000000", + "c090a838 c0753384 -> c090a838 c0753384 00000000", + "c117144a 409d1704 -> c117144a 409d1704 00000000", + "3fe4c3d0 4081af24 -> 3fe4c3d0 4081af24 00000000", + "c0b175a8 3f2cc738 -> c0b175a8 3f2cc738 00000000", + "c13bd452 3fbea680 -> c13bd452 3fbea680 00000000", + "beac5480 bff45bb4 -> beac5480 bff45bb4 00000000", + "40208ef0 bf60d888 -> 40208ef0 bf60d888 00000000", + "4173ce24 bf71eb20 -> 4173ce24 bf71eb20 00000000", + "c12a8d1b bfd3c7e0 -> c12a8d1b bfd3c7e0 00000000", + "40fc4c40 408f0280 -> 40fc4c40 408f0280 00000000", + "41293330 bff368d0 -> 41293330 bff368d0 00000000", + "41943d08 3f2af5a8 -> 41943d08 3f2af5a8 00000000", + "4015b548 4097039c -> 4015b548 4097039c 00000000", + "4154aa10 bcdb0e00 -> 4154aa10 bcdb0e00 00000000", + "c16eb17e be8386c0 -> c16eb17e be8386c0 00000000", + "4033fe98 3fddb520 -> 4033fe98 3fddb520 00000000", + "4190e6e8 c05659d8 -> 4190e6e8 c05659d8 00000000", + "c1435daa c03a39b7 -> c1435daa c03a39b7 00000000", + "4130259a c04d8577 -> 4130259a c04d8577 00000000", + "c099ba74 4009bd56 -> c099ba74 4009bd56 00000000", + "410f00f8 be8dff60 -> 410f00f8 be8dff60 00000000", + "c13362b6 c0215ac8 -> c13362b6 c0215ac8 00000000", + "405009c0 c01120bc -> 405009c0 c01120bc 00000000", + "415ed324 c0880c20 -> 415ed324 c0880c20 00000000", + "c0a138e8 40004824 -> c0a138e8 40004824 00000000", + "417e9224 408dade0 -> 417e9224 408dade0 00000000", + "3edc5800 c0793590 -> 3edc5800 c0793590 00000000", + "3fd29250 c0874fda -> 3fd29250 c0874fda 00000000", + "c15dad96 3ffc5c98 -> c15dad96 3ffc5c98 00000000", + "c02cfd20 3f3a4508 -> c02cfd20 3f3a4508 00000000", + "c188a5e5 4068fb48 -> c188a5e5 4068fb48 00000000", + "c01872d0 40149c88 -> c01872d0 40149c88 00000000", + "c186d632 404bc2cc -> c186d632 404bc2cc 00000000", + "c0b7f754 bf8304dc -> c0b7f754 bf8304dc 00000000", + "c134d879 3fdc0fb8 -> c134d879 3fdc0fb8 00000000", + "40a6bf38 c08e83d2 -> 40a6bf38 c08e83d2 00000000", + "41917ca4 409a3464 -> 41917ca4 409a3464 00000000", + "c130fbf7 c0050ce4 -> c130fbf7 c0050ce4 00000000", + "c18328d9 4009d4da -> c18328d9 4009d4da 00000000", + "419b5d7e c00971f4 -> 419b5d7e c00971f4 00000000", + "c170cbec 4024c252 -> c170cbec 4024c252 00000000", + "c17119a8 bf3be688 -> c17119a8 bf3be688 00000000", + "4043f410 3fa347c8 -> 4043f410 3fa347c8 00000000", + "c1580830 405b0360 -> c1580830 405b0360 00000000", + "409ea7b8 c074bdf4 -> 409ea7b8 c074bdf4 00000000", + "c18e4610 c0870458 -> c18e4610 c0870458 00000000", + "411db9d0 40449184 -> 411db9d0 40449184 00000000", + "40b61240 406605b0 -> 40b61240 406605b0 00000000", + "c195d19c bdffe7c0 -> c195d19c bdffe7c0 00000000", + "c153ae64 40283fa4 -> c153ae64 40283fa4 00000000", + "41856d78 befd5370 -> 41856d78 befd5370 00000000", + "4164c038 40015bb0 -> 4164c038 40015bb0 00000000", + "c1898bbe 405b99c8 -> c1898bbe 405b99c8 00000000", + "419c313e 3e93cfc0 -> 419c313e 3e93cfc0 00000000", + "4177c668 c020a0d1 -> 4177c668 c020a0d1 00000000", + "419bd8fe bf1e09c0 -> 419bd8fe bf1e09c0 00000000", + "41077350 c061cd54 -> 41077350 c061cd54 00000000", + "c0028b48 bfec1c38 -> c0028b48 bfec1c38 00000000", + "411a14f0 c09b918a -> 411a14f0 c09b918a 00000000", + "c15c607d befa8240 -> c15c607d befa8240 00000000", + "40c795f8 3e100f40 -> 40c795f8 3e100f40 00000000", + "412f29b6 c0800d42 -> 412f29b6 c0800d42 00000000", + "409527f8 3f05a138 -> 409527f8 3f05a138 00000000", + "c0f582e8 c061983c -> c0f582e8 c061983c 00000000", + "c0cd40b0 bfae034e -> c0cd40b0 bfae034e 00000000", + "c11969ab 3f835b18 -> c11969ab 3f835b18 00000000", + "3fec19f0 c03184a7 -> 3fec19f0 c03184a7 00000000", + "c0a899c2 3fddbd04 -> c0a899c2 3fddbd04 00000000", + "4180010c bf49d560 -> 4180010c bf49d560 00000000", + "c0b7c44a 4015633e -> c0b7c44a 4015633e 00000000", + "40b282d4 3faafbb0 -> 40b282d4 3faafbb0 00000000", + "41919e7c bfa016a4 -> 41919e7c bfa016a4 00000000", + "c16c6ff2 3f95eca0 -> c16c6ff2 3f95eca0 00000000", + "c0aaf800 3f7e1c90 -> c0aaf800 3f7e1c90 00000000", + "c0b59754 bfddd188 -> c0b59754 bfddd188 00000000", + "c1740f40 c09a4012 -> c1740f40 c09a4012 00000000", + "c0d73d7e 400ba1e4 -> c0d73d7e 400ba1e4 00000000", + "c17b8b58 bf789818 -> c17b8b58 bf789818 00000000", + "40e257bc 4024dabe -> 40e257bc 4024dabe 00000000", + "c1849c26 3e380360 -> c1849c26 3e380360 00000000", + "3fb2d9c0 4059fab4 -> 3fb2d9c0 4059fab4 00000000", + "c146e5b0 3e6c8100 -> c146e5b0 3e6c8100 00000000", + "419c646c bf93f71c -> 419c646c bf93f71c 00000000", + "417afa54 bde10140 -> 417afa54 bde10140 00000000", + "41327f8e 40307eac -> 41327f8e 40307eac 00000000", + "c05840d0 bfe6eb70 -> c05840d0 bfe6eb70 00000000", + "40c97e20 bf470ef0 -> 40c97e20 bf470ef0 00000000", + "41476e50 c098d96e -> 41476e50 c098d96e 00000000", + "40170fe8 bf8195f2 -> 40170fe8 bf8195f2 00000000", + "41962ab8 409fced8 -> 41962ab8 409fced8 00000000", + "c1491bb2 4005b768 -> c1491bb2 4005b768 00000000", + "4139336c c09bdbca -> 4139336c c09bdbca 00000000", + "c122f268 c024e987 -> c122f268 c024e987 00000000", + "c112b3b5 4051e158 -> c112b3b5 4051e158 00000000", + "c175f94f 4094ec92 -> c175f94f 4094ec92 00000000", + "419ed952 c064d8a6 -> 419ed952 c064d8a6 00000000", + "c0d27a18 40589da4 -> c0d27a18 40589da4 00000000", + "c140f7fb 403f46b6 -> c140f7fb 403f46b6 00000000", + "c00e71e8 bf2e2c58 -> c00e71e8 bf2e2c58 00000000", + "c091a858 401d8b9c -> c091a858 401d8b9c 00000000", + "410ffbfc 40718c08 -> 410ffbfc 40718c08 00000000", + "414d0798 3f52d6a8 -> 414d0798 3f52d6a8 00000000", + "418393b8 403f6678 -> 418393b8 403f6678 00000000", + "c0b27e6a 4083127e -> c0b27e6a 4083127e 00000000", + "418219cc c084ec3e -> 418219cc c084ec3e 00000000", + "415c1600 c00e092a -> 415c1600 c00e092a 00000000", + "41388550 c0846afd -> 41388550 c0846afd 00000000", + "c0d6c82e 4050a3d4 -> c0d6c82e 4050a3d4 00000000", + "c18f9c5c c04f2e1c -> c18f9c5c c04f2e1c 00000000", + "c10e458a 401db262 -> c10e458a 401db262 00000000", + "c17747da c08b7383 -> c17747da c08b7383 00000000", + "c0a21d74 4048a668 -> c0a21d74 4048a668 00000000", + "4132cc08 c011378d -> 4132cc08 c011378d 00000000", + "419ae3aa 404c216c -> 419ae3aa 404c216c 00000000", + "c130cbc2 c09bec34 -> c130cbc2 c09bec34 00000000", + "4047a678 c06eb0f0 -> 4047a678 c06eb0f0 00000000", + "c19bfb67 c0313630 -> c19bfb67 c0313630 00000000", + "c1105e4e bff54f78 -> c1105e4e bff54f78 00000000", + "4178afd0 c03b823c -> 4178afd0 c03b823c 00000000", + "c17edf04 bf096380 -> c17edf04 bf096380 00000000", + "417455ac bfb1b162 -> 417455ac bfb1b162 00000000", + "4186d6e0 c0961069 -> 4186d6e0 c0961069 00000000", + "c1723cda 400bbbce -> c1723cda 400bbbce 00000000", + "c0babbf0 c0206713 -> c0babbf0 c0206713 00000000", + "3e803780 bfa3379a -> 3e803780 bfa3379a 00000000", + "411e7f6c c0146e5d -> 411e7f6c c0146e5d 00000000", + "c15d458c 3fb2f2e8 -> c15d458c 3fb2f2e8 00000000", + "40a9a6ec 3fcdc600 -> 40a9a6ec 3fcdc600 00000000", + "3e26b100 c09dcf69 -> 3e26b100 c09dcf69 00000000", + "4141e9a0 3fb0bdf8 -> 4141e9a0 3fb0bdf8 00000000", + "41499f24 3ed4fae0 -> 41499f24 3ed4fae0 00000000", + "40ede17c 407c0c00 -> 40ede17c 407c0c00 00000000", + "3e87bcc0 400ac6f0 -> 3e87bcc0 400ac6f0 00000000", + "c116125c 3f26ee08 -> c116125c 3f26ee08 00000000", + "41478854 c0985e2d -> 41478854 c0985e2d 00000000", + "418bfa00 405c4984 -> 418bfa00 405c4984 00000000", + "c188278c 40541358 -> c188278c 40541358 00000000", + "41695cb0 402190d8 -> 41695cb0 402190d8 00000000", + "c16494c7 bfc9dbd0 -> c16494c7 bfc9dbd0 00000000", + "40f5c684 c08c77e3 -> 40f5c684 c08c77e3 00000000", + "41861336 40162e66 -> 41861336 40162e66 00000000", + "c0efc09c 4060d00c -> c0efc09c 4060d00c 00000000", + "c11657d4 bff3500a -> c11657d4 bff3500a 00000000", + "41974a4e bfa2e084 -> 41974a4e bfa2e084 00000000", + "4067d968 3f917d60 -> 4067d968 3f917d60 00000000", + "4142bd80 40744ea4 -> 4142bd80 40744ea4 00000000", + "c075af70 beb9f000 -> c075af70 beb9f000 00000000", + "40a80ff8 bfcc935e -> 40a80ff8 bfcc935e 00000000", + "40e06d20 c0972f1e -> 40e06d20 c0972f1e 00000000", + "3f7a1480 bf740f20 -> 3f7a1480 bf740f20 00000000", + "bff18eb0 4038e688 -> bff18eb0 4038e688 00000000", + "c01fbda0 be341ea0 -> c01fbda0 be341ea0 00000000", + "c0263f40 40811af4 -> c0263f40 40811af4 00000000", + "c1441f46 bfe2b1c4 -> c1441f46 bfe2b1c4 00000000", + "41301cc2 407342a0 -> 41301cc2 407342a0 00000000", + "3f377c80 c0587754 -> 3f377c80 c0587754 00000000", + "c0ed0428 401ad968 -> c0ed0428 401ad968 00000000", + "410c3892 c08647a0 -> 410c3892 c08647a0 00000000", + "c12d8054 405620b4 -> c12d8054 405620b4 00000000", + "40a4c7a8 c0421a14 -> 40a4c7a8 c0421a14 00000000" + ] + }, + { + "glsl": "vec3 cultmath_af_from_iv(vec2)", + "tolerance": "ulp-bounded", + "cases": [ + "4104ebdf 413378c5 -> 411c3252 00000000 3fba33f2", + "4197e194 4197e394 -> 4197e294 00000000 3a04bf1d", + "c1337a3e c1337976 -> c13379da 00000000 38de6f48", + "c01cca84 c0158a7c -> c0192a80 00000000 3d680373", + "c1404a76 c13f9ed2 -> c13ff4a4 00000000 3cabbc09", + "c187533f c1875279 -> c18752dc 00000000 3956ea68", + "414b6b37 414b7669 -> 414b70d0 00000000 3ab4b6ed", + "408a824b 408a85f5 -> 408a8420 00000000 396ed430", + "c10d3886 c0ed17bd -> c101e232 00000000 3f3565c9", + "c056b53c c0564be4 -> c0568090 00000000 3b52e5ad", + "c176f478 c1741d08 -> c17588c0 00000000 3db5e3b8", + "406e6178 40829b34 -> 4079cbf0 00000000 3e36a885", + "c1820399 c180b57f -> c1815c8c 00000000 3da71520", + "4024b8c4 4031ccec -> 402b42d8 00000000 3dd143e4", + "411c1508 411cfdd4 -> 411c896e 00000000 3ce8dfa0", + "c182c9d8 c16bff08 -> c178c95c 00000000 3f4ca646", + "c08195f2 c081724e -> c0818420 00000000 3b0ed0cb", + "40cc912d 40ce966b -> 40cd93cc 00000000 3d0155f5", + "41491e26 4149249a -> 41492160 00000000 3a51a492", + "c117d836 c117b7ea -> c117c810 00000000 3b817bec", + "bfc13aab bfb6bcd5 -> bfbbfbc0 00000000 3d27dee2", + "c17167ea c1716092 -> c171643e 00000000 3a6ec5a0", + "4141d33c 4141fd74 -> 4141e858 00000000 3ba940ff", + "c0c7d0d8 c0c7cfb8 -> c0c7d048 00000000 389c7d0e", + "c16b1d21 c169b42d -> c16a68a7 00000000 3d3488b2", + "41353eca 4135671a -> 413552f2 00000000 3ba19ab4", + "c0510e5d c050eae3 -> c050fca0 00000000 3a8e5087", + "c18648f9 c186485d -> c18648ab 00000000 392cc91f", + "c1031961 c10310cd -> c1031517 00000000 3a8a4633", + "c17f1a84 c17ddffe -> c17e7d41 00000000 3d1d52f2", + "bf562c5e bd73e218 -> bee56a80 00000000 3ec6ee57", + "4168fcdb 416bce1d -> 416a657c 00000000 3db457de", + "c05e4d83 c05e349d -> c05e4110 00000000 3a480e4e", + "410a51af 410ac559 -> 410a8b84 00000000 3c6776b1", + "40a23995 40a23b03 -> 40a23a4c 00000000 38c123b0", + "c0ebc3c1 c0ebc22b -> c0ebc2f6 00000000 38d9bc3c", + "402ca56f 402ca7f1 -> 402ca6b0 00000000 38a5e540", + "417e925a 417f511e -> 417ef1bc 00000000 3cbee3ea", + "3f8a3515 3f8ae6cb -> 3f8a8df0 00000000 3b31c75d", + "41813a8b 418390d9 -> 418265b2 00000000 3e15979d", + "c10148fd c1007d15 -> c100e309 00000000 3ccbf829", + "3f91114d 3f9124f3 -> 3f911b20 00000000 399dc125", + "4128571f 412e3175 -> 412b444a 00000000 3e3b4d79", + "c046afa3 c046a82d -> c046abe8 00000000 3971dabf", + "c1223e78 c1212826 -> c121b34f 00000000 3d0b3324", + "c116edf2 c116075a -> c1167aa6 00000000 3ce6aade", + "c16a61a6 c16615f4 -> c1683bcd 00000000 3e0979ea", + "4177d404 417c1844 -> 4179f624 00000000 3e088bf0", + "410a5c85 410abc2f -> 410a8c5a 00000000 3c3f76af", + "41577b2e 4158921a -> 415806a4 00000000 3d0b8389", + "40d0928a 4105fc53 -> 40ee4598 00000000 3f6d98f6", + "c07e1b42 c07e16de -> c07e1910 00000000 3910786d", + "c1148d0b c10dbea5 -> c11125d8 00000000 3e59cf12", + "4153e3da 415404f6 -> 4153f468 00000000 3b84da02", + "c0a8de2f c0a72281 -> c0a80058 00000000 3cdde18e", + "bd972560 40ae28ee -> 402bcc58 00000000 4030859a", + "c0b7a179 c0b7465b -> c0b773ea 00000000 3bb669e8", + "c13a76d0 c0c240e9 -> c10dcba2 00000000 4032ace6", + "c123050d c11d4c6d -> c12028bd 00000000 3e37168c", + "4171a43b 4176bb75 -> 41742fd8 00000000 3e22eb1b", + "4148438c 41499b3c -> 4148ef64 00000000 3d2be49a", + "c12497a5 c12496d3 -> c124973c 00000000 38e692f5", + "412fe85d 413ab07f -> 41354c6e 00000000 3eac8395", + "4178971c 4190c71e -> 41868956 00000000 3fa3dd11", + "c162637e c1625eba -> c162611c 00000000 3a1c098e", + "414af8b3 414af975 -> 414af914 00000000 38db5f2f", + "3ebd7340 3ec00ac0 -> 3ebebf00 00000000 3b25e600", + "c19dd647 c19db3f1 -> c19dc51c 00000000 3c09a6eb", + "c167616a c1671e66 -> c1673fe8 00000000 3c0641d8", + "4174b643 4174b6ed -> 4174b698 00000000 38c896de", + "417423d8 41742480 -> 4174242c 00000000 38c68490", + "4125c05f 412ad8b9 -> 41284c8c 00000000 3e230deb", + "419fc511 419fcce7 -> 419fc8fc 00000000 3afd3f34", + "c147b2fc c140b936 -> c1443619 00000000 3e5f3bdf", + "c0c7e582 bf954638 -> c06d3710 00000000 4022940d", + "c0254660 c02544a0 -> c0254580 00000000 386a5466", + "406bdb29 406be317 -> 406bdf20 00000000 3980b7c6", + "4121b3e4 4121b534 -> 4121b48c 00000000 39321b53", + "408ddea6 408e07ea -> 408df348 00000000 3b255704", + "c0fd9af4 c0fc8090 -> c0fd0dc2 00000000 3c8d41da", + "4116ac26 417026b2 -> 4143696c 00000000 4032f554", + "c1297450 c12971dc -> c1297316 00000000 39a24ba2", + "4112a44e 41133c02 -> 4112f028 00000000 3c97c668", + "c01c1fa4 c01c1dac -> c01c1ea8 00000000 3882e0fd", + "c094a2c1 c04b175d -> c07a2e70 00000000 3f3c5c94", + "413e2f18 414f8668 -> 4146dac0 00000000 3f0abb50", + "41457ba4 415ec09c -> 41521e20 00000000 3f4a289f", + "414654b3 41521145 -> 414c32fc 00000000 3ebbcac4", + "414d894b 414e5be5 -> 414df298 00000000 3cd2b3cb", + "c18c9306 c18c66de -> c18c7cf2 00000000 3c30e64a", + "40390b34 40391a3c -> 403912b8 00000000 39f1f234", + "bf838ad0 bf836810 -> bf837970 00000000 3a0b41c5", + "41289fae 413adac2 -> 4131bd38 00000000 3f11d95b", + "bfc6e47c bfc5f644 -> bfc66d60 00000000 3b6e50dd", + "3febc758 40a6d9da -> 4061cbb0 00000000 3fd7d032", + "414dea2c 415d97ac -> 4155c0ec 00000000 3efad9bb", + "c1662aa0 c1660bf4 -> c1661b4a 00000000 3b76462b", + "c14d2d19 c1041555 -> c128a137 00000000 40122fbb", + "4142bd5c 41447b44 -> 41439c50 00000000 3d5f0048", + "41969e37 4196d515 -> 4196b9a6 00000000 3c5bc36b", + "41719d23 4171a04d -> 41719eb8 00000000 39d20d02", + "c0e06397 c0de05d5 -> c0df34b6 00000000 3d177783", + "c16d9b9c c16b6344 -> c16c7f70 00000000 3d8e1d6d", + "c1530702 c149c5fa -> c14e667e 00000000 3e941226", + "c133c0db c0551893 -> c0e90700 00000000 407cf599", + "c10d0f63 c10d0e7b -> c10d0eef 00000000 38f9a1ec", + "c11930e9 c1011801 -> c10d2475 00000000 3f40c7d9", + "418acccc 418bba1c -> 418b4374 00000000 3d6d6177", + "bf7dfc63 bf7df31d -> bf7df7c0 00000000 38965bf9", + "c0dac064 c03bc918 -> c09c5278 00000000 3ff9b7e7", + "c18f34e5 c18f3473 -> c18f34ac 00000000 3903e69d", + "c03a4ab9 c03857f7 -> c0395158 00000000 3c796ca5", + "419fdc52 419fdc8e -> 419fdc70 00000000 389ff724", + "40d4ddad 40d5753b -> 40d52974 00000000 3c17a8af", + "4030e6b6 40afb00d -> 408411b4 00000000 3fae7990", + "419593b9 41959623 -> 419594ee 00000000 3a1f2cb1", + "4033df13 4038b0fd -> 40364808 00000000 3d1a4023", + "40640da1 4064a7bf -> 40645ab0 00000000 3b9a3a95", + "41515e4a 418223d7 -> 416ad2fc 00000000 3fcba612", + "c1881f10 c1881dec -> c1881e7e 00000000 399a81f1", + "40a5b67e 40b0e952 -> 40ab4fe8 00000000 3e332ea2", + "c18f0c17 c18e29eb -> c18e9b01 00000000 3d623de2", + "409686be 40abfd7a -> 40a1421c 00000000 3eabb68c", + "3f3fc0a1 3f409bdf -> 3f402e40 00000000 3adb5613", + "c01ea3fb c0163b25 -> c01a6f90 00000000 3d868e9d", + "c006d9be bfad2743 -> bfdd6d60 00000000 3ec118b5", + "c12d1e94 c12d1dd4 -> c12d1e34 00000000 38d5a3d2", + "c04f7580 c006f7d0 -> c02b36a8 00000000 3f10fb94", + "4114d64d 412b2903 -> 411fffa8 00000000 3f32965b", + "403dfb10 403e0170 -> 403dfe40 00000000 394ef806", + "4192250c 41957b68 -> 4193d03a 00000000 3e559bac", + "c16095e9 c1605b7f -> c16078b4 00000000 3bea184b", + "c19c1df6 c19c1d86 -> c19c1dbe 00000000 390383bf", + "be8891c0 3d849902 -> bdced6ff 00000000 3e29b811", + "c122ba2d c121ebad -> c12252ed 00000000 3cce9457", + "411cc4d2 411df6c6 -> 411d5dcc 00000000 3d1903df", + "4191c1bc 4191c494 -> 4191c328 00000000 3a3a8e25", + "c10a03f4 c109ee8a -> c109f93f 00000000 3b2bda04", + "c1a1064b c19711db -> c19c0c13 00000000 3f1f4842", + "416d3e2e 416d4442 -> 416d4138 00000000 3a463511", + "be827828 be7e5ab1 -> be80d2c0 00000000 3b52b7f4", + "c1bb2c17 c17809d6 -> c19b9881 00000000 407c9d0e", + "c032503f c0324151 -> c03248c8 00000000 39f044a0", + "419b0415 419b542b -> 419b2c20 00000000 3ca052d5", + "c16b31f2 c15eb87e -> c164f538 00000000 3ec79916", + "40b932c6 40b9340a -> 40b93368 00000000 38ad9341", + "c11fb5df c07a61ec -> c0de4e5a 00000000 40423af0", + "c1646837 c12bc8a5 -> c148186e 00000000 3fe27eba", + "4014398f 401c9d81 -> 40186b88 00000000 3d864059", + "4074f671 407909cf -> 40770020 00000000 3d026fa4", + "c11e0eb3 c10b1cbd -> c11495b8 00000000 3f17904e", + "c199cd1a c180bf00 -> c18d460d 00000000 3fc8716a", + "415d3a58 41833304 -> 4171d030 00000000 3fa4af43", + "414ccc73 414cccdd -> 414ccca8 00000000 3883999c", + "412bbd47 412bbe6d -> 412bbdda 00000000 391dbbe7", + "411ade0b 41632c01 -> 413f0506 00000000 40109c25", + "41929bf3 4192a635 -> 4192a114 00000000 3b25454c", + "409d7acc 40a0b094 -> 409f15b0 00000000 3d4d7706", + "410695f6 410698da -> 41069768 00000000 39bd34c7", + "c1a4e4d7 c19acc9b -> c19fd8b9 00000000 3f21850a", + "c1a5bc8f c14dc94a -> c186509a 00000000 407b5ffb", + "401d9b65 401f01cb -> 401e4e98 00000000 3c333cf0", + "417a8e91 41800c94 -> 417d53dc 00000000 3e3156e0", + "c17008f6 c0a430df -> c12110b3 00000000 409df0a4", + "c192d411 c192c4c5 -> c192cc6b 00000000 3b75e5a8", + "4135c796 4135eb9a -> 4135d998 00000000 3b906af6", + "c131a02c c13191a8 -> c13198ea 00000000 3ae9a340", + "419442a1 41963f5f -> 41954100 00000000 3dfe6864", + "c0fc01d4 c0fbfec4 -> c0fc004c 00000000 394be00f", + "c17f663c c17e8474 -> c17ef558 00000000 3ce1e7ed", + "c114836d bf768210 -> c0a3eb8e 00000000 40851b5f", + "41200f2e 41200fee -> 41200f8e 00000000 38d401fe", + "41981756 419b0d6e -> 41999262 00000000 3e3d8ad8", + "c1a8a951 c1489276 -> c1867946 00000000 4088c056", + "c0d7f925 c0d5372f -> c0d6982a 00000000 3d308440", + "c1406810 c14043aa -> c14055dd 00000000 3b91f834", + "409150d7 40a9ab79 -> 409d7e28 00000000 3ec2d5ba", + "bf47a777 bf46f149 -> bf474c60 00000000 3ab646f5", + "c11ee6d3 c11d7787 -> c11e2f2d 00000000 3d37afee", + "c19eef5e c19eedec -> c19eeea5 00000000 39c2eef6", + "4096ce09 40bf9d97 -> 40ab35d0 00000000 3f233e98", + "41930c6e 41931f7a -> 419315f4 00000000 3b98f31f", + "c0f9653c c0f909b4 -> c0f93778 00000000 3bb74e59", + "41040c46 4106d69e -> 41057172 00000000 3db29a37", + "c0f70d58 c0195af9 -> c0a1dd6a 00000000 402a5ffb", + "c102960b c101103d -> c101d324 00000000 3d42ef29", + "c18a41c9 c18a1c6d -> c18a2f1b 00000000 3c15b521", + "c00474db c0046fe5 -> c0047260 00000000 3920d1d3", + "c12dc27f c12d32c9 -> c12d7aa4 00000000 3c8fcbb8", + "bef4649c bef3f9e4 -> bef42f40 00000000 39d5ad19", + "40a3f920 40a409d8 -> 40a4017c 00000000 3a86640a", + "40d4dad7 40d573b1 -> 40d52744 00000000 3c18f4ae", + "c14ff822 c14fd1ba -> c14fe4ee 00000000 3b9a07fc", + "4129f846 412a3e02 -> 412a1b24 00000000 3c0ba290", + "4150d776 4154c07a -> 4152cbf8 00000000 3dfa47a6", + "c048a23e c0313cd2 -> c03cef88 00000000 3e3b2c29", + "c18e9d3c c1545a17 -> c178ca48 00000000 4011c109", + "41846a38 4184de78 -> 4184a458 00000000 3ce8a138", + "c1983990 c1982d2a -> c198335d 00000000 3b479073", + "411dd735 411efa1b -> 411e68a8 00000000 3d117cf0", + "c19e66da c19e496e -> c19e5824 00000000 3bebfe67", + "40904532 40908bce -> 40906880 00000000 3b8d5c23", + "c10747e9 c1059c19 -> c1067201 00000000 3d55f074", + "c0e469a3 c0e463c9 -> c0e466b6 00000000 39bed1a7", + "c0dc442f c0dbffd1 -> c0dc2200 00000000 3b88f311", + "c18ef03d c18abfc3 -> c18cd800 00000000 3e86117c", + "c18451e6 c1743803 -> c17e6de8 00000000 3f235f51", + "3fd593aa 3fd68076 -> 3fd60a10 00000000 3b6ce6d0", + "c15d537c c15cc98c -> c15d0e84 00000000 3c8a0baa", + "c150d04f c1177491 -> c1342270 00000000 3fe56f60", + "40b816f0 40b827b8 -> 40b81f54 00000000 3a86f828", + "4149b509 414e88f7 -> 414c1f00 00000000 3e1a80fa", + "3f38657b 3f4dd385 -> 3f431c80 00000000 3d2b711e", + "bf1f7bc5 bf1f5c3b -> bf1f6c00 00000000 397cef7c", + "bcdd8000 4042cd20 -> 3fc11220 00000000 3fc48838", + "c192388d c17f8e35 -> c188ffd4 00000000 3f938c26", + "410850a0 41466b80 -> 41275e10 00000000 3ff86be3", + "c14ebf01 c14ebb0f -> c14ebd08 00000000 3a017afc", + "40bee0cf 40bef541 -> 40beeb08 00000000 3aa44ef5", + "c1577504 c1564e60 -> c156e1b2 00000000 3d135f77", + "c189115f c188501b -> c188b0bd 00000000 3d415522", + "c1747cba c17437e6 -> c1745a50 00000000 3c09e51f", + "409297c4 409adacc -> 4096b948 00000000 3e0431b6", + "408ceebe 408d05da -> 408cfa4c 00000000 3ab96d06", + "c08dc506 c08c1d62 -> c08cf134 00000000 3cd3dadc", + "c08579c6 c08437ba -> c084d8c0 00000000 3ca10e58", + "c173f891 c15b94b9 -> c167c6a5 00000000 3f431fb4", + "41866291 418a87b7 -> 41887524 00000000 3e84a6ea", + "40252358 40cd7c44 -> 409006f8 00000000 3ff5d563", + "4168db49 416e3d37 -> 416b8c40 00000000 3e2c4179", + "c1739fd2 c1739c5e -> c1739e18 00000000 39e49cff", + "40aa2983 40aa9dfd -> 40aa63c0 00000000 3be91ea7", + "c0f02106 c0f019f2 -> c0f01d7c 00000000 39e64084", + "419a5528 419a56e0 -> 419a5604 00000000 39e5a56e", + "40b88455 40b8afe3 -> 40b89a1c 00000000 3b2e9458", + "40f6f313 41217dba -> 410e7ba2 00000000 3f981113", + "c0cdfe51 c0c87c83 -> c0cb3d6a 00000000 3db03cf8", + "41585a38 41585af0 -> 41585a94 00000000 38d30b5e", + "c163d816 c163b9c6 -> c163c8ee 00000000 3b7363d8", + "c18d15d9 c180e221 -> c186fbfd 00000000 3f433c9a", + "4197cf46 4197d132 -> 4197d03c 00000000 39ff7d13", + "40a49c60 41695544 -> 411dd1ba 00000000 40970731", + "3ca6fad8 3da0eb4a -> 3d4aaa00 00000000 3cee5950", + "40e906b6 4137e6d1 -> 41163516 00000000 4006c71a", + "419d2dfb 419d4c4d -> 419d3d24 00000000 3bf32d4c", + "4048bf51 406616ef -> 40576b20 00000000 3e6abdd6", + "c0637be9 c0623937 -> c062da90 00000000 3c216738", + "c052ad0c bf0b8bd2 -> bff59000 00000000 3fafca32", + "4192b1be 4194e366 -> 4193ca92 00000000 3e0c6ea7", + "c16b33f9 c16af9d3 -> c16b16e6 00000000 3be90d9a", + "40ffe513 41002977 -> 41000e00 00000000 3bdbf615", + "40bab098 40bab1f8 -> 40bab148 00000000 38bbab20", + "414c8967 414d2e71 -> 414cdbec 00000000 3ca523a6", + "c14e81cd c14e0d6f -> c14e479e 00000000 3c68efa0", + "4195c5e3 41a94f19 -> 419f8a7e 00000000 3f9c4a59", + "407ce2c7 407cfc59 -> 407cef90 00000000 3a4d8cfc" + ] + }, + { + "glsl": "vec2 cultmath_af_range(vec3)", + "tolerance": "ulp-bounded", + "cases": [ + "4158c790 c095f5c1 3ee73bea -> 410692bd 41957e31", + "4187ff2c bf1d78e0 3e1ed5f4 -> 4181d5b0 418e28a8", + "41896304 40260a98 3c962126 -> 4168f83e 419e49e9", + "c18da13d c0444cc2 3a934ca7 -> c1a62d2c c16a2a9a", + "41270838 409ceada 3caabb56 -> 40b07abc 4175d311", + "4174b35c c043205a 40b48d39 -> 40d34922 41bfe114", + "c184263f bedeef60 3ad295eb -> c187a54f c180a72f", + "40bd19e0 c068963f 3d63848c -> 400e0f49 4119960e", + "3ff85750 40012d54 3a4457c0 -> bda1c01b 407d6551", + "4199a976 bf9fa36e 3ad9881f -> 418fabcf 41a3a71d", + "41886cc2 c077275a 3fc02270 -> 413b0b4a 41b353df", + "c0c5c820 407bde5c 3c022872 -> c121fc3b c00f2f94", + "4111de08 3e8a7d30 3c907749 -> 410d41da 41167a36", + "c0e1a408 40254ef8 3bf0c6fe -> c11a43e5 c08ec047", + "4198e74c 3fb6a0f4 3abde733 -> 418d7a3b 41a4545d", + "c02491a0 c00a63d0 3fb43bf6 -> c0c489c2 3f7fc10e", + "c18dad62 c002fafa 3b44f6c9 -> c19e12f3 c17a8fa2", + "c1908d49 4039ee20 3a06fa86 -> c1a7cc25 c1729cd9", + "419f31a8 c045c560 403e9b9e -> 415d4af6 41cfbdd5", + "c14b80f8 bf9e1920 3ef6e4fb -> c166fb52 c130069e", + "bfbc3d10 408fb98e 3c0885df -> c0bf0d21 4041dd32", + "c13a7ce8 c06c1bff 3982e0b5 -> c17584fd c0fee9a6", + "c0a02e28 bf584ee0 399fd698 -> c0bb3a8f c08521c1", + "c185312e c05d2aac 3e6032b9 -> c1a296f3 c14f96d2", + "3d265400 40288586 3e9e8577 -> c039bcf1 403eef91", + "416b7d08 3fd97cb0 3cf17f31 -> 414fd4a2 418392b7", + "c0ea9704 c077d0ae 3c49bcd1 -> c1337228 c05c9370", + "c0c62ed8 408756d8 39839a6c -> c126c3e9 bffb5775", + "418f7eaa 4071f418 3b87836f -> 41626f48 41adc5b0", + "40f6f4a4 bfc28ebc 3ea6c151 -> 40bbe4cd 4119023e", + "bf175de0 3eeaf800 3a145f69 -> bf867f84 be06f2dd", + "410814ca 4066b5a0 3f8e3a1d -> 40728045 41538983", + "419c811a 402c4b84 3d14085f -> 4186ad9a 41b2549a", + "c15e3e39 c08c86aa 400c7037 -> c1a3ced8 c0e9bd83", + "41150ede 409783cc 3a04c6be -> 409295ae 4160d2e5", + "40a33820 3ef1f8e0 390f6b11 -> 40941768 40b258d8", + "c00ee540 c06d19bb 3d251680 -> c0bf49b6 3fc191da", + "c07dba28 4069ca88 3b087282 -> c0f3d375 be9e6b24", + "c0e8395e c0747036 390dc884 -> c1313955 c05c0023", + "c0fa99ea c0197a3c 3af15922 -> c123b319 c0adcda2", + "4123f010 3fe8dd38 39641810 -> 4106d379 41410ca7", + "c08758ca 3f7c4e10 3de1f207 -> c0aa6a5f c0488e6b", + "3f333c80 4038918a 40511789 -> c0ae6d08 40db3c28", + "4178f744 40727ad4 404c0672 -> 410956db 41b44bd6", + "419a5668 c0570d49 393ea966 -> 417ee8a8 41b5387b", + "3fff09e0 3f8fb36c 40b2d0e2 -> c096fb56 410b4023", + "418c7cc8 bfe53d76 3b93b5d3 -> 417c3f57 419ad9e5", + "c1027e12 c020cc5f 3aa221a2 -> c12ab646 c0b48bbd", + "c14bd88c c08503aa 3e817d9a -> c1893330 c1054ab9", + "41779818 bf879d12 3f03ed2c -> 415e6592 4188654f", + "4135166c 3fec9e50 3d0fb33f -> 4116f2e2 415339f6", + "c11a8370 bf845c98 3c008d09 -> c12b2f31 c109d7af", + "41407c40 bfce8978 3bf58502 -> 41268c52 415a6c2e", + "3fa5cb10 c07997ba 3f1f70cb -> c04e8e7b 40ba2cc6", + "c1706750 3f0d3db0 3aead95f -> c1794292 c1678c0e", + "409a12f0 bf506d68 3ea46203 -> 406b7e2e 40be66c9", + "c1387889 402c0214 3d0a782f -> c1640394 c10ced7e", + "c1877054 bfac8d08 409f10ce -> c1b9fd64 c129c689", + "41871d76 becc8ef0 3c700a0c -> 4183cd30 418a6dbc", + "415f614c c037d3d9 3d1893f7 -> 4130d3b1 4186f773", + "4188f0ee 3fa73760 3c197f6c -> 417cd47e 4193779d", + "c15955ec c07c1f27 3a74b181 -> c18c30cd c11a4a3e", + "c08aa5dc 400f3040 3f2fddf9 -> c0e839ca bfb447ba", + "40dd7130 401f64fc 3b18f772 -> 408dab80 41169b6f", + "403ec808 bf6eff48 3a949689 -> 4002f593 407a9a7d", + "40381db8 c05216c1 3e965fab -> bf33145f 40ce8044", + "4195b374 3f1c77f8 3e5a4a7d -> 418f1b15 419c4bd3", + "41995e52 3fe02144 3e09020d -> 418a4a2f 41a87275", + "c1289040 4003f714 3b81f288 -> c1499e50 c1078230", + "40328500 bfd17cbc 409a204c -> c06a7a1b 4113e107", + "418ea48c 40908d7a 39df5331 -> 41550086 41b2c8d5", + "4166d93c 3f85af0c 3ccc4b93 -> 4155bd26 4177f552", + "c0ad6508 4094774e 3c90320e -> c121364e bf42eb9f", + "c18de01b c02da39c 3e9896fe -> c1a5f6f4 c16b9282", + "418adb68 3ec676f0 390390f6 -> 4187c141 418df58f", + "c12b685c 3cb6ac00 40c6920a -> c1878664 c08f87e0", + "41122b80 bfbf9904 3b352617 -> 40f45a05 412a29fe", + "411956c4 4073fa60 3e3995f6 -> 40b2e38d 41593bc2", + "415f06d4 408f929c 40bfcad9 -> 405d6006 41c35ad3", + "412c30b2 40879d2a 3c239595 -> 40d07251 4170283b", + "416f95fc 404e0bd4 3c0db4d6 -> 413bef88 41919e38", + "4153bd70 40116f80 3f783dbb -> 411fdda4 4183ce9e", + "c16170ac 4008d8c6 3f834917 -> c18a0809 c12ed147", + "41142c6a c03b460e 3a8f5a8b -> 40caacc0 41430274", + "c1238a5c 40839390 3b96865c -> c1656703 c0c35b69", + "4196afce c0917d17 39fbebc0 -> 41649f01 41bb101c", + "4087248c c09e93f2 3a8931eb -> bf3bc05b 4112e091", + "c0b1c740 405a75c0 4108382c -> c18bdca7 40cbe41b", + "40bdbd5c c03c0dae 396c220a -> 403f6936 410de30f", + "bf83f860 c09c16cc 3e429dbc -> c0c329de 40812dae", + "3f741880 3f747320 39267ffe -> baca4d13 3ff44b13", + "c1903d9a c04b5632 3c31103a -> c1a9be8d c16d794f", + "c0f6746e c07e069e 3f922728 -> c14d00d1 c025ce77", + "c131ace9 c084fbd8 3f60036d -> c181158e c0c25d6c", + "c15bef5c 409eff8c 3a07e3c7 -> c195b8aa c10c6d63", + "c1863adf bd47b800 40e2a536 -> c1bf4814 c11a5b53", + "c0e766e0 4071a738 40524366 -> c164ae26 be2e2e8d", + "414c494c c0576194 407ce334 -> 40ae700c 41a0ad49", + "c1429ff6 3e0799c0 40fff4fe -> c1a25c78 c0810df7", + "c07c7e18 c04b213e 3d251931 -> c0e519ec bf3b2161", + "41112970 3fde1cc4 3ea4ebe1 -> 40e07cdb 41321473", + "c16249a4 c03b151c 3deadcc6 -> c189725b c131ae93", + "c154b9a5 c0430f28 3ee1cc33 -> c18645f0 c11ce769", + "41883884 3fd133a8 40a49d36 -> 4123fbe0 41be7318", + "c0e03b36 bf9581a0 407ed79e -> c14283c2 bfedbb9b", + "c0582b50 c03cf444 3fd5970f -> c0fff59e 3f9f2938", + "415b9498 3f2d51b0 3eb7db9d -> 414b0091 416c289f", + "c17fb2ac c05ccb9c 39f7099a -> c19b73ca c1487dc4", + "412192e6 4002f838 3c6ce0ff -> 41009994 41428c38", + "c172adf0 406224c4 3e62bc7c -> c1976113 c13699ba", + "c03d1600 4081f2a8 408017c7 -> c1304ac3 40a37f86", + "c1061dd2 bd359600 3c39bf4d -> c10701e0 c10539c4", + "c18438e7 c09cd2dc 3f62080a -> c1b27de9 c12be7ca", + "3f9ee310 408421f2 3b4f03a6 -> c0390631 40abf4a1", + "c125aa12 bf8848f0 3a9a4084 -> c136b80d c1149c17", + "4193897c bd664600 3f4bfa65 -> 418cb67c 419a5c7c", + "419fc934 40336adc 3b648142 -> 418954a9 41b63dbf", + "c168e099 3f152f88 3df324f2 -> c17419ea c15da748", + "41466c00 3fa1b838 3d204e57 -> 4131949d 415b4363", + "402b6cb0 bf87b914 3d062565 -> 3fcaef03 407161df", + "c14ce136 c09105da 3ba7cde2 -> c18abc97 c104493f", + "c18cd970 bb92f000 3f1cacbf -> c191c80e c187ead2", + "c1990718 c094ffe5 402f8633 -> c1d437e5 c13bac96", + "3ebabbc0 bfdb320c 3a770139 -> bfaca20d 4004fff6", + "c18e4ec4 bff08f98 3e406a95 -> c19ed89d c17b89d7", + "c0637c58 4065f764 3f244cf6 -> c0f9438d 3f2e39a5", + "41868ee8 403a7ff0 3a93826f -> 415e7924 419de13e", + "40c23e44 c072405a 397305fd -> 4012383b 411db036", + "c0b27b58 c00664e7 3bb3aee0 -> c0f5dac6 c05e37d3", + "c0f8df6c be489d00 400e00af -> c1231260 c0ab9a18", + "417b1ff4 3e4b3040 3df61bf8 -> 417606eb 41801c7e", + "c108de78 bfbe84ce 4009011f -> c142ef66 c09d9b15", + "4192921a c04bf732 403cd329 -> 4142f185 41c3ab72", + "bfdd3910 408c9094 3ca37487 -> c0c48259 402bcba3", + "417b49b4 bdc86e80 3c945c85 -> 41796e99 417d24cf", + "c1774395 c0862188 3adf2414 -> c19d2db3 c1342bc4", + "41933ce8 bf8169d6 3f2f1fa4 -> 4185ad44 41a0cc8c", + "41248120 3fbb7ab8 3dd2ed63 -> 410b6be2 413d965e", + "c1049ea9 3e8ffd40 40877170 -> c14cd758 c07197e9", + "c113edd2 3fff1598 3a924ae0 -> c133d522 c0e80d03", + "c06793a8 400a6188 3d71199e -> c0badcd7 bfb2db45", + "c19a1c02 3f956bd8 3f8dd432 -> c1ac500e c187e7f6", + "4049ea90 402c40f0 40cc0526 -> c0bd306e 41438d7f", + "4138c0ea bef01a50 3e7b6a9f -> 412d5261 41442f73", + "416b33dc beb9d860 3fe94ce6 -> 41483b6b 41871626", + "41202aa4 4025d570 3e12c5c1 -> 40e8d449 414beb24", + "409b6c08 c021fa17 3a91a9fe -> 4014cba6 40ec723d", + "c17efa3f bfc5d30c 3986026d -> c18bdadf c1663ec1", + "c17e9728 c069612e 4014f431 -> c1af164b c11f01ba", + "c19bb65a 409de01a 3fa6df13 -> c1cd9c5f c153a0aa", + "418d0d7c 3ff2240c 3cd29ced -> 417b6d14 419c646e", + "40ec1328 c090e11d 3aaae573 -> 40364e8a 413e7f86", + "4181ee56 406f81b8 401521b0 -> 4122b3bc 41b282ce", + "40b69fdc 3eb20e40 3b6d1c67 -> 40ab6148 40c1de70", + "c10966ed c052471b 40ab6fd8 -> c189d859 3d62d73b", + "c18384f2 40794d48 3a56f771 -> c1a2b053 c148b322", + "412044bc 3ed9b5b0 3f332ce7 -> 410e4435 41324543", + "c103ce75 3fbcb858 3c95b664 -> c11bb065 c0d7d90b", + "c19fe15e bff5d554 3bea0c03 -> c1af4d5f c190755d", + "bf8d08a0 40852f92 3cd1187b -> c0a942dd 40457d19", + "418b5f28 3f279d58 3f41b7a7 -> 41801477 4196a9d9", + "c167c357 405ea5f4 3f290188 -> c194fe7f c12589af", + "c1428f69 c04ad5ba 3c460091 -> c1757667 c10fa86b", + "40a53e58 405c4df8 402a1022 -> bf6f865c 413436bd", + "4124fc7a 4099330c 3eba2f25 -> 40a522d7 41776788", + "417e6228 404a6cb0 3a425972 -> 414bc3df 41988039", + "c1692344 c09bc6a2 3ade1b3c -> c19b86cd c11b38ef", + "40f3fa78 c017b16f 3c931741 -> 40a78e95 4120332e", + "c18eb623 3fbbd2d4 39298a7b -> c19a73af c182f897", + "c1968372 3feb98a0 3efe9351 -> c1a93754 c183cf90", + "41617490 bf2a0908 40c55d29 -> 40e84aac 41a761e4", + "c08a20f4 4004919e 40e0bacb -> c1569254 4098e2c1", + "3fb7da20 c061a396 3b87e612 -> c005fa8d 409eea56", + "c14b9c51 407b8584 3eae6aad -> c187f88c c1074789", + "412a713e c0725a72 3e9ec32d -> 40d1c8f3 416bfe03", + "4175d680 3eb9a880 3e61cfe4 -> 416c81ec 417f2b14", + "4119c0c6 c07e24e6 3ceb5259 -> 40b383ab 4159bfb7", + "c1421287 bfd0fba0 405182d7 -> c1884961 c0e72498", + "c1293a70 c0233eee 3b55483a -> c152178d c1005d53", + "c0e01180 c0852b89 3a84d1f0 -> c132a2b6 c035bb27", + "410e34d6 c0558885 3d000f77 -> 40b0a531 41441713", + "40e2bd78 40312a9c 3c393422 -> 4089cb7c 411dd7ba", + "41889892 bfabe616 3b1575e9 -> 417baaf8 41935ba8", + "411b69ac bfb13234 3e623a5e -> 4101ba71 413518e7", + "41604aa4 40913c88 39d132df -> 4117aaab 4194754e", + "3fc97460 4087c5fe 3a05e9bd -> c02ada41 40ba2751", + "4121a0c0 c05342c7 39bb0bd2 -> 40d99d15 415672f5", + "c0a28074 c006c7a8 3da4dd6f -> c0e877cd c0391238", + "c1218037 3fc70e30 399fe2b4 -> c13a6349 c1089d25", + "c18e64ea c05bd60a 3ca7aa28 -> c1aa09a1 c1658067", + "40d5835c c038fbfc 395a0f88 -> 4072072e 41190191", + "c171a99b bece8940 3edc0b79 -> c17efe51 c16454e5", + "c13b3af6 c08376be 3bacf66c -> c17d0c04 c0f2d3d0", + "c107e77d 409252b0 3a3ac740 -> c15113cd c07aecb4", + "c083bccc 409f2b8c 3f024a40 -> c11998da 3faee06d", + "414e9864 bfc4ba58 3c06831a -> 4135df6a 4167515e", + "4134ff32 3ffb2e0c 3b836942 -> 411588f6 4154756e", + "c159103e 3f395a00 406d9d54 -> c19006a3 c1121337", + "c182fbe8 bfc115e0 401b35bb -> c1a27407 c1470791", + "4185b660 bf348438 39d3e559 -> 41801161 418b5b5f", + "c0ee56a8 406dfaf8 3a91b6ab -> c132aeab c06e9ff4", + "414603f0 c087ac00 3a147139 -> 41022b8d 4184ee29", + "4086b990 4011f440 3cf25417 -> 3ff3343c 40d0a611", + "c08841a8 bfd72d00 3fbfc672 -> c0edfe93 bf8a12f3", + "413e2e10 beb3be20 3a979d0d -> 41388b56 4143d0ca", + "418cc61e 40984f16 3b3d01eb -> 414d58cb 41b2dfd7", + "41597d00 bf406268 3b6669f1 -> 414d6865 4165919b", + "4108f26c c011c906 3ca3f6d8 -> 40c85c48 412db6b4", + "c165648a 3d747b00 3e1f9f77 -> c168d792 c161f182", + "c160383c c095d8e6 39bb1e17 -> c195931b c1154a40", + "415fcee8 c006a7b7 3e4b9617 -> 413af692 4182539f", + "c1312892 4097a1ec 398c2ab6 -> c17cfab0 c0caace7", + "4193a28e 3f6076a8 40816fae -> 415885c3 41bb023b", + "40684548 c034c466 3dfcb1b3 -> 3f2e6ce7 40d277ab", + "4178bef0 bf8dbff4 3bd9588d -> 4166ebb5 41854915", + "bf144660 bfae7c9c 39b4c3d9 -> bff8ab28 3f48c98f", + "4100d3dc c05f840e 3bf0a5fa -> 4091a971 4138d300", + "40597790 c0870820 38dd365f -> bf526a22 40f3c4d4", + "4147bf40 3f7f8900 3dcbf030 -> 41362ec2 41594fbe", + "411f5010 3fd6d0f4 3d9114f4 -> 410353bc 413b4c64", + "3ef6ab80 bee357e0 3e9b33f1 -> be87e077 3f9d4dde", + "c094df28 3ee08220 392aa209 -> c0a2e8a9 c086d5a7", + "c14fba16 3f3d5a88 3ce9ab6e -> c15c04a2 c1436f8a", + "c16e928e 40992390 3c0f2aa1 -> c19da41a c121dce8", + "c0a59bb0 c0209452 3a8e8fc9 -> c0f5eed1 c02a911d", + "40f46b60 4050337c 402651c5 -> 3fe4a294 4157d70d", + "c1814d4a 3fc7bfb4 3a06e346 -> c18dca5c c169a070", + "3f9efa10 bf043060 40229155 -> bfe840ec 40898d43", + "4117e2e4 c0168efe 3a95c1d2 -> 40e474d5 413d8b5e", + "40c67808 bf639f50 3bf3015c -> 40a9c750 40e328c0", + "bf119960 c0380af2 407e5dbe -> c0ed6793 40c9013b", + "c0be4df8 c086d205 3f253352 -> c12ce33f bf8b55ce", + "41870444 3d4ba100 3fd7e676 -> 41724005 4194e885", + "c0e3c766 bff314a0 3f636a54 -> c11e7cf6 c08a94e0", + "409513bc 3f3d99a8 3a3e56a8 -> 407ab512 40acccef", + "419b2b32 c036d444 3fb297e3 -> 41724e3e 41bd2f45", + "413e4bc0 c072766d 3f3ee155 -> 40eb7ffe 41836bc0", + "410aad0c 3fc3d548 38f026f9 -> 40e463c2 41232837", + "c038e898 4099eb0e 38e3ed6b -> c0f6604d 3ff5ded6", + "4149d3c8 3fbb5b10 3b22eaa6 -> 41325e29 41614967", + "4153a8ac 3e509e60 3a2afaf5 -> 4150637a 4156edde", + "c1751d32 3fb50e10 3f2d5414 -> c18b4a24 c153a61e", + "bfa1a580 c099d36e 3a3a755e -> c0c242ae 4062dfdc", + "c00beb20 402c2d26 3b82f7bd -> c09c2ceb 3f020e56", + "c193fc68 402f491e 3bafe505 -> c1a9f095 c17c1077", + "c0bcb5ac 4084aab2 40168d32 -> c1465388 3f19ddbe", + "41059d76 40218c80 393b6042 -> 40ba731f 412e015d", + "c0963712 4064f044 3b389010 -> c104632b bf8e9f36", + "407ecb50 402c395e 3d80632f -> 3f9d1d7c 40d783f1", + "4182964a 40878996 39a1726f -> 41416671 41a4795b", + "4123057a c07bcb40 3ffb4f24 -> 4089516b 4180b11f", + "4029c1f8 bdb18c00 3f636d55 -> 3fd6b469 406829bc", + "40a25930 bbb1e000 40934e11 -> 3eede93a 411ae9e6", + "c0707518 3e28b140 3a7794a7 -> c07b0fb5 c065da7b", + "40cae8b4 bfbd72ec 3bbcd5cf -> 409b5cb4 40fa74b4", + "bfeab1b0 3fe54bb4 401f1fa2 -> c0c38f36 401c6cbc" + ] + }, + { + "glsl": "vec3 cultmath_af_add(vec3, vec3)", + "tolerance": "ulp-bounded", + "cases": [ + "40d43ce8 c087efa9 39613e6a c16aec62 c07bc9ac 403cc5ae -> c100cdee c102ea40 403cc980", + "c12da602 bf04ef28 3e52a901 416eafe0 3e32ffc0 401334ae -> 408213bc beb05e70 40205f5a", + "c13bb855 3f1a7af0 399ddbc6 c188bb12 3ff3ff78 3e522b2e -> c1e6973c 40209e78 3e5281fe", + "bfc3dff0 be934a60 3b29878b 400ff980 40107892 3c43020f -> 3f382620 3ffc1e8c 3c6d6ec2", + "418d570e bffc8a8c 3ac6433a 416372b0 400f701e 3d0a596e -> 41ff1066 3e8956c0 3d10abb8", + "418c1916 c062f6f7 3993b387 40e6d1c4 c024f1e0 3fdf597a -> 41c5cd87 c0c3f46c 3fdf63ba", + "c195fd6e 3f9be3b8 411ffb75 4194aff8 bfcfd300 392c8a53 -> be26bb00 becfbd20 411ffc2d", + "c16eadd8 40227cbc 410204a5 c18cb723 c04064f0 40c11ba2 -> c2020708 beef41a0 416292a5", + "c18039b0 3f5ba9f0 3e4f8fe4 40f2c80c 409d92e4 3a8f8f7d -> c1070f5a 40b90822 3e50b29e", + "40773020 c079dfec 3bb7222b 40c1a4c0 c0459a7c 3f85d8af -> 411e9e68 c0dfbd34 3f869061", + "40ca3f3c bfa23160 3ed550f9 c15b9017 408b3ed8 3bedb0ff -> c0ece0f2 40456500 3ed9091a", + "4134ba24 3e86f640 3b7f2baa c121ac83 c092bda4 39679e62 -> 3f986d08 c08a4e40 3b86feeb", + "40cff3f0 bcedf700 39f4164e c126d9ec bf354f20 40e76b14 -> c07b7fd0 bf3cbed8 40e76efc", + "c02b2300 40741c60 39ac403a c10ceaec 4023e60e 3a00dc23 -> c137b3ac 40cc0137 3a5b731f", + "415c62ec c02131da 3a188197 4174845c c0853193 39f197d6 -> 41e873a4 c0d5ca80 3a8d1e63", + "bd35a200 bf999a7c 3c68e268 c1427d2d bff77f5c 40d446dc -> c14332cf c0488cec 40d4bb79", + "c0e7495e 40967120 3f9ed448 408ca048 c00e286b 3e6d5e06 -> c035522c 401eb9d5 3fbc803f", + "40b0b778 40613920 3d57e1b7 406ce3b0 c0974af9 3cf57942 -> 411394a8 bf9ab9a4 3da9546e", + "c1883885 bfb6b418 3e8131bb c09ca770 c03464df 3b680d8b -> c1af6261 c087df76 3e830524", + "41005bee bf9beaee 3df9a294 405b39e0 bfefe608 3e8b6aa6 -> 41372a66 c045e87b 3ec9d529", + "c133bc4f 40801ae6 3c85c395 c1495d70 401f091e 3e4fb998 -> c1be8ce0 40cf9f75 3e6079ad", + "40b10314 beac46d0 3a6adb4b c153aefe 40866562 3ca1aa3e -> c0f65ae8 407741ea 3ca91842", + "c15e0335 404ee720 3dd554cc c0327270 3fb572b0 40545d57 -> c1854fe8 4094d03c 405b0860", + "40707f70 c01ed5fd 3aa6c604 40c9c314 4044f564 3edbafeb -> 41210166 3f187d9c 3edc5814", + "bff42e80 c07261e9 3a28fcd5 c172bccc bd364700 408c6a78 -> c188a14e c0753b05 408c6ff3", + "c0c3a420 40158cd8 3e46a2db c16bb081 bf8be71c 3bfd0bde -> c1a6c148 3f9f3294 3e4e90cd", + "c12c85a6 bef236a0 39ac72e6 417513d0 40915bb0 3d828091 -> 40911c54 40823846 3d83315a", + "c0f9c412 40819efa 3c7ba656 c132561c c0305084 3c1a67e2 -> c1979c12 3fa5dae0 3ccb2fa7", + "41040458 c0130774 3ef6488c 41085452 409d574e 39530ecd -> 41862c55 4027a728 3ef6656a", + "bfada460 3f1d8898 39983863 410a4b08 40152860 3e0c14b3 -> 40e92cf8 403c8a86 3e0c6367", + "bc8b5800 3e33e540 40e4a528 c118520b 404e2310 3a5feeb0 -> c11897b7 40596164 40e4ac4f", + "4140f4b4 c099295a 3c6e6b9a c12f5d00 c04016f6 3a672177 -> 3f8cbda0 c0f934d5 3c7d014d", + "41407ce8 40070c0a 3b49894a 40cce260 bfbc78dc 4050bb94 -> 4193770c 3f233e70 4050ee4f", + "c106e449 c02c2fb4 4042fec1 412ae39e bf5b0490 3aaf5b82 -> 400ffd54 c062f0d8 404314cf", + "c1451b66 3fabb33c 3f0343b1 4077c928 3fd40818 3fb312e4 -> c107291c 403fddaa 3ff4b527", + "c1099e9a 3e12f540 3ecaf927 bf73b040 bf954d06 3c79eb8e -> c118d99e bf82ee5e 3ed2c9e3", + "40668a98 c09247ff 39884958 c11f62ba bf4d2b40 3a87e4fa -> c0cb8028 c0abed67 3aab6ec8", + "4193ecac be49ea80 396bd992 3ff28820 4083ebb4 3b25e98f -> 41a3152e 407b38c0 3b362c2c", + "41287054 4043da44 3a9103bf c146770c bf3b42c0 3d04b260 -> bff035c0 40150994 3d093ebb", + "41137c6c 4014d6b4 39abdc8e 411d21e8 c0899b52 3a642e9d -> 41984f2a bffcbfe0 3a9faee8", + "c00e8148 c03cfd96 3bc4cffc 408460c0 408d20ce 3e89076b -> 3ff48070 3fba880c 3e8c1b20", + "bf9c84f0 bfe09024 3b8e7751 4166f998 c08998bb 3b683910 -> 415368fa c0c1bcc4 3c019706", + "414c9c90 c093c0c8 3f917426 40f7d8d4 40430a48 4095f582 -> 41a4447d bfc8ee90 40ba52c4", + "c15ce12a 3fb92730 3ed04fa5 bfc622b0 405581f4 3ffe4ed9 -> c175a580 40990ac6 401931bb", + "409a176c c08193c7 398f927d c134c749 401d4890 3e37c6e7 -> c0cf7726 bfcbbdfc 3e3810c0", + "c0d50cdc 405c80bc 40ea3685 417cd170 40443d98 3a9c0e1a -> 41124b02 40d05f2a 40ea4074", + "415e58f0 bfb4ae0e 3d4b892d c0935466 c0870f2f 3db1be07 -> 4114aebd c0b43ab2 3e0bc513", + "4190f802 4080bb60 3ef51586 417f4b04 bfa62738 3c0fa76d -> 42084ec2 402e6324 3ef9976a", + "40f479ec c09437ff 408b615c c153de0c 3f680c50 3c5b4fb0 -> c0b3422c c06e6cea 408bcf1f", + "bfadc600 409a9202 3beaed49 be202280 409a9254 3fd942cb -> bfc1ca50 411a922b 3fda2e1f", + "407ae6c0 c01ec955 3da07483 40e25f8c 3e8c0040 3aafbedc -> 412fe976 c00d494d 3da33a22", + "c0d0a62e 4087e590 3abcaf85 4075a160 405d9378 3c8af860 -> c02baafc 40f6af4c 3c96d82a", + "c086846c 402cc908 3d8b999e c16a1d46 3da97480 40912003 -> c196afbe 403214ac 40934e9d", + "c170e1ef 400dc0b0 3ed79b31 40fe9e48 3e87de00 3eb3756a -> c0e32596 401ebc70 3f4588f4", + "410eb658 c056a862 3af3c399 3ff38710 bf35d748 3fa7f302 -> 412d273a c0820f1a 3fa83075", + "415c3250 bfbe4d0a 3bf80550 40f76d28 40379c1c 3facc0f7 -> 41abf472 3fb0eb2e 3fadb9be", + "c19d27f2 c00f3fc2 3bf590ac 3ed7bd00 3f1e7870 3c8a58a9 -> c199c8fe bfcf434c 3cc7e690", + "c15b8ff8 c08195b1 3a6b441d bf394dc0 402f642a 3fa2625f -> c16724d4 bfa78e70 3fa28050", + "416583d0 bfd736f4 3cc1b901 c01a2b08 bfa33d36 3b051cb7 -> 413ef90e c03d3a15 3cd27a6e", + "c17282e4 407f6a8c 3ed8cdee 416cb6d0 40913cb6 3bd56500 -> beb98280 410878fe 3edc24ac", + "4196e814 40393c0a 3cf69113 c06b62f0 40213608 3cf43d53 -> 4172f76c 40ad3909 3d757bdc", + "c1899183 40010736 4103b0b0 411a0700 c06fcd62 3abdad44 -> c0f2380c bfdd8c58 4103b6af", + "401051f8 3f90c050 3a249d89 c10d3537 3ed6f8e0 3c6b6e69 -> c0d24172 3fc67e88 3c75d8cd", + "3f3bc040 c025b4c9 3c12f286 c02af188 3ff1da20 39a05bfa -> bff802f0 bf331ee4 3c17fffc", + "c0c17d98 bf750bb8 40fadd5a c176dfa4 3f38e5e0 402fdecc -> c1abcf38 be709760 41296680", + "c17f8ff4 c06706b6 3b85f69d c19c18d9 3ec76e10 3f3a7832 -> c20df06a c04e18f4 3f3b8696", + "418ff028 401f48c2 3baa4e07 40fa8518 bf61c760 397170fb -> 41ce916e 3fcdadd4 3bb2b506", + "41580d20 c074832d 3a8d5887 c124b308 407560c0 3f206182 -> 404d6860 3c5d9300 3f20a86c", + "c17ebea1 405ab254 3bc7f729 c103ec38 bfbe0a62 3ebb93c1 -> c1c1556c 3ff75a46 3ebeb6ed", + "41971a7c 3ec46450 39a6b792 c081c96e c08d0cf8 3bb34612 -> 416d5041 c080c6b3 3bbe4871", + "3f241e80 4091de50 39e886fc c1818253 3fb32de0 4108a229 -> c178c2be 40bea9c8 4108a418", + "4183224c 3ee23190 397631ca c0907ef8 bfda2b14 3f6c76e5 -> 413e051c bfa19eb0 3f6c8729", + "c183bbba 404e7978 3fb34f54 c0e8d304 3f74d9f0 3d92a978 -> c1bdf07b 4085d7fa 3fbc7ad7", + "4179f3c0 c07c0bc3 3d24b7b1 c1338928 c09c1081 3ee60e13 -> 408cd530 c10d0b31 3efaa6c0", + "3f9d3120 40445c44 3f5da9cb 408c3138 c0933354 38d32724 -> 40b37d80 bfc414c8 3f5db0e4", + "40dc2634 3f031430 3e8fbedc 419e92d0 c04cff5e 3c9c08f7 -> 41d59c5d c02c3a52 3e998321", + "418dc79c 401b11c6 3cc90297 c133076b bfd5ef3c 3ba2c4f9 -> 40d10f9a 3f4068a0 3cf1c276", + "40ed1db4 400980dc 3d14fc8b 4023d450 c03c74c5 3d34b070 -> 411f83ee bf4bcfa4 3da4dbea", + "41757128 3fc53cf4 40a577c0 3f96f5f0 3da4f100 40ba5b0a -> 418427f3 3fcf8c04 412fe982", + "41511ca4 c00234ee 40a70d3d c08172c8 3f2554d8 3fbb86e0 -> 41106340 bfb1bf70 40d5ef17", + "c140c528 bfffa80c 3d07e9a0 c188776a 4009ad6c 3f47878a -> c1e8d9fe 3e1d9660 3f500805", + "c17dcc58 c0824e89 3e27d9f8 c0a228fa bf9ff16e 3fec9147 -> c1a7706a c0aa4ae4 4000c6b4", + "c1823ad3 3fd9f658 3f3c489d c14bbe3b 3eaf4000 3b493f05 -> c1e819f0 4002e32c 3f3d13d9", + "c13440b9 40955888 3dcfd49d c11a82dc 4010a772 3c0b4ccb -> c1a761ca 40ddac41 3de14c31", + "c18d7a6a c064cba4 392c52c8 411770de c053f705 4026ad73 -> c10383f6 c0dc6154 4026b06b", + "3efe7d80 3ee244c0 3a4ad175 40b95668 c0992122 39189b8c -> 40c93e40 c08afcd6 3a73a0dd", + "41566db4 3f8efb58 411f7ff0 4115a272 bec9f4b0 39bfffda -> 41b60813 3f38fc58 411f8191", + "41770880 3dd0b4c0 3b22d2fb c1836754 bf22ae78 3fc5b9cb -> bf7c6280 bf0897e0 3fc60b4d", + "c14cfa08 3fb9b5bc 3f25e773 40bafef8 c04c78aa 3dc7dad5 -> c0def518 bfdf3b98 3f3ee365", + "c12b965c 408cdeda 3d98406c c1744e06 c09afd00 3d956aaa -> c1cff231 bee1e260 3e16dc30", + "411687e8 bfaa7a6e 408560a5 41264346 4066f798 3fb695f1 -> 419e6597 4011ba61 40b30658", + "40e0d048 bf048368 3e339f89 411a0f74 bfc457de 4083267c -> 41853bcc c0034cc9 4088c3a6", + "c101754f be1e32a0 3d0b07e1 beeae880 bfa316bc 3a9d4709 -> c108cc93 bfb6dd10 3d0ffc1d", + "c17b9c51 4095ecea 3ead98db c13c8ed8 3f11ab98 3d2a32b4 -> c1dc1594 40a8225d 3ec2e357", + "c172ad82 406a746c 3bd7b2eb 404c61a0 bf5ef9c0 39859ac1 -> c13f951a 4032b5fc 3be082c6", + "bfd54100 400abbbc 3dc3addd c03a9678 409f8d44 399a44d2 -> c0929b7c 40e4eb22 3dc44e0c", + "c14731a6 c06f3ac5 3e34a4be c0049e00 bf5cdb30 410bb34e -> c1685926 c09338c8 410e85fd", + "c1471115 bf1c25f0 3b4ba548 415978c0 bfc21666 3e9a8e09 -> 3f933d58 c00814af 3e9c25c7", + "c182a5bf 401bc0e8 3d8d171e c0b53a8c c09c58db 39346f97 -> c1aff462 c01cf0ce 3d8d7d98", + "416a9914 3e98b610 3f8f78d6 c07f50a8 3fb1e190 41164530 -> 412ac4ea 3fd80f14 41283462", + "bfa814a0 c0815822 391d8053 402cf5a0 c00b486d 3e105efd -> 3fb1d6a0 c0c6fc58 3e10884d", + "c0de1ab4 406d304c 3cdc5e1c c0603bf0 c000259f 3adf937c -> c1271c56 3fda155a 3cea6faf", + "402a4f08 3fcb1660 3d73642c 410b6abe c09abee2 39cbbb61 -> 4135fe80 c04ff294 3d750a52", + "c12fd8b2 c00c2643 3d1f4a97 40d6ad88 bdf38300 3e640588 -> c08903dc c013c25b 3e85ecf2", + "3f0b6fa0 c080056c 3eac34a1 3ffa5c00 3f1b0540 3cb4d2d1 -> 402009e8 c0594988 3eb78296", + "c10a5940 4038a284 3cdfa554 40c615c0 4099ed8a 3b7ec187 -> c01d3980 40f63ecc 3cff91e3", + "4190ebd4 3f9b630c 3a1f1d70 40d7dcf0 3e0861a0 3cc889fe -> 41c6e310 3fac6f40 3ccdb761", + "c13997ce 3fe586fc 3c8751fa c0e4ccac 403acef8 3bbe33e5 -> c195ff12 4096c93b 3cb70deb", + "be9c8e80 c060d1a9 3d172502 40f5c700 c04ffb78 3bc9efdb -> 40ebfe18 c0d86690 3d30712b", + "bf6fc740 403fa6a8 3ddfc1ca c0abc622 c08ebcc9 3a56270c -> c0c9bf0a bfbba5d4 3de17209", + "4187d572 4031a724 3b68abcb c19a80bc c06db5bc 3d137c88 -> c0155a50 bf703a60 3d220a95", + "4125daf4 4083c470 3e7a509b c12868d8 40299318 3c1ab7b0 -> be237900 40d88dfc 3e81fef1", + "41939ab8 3f10e350 3f307e78 c1932270 bed88b70 3f8da25b -> 3d709000 3e127660 3fe5e1a7", + "c12f65e1 c044e2b1 3b635b53 c0bcab58 c08de806 3c3bc166 -> c186ddc6 c0f0595e 3c74f9c4", + "4191b7a0 3f6b0cf8 3def23b4 c16d4298 c04df8ad 3a5406c2 -> 4058b2a0 c013356f 3df0cea9", + "41068da4 3f5eeae0 3d8ccb22 4197be44 401c4394 3ef45081 -> 41db0516 4053fe4c 3f0bc399", + "c154aa82 c0449657 3d684e2d 418a5754 3f1efad0 4020f96c -> 4080084c c01cd7a3 40249ac9", + "40e02608 c01c7b9c 3c07e673 c1743c34 4013ae14 3dfd363a -> c1042930 be0cd880 3e071ba6", + "40670da0 bf22de50 3ba7918e 413f4b66 c0667d31 40020a1a -> 41790ece c0879a62 40025e3a", + "416f9c3c 3f380248 3ece4938 417edc08 c08c385c 3d45c609 -> 41f73c22 c06a7026 3ee7065a", + "417e99e0 3fa63b18 3eee6394 be211500 3e884c20 3a4b3707 -> 417c158c 3fc84e20 3eeecb69", + "4151a5a0 409bdc74 3f93b35d c195c5a9 406e65a0 3b0a4ef5 -> c0b3cb64 410987a2 3f93f8ff", + "c11ad8cd c004148b 3e1115f3 c0387f98 3fe32ea0 3f818d6e -> c148f8b3 be93e9d8 3f93b09c", + "c1970424 4057ab74 40014654 419e6924 3ec7cff0 3f02f9cc -> 3f6ca000 4070a572 402204e4", + "41597c58 4083a120 3c8d14f2 4103c1f8 c0800fda 3a3e8742 -> 41ae9f28 3de45180 3c933516", + "c19b4c1d c013868a 39ecf54b c19b86e7 c094f59c 4092a983 -> c21b6982 c0deb8e1 4092ad9c", + "c1623b78 c04d68a8 3ee1c245 c13c3d22 c005e1d5 3f05dcd4 -> c1cf3c4d c0a9a53e 3f76bff9", + "4143d670 3f153ae0 3d872138 c187666a 3ed12190 3fd9f7a9 -> c095ecc8 3f7dcba8 3fe269f8", + "c12bf57d bf090f78 403dbb8b 418665d0 406ce450 3aae7116 -> 40c1ac46 404aa072 403dd18a", + "40e0e3b4 4080e3cc 3995342e 419d6154 3fc7fea0 3df43005 -> 41d59a41 40b2e374 3df4d56d", + "4113a078 c0721c57 402ec640 40b36488 c09a798c 3dd401bf -> 416d52bc c109c3dc 403566b7", + "c16a5576 c023caff 3b017295 40fd7ae8 c00be46d 3cfe6f57 -> c0d73004 c097d7b6 3d075a56", + "c188e7a8 404e34d8 3d5c6950 c174dffe c0470426 41002b59 -> c201abd4 3de61640 410107eb", + "c14e23e4 4028492e 3be94ed1 40814118 be824450 3e3d3f37 -> c10d8358 401800a4 3e448c88", + "c10e2641 c081b1ec 3c229d46 4181fb44 bfceff50 39eee0ad -> 40eba08e c0b571c0 3c2a4878", + "415a7d9c c05cfe22 3d28699a c1774c62 be7dd9c0 3a37af8d -> bfe67630 c06cdbbe 3d2b4de3", + "41641c90 c055dbd5 3d4fba2a 41937b4e 408dc450 3b47aefe -> 4202c4cb 3f8b5996 3d5c56f0", + "405f02b0 3eaa6870 3a2f49c0 c0852d70 405e9aec 39361bb4 -> bf2d60c0 4073e7fa 3a5deffb", + "40e4ad1c 3f154760 3ebdaa52 c17987b8 4048e690 403b5c0f -> c107312a 406e3868 40531197", + "c12356c8 c06d0014 3e303512 c12a4b90 bfb04ef6 408335ad -> c1a6d12c c0a293c8 4088b792", + "c13da8f9 c02e4c01 40585641 c16e7682 40643fdc 39e182b2 -> c1d60fbe 3f57cf6c 40585dc9", + "419833fa 4092be8a 3bb1fcad c09d5a76 c018fc8e 3908f2cf -> 4161bab9 400c8086 3bb6c6bc", + "c01a5a30 c0351a82 3fa100f7 c1886894 c09b9010 40a2c01f -> c19bb3da c0f61d51 40cb00a0", + "beef8c80 40747078 3ae0dd8a c0be34a4 c02e6003 3f4b47e8 -> c0cd2d6c 3f8c20ea 3f4bb8dc", + "c051fa40 40140552 3ba5bb98 40a5c3c4 406c0a64 3baf02b6 -> 3ff31a90 40c007db 3c2a7ecb", + "3f60e500 3f9cd03c 3f856575 4097a224 40249f96 3ec13a8f -> 40b3bec4 407307b4 3fb5b470", + "c08b8aec bece4690 3e77774f c193bc92 bf4bfab0 4032d09d -> c1b69f4d bf998efc 4042487e", + "41009d00 bffdf870 3b490617 40f3e820 3fda1da8 3dc593d4 -> 417a9110 be8f6b20 3dcbe40a", + "c1888534 c05285a2 39c6eadf 4103c1ce 401d30d4 3cce8a17 -> c10d489a bf555338 3cd1b923", + "40e0df08 40617158 38d202a0 c16f79d7 4006080c 39de006a -> c0fe14a6 40b3bcb2 3a0ca434", + "4136fe64 bfc4f3da 39935c5c c19f8801 40634474 3e1f6b41 -> c108119e 4000ca87 3e1fb79a", + "c13373f6 404d9860 3ca3143a c17d1417 3fe9e89c 3d719ec1 -> c1d84406 40a14657 3da1a482", + "c116a2cf bf000d00 3ca6855f c0b63940 3fc0915c 40886eaf -> c171bf6f 3f808adc 4089155d", + "bf31fe60 40118d60 395358fa 411c1a42 3f0fb120 3e9dc69c -> 4110fa5c 403579a8 3e9de28e", + "c12cfd94 404a4c80 401b1fcb c0e238d0 408db0c8 40564da7 -> c18f0cfe 40f2d708 40b8b6f7", + "4109c0a0 c0164b9f 3f106fd8 40034ab0 c01e7789 39765913 -> 412a934c c09a6194 3f10803f", + "413fab2e 3fff5a04 3c8be434 40cb6628 be94bfe0 3fe16e1b -> 4192af21 3fda2a0c 3fe39e5b", + "c16844d6 409604d8 3ce27463 401ddd68 bf166d90 3b451339 -> c140cd7c 40833726 3cfb3727", + "c19a038a bfe8e7b6 3a85d543 c188441a 3fb74c20 41020d0d -> c21123d2 bec66e58 41021169", + "c18b5dfb c0928def 3f6429cc c08a3ab6 402ded90 39244fa3 -> c1adeca8 bfee5c9c 3f643599", + "41276408 407da870 401def58 400c5ea8 bf8da970 3c4896f0 -> 414a7bb2 4036d3b8 401eb837", + "41427990 40088a84 3a7ef47b c090183e 3f14e0c8 40b5eda4 -> 40f4dae2 402dc2b6 40b5f5bc", + "bf5231c0 404f2d04 3afb217f c1387ff4 3f88be2c 3db0fbc7 -> c145a310 4089c60d 3db4f0ac", + "c139f609 3fd71628 3bf54f3d 40500ce0 bffa2aa8 3e082503 -> c105f2d1 be8c5200 3e0fd1af", + "41020594 3fb491a8 3e03dc18 412bb474 be0715c0 3a93b4e2 -> 4196dd04 3fa3aef0 3e050893", + "c10a1e89 bf6cdde0 408b40d1 c06ce288 407e4f0c 3d98e27a -> c145572b 40431794 408da483", + "41969fd0 3f528e08 3da1b646 c1895cee 4024fdc8 3edf38fc -> 3fd42e20 4059a14a 3f03d3a0", + "c180a01a 3fa80a18 3f2150cf c11f7300 3f726ba8 3fa0325b -> c1d0599a 40109ff6 3ff0dbb3", + "4110554e 4061c3fc 3c75abf7 c13e48cf c04b2752 39ad0fca -> c037ce04 3eb4e550 3c7b216b", + "40d2201c c04c9f2d 3dba507b c11d9e95 40611214 3f0d197f -> c0523a1c 3ea39738 3f2463d2", + "413d319a c0949d72 3ab1e889 419be7fa c07ded97 3a14935e -> 41fa80c7 c109ca1f 3b0097f0", + "c19891e4 c088f53d 4089106d 40e8cb90 c025bf92 390b28c5 -> c13cbe00 c0dbd506 408911b1", + "c19d7de7 c0673c2c 4063b3ac 40bc1b9c be2bc960 3e6e0ed1 -> c15cee00 c071f8c2 407294ef", + "41698178 bef43b60 3932d79d 41419548 c092506e 3e178c01 -> 41d58b60 c0a19424 3e17c0b0", + "41301f48 c085eadd 3aae96d5 414f5cf4 401bb750 3ec51711 -> 41bfbe1e bfe03cd4 3ec5c8eb", + "419a80e6 400ade32 3ee638a3 4153fbf4 c00feb78 3ed11c1e -> 42023f70 bda1a8c0 3f5bac78", + "3d5e3a00 c0791028 3d32e2dc 408bc890 c02e4f38 3d28312a -> 408d8504 c0d3afb0 3dad8f93", + "417b9b10 c092083d 3dea70bd c064da30 c0434d00 392dc64b -> 41426484 c0f3aebd 3dead191", + "c1442aca c09a7edf 3e9c8296 4185ec12 bf8e4018 3c22aa07 -> 408f5ab4 c0be0ee5 3ea1993e", + "41013486 409dce28 411349e6 406cc4d8 c0384f55 3906a997 -> 413c65bc 40034cfb 41134a84", + "41883fe8 c04080bc 3b9031ed c13a103e 3ffec7fc 3d2ad296 -> 40acdf24 bf82397c 3d3cdf4b", + "c103d5d7 c04ef7c4 40050df5 c147ab64 c062d06e 3bee701b -> c1a5c09e c0d8e419 400585a3", + "c1630d02 c01c62d0 3aae1930 3f7288a0 4082fd1c 411e628e -> c153e478 3fd32ed0 411e6818", + "c198dc74 bfad5458 402d954c c14c2500 3fa70f7c 3edaac56 -> c1feeef4 bd489b80 4048eb63", + "c1999798 c08b6b2d 3b8f39cf c0d81026 c06d3892 3b286bff -> c1cf9ba2 c10103bb 3be47ffa", + "4112d2b6 4082d682 4047a976 3e815000 3feac8f0 3911751c -> 4116dd36 40bd88be 4047ac06", + "c143d8d4 3feb5698 3bad2a08 c183ef56 c07560b4 39015841 -> c1e5dbc0 bfff6ad0 3bb22aa8", + "c14c9a72 c09e3fb2 4011f073 c0e06c54 402ee528 39f3091f -> c19e684e c00d9a3c 4011f86c", + "c1502e4c c04df403 3989966e 419dfb2c 3f33a880 3fdcfb72 -> 40d79018 c02109e3 3fdd0463", + "c1698f24 401ca4d0 3d8ed4a8 40835250 c046e10e 3e18bda9 -> c127e5fc bf28f0f8 3e602ad5", + "c1429c03 4094191e 3934b29c 4050dc60 bed3f0d0 3ac811da -> c10e64eb 4086da11 3ae04be0", + "bf875cd0 3fb1d100 3d94be26 c14269b8 bfd9fd44 3aafc265 -> c1535552 bea0b110 3d9783fc", + "40c5c82c 408b12e2 3ae3bf97 416bcc84 c09fc547 3e089ef9 -> 41a7584d bf259328 3e0a6be5", + "4162e25c 3ffa0df0 3b3aefdc c1434486 c05bd9c7 40b655c5 -> 3ffceeb0 bfbda59e 40b66d35", + "c14129e4 4016dc5c 3ba1c513 c18f8dd5 bfa4ccb8 409df144 -> c1f022c7 3f88ec00 409e19fd", + "bfce3010 3f69f810 3cf37527 41065132 407abd58 3a19fd82 -> 40d91660 409a9dae 3cf85c5e", + "4196b4fc 3f7cb9e0 3d5e2cdb 40aca044 4060b570 39f63b31 -> 41c1dd0d 408ff1f4 3d60361a", + "401c2840 40991db6 3fdfba8d 4165532c bfcddb78 3e1a3ab9 -> 41862e9e 404b4db0 3ff30293", + "407e9138 c0155ba6 3f066940 418df834 401ec6ca 3cf1797e -> 41adca5b 3e16b240 3f0df673", + "c0c81450 c09fabbe 3f89aa7e c1813d52 bf953e0c 3ed2b735 -> c1b34266 c0c4fb41 3fbe593b", + "417ebfa8 bf473fc0 3b0ed4d9 40aaf218 40175c4a 3efba3aa -> 41aa1c5a 3fcb18b4 3efcc43f", + "4113c290 408e4972 392bae87 c181db44 407e628c 3c27a5e6 -> c0dfe7f0 4106bd5c 3c2a9257", + "c186fbca c0336dc5 3fd5089a c15bb82b 3fab62d4 3a4b7a6a -> c1f4d7e0 bfbb78b6 3fd52317", + "c17e4f2e bfccfa24 3c1d73e4 4101fe88 bef5ffc0 39aedb23 -> c0f8a14c c0053d0a 3c23122f", + "c1436fb3 3e249980 3fd2bc6f c19fc5cd bfab494c 4035f1f8 -> c200bed3 bf96b61c 408fa864", + "c0c33dea 406fe498 3a9c592e 414f592c 4017907a 3c649edb -> 40db746e 40c3ba89 3c785df6", + "416aadd0 4088c880 3ad48ab5 c17978ba c06e6f60 3c19d5f0 -> bf6caea0 3f0c8680 3c346d37", + "406d6fc0 3d56b580 3be5b98d c0ca2f3a 4069e7f8 3d33b783 -> c026eeb4 406d42ce 3d507513", + "40bf8cdc c04851fc 3be072ad 40a44270 409dea46 392d1b53 -> 4131e7a6 3fe70520 3be642fb", + "c0de4924 c0433ee2 40e31aac c157a670 bf1c5d00 3aa5d283 -> c1a36581 c06a5622 40e32547", + "417ccc20 40134e2a 3d9fd3aa c15eac68 401de870 39081670 -> 3ff0fdc0 40989b4d 3da01b12", + "c1488683 c01d50c1 3cd49268 c058cf60 c00665fb 3eae628a -> c17eba5b c091db5e 3ebbae4b", + "418b85d4 409d94c6 3a9736c8 c19f5aa3 c0795d65 40ad436b -> c01ea678 3f83984e 40ad4cf0", + "4116b37e 4061a7e8 3e37596b c0ecfaac bfbea410 3a4e476b -> 4000d8a0 400255e0 3e3828c1", + "3f71dbc0 400c3fd4 40234672 c1521090 bcc12100 3d51cdc4 -> c142f2d4 400abd92 40268ded", + "bfd35540 40025440 3ce2ffdb c19919c8 c03644db 3c1eca44 -> c1a64f1c bf4fc26c 3d194821", + "414a47d8 bf17dea0 3f1ec780 c0c4784c c05c911a 3de935f5 -> 40d01764 c0814461 3f3beef3", + "c1916699 c02a05bc 40438224 40cdad60 3ff62b18 38d44017 -> c13bf682 bf3bc0c0 4043840b", + "4153ebc0 bfe0de94 398acbb2 c1987a94 bf3967b8 3daefbac -> c0ba12d0 c01ec938 3daf8aa9", + "4032c440 3f4d25d8 3c00ce23 c080eb9c 3fa46bb8 39101093 -> bf9e25f0 40057f52 3c031bb6", + "c165921b c09b3af9 3f1b3d2f 418d1930 40921b68 3daf0b37 -> 40528114 be91f910 3f311eda", + "c13487da 40849f9e 3f44fad7 4149afd8 c048fd20 3e646eb4 -> 3fa93ff0 3f808438 3f7e16b9", + "c1478d36 bf8e20b8 3e9cd065 c1936b8d c013b024 39c8fdfb -> c1f73228 c05ac080 3e9d06f8", + "c10b66a8 4068470c 390ecaa8 4173650c bfcbf8a0 3a40d185 -> 40cffcc8 40024abc 3a66a682", + "bfda7920 bf25d600 3b1d52f8 4180b3b6 bef72500 3e19560b -> 41661848 bf90b440 3e1bcf41", + "c19a2ede bf550960 3f935710 4113c9cc c03caa0c 3a041469 -> c12093f0 c071ec64 3f93680b", + "41630738 c07cff94 3b206b65 c0f26390 bf1285d8 3f466c67 -> 40d3aae0 c090d085 3f470d91", + "c130c367 c09ae20c 3a00ef29 c059ef00 40569810 39238e68 -> c1673f27 bfbe5810 3a2dcef6", + "417907d0 3e9c4150 3b6af2b6 c1181382 c0750f27 3f2c1fa5 -> 40c1e89c c06186fd 3f2d0b3c", + "c19fc0d7 c018f729 3e70c30d 405511e8 be27e7a0 3d25a562 -> c1851e9a c02375a3 3e8d18a2", + "c0f2d0ca bfebde8c 3ca8a632 c1650349 403e0d94 3f338403 -> c1af35d7 3f903c9c 3f38cab1", + "4164c7c8 bedab820 3a596aa3 401150b0 c049de51 3a2e4aac -> 41848dfa c0653555 3ac65f87", + "4158f040 4096e590 3c4eaa0a c120ac0a 3fc850ec 3b89a304 -> 406110d8 40c8f9cb 3c89d167", + "c194d066 c09269b2 3dcc9aea c1280842 c09bd033 40812ecd -> c1e8d487 c1171cf2 4084618e", + "c0e10fd4 4023b5c6 3b4e7fb3 c1910e2b 40134b7c 40a85d93 -> c1c95220 409b80a1 40a877aa", + "bfad0c20 c0123d43 3e028996 41241724 c09fe4b1 39e25679 -> 410e75a0 c0e90352 3e02fed5", + "c0ec9b16 c085e00f 3aa32ebc c0a76ecc 3f12f1d8 3bfe6c93 -> c14a04f1 c06703a8 3c13dd1c", + "41631cc0 bfd698b6 3bafd95b c16255d6 bf440f08 3a0c0a5b -> 3d46ea00 c01c501d 3bc16ea0", + "c0c5f360 40212a5c 3dd7ad27 c0c082d8 3ef72a40 3c838e00 -> c1433b1c 40400fa4 3df89851", + "4186dcf0 c088b012 3b55a5c4 3fd48540 404b84a0 39286c92 -> 41942544 bf8bb708 3b61665c", + "bf9c3ca0 4046e420 3edd46e3 c126b36d c0830527 406b469a -> c13a3b01 bf7c98b8 408377dc", + "3fde8450 c0859371 40e5be8d c1046550 408ad0c0 3d995366 -> c0d1298c 3e27a9e0 40e823f7", + "c17c668c 3fd42d30 3cda1daf c131fca0 bf964a00 3e81b45f -> c1d73196 3ef78cc0 3e8f59af", + "4040b2e8 c04e3214 3b967089 c15d58eb 4000dd5c 40f95625 -> c12d2c31 bf9aa970 40f97be9", + "c0cd2686 c0756019 3ed38ba1 c14cd605 40565d44 408adb00 -> c199b4a4 bef816a8 409813eb", + "414f9cb0 402f21cc 3b085cc0 c14e2976 407fd030 3f0a7c53 -> 3db99d00 40d778fe 3f0b0526", + "40c12320 3f93a058 3ad40bff c02be1d8 3f83b340 3ba5d2e5 -> 40566468 400ba9cc 3bdb0234", + "c1506eca c08f4a30 3f9d840b c196841a c023dae1 40257a1b -> c1febb7f c0e137a0 40743ccb", + "4191f262 3fff6d00 3e90bd93 409c0a54 409531f8 3e22a01a -> 41b8f4f7 40d50d38 3ee21167", + "41059e42 bfec5d54 39b8c74a 411f3c64 407fdfd8 40ed1cec -> 41926d53 4009b12e 40ed2007", + "c1467e86 bfaeb2e2 3ae2cfae c192011f bea01000 3a25b22f -> c1f54062 bfd6b6e2 3b1cd9c4", + "c0f1a556 4088e176 3f2d2fb3 411897d0 3f35c740 3f2f97e2 -> 3ffe2928 409f9a5e 3fae640d", + "c01c47f8 c064dad2 3d9973bc 419d4b34 40998af0 3e3448d4 -> 4189c235 3f9c761c 3e8083af", + "c15967be bff3161e 4013ac5d 415a0ec8 c030bac9 39f9312b -> 3d270a00 c09522ec 4013b443", + "c1282e1b c03e6878 3f8b65b8 409e0918 be9e3dc0 3dab1186 -> c0b2531e c0523030 3f961720", + "40afa324 4080aa96 3ec19aec c11497d8 c02479f0 3ace3e64 -> c0731918 3fb9b678 3ec269de" + ] + }, + { + "glsl": "vec3 cultmath_af_sub(vec3, vec3)", + "tolerance": "ulp-bounded", + "cases": [ + "c1822c5d c06c6c8c 3da954e2 40ca1250 400bca78 3b79ce30 -> c1b4b0f1 c0bc1b82 3db1319b", + "c17577bb c034b30b 39e64c6a 41875d04 c04ae8d8 3fa86375 -> c2010c71 3eb1ae68 3fa872e9", + "c18b1ecb 400b07d6 3b9bf8c7 bfcb57b0 3f629090 3db5ee20 -> c17cd2a0 3fa4c764 3dbfb643", + "c17f24fc bfc11a9e 3a02dd69 c1782aea 40442bdc 3b0c849d -> bedf4240 c0925c96 3b2d8c2a", + "c0bc7c56 3ff05194 3ca0f276 c14bb870 c07f39c4 3e626cc0 -> 40daf48a 40bbb147 3e768e4c", + "c18f72c0 c0527a26 40490343 c0d27d8a 407eef24 3fe19eb9 -> c135a6bb c0e8b4a5 409ce97f", + "c18afdf3 3cc78d00 3dba0fda 40106250 400f8684 38e13d31 -> c19d0a3d c00df76a 3dba5321", + "c19dcef6 bfedd046 39034d06 419abf8e 3f150888 3f131283 -> c21c4742 c01c2a45 3f131d59", + "c120bc9e c04ec377 3b7250f4 c16f7a8f c05572db 403c9754 -> 409d7be2 3dd5ec80 403cd408", + "c1801724 4099e918 3b88b5a5 c0e9f1ae bff97158 39154507 -> c10b3571 40d8456e 3b8ddb82", + "4164d790 408b205a 39c85b9f bfcebfa0 406d9f88 38f04c84 -> 417eaf84 3f2284b0 3a065ac7", + "4187cfd2 40130130 3f9f1db2 bf0c6f40 bf86de20 398d3893 -> 418c334c 40567040 3f9f2737", + "419f6034 3ee61380 3e085ccb c16a96e9 c07c8cfe 3ced1a79 -> 420a55d4 408ca7b7 3e2609e3", + "c12219e9 bfa77a90 39e8ef41 41045a32 408a8218 4085d368 -> c1933a0e c0b460bc 4085d744", + "c0b928ca 40884a88 38dd3af9 c19e3a20 c029107d 40772c4b -> 415fdfdb 40dcd2c6 40772e68", + "3f80d690 401efcf0 3a7e28ac 401c2338 3ffe8200 3f62d21b -> bfb76fe0 3efddf80 3f6311d2", + "41797188 3ef1d050 3c8646c0 c19d8877 c08d1043 3fb1c0cb -> 420d209e 409c2d48 3fb3db33", + "418a8be0 40280a58 3f50cc52 c12e63ff c06e3af8 3e033c11 -> 41e1bde0 40cb22a8 3f719d8e", + "c113d8cc 40953450 4018c94a 41545398 c0896f77 409919de -> c1b41632 410f51e4 40e57ed0", + "41805d82 c05f82a2 38d66e9a c14502f9 c07d4390 40aa9ef9 -> 41e2defe 3eee0770 40aaa013", + "c19db3b3 408609f2 3cf15e1c 41807dde 40938f1c 3e2e5537 -> c20f18c8 bed852a0 3e4c8a13", + "4178cd44 bf212190 3fd9a1c5 4156a65c c0240871 3d7d0609 -> 40089ba0 3ff7801a 3fe18a24", + "c190322a c02708fc 3b687a9a c0feb49a 402f4274 3a3a2c0d -> c1210a07 c0ab25b8 3b8bfe26", + "c16521e6 c06e3066 3f7a9704 c095a400 4017648c 3b11ff6e -> c11a4fe6 c0c2ca79 3f7b2a0e", + "c192a11c 3ff78e78 3e7b15f2 c118bcd6 3fcc9b8c 3d61455d -> c10c8562 3eabcbb0 3e99b4d2", + "41380948 3ffce900 3c86537d bdc91400 c09fd4c3 3a03d879 -> 41399b70 40df0f03 3c8a976e", + "c0d3a16c c095f1ff 3f822432 c168e869 bfce6286 3e917afd -> 40fe2f66 c044b2bb 3fa68354", + "41304944 bf29fca0 3ed1f0b1 c0e9b394 4034b7ea 3fb7f7e6 -> 41929187 c05f3712 3fec74cf", + "40a142bc 3f802aa4 3bfb393b 415442b0 c08ff08a 40f3fe48 -> c103a152 40affb33 40f43d41", + "c161c58b 3f9ed530 3a55dd9a c1307537 40663df4 3b0ee9c0 -> c0454150 c016d35c 3b44b837", + "4195dc12 3fa31434 3f44ea9f c1014234 c090b175 3d5c40cf -> 41d67d2c 40b97682 3f52b0c3", + "c17644a2 3f04b750 3acc2198 c153685a 40953b36 3f03eea3 -> c00b7120 c084a44c 3f045521", + "c0a7e48e c09e1cbe 3f61b979 c17b3d64 3f89c974 3b56687f -> 41274b1d c0c08f1b 3f6290f7", + "3ff0ae70 400f9792 4065cabc 415db1d0 3fad439c 39508b4c -> c13f9c02 3f63d710 4065ce40", + "c1527746 4091e34c 3e30b86a bfab79e0 bfbf993a 3bf9d506 -> c13d080a 40c1c99a 3e388b95", + "41928e20 c0795b7f 3a3d2e79 3faca900 c054a08c 3eb88613 -> 4187c390 bf12ebcc 3eb8e6e7", + "418451cc be674880 3c0b12e4 c14f11fc c03e9528 3aba4722 -> 41ebdaca 403020a0 3c22dcc2", + "406e2610 bedfb250 3e1714eb 40ac28f0 bebe5840 392e99ff -> bfd457a0 bd856840 3e174109", + "c1481407 c0934413 38fd6b6b 4172cba4 3fe2e9f0 39f92f7d -> c1dd6fd6 c0cbfe8f 3a24c8b2", + "c0b770ac c071f9aa 4111fe63 410b3160 c06ecaee 39971781 -> c166e9b6 bd4baf00 4111ffa9", + "417f2d7c 3fbe5538 3a85662c 4167b4a8 c0388499 3ff3cf9e -> 3fbbc6a0 408bd79a 3ff3f136", + "403521a0 40567898 3fc68b14 c18721e4 c043e0fa 39ead170 -> 419dc618 40cd2cc9 3fc69a9e", + "41857ffa bfa5b06a 3d2725c9 40e70a7c c01dd7b4 3a4df89f -> 41177ab6 3f95fefe 3d2a6859", + "c03d99c8 3ebba210 3cef34a2 4083fa54 c00d5b36 3b69ed03 -> c0e2c738 4024cf78 3d0642d3", + "c0b333f8 c086d7b2 3c6fd8a5 c194642b c04eb288 3931c797 -> 414f2e5a bf7bf370 3c72d78e", + "410c5c58 c0428dca 3bb963df c103f2bf c00e3f76 3d6fda69 -> 4188278c bf513950 3d838c65", + "c196b82e 4013d4da 3adb1482 4131ccfe c08b99f8 3e0ce244 -> c1ef9ead 40d58465 3e0ea19e", + "c13c0ea6 c079d040 3fd90303 41772c98 4093e5d8 3b5fea8c -> c1d99d9f c10866fc 3fd97423", + "4090f354 3fe184d8 400ce4f9 41485034 c005682e 3db70826 -> c0ffad14 40762a9a 40129d73", + "407d1388 c002cfb1 3be6d34c 417a3ee8 3ca75300 39f2a062 -> c13afa06 c0041e57 3bf66b62", + "c0457678 3eb085f0 3e47d69c 413b0df2 bfb5403e 411c3b48 -> c16c6b90 3fe161ba 411f5abc", + "413302de 3f9de564 3998a2f1 4195f63e 3fc1eb34 3d279a51 -> c0f1d33c be901740 3d28d378", + "418b04e4 3fbd9920 3cbda048 4169d53c bf83e85e 3a38715a -> 4030d230 4020c0bf 3cc36e6c", + "bfb4ea50 be8634c0 39c85525 c16f1da7 bf450800 3d02cc76 -> 4158805d 3f01eda0 3d046b32", + "c14e84ec 40820612 40550b51 41269596 3ff990c0 41023f89 -> c1ba8d41 400743c4 41378282", + "c14a4593 c067af45 3ab6e835 c1789c7b bfb9aac8 3d356d45 -> 40395ba0 c00ad9e1 3d3b29a4", + "419648a0 bf5c7260 3f1213e2 415eac04 c07bd642 3ba20571 -> 409bca78 4044b9aa 3f135875", + "c1371a11 3fb91a58 390b53f5 c0c4eacc 4072466c 3e28d390 -> c0a94956 c015b940 3e28f858", + "c174e764 40156fa2 404749c1 c13fef39 bf3c8f38 39112b2f -> c053e0ac 40449370 40474c2c", + "c19aad12 c0023ff7 39bd23b7 c15b5fea 3f9325b8 3c953241 -> c0b3f474 c04bd2d3 3c983877", + "c18216ca c099c95d 408c4d83 c17928fc 40363210 3d17517e -> bf304980 c0f4e265 408d7c40", + "c10432c2 404ea640 3b3f9593 418df670 407d4400 4103a5fe -> c1d00fd1 bf3a7700 4103b21a", + "c0516b58 3f956d68 3be0d5ff 4131abf0 4083fe36 3f47b07f -> c16606c6 c03d45b8 3f49734d", + "bf741ca0 4053f07c 39105c18 c118f0df 3fb6f9f4 40cd5fbe -> 4109af15 3ff0e704 40cd6101", + "c1854a21 409ad4d4 3eb9254f c13f8f08 c081a12c 394d8c7d -> c0960a74 410e3b00 3eb940bf", + "c0525da8 4047f128 3f7cdce6 c10083dd 4022d998 398ee472 -> 4097d8e6 3f145e40 3f7cef28", + "c0654550 c07df114 3fff30e7 3ee4fe80 c08f2152 39470877 -> c080f290 3f014640 3fff3753", + "c0bfad62 c050d444 3fc5b759 4184fe0a 40346b9e 3f8ef703 -> c1b4e962 c0c29ff1 402a57ab", + "40ce0b20 c0270613 3c084950 c0d5bc2e 3fe7c3c0 3f547363 -> 4151e3a7 c08d73fa 3f5695ae", + "c041c7e8 4054c34c 3bf9e8e6 419568ca c0974a5e 3f813ba2 -> c1ada1c7 4100d602 3f823681", + "41413054 c08251dd 3edf733d c129c8a0 3e842250 3988d2c5 -> 41b57c7a c08a9402 3edf98e0", + "c14a2820 c073d6f8 40031a91 c10caccc bfaa057a 3f8e6746 -> c075ed50 c01ed43b 404a4e5a", + "4185eadc 4094cb20 3f47b8bf 419712c6 408372e8 3cb5cf85 -> c0093f50 3f0ac1c0 3f4d6773", + "c170d2e8 c0549055 3a8608be 4180a786 3ecb5b40 3ff3917e -> c1f910fa c06dfbbd 3ff3b426", + "40df4dec c084dca9 39edd73b c19b28ed 406fee1c 408260da -> 41d2fc68 c0fcd3b7 408264de", + "4106cc2c bf108ba0 397b8850 c18d16a6 4005f7a0 40c9d4b3 -> 41d07cbc c02a1a88 40c9d6f0", + "c0dc90e0 c0760e10 3e31bd6b 4122d8d0 40142434 3e534465 -> c18890a0 c0c51922 3ec283dc", + "418f2430 bf989b46 40125ed6 403f5880 bf5acf38 3ec2d9e1 -> 416e7240 beaccea8 402aba5a", + "c19e485e bef4bc40 4081dbdf 41824f88 405ba47c 3ffd7bf8 -> c2104bf3 c07a3c04 40c13b39", + "407d2988 407fe14c 3d6e84e3 416e24e0 3f1a41b0 3c8356e5 -> c12eda7e 405950e0 3d981f5e", + "4190a1c6 c053d646 3ad9ccd3 418ce36a 3f321c20 4084d641 -> 3eef9700 c0802ea7 4084e3ef", + "411549d8 40660300 3af8c3ba 414bd208 3fb8dbfc 3f72581f -> c05a20c0 40099502 3f72d4e9", + "414da9c8 bfa75770 3c791ef9 3fc7dba0 40918356 3a236624 -> 4134ae54 c0bb5932 3c81cd01", + "419ae388 c0801796 3e65fb66 c058f158 c09ce08f 3c54c079 -> 41b601b3 3f6647c8 3e734d67", + "c19fe398 40422d1c 3dcd9dcd c13557f3 becbbd70 3f8ed003 -> c10a6f3d 405ba4ca 3f9baa4a", + "c06e3f20 bee36a80 3d74c750 c13b9629 40419544 3d84f112 -> 41000661 c05e0294 3dff5a86", + "c04fe150 c09dbf38 3f919e9c 418f6e40 405a8c48 3d745c48 -> c1a96a6a c10582ae 3f994274", + "bfca9810 c04862c5 3f8bfe07 c16f1668 c084855e 3cbfe0ba -> 4155c366 3f814fee 3f8efe06", + "c0453f70 c08cd889 40aa0c53 c0879056 40785dd0 3ca47abe -> 3f93c278 c10483b8 40aab0ec", + "412d4320 3ebbff00 3a23f8cb 414db548 408cc700 3f2d6c8a -> c001c8a0 c0810710 3f2d95f4", + "c0e230ea 3e2a1cc0 3c224ff6 c009ded0 bf6278b8 3a668858 -> c09d4182 3f867ff4 3c30d063", + "c1266da2 3f927fe8 3d8bba70 416f50fc c0730c38 40af38a1 -> c1cadf4f 409e2616 40b167d3", + "c0c2e494 c0948ab9 3b00200d c1764eb5 40918b56 3fa68a38 -> 4114dc6b c1130b08 3fa6cae6", + "4093164c 406f4f38 3ec61cc7 c16f1796 403bf7ea 3c248088 -> 419c515e 3f4d5d38 3ecb4363", + "41693bb4 c09be3cf 3d8ce3ea c181af09 bf236530 3b0bb83b -> 41f64ce3 c0877729 3d915338", + "c077c178 3f794988 396311a2 c191974c 3f9fe9f0 3b89f906 -> 41653e3a be8d14b0 3b918670", + "c15c3180 c09b6da7 3f40c6a6 40983e60 40775738 3cba6acb -> c1942858 c10b8ca2 3f469bbc", + "41824898 3efaf940 409fad80 419f2ac4 c0912af1 3fe9a2cc -> c0671160 40a0da85 40da1652", + "c13e84f8 3fc723ac 3da642d4 c0d16da0 c03bd5e0 396033e4 -> c0ab9c50 408fb3db 3da6b7e6", + "c1906ef6 c04200de 3ed2c7ad 419fa0a6 3f513258 3c216367 -> c21807ce c0764d74 3ed7d811", + "409b4cd8 c0630f8f 3a4c24b3 c0feacba be95f530 3d2885fd -> 414cfcc9 c05050e9 3d2bc6ac", + "41739190 bffc8764 3c39b0ca 418aad08 4092f166 40a1610c -> c0072200 c0d2133f 40a1bdff", + "c190cce0 409eb5fc 410a79f7 c1227d79 c033a8c5 3adb3db8 -> c0fe388e 40f88a5e 410a80e9", + "bfc90e00 406ae630 40d9942c 4109c1a2 407b26e4 3f4cbf9f -> c122e362 be8205a0 40f32c44", + "40eb585c c08a9cc7 3c137e64 c19fa59e 409ed7aa 3c625377 -> 41da7bb5 c114ba38 3cbb3230", + "40828134 c05b72cc 3c6e60a6 c09ddc10 409f3c62 3f022ad3 -> 41102ea2 c1067ae4 3f05e575", + "c10463ee 3f87cf18 39812726 c0551900 c0753c07 3f05cb21 -> c09e3b5c 409c91ca 3f05dbec", + "c0160960 c013df0b 3e9b4d0e 40b4ee74 c0071d0f 3a93f998 -> c0fff324 be4c1fc0 3e9be218", + "c186dc9c 40419f8c 40f738c1 bf5eee20 3ec31860 3ebfbcd0 -> c17fca56 40293c80 41019a62", + "c02d9358 4081f264 3af0b2c8 c082f740 3f95faec 3c2b50ad -> 3fb0b650 4038e752 3c497827", + "c16b86de 4072dcc8 3b5ee874 419a7fe2 bfd7cd0c 3f831c11 -> c20821a8 40af61a7 3f838cc9", + "3ea33bc0 3fdaa88c 411d018f c1226ec8 4090f33a 3dee5cd5 -> 412788a6 c034922e 411ede60", + "c155850a c091edc0 3f104e4e c1315f8c c010bc06 3e062c91 -> c01095f8 c0131f7a 3f31d9c6", + "c1850d16 c0926586 3f469b78 c14c95e8 408736fe 3a7436ec -> c0761110 c10cce42 3f46d95d", + "bffd9090 c070d086 3dfee5e5 410b71b8 408e3b66 39966329 -> c12b23ca c10351d4 3dff85cc", + "beec5880 407d17b4 410cda31 40df45c8 c009bf62 3ac97d59 -> c0ee0b50 40c36b8b 410ce093", + "c0207ca0 bed410e0 409be95e 417c5970 c06aa2a4 3dfc20ca -> c1923c4c 40502088 409fda16", + "c1383507 4054b590 3fe27d0c c0ad8f92 bff0d3ae 3cf2214c -> c0c2da7c 40a68fb4 3fe645fa", + "4194a218 c03a0548 3e146873 40a58738 bfeec2e6 3d98fb0b -> 41568094 bf8547aa 3e60e9a3", + "40a12ed0 40908784 40c2e2e2 c17802cf 3ea58ea0 3a8f2fa8 -> 41a44d1c 40862e9a 40c2ec13", + "c14ad490 4039a828 3edb4b83 41822f98 4024f3fa 394b4f78 -> c1e799e0 3ea5a170 3edb68a3", + "409f0b2c bda4fdc0 3f200cea 3f03b060 4036bf9c 3f989acb -> 408e9520 c03be78a 3fe8a18a", + "40edfcec c04673a0 4060d030 c159ab2a 4095cf90 399415b1 -> 41a854d0 c0f90960 4060d552", + "c0c752a4 c0847524 3f92bfd6 c1701464 40222d88 3ef29f6d -> 410c6b12 c0d58be8 3fcf683a", + "bfd06690 4065d9a4 3da67285 c05b79a8 c0907a42 3b97b6c4 -> 3fe68cc0 4101b38a 3daff2f0", + "c10c879b bfe038b4 4088e44e 41131d7c 40072e04 3baa936b -> c18fd28c c0774a5e 40890f27", + "417ab134 bff92874 3977918b 413aea58 4019b2a0 3fa0f9d0 -> 407f1b70 c08b236d 3fa101da", + "41165aaa bef7b6b0 3c577b6f 414adab8 408de292 39d96d1f -> c0520038 c09d5dfd 3c5e67b2", + "40d3e1d8 beadeb70 41044fc8 416ab664 402ba682 3c8b8bb9 -> c100c578 c04163f0 410495a1", + "c0855ea0 c08d6016 391ff77b 419d30b2 bf064ff0 4047255b -> c1be885a c0792c30 40472856", + "bf7aa660 c0399bfe 3c8712e0 c03d36c8 c02b3d60 3ca4cec8 -> 3ffd1a60 be65e9e0 3d15f311", + "4178fa08 4091f07c 3ccab81b 40288990 c0145c5c 3e530a87 -> 414ed7a4 40dc1eaa 3e6c668c", + "c19b0497 3f05b158 3bafcea1 419d6d9e bf8cde1c 3b538b75 -> c21c391a 3fcfb6c8 3c0d6cee", + "c0f17e0c 3fb630ec 40729f4a bfd4c840 401f30ac 3da76207 -> c0bc4bfc bf88306c 4077da85", + "c0c07cac bf5cf7f0 3d7895e4 bf4d0100 bfd46668 3e3c87c8 -> c0a6dc8c 3f4bd4e0 3e7aaed1", + "c0f78ad4 408027e8 3ff909af c14f58be bfead686 3d37507f -> 40a726a8 40badd8a 3ffec49b", + "c09e1d00 bff4af4a 3aa375de 41942f94 4088f058 410a489b -> c1bbb6d4 c0c61c2a 410a4ddd", + "40b9679c c08018bc 3a7fb86a c0e586de 4060bf90 3a9d8c09 -> 414f773d c0f07884 3b0ffbdb", + "c17fc6fa bf762a30 3e60af69 41165642 3fe640d4 3f87e4a6 -> c1cb0e9e c030aaf6 3fa3fb7e", + "419a82ec c09a88af 3ca5d29f 416703ac c08787ae 3f9ed8ca -> 409c0458 bf180808 3fa1704a", + "41814f68 bf68d7a0 3e03e9ff 41507030 c01a1834 3be1ea52 -> 4048ba80 3fbfc498 3e0afa83", + "41433398 40594f90 3ac5ae44 414a147c 3f93d378 3985626f -> bedc1c80 400f65d4 3ae75c63", + "c1859116 bfc23c58 3ebf1c49 4189182a 4048b328 3e341649 -> c20754a0 c094e8aa 3f0c9628", + "c093ac0c c096bd73 3d85042b c0cbfa10 bf9a122e 408f4d60 -> 3fe13810 c06071cf 40916185", + "419b1fa6 3f379150 3aa62838 c18f29e6 c0445e97 3b4df027 -> 421524c6 407242eb 3b91cabd", + "c189f25c 3e1edd00 3abe44b3 c07ee848 c08d1fd6 3d20fd7e -> c1542aa6 409216be 3d270182", + "c0ac600a 404c43a0 3fcbf809 41534b1c 3e208fa0 38efbcb7 -> c194bd90 40423aa6 3fcbfc82", + "4199b14a c009db74 3f5fdfc5 405f1928 3fc2f0dc 3b95253a -> 417b9c4a c06b53e2 3f610b54", + "c182f21c c08e953d 3a88714a 4147d130 be7264c0 39353dba -> c1e6dab4 c0870217 3aa33b78", + "c11c01d6 3f226f68 39131e8f 40f579b0 405a30e4 3e8c3dc4 -> c18b5f57 c031950a 3e8c52b7", + "c161db92 40218764 3e2ea74c 418b98c0 401df930 39a5bf30 -> c1fc8689 3d638d00 3e2f021f", + "415f30dc 3fe92fcc 3b37273f c142ded0 3f218160 41161021 -> 41d107d6 3f986f1c 41161bb8", + "41457b68 3f0b7b78 3d16e2ba 400249d0 bfb2bf38 399c94ce -> 4124e8f4 3ff87cf4 3d18282d", + "c1653460 bf830dfc 407d0f49 4178f0d0 4093c836 3a017f6c -> c1ef1298 c0b48bb5 407d17ff", + "c1814aae c01bb5e8 3924a2fd c149b3d2 c063b83d 3cf612f6 -> c0638628 3f9004aa 3cf765a8", + "bed79300 3fde264c 40f915b4 c15e1cbd 3fea4d58 3f27c326 -> 41576025 bdc270c0 41070722", + "c0ac7c52 c091e775 3d481209 40f11fa0 4018956c 40e2db62 -> c14ecdf9 c0de322b 40e46bbc", + "c1927175 407cfa98 3cea78d5 c13a3412 c09ed51d 4082207e -> c0d55db0 410ea934 40830b1e", + "3faf7b40 40036bb6 3a87d489 418b0068 3eff89c0 3c86e092 -> c18008b4 3fc6f4fc 3c8f8102", + "412b9792 c01acc62 3f65a980 3e9c48c0 4059c6d8 3ce73b1d -> 4126b54c c0ba499d 3f6ce46c", + "c18c8c1b 3fd1c960 3d48969c 413bfd02 4003ddc8 40336929 -> c1ea8a9c bed7c8c0 40368c05", + "c19ad19b 3fc73050 4011aaf2 411ffa8a c0492f63 3b7a0920 -> c1eacee0 409663c6 4011ea05", + "40129c08 beee2180 3e4a7203 418fc5b0 405dd3b8 3961940b -> c17ae45e c07b97e8 3e4aaf5c", + "40bcad6c 40779eb8 3af16c3c c174e6b6 3fc0b988 3c801dd6 -> 41a99eb6 401741f4 3c8f63c5", + "c104d6fa 3f0d0a50 39d9aac2 3fb2d710 bf93689c 3c5a491e -> c11b31dc 3fd9edc4 3c61441e", + "4197454e 4064df94 3a88e443 c190c1df 3fa35008 39774681 -> 42140396 40133790 3aacb6d6", + "c02dbe68 c020dfed 3a6798aa 40824ff8 c04490dc 3bb42405 -> c0d92f2c 3f0ec3bc 3bd151e9", + "c18fdeae 4035f4dc 3d227f08 41778c28 bf9d4c60 399f196b -> c205d261 40824d86 3d23e2cc", + "4135d448 404c0c34 3c3b1530 410f6126 40668b90 40bb1176 -> 4019cc88 bed3fae0 40bb6f12", + "c1284539 408de740 38f47fd4 c18045f5 3c53fe00 3a1507ea -> 40b08d62 408d7d41 3a361404", + "412a2b46 be3fe160 3b1409f0 c008eb80 40316ff0 3b8f17d3 -> 414c6626 c03d6e06 3bd99ab9", + "c177faec 3dc55800 3fac7903 4058cb68 3e2faf80 3dc8128f -> c19716e3 bd9a0700 3fb8facf", + "41717904 c09168a2 3e972c0a 41819b58 c08a94ac 40093d0e -> bf8ded60 be5a7ec0 401c229e", + "4186207a bf718c88 3a4b7302 c19749bd 40208698 3d8a7b50 -> 420eb51c c05ce9ba 3d8c25cf", + "c19bb66a c020d688 411b4280 c11fbdea 4046c17c 3e86e31b -> c117aeea c0b3cc02 411f79b2", + "c18c6b97 bfd5c762 3f0b9dd2 c193bf31 4056fcb4 3eb90cfc -> 3f6a7340 c0a0f032 3f6824be", + "c187476a 3f84fb08 38e379fe c14555f3 402f8714 400b2c2f -> c09271c2 bfda1320 400b2e18", + "c112de67 c0405e91 3b5342b3 4148ac3c 405de820 3f977ae3 -> c1adc552 c0cf2358 3f97e56f", + "c11a076c bfd6f7d4 401a6f2a c161dffd c04cccde 399db19f -> 408fb122 3fc2a1e8 401a743a", + "41975938 4018bec0 3ee5dd2d 41030cf8 bf2b3830 3d33f177 -> 412ba578 40438ccc 3efc5d25", + "c0d8c5ec 4067c300 3ea033e6 c16545b6 3e70a0a0 3a5a7911 -> 40f1c580 4058b8f6 3ea0a28b", + "41606790 4058f768 3a928667 419a612a 4051c584 3ca9482b -> c0a8b588 3de63c80 3cb27b61", + "4165e2f8 401196e6 3e12f720 c0f28c94 40244aec 3d6a12fc -> 41af94a1 be95a030 3e4d817b", + "c116d1bf c0401912 3c2794e4 c14b08d0 409d24c2 3cca4004 -> 4050dc44 c0fd314b 3d0f1071", + "4144e1ac 40848468 3a1891d7 40c229ec c08b1868 39a0a220 -> 40c7996c 4107ce68 3a6c9162", + "4199f8f4 bfa33796 3a3b3e7e bf5c1880 40670d04 4068babc -> 41a0d9b8 c09c5468 4068c6e3", + "c1543a35 c06d98b0 3e756d03 c174cd6e 40942256 38e534c7 -> 40024ce4 c1057757 3e758c52", + "40865590 c00ec9c1 3d0814fe c05d8530 c007c2f7 3c95f54f -> 40f51828 bde0d940 3d531778", + "bfb6f560 3fcff16c 4101fd02 c14d69d6 bf188528 3ac03f7a -> 41368b2a 400e1a00 4102031a", + "c16efae0 c0615a27 40110700 c132d421 bfd44ec4 3de42961 -> c0709afc bfee658a 4018286b", + "c1960541 c040ad63 3f3d1583 c1780beb c09f97ca 3c9acf32 -> c04ffa5c 3ffd0462 3f41ec5d", + "c104f269 bf98b6fc 4009910e 410c51b6 c0650c14 3e0d45d6 -> c188a210 4018b096 401265c2", + "41704dc4 c0620450 3ac88423 c0875ec8 3f473198 3ebe9ee9 -> 4199fe94 c089e85b 3ebf6a6b", + "4197bf6a bf877a10 3e0a859a 41004cf0 c0480c61 3c2958db -> 412f31e4 40044f59 3e151e72", + "c17635df 40415bc0 4092609e 418c0044 bfbd8d52 3ebc545f -> c2038d9a 40901134 409e2639", + "c192dab4 408f4cf6 3f70a08f c18ab4ea c0846176 3f2917f5 -> bf825ca0 4109d736 3fccdc9c", + "c12a377b bd992980 3b1c86b4 c190515f c0726ee4 405bd425 -> 40ecd686 406da598 405bfb81", + "405c4968 401ccfa0 3d23b525 c0badf98 bffb6f32 3b40765f -> 41148226 408d439c 3d2fca48", + "419ed462 4014b484 3d911dc4 c15b99c6 c02252f6 40a055e2 -> 420650a2 409b83bd 40a29ab0", + "419a9aba bfd8aee6 38ef7369 c0a55c88 409f14fa 3a81cb85 -> 41c3f1dc c0d540b4 3a94a7cd", + "3f6dbd20 bfb9e5e2 39120c48 c15ddd48 3fe40e40 3a757dd5 -> 416cb91a c04efa11 3a8f416c", + "415554f0 4045b798 395284ef bf8a7e60 c05977e9 398bf682 -> 4166a4bc 40cf97c0 39ffac8e", + "c01d4a38 408810ca 3b17cdc9 410c3340 c076c568 3c857ba8 -> c13385ce 4101b9bf 3c989c12", + "4174d1b0 402739ec 3cfec80c 40b144f0 408d2be4 3e00437e -> 411c2f38 bfe63bb8 3e201f6e", + "c1647248 4075e460 3ea4a051 4162f908 c0854fa4 3f198933 -> c1e3b5a8 410020ea 3f6bdbb2", + "41219310 406876c8 3cf5a9fd c187876e 3e840b80 3f2df027 -> 41d850f6 4057f558 3f359f69", + "41722d5c bf2c4178 39626208 c0deaa6a 3e53b480 3cf388a8 -> 41b0c148 bf612e98 3cf57b6e", + "c12adb94 3e6f8900 3d949d48 bf1bc3e0 400bb8c8 3b843615 -> c1211f56 bff98070 3d9ce6b5", + "40d5c794 c016fdcb 3a9e1f14 4030ebf0 bdeffe80 3ac6053e -> 407aa338 c00f7dd7 3b3274bc", + "41006154 40763958 3c3fa001 c06866a0 c09cb580 3f6ecf09 -> 413a7afc 410be916 3f71cedf", + "be925c40 40992df0 3ed93b06 c0f9abd8 400d1d24 3b4829a3 -> 40f08614 40253ebc 3edaccaa", + "406623c8 bfb08764 3c22bd70 3e85b9c0 405a05e4 3e83296b -> 40556c90 c09924cb 3e884062", + "c09f48fe 3f109b90 3ad1371e c19ebebc bf7207d0 3c4907f7 -> 416dd8f9 3fc151b0 3c63706a", + "c190d3cd 3fca93ec 3a31024c c0993d28 c047eac0 3f9fe603 -> c1550906 40969a5b 3f9ffcbd", + "3fc93480 c04aa7ed 3bf590be c119e348 bf9ee062 3e244b97 -> 413309d8 bff66f78 3e2bfb6f", + "c0bc6278 be9dc7f0 3c541c1f c1115a40 40553dac 3d520a46 -> 404ca410 c068f6aa 3d838c1a", + "412c39f4 4088fd2c 3f845927 c10fe4e1 407cbad0 3bc5f77a -> 419e0f6a 3ea9fc40 3f851fc7", + "4186f5ac c0594a04 3c56888d 3fb11800 c0442084 4091ec09 -> 4177c858 bea94c00 40925776", + "419e78b2 3ffbbd9c 3ccc0c1a 419eab08 bfb0e41a 3bbc9acc -> bcc95800 405650db 3cfb399c", + "40a3ee50 c077f44e 3f66f38c 4181fc2e 402a0384 3d46a33d -> c1320134 c0d0fbe9 3f735eea", + "41870d4a bfd9278e 410882a5 4125580a c05da0f6 3bebf5b3 -> 40d18514 3fe21a5e 4108a035", + "c11a0fab c07e4a40 3ed3b85d 40fe3e7c c029b76b 3cc7a0d3 -> c18c9774 bfa925aa 3ee034d5", + "4093079c 402e3d46 3f27ec19 41462f20 bed44f50 390f5302 -> c0f956a4 4048c730 3f27f5c7", + "c0853b98 c09e7e38 410e4507 c01189b0 c016d4ad 3e7edfef -> bff1db00 c02627c3 41124095", + "c1587334 bfb70e98 39b47efa c1897834 3fb95d30 3ecd490a -> 4069f4d0 c03835e4 3ecd7708", + "412098c4 40684198 3a2a1f29 c115e9a7 bfbbfee4 3d882845 -> 419b4136 40a32085 3d8988cc", + "c0a6600c c0576cb6 38e2be20 c18f614a c00fcd54 399544c5 -> 414b928e bf8f3ec4 39d4e02d", + "c13a900c c0022ddb 3ca6fa9d c19f6435 bf985b78 3db5985f -> 4104385e bf58007c 3ddf5ba2", + "4062ef98 404bc808 3e3ec2f0 4128e75e 3dd2c500 3f234703 -> c0e056f0 404531e0 3f52f86e", + "3ff263d0 be583000 39603c6f c0f70904 3f772d00 3f768228 -> 4119d0fc bf969c80 3f7690e8", + "c116fa2c c0471f42 3c9f778e c19f2463 408f433e 3bed2886 -> 41274e9a c0f2d2df 3cdae5d5", + "c0ece638 bfc12516 3c4c2f40 c140bf74 bf1a8360 3b43f3e3 -> 409498b0 bf67c6cc 3c7d427b", + "c1997bd7 bfc7b670 3ee320a6 417f4ed8 c029d260 40dcb41a -> c20c91a2 3f8bee50 40eae67b", + "c134e537 c059aaca 40f2ca05 3f368620 be3ac500 3fad7a8d -> c1404d99 c04dfe7a 410f146c", + "c18ab1d0 409bd8b2 3bbde2f7 c07c2488 400bc9de 40167256 -> c1565a7e 402be786 4016d191", + "bd724e00 c08bd5c2 3d8bd125 c1398f65 40383798 398dd39d -> 41389d17 c0e7f18e 3d8c6866", + "40aa4528 c03c4747 3bb5f29b c1833863 3fef5564 3917549c -> 41adc9ad c099f8fc 3bbb8194", + "c152ec0c bf8d094c 3c004e3e c0b49a5e 401bdf0c 3a31e10b -> c0f13dba c06263b2 3c0b98a6", + "c195e878 3fbdfb40 3ada1d05 c1808c24 c0834fa6 3c79de12 -> c02ae2a0 40b2ce76 3c8aa166", + "414d3fe8 401d9d74 3eafb8a6 40a84130 408b8d6e 40cb7ba4 -> 40f23ea0 bff2fad0 40d6774e", + "c199d239 40844b04 3e11ca63 3fe5c950 400032dc 3bfa5f94 -> c1a82ece 4008632c 3e19a333", + "41121abe 404697b4 3d77315c 40b6fdec c015d3bc 391225d8 -> 405a6f20 40ae35b8 3d77cc6d", + "40981b08 c06e7471 3cbda82b 418e20d0 bfc35726 39971c7f -> c150341c c00cc8de 3cc02316", + "c0060948 4004e98e 3ad83497 c148c5ef bfa67066 396abcdc -> 4127439d 405821c1 3af746d9", + "40a66860 bf7f6818 40eb7c48 c16cadfd bf8d9e98 3929360c -> 419ff116 3ddea8c0 40eb7dd1", + "c09a7640 c09f7f01 3b123bd8 c08f12dc c089662a 3f3cd30b -> beb63640 bf30c6b8 3f3d6564", + "c07cfb18 3fa0b594 3e8f7b30 c19f50e5 3fa73024 40946289 -> 417f6304 bd4f5200 409d5a66", + "408d085c bf8e9fcc 3c05fc64 c1362588 be525cc0 40cf2d5f -> 417ca9b6 bf68a868 40cf708b", + "419465c2 4070fe48 3c21bf34 4093696c c04a5d17 3ab5d1ac -> 415f16ce 40ddadb0 3c38ccf1", + "410b02c0 4055217c 3d559847 401b5d68 4084c27c 3b4f6e3c -> 40c856cc bf518df0 3d62964d", + "c077a470 3f7b22f0 3c508273 3f1530c0 c01745eb 3b096e29 -> c08e7850 40560ea7 3c72fd3c", + "4125d70a c02c690f 39499e8b 40ff2bb0 3edc0330 3ed02f82 -> 401904c8 c047e975 3ed04973", + "c079c0a0 bf0b8220 397e9fb6 c0f2ea00 3f42d788 41012652 -> 406c1360 bfa72cd4 4101275e", + "40eff5c0 bf5a9920 4080893e c19c70af c0557982 3d161b32 -> 41d86e1f 401ed33a 4081b5b7", + "40b6a23c 4029102a 3f9d0dd3 c192c9b8 3fdae7a8 3aca1970 -> 41c07247 3f6e7158 3f9d412b", + "c12cb990 4059e9e0 3eb717a5 c08d88c0 3f7d1d40 391877af -> c0cbea60 401aa290 3eb72bd9" + ] + }, + { + "glsl": "vec3 cultmath_af_neg(vec3)", + "tolerance": "ulp-bounded", + "cases": [ + "3efc0ac0 c081676f 39dcef73 -> befc0ac0 4081676f 39dcef73", + "c17458ea bedfe490 3ae60300 -> 417458ea 3edfe490 3ae60300", + "4181bea0 c016ca53 3b988793 -> c181bea0 4016ca53 3b988793", + "40e29ec4 c037db20 3efba417 -> c0e29ec4 4037db20 3efba417", + "c15f2030 bf6e2100 3afc5f9a -> 415f2030 3f6e2100 3afc5f9a", + "401eda20 406edc78 3d01ebd0 -> c01eda20 c06edc78 3d01ebd0", + "c09a7dcc bf8a0f7a 3c6be155 -> 409a7dcc 3f8a0f7a 3c6be155", + "41829a4a 406e01f0 3d00bee8 -> c1829a4a c06e01f0 3d00bee8", + "c008d338 400b6a78 3d809c4e -> 4008d338 c00b6a78 3d809c4e", + "414c45b4 be9d84c0 3ea699b5 -> c14c45b4 3e9d84c0 3ea699b5", + "c19c6e18 c09bd5e4 3fdaf64f -> 419c6e18 409bd5e4 3fdaf64f", + "c191beec 403eaaaa 3ccef4d8 -> 4191beec c03eaaaa 3ccef4d8", + "416421b0 3f314f30 40c6150f -> c16421b0 bf314f30 40c6150f", + "40dc1108 c07c5ea0 404f3bad -> c0dc1108 407c5ea0 404f3bad", + "c14d4b76 409500de 3d6a2cd1 -> 414d4b76 c09500de 3d6a2cd1", + "c186922e 40798cc0 3b08fc23 -> 4186922e c0798cc0 3b08fc23", + "4199d34a 3fd71fd0 40db8808 -> c199d34a bfd71fd0 40db8808", + "c139556f c09cc2be 3a8da2bd -> 4139556f 409cc2be 3a8da2bd", + "40811a2c 3f855780 3aac3dd7 -> c0811a2c bf855780 3aac3dd7", + "40a09834 3fdbb7c8 3efc058f -> c0a09834 bfdbb7c8 3efc058f", + "40f24048 c0846439 3f721cc9 -> c0f24048 40846439 3f721cc9", + "40a7cebc 3fd45724 3a82a2fd -> c0a7cebc bfd45724 3a82a2fd", + "417aab4c c07ba257 3de353b4 -> c17aab4c 407ba257 3de353b4", + "4130e6d8 400394d8 3b58e883 -> c130e6d8 c00394d8 3b58e883", + "3fd6a7c0 4004b898 3b315857 -> bfd6a7c0 c004b898 3b315857", + "3f978b60 4012800c 40597cfb -> bf978b60 c012800c 40597cfb", + "418f5f50 bfaa20e2 3fb08496 -> c18f5f50 3faa20e2 3fb08496", + "c102d8f6 40589f20 3f81bbed -> 4102d8f6 c0589f20 3f81bbed", + "4116891c 40407940 38d8d03d -> c116891c c0407940 38d8d03d", + "40b31230 bf102bc0 3a73ece7 -> c0b31230 3f102bc0 3a73ece7", + "c11baa72 c02c2a6c 4044e2bf -> 411baa72 402c2a6c 4044e2bf", + "c13342ac 409a742e 390317c6 -> 413342ac c09a742e 390317c6", + "c17b9572 3ef66960 3b4ea273 -> 417b9572 bef66960 3b4ea273", + "c0bdcaba c047cfa8 3f52bc88 -> 40bdcaba 4047cfa8 3f52bc88", + "40c65e00 c03d4043 40cfbae4 -> c0c65e00 403d4043 40cfbae4", + "40d10974 3fd7ab38 3b55e65c -> c0d10974 bfd7ab38 3b55e65c", + "41867c16 c089322a 40183c16 -> c1867c16 4089322a 40183c16", + "c0e9a1fe 3fe31af8 3b5a6eb3 -> 40e9a1fe bfe31af8 3b5a6eb3", + "41835da0 bf880dc6 3ae3446e -> c1835da0 3f880dc6 3ae3446e", + "419e1362 407d5e40 40077fa0 -> c19e1362 c07d5e40 40077fa0", + "c142090c c01eddcc 3e53d510 -> 4142090c 401eddcc 3e53d510", + "c058c758 3e491480 3cc161a2 -> 4058c758 be491480 3cc161a2", + "c06fbce8 bfe3d0b8 3ed72f65 -> 406fbce8 3fe3d0b8 3ed72f65", + "c11512ed bf45e480 3a521ad5 -> 411512ed 3f45e480 3a521ad5", + "40885414 3f768ff8 3a4e89ca -> c0885414 bf768ff8 3a4e89ca", + "c191a8b0 3fb3f014 3e17c487 -> 4191a8b0 bfb3f014 3e17c487", + "4158c470 c088fdd0 39f0554f -> c158c470 4088fdd0 39f0554f", + "410017c8 400c90ec 3bee8913 -> c10017c8 c00c90ec 3bee8913", + "c0e85992 408c9346 3d6ffc55 -> 40e85992 c08c9346 3d6ffc55", + "406e2e50 bff3fbd6 3db45b36 -> c06e2e50 3ff3fbd6 3db45b36", + "412dd162 c077bc38 3e3ada16 -> c12dd162 4077bc38 3e3ada16", + "c02770a8 c05cdc50 40993420 -> 402770a8 405cdc50 40993420", + "c1733982 c0269f2a 3e87b1cd -> 41733982 40269f2a 3e87b1cd", + "bef886c0 c01b8c0b 3bb0eef3 -> 3ef886c0 401b8c0b 3bb0eef3", + "4047c6c8 3fc41630 3e331c67 -> c047c6c8 bfc41630 3e331c67", + "40f8f1ac 400704f8 3985649c -> c0f8f1ac c00704f8 3985649c", + "c16fd2e3 406a0ca0 3f153e8d -> 416fd2e3 c06a0ca0 3f153e8d", + "40917218 c05c9daa 3fa8b5b4 -> c0917218 405c9daa 3fa8b5b4", + "41150eae bd083500 3d3b84ad -> c1150eae 3d083500 3d3b84ad", + "c105f0cb c0810286 3da35083 -> 4105f0cb 40810286 3da35083", + "4186bb96 40488534 38f28adc -> c186bb96 c0488534 38f28adc", + "c1691f07 bd391380 3b67f446 -> 41691f07 3d391380 3b67f446", + "3f921f90 c051f9cd 3c83f4d7 -> bf921f90 4051f9cd 3c83f4d7", + "c1244eec 4087f412 3c25970f -> 41244eec c087f412 3c25970f", + "418fd878 bfbeeeb8 3f9807bb -> c18fd878 3fbeeeb8 3f9807bb", + "4136e5ae c071d9ea 39615327 -> c136e5ae 4071d9ea 39615327", + "410f4498 3f9eb4c0 3ab59e4a -> c10f4498 bf9eb4c0 3ab59e4a", + "bf8d0b20 bea50c50 3c87765c -> 3f8d0b20 3ea50c50 3c87765c", + "4156ee2c c05ae7ae 40c4766a -> c156ee2c 405ae7ae 40c4766a", + "4152150c 40011528 3e4831f0 -> c152150c c0011528 3e4831f0", + "c0760c98 4052b74c 3a07b3be -> 40760c98 c052b74c 3a07b3be", + "c17d3c4a 3f2b1280 3c876602 -> 417d3c4a bf2b1280 3c876602", + "c0d4af2c bf13bd18 3a21199f -> 40d4af2c 3f13bd18 3a21199f", + "c0ce5c96 c055cb1d 3d4b109b -> 40ce5c96 4055cb1d 3d4b109b", + "40b2d260 40747c08 40a1f3e9 -> c0b2d260 c0747c08 40a1f3e9", + "c0bd2c44 3fd21040 3f86bc07 -> 40bd2c44 bfd21040 3f86bc07", + "c19d1b45 408f1480 3bf48100 -> 419d1b45 c08f1480 3bf48100", + "bfafa460 3f2b2a70 3cbd8640 -> 3fafa460 bf2b2a70 3cbd8640", + "c0bd6db0 bf039dc0 3baf102b -> 40bd6db0 3f039dc0 3baf102b", + "40308058 c0000d36 3b9f9b7c -> c0308058 40000d36 3b9f9b7c", + "c0bd3cb4 40002ee8 3cb7dd1f -> 40bd3cb4 c0002ee8 3cb7dd1f", + "41353bca 4059343c 399d9fce -> c1353bca c059343c 399d9fce", + "c11aceae 405ced38 3f54f414 -> 411aceae c05ced38 3f54f414", + "c0a4bcac bef615e0 3e8056d5 -> 40a4bcac 3ef615e0 3e8056d5", + "4182cb38 bf931a3c 3c9fceab -> c182cb38 3f931a3c 3c9fceab", + "40f245a0 4054f56c 3dadd37d -> c0f245a0 c054f56c 3dadd37d", + "3de87e00 be27bb00 3cf157c5 -> bde87e00 3e27bb00 3cf157c5", + "c15df840 404c4b94 3f17baeb -> 415df840 c04c4b94 3f17baeb", + "c10a7494 408c38b4 3b0665ac -> 410a7494 c08c38b4 3b0665ac", + "40d98aa0 40217a8e 39cc87d7 -> c0d98aa0 c0217a8e 39cc87d7", + "c1809bfe 40298852 3a07048f -> 41809bfe c0298852 3a07048f", + "414cec08 409bb808 3cae75e9 -> c14cec08 c09bb808 3cae75e9", + "c189f923 40642d68 392b49e8 -> 4189f923 c0642d68 392b49e8", + "415cf3a4 c081e3ba 3f9f2401 -> c15cf3a4 4081e3ba 3f9f2401", + "c199752a 3fb04f9c 3cf5184a -> 4199752a bfb04f9c 3cf5184a", + "40f94778 3f11d420 3a548d8e -> c0f94778 bf11d420 3a548d8e", + "c19b4276 3fd04cb0 3df5c4e7 -> 419b4276 bfd04cb0 3df5c4e7", + "41264450 c08ad872 3f9a6bce -> c1264450 408ad872 3f9a6bce", + "415bf32c c040a61b 4022c343 -> c15bf32c 4040a61b 4022c343", + "410a04e4 bb9f4000 404d2cf2 -> c10a04e4 3b9f4000 404d2cf2", + "4155442c 3f61b010 3998c251 -> c155442c bf61b010 3998c251", + "40d42fec c011265a 3e8b7831 -> c0d42fec 4011265a 3e8b7831", + "3fbfbb60 c083aacc 3c20f371 -> bfbfbb60 4083aacc 3c20f371", + "419d8fc2 3f9a2a2c 40f35cea -> c19d8fc2 bf9a2a2c 40f35cea", + "3fdcd500 3fc9baac 3c1139e1 -> bfdcd500 bfc9baac 3c1139e1", + "40dfea98 c06b0ded 3a53c74c -> c0dfea98 406b0ded 3a53c74c", + "c119ff97 3f639ec8 3c03fedd -> 4119ff97 bf639ec8 3c03fedd", + "c1863586 3fc69470 39930bb8 -> 41863586 bfc69470 39930bb8", + "3dc47600 bfcef89a 3e9fa9ab -> bdc47600 3fcef89a 3e9fa9ab", + "c0dea116 40987c6c 3f5e0854 -> 40dea116 c0987c6c 3f5e0854", + "40588aa8 bfbbee54 3bc7c364 -> c0588aa8 3fbbee54 3bc7c364", + "c1657b07 c09c2844 3d0ab028 -> 41657b07 409c2844 3d0ab028", + "4110a83a bfba0494 3bd9d851 -> c110a83a 3fba0494 3bd9d851", + "414536d0 c05b53b2 3cb4194b -> c14536d0 405b53b2 3cb4194b", + "413554e8 c098d237 392aa221 -> c13554e8 4098d237 392aa221", + "40b2b63c c035564f 3c34b2ca -> c0b2b63c 4035564f 3c34b2ca", + "c03b7528 c0540222 3960d9a6 -> 403b7528 40540222 3960d9a6", + "40ff7f18 bedfdc30 4064a773 -> c0ff7f18 3edfdc30 4064a773", + "3f4c6760 4021af38 4093fe72 -> bf4c6760 c021af38 4093fe72", + "c19ef0c6 c05770f5 40973bc8 -> 419ef0c6 405770f5 40973bc8", + "419c3ddc c038b809 39be6763 -> c19c3ddc 4038b809 39be6763", + "408dd94c bf6c0c98 3b0e2a1f -> c08dd94c 3f6c0c98 3b0e2a1f", + "40989600 bf7ba960 3e2d1611 -> c0989600 3f7ba960 3e2d1611", + "41757f3c 408fa68a 3f46d7bd -> c1757f3c c08fa68a 3f46d7bd", + "414ad5a4 bef9b330 3c6bc43c -> c14ad5a4 3ef9b330 3c6bc43c", + "418418e0 c001bff5 3e1538dc -> c18418e0 4001bff5 3e1538dc", + "c190de85 40682dcc 39d4b575 -> 4190de85 c0682dcc 39d4b575", + "c106388d 407f8cc0 3f8bff54 -> 4106388d c07f8cc0 3f8bff54", + "410dae5a 40766320 40434b1b -> c10dae5a c0766320 40434b1b", + "c0092ea0 3f9b7d08 3b01d381 -> 40092ea0 bf9b7d08 3b01d381", + "403aab00 400e89c6 3a6d2917 -> c03aab00 c00e89c6 3a6d2917", + "c0341f30 404cbb14 40abb3a6 -> 40341f30 c04cbb14 40abb3a6", + "c0b7cdde c05197bf 3df26676 -> 40b7cdde 405197bf 3df26676", + "41300500 c09977f1 3f9811a6 -> c1300500 409977f1 3f9811a6", + "412843e4 3edd1bd0 3aa4af7c -> c12843e4 bedd1bd0 3aa4af7c", + "c1078c8a 4074e9e8 38de42ee -> 41078c8a c074e9e8 38de42ee", + "bf572b40 3f892968 3c9af5d5 -> 3f572b40 bf892968 3c9af5d5", + "c0ff99be 4091edbe 3b50a9fc -> 40ff99be c091edbe 3b50a9fc", + "41487bf0 3ffd6888 3b6e2aa8 -> c1487bf0 bffd6888 3b6e2aa8", + "c0fa92e0 40793e4c 3d286ec9 -> 40fa92e0 c0793e4c 3d286ec9", + "c0a5f032 c0516959 3f4bdcb4 -> 40a5f032 40516959 3f4bdcb4", + "c091a4bc bff25706 3ab133d9 -> 4091a4bc 3ff25706 3ab133d9", + "405e16c0 bf904522 3e12e3e7 -> c05e16c0 3f904522 3e12e3e7", + "bf128f40 3f1062c0 39dc7772 -> 3f128f40 bf1062c0 39dc7772", + "c19a3807 c011c20a 3fd07989 -> 419a3807 4011c20a 3fd07989", + "c0f77410 3e8c1c80 3bc5b323 -> 40f77410 be8c1c80 3bc5b323", + "406128c8 bf2b1478 38d7592b -> c06128c8 3f2b1478 38d7592b", + "417ac978 40959388 38f74bf8 -> c17ac978 c0959388 38f74bf8", + "41429110 c0118863 3f2e7486 -> c1429110 40118863 3f2e7486", + "417a0d6c 3f94f880 39d5612b -> c17a0d6c bf94f880 39d5612b", + "40ab5240 c08f4ec0 3b2a89e3 -> c0ab5240 408f4ec0 3b2a89e3", + "c1829d82 bf740b98 3fdd7551 -> 41829d82 3f740b98 3fdd7551", + "c194e18c bf9dd37c 3abe9466 -> 4194e18c 3f9dd37c 3abe9466", + "419df6c0 4000edc6 3cf5a8d0 -> c19df6c0 c000edc6 3cf5a8d0", + "c0a1a748 409ddb20 393c1922 -> 40a1a748 c09ddb20 393c1922", + "c0d8d13a c00b5cd0 3e8651dc -> 40d8d13a 400b5cd0 3e8651dc", + "bfd5e390 3d86bf40 3a3ae422 -> 3fd5e390 bd86bf40 3a3ae422", + "4191e8fa 400a6c5e 3c9c9534 -> c191e8fa c00a6c5e 3c9c9534", + "3f520880 be5c1920 3a8b6dd1 -> bf520880 3e5c1920 3a8b6dd1", + "412a2d52 3f527250 392287bf -> c12a2d52 bf527250 392287bf", + "bfcf57f0 3ffaf088 3f372230 -> 3fcf57f0 bffaf088 3f372230", + "3ec0f680 40612dec 3d080075 -> bec0f680 c0612dec 3d080075", + "415aadd8 4088657c 400b58c4 -> c15aadd8 c088657c 400b58c4", + "419e521c c011f6f0 3a148d10 -> c19e521c 4011f6f0 3a148d10", + "c170e5b8 3f7172c8 3ee9471d -> 4170e5b8 bf7172c8 3ee9471d", + "3e8d9740 3e39d1e0 40ed9054 -> be8d9740 be39d1e0 40ed9054", + "419456e6 be8832a0 410f30e2 -> c19456e6 3e8832a0 410f30e2", + "c185779f bf2cd888 39b99adc -> 4185779f 3f2cd888 39b99adc", + "41906b0e 3fe3ae58 3b9c9db7 -> c1906b0e bfe3ae58 3b9c9db7", + "41512688 3f3691d0 4088c358 -> c1512688 bf3691d0 4088c358", + "40b613bc c013f06b 3f7045b1 -> c0b613bc 4013f06b 3f7045b1", + "c1438de3 3eaec3f0 38df2dd8 -> 41438de3 beaec3f0 38df2dd8", + "410fab4e 406924a8 3a059b16 -> c10fab4e c06924a8 3a059b16", + "3fee2eb0 bde2ce80 3bc57694 -> bfee2eb0 3de2ce80 3bc57694", + "414a5cac bebb0550 3ac352f8 -> c14a5cac 3ebb0550 3ac352f8", + "c1999e4c 4054b7ac 39d5832f -> 41999e4c c054b7ac 39d5832f", + "410fb64e c09ba619 40cabe72 -> c10fb64e 409ba619 40cabe72", + "c12c0135 40417f88 39b77c79 -> 412c0135 c0417f88 39b77c79", + "c190bbc6 bf9df77e 3b083d3e -> 4190bbc6 3f9df77e 3b083d3e", + "40ce35d0 3f14c660 3a424194 -> c0ce35d0 bf14c660 3a424194", + "c1543ff3 c09b3c58 3bc30338 -> 41543ff3 409b3c58 3bc30338", + "418d4118 bfbccb4a 38d94c4d -> c18d4118 3fbccb4a 38d94c4d", + "c0b818c2 40895416 3951a833 -> 40b818c2 c0895416 3951a833", + "4110d79c 4034c16c 3e3e5efd -> c110d79c c034c16c 3e3e5efd", + "4154e87c bf546028 4105cf3e -> c154e87c 3f546028 4105cf3e", + "40abf864 c0154be8 40dff963 -> c0abf864 40154be8 40dff963", + "41701958 408a095a 3fd823aa -> c1701958 c08a095a 3fd823aa", + "c137198b c07e8fc5 3d7bbdb2 -> 4137198b 407e8fc5 3d7bbdb2", + "41950eae bf63d3b8 3cb02956 -> c1950eae 3f63d3b8 3cb02956", + "40edad98 c0621656 3a20cafa -> c0edad98 40621656 3a20cafa", + "411ac71e c08bb7cc 3b5e80ab -> c11ac71e 408bb7cc 3b5e80ab", + "419598ca bf9bf470 3cc01f08 -> c19598ca 3f9bf470 3cc01f08", + "c19fa724 406fda78 39bc1be6 -> 419fa724 c06fda78 39bc1be6", + "c12bcc8f 3fd19f94 40520744 -> 412bcc8f bfd19f94 40520744", + "c18c3a44 409f6f58 3abba5e9 -> 418c3a44 c09f6f58 3abba5e9", + "c0f7ae4c bf81d362 3c0a54cc -> 40f7ae4c 3f81d362 3c0a54cc", + "c1706d3d bf5dcf00 39b9bbde -> 41706d3d 3f5dcf00 39b9bbde", + "c1037c23 4037018a 3e0b5558 -> 41037c23 c037018a 3e0b5558", + "c0db917e 40916ccc 406a173c -> 40db917e c0916ccc 406a173c", + "413e35dc 4098326a 40a32c38 -> c13e35dc c098326a 40a32c38", + "3fbb4b70 3fda39a4 3f511369 -> bfbb4b70 bfda39a4 3f511369", + "4180d7fc 409b0840 3ec99c9a -> c180d7fc c09b0840 3ec99c9a", + "c0c2f534 4069b6e8 39240f54 -> 40c2f534 c069b6e8 39240f54", + "3ed5bc80 3faa94f0 3c1557ea -> bed5bc80 bfaa94f0 3c1557ea", + "4165258c 3fb7cbd8 3b0eb630 -> c165258c bfb7cbd8 3b0eb630", + "418d8800 c05ea503 39ada273 -> c18d8800 405ea503 39ada273", + "c0de0376 3fcbe908 38f1fbf6 -> 40de0376 bfcbe908 38f1fbf6", + "c18603c2 3f1d1020 3eedc8ca -> 418603c2 bf1d1020 3eedc8ca", + "c190bdf6 bf06f300 3b1fd3e7 -> 4190bdf6 3f06f300 3b1fd3e7", + "40e5a858 3fb1f4a0 3c57419f -> c0e5a858 bfb1f4a0 3c57419f", + "401d2ca0 c03d8422 40486d79 -> c01d2ca0 403d8422 40486d79", + "c1538a4d 3ff99408 41141678 -> 41538a4d bff99408 41141678", + "c172af00 c05a2480 38f5b5b6 -> 4172af00 405a2480 38f5b5b6", + "3ef82c00 4092d1ba 3d3e4106 -> bef82c00 c092d1ba 3d3e4106", + "c0c4953c 3f91c9bc 3f4b8e6f -> 40c4953c bf91c9bc 3f4b8e6f", + "410b768a bf2286c0 3a06fef1 -> c10b768a 3f2286c0 3a06fef1", + "c0b4480c bf91b17c 3f837ce7 -> 40b4480c 3f91b17c 3f837ce7", + "4145858c be964360 3b7a0d9f -> c145858c 3e964360 3b7a0d9f", + "41894d42 bee47880 39b82ebd -> c1894d42 3ee47880 39b82ebd", + "c1904f74 40632be4 3d46c079 -> 41904f74 c0632be4 3d46c079", + "c0157768 3fd76274 3ab11e7c -> 40157768 bfd76274 3ab11e7c", + "40936bfc 408fb15e 3c65ccd1 -> c0936bfc c08fb15e 3c65ccd1", + "be9380c0 3ff15fb4 40b11687 -> 3e9380c0 bff15fb4 40b11687", + "c110700c 404dbf00 41138d90 -> 4110700c c04dbf00 41138d90", + "c1831e55 40493ce8 3ea2d561 -> 41831e55 c0493ce8 3ea2d561", + "bf56aa20 4016431a 40aec135 -> 3f56aa20 c016431a 40aec135", + "41910dd2 bfd9e4ec 38ed67e1 -> c1910dd2 3fd9e4ec 38ed67e1", + "c1698cc3 bfe60f9c 3af0dbee -> 41698cc3 3fe60f9c 3af0dbee", + "409b5b2c 4021c59c 3eed5d61 -> c09b5b2c c021c59c 3eed5d61", + "40541550 c02624d2 3acecb9f -> c0541550 402624d2 3acecb9f", + "4175da2c c089080c 3b59809f -> c175da2c 4089080c 3b59809f", + "bf6ff0e0 c0897064 3b39dbfd -> 3f6ff0e0 40897064 3b39dbfd", + "c0f6ba64 4013ff64 3e581088 -> 40f6ba64 c013ff64 3e581088", + "c0bf19ac c02fed6f 3f0c95c5 -> 40bf19ac 402fed6f 3f0c95c5", + "c1440046 c084d9fe 3d886735 -> 41440046 4084d9fe 3d886735", + "4145ff20 400bc234 3e439bc3 -> c145ff20 c00bc234 3e439bc3", + "40943e80 beffd6c0 3ecd39b9 -> c0943e80 3effd6c0 3ecd39b9", + "c1588660 c0815d46 3d2d1878 -> 41588660 40815d46 3d2d1878", + "3f874290 bff7a22e 3bb41522 -> bf874290 3ff7a22e 3bb41522", + "419774c6 409e0434 3a7b211e -> c19774c6 c09e0434 3a7b211e", + "40fb14a4 40832f52 3c8ea506 -> c0fb14a4 c0832f52 3c8ea506", + "4105c2d2 bf8c946e 3f0f772b -> c105c2d2 3f8c946e 3f0f772b", + "40f19fc4 3f9184cc 3f063e8c -> c0f19fc4 bf9184cc 3f063e8c", + "c06684a0 c0520d70 38da6f3d -> 406684a0 40520d70 38da6f3d", + "c10b22f7 3f8350f4 39aeed08 -> 410b22f7 bf8350f4 39aeed08", + "c117c988 408d39c2 3ac5141e -> 4117c988 c08d39c2 3ac5141e", + "40f0d594 4047ae78 391bcdd9 -> c0f0d594 c047ae78 391bcdd9", + "c1875b32 4095730a 3f65218c -> 41875b32 c095730a 3f65218c", + "3f42b3e0 400e608c 40f2db8f -> bf42b3e0 c00e608c 40f2db8f", + "41329144 40900c50 3f2912a2 -> c1329144 c0900c50 3f2912a2", + "418d9c98 bfd48d7c 3ef92b62 -> c18d9c98 3fd48d7c 3ef92b62", + "bfa0d7f0 3fc5d1b0 3b76c3ca -> 3fa0d7f0 bfc5d1b0 3b76c3ca", + "c1002637 be1c3180 3e984442 -> 41002637 3e1c3180 3e984442", + "41499930 c072d0c4 39aefc40 -> c1499930 4072d0c4 39aefc40", + "4183ee7e c03c3f97 3b215064 -> c183ee7e 403c3f97 3b215064", + "410aea1e 40729c98 3cef6591 -> c10aea1e c0729c98 3cef6591" + ] + }, + { + "glsl": "vec3 cultmath_af_scale(vec3, float)", + "tolerance": "ulp-bounded", + "cases": [ + "c04b7280 c0174e26 3bdb8610 c0465c1e -> 411da3c4 40ea79c6 3caa3b15", + "c109813e c04ba610 3a29032c 3f740248 -> c1031069 c0421c18 3a23e6ef", + "418dbc48 3fc9fd4c 3a699a57 3fabbb98 -> 41be2942 40078036 3a9ff1b6", + "412f24a4 40831f50 3e54cdf3 4075bde8 -> 42281ffe 417bbc79 3f4c4a9f", + "bfa6d2d0 40583950 3b0c439a 4020c66c -> c0518a1f 4107cb6a 3bb08c11", + "418f5fd0 4039470e 40d128f4 beb1d5b8 -> c0c731e1 bf80b4ce 40114c14", + "3fd0af80 406ed19c 3e92a609 405c02e0 -> 40b3592a 414d3ed1 3f7c11e2", + "c1569360 402725ac 3d14f0a7 401a9a7e -> c2019626 40c9e311 3db3f8b3", + "415bc3f4 be2f2880 3bcb2c4b 3fbf0ad8 -> 41a40082 be82b6a3 3c17f1b6", + "c081962e 40532c58 3937e56c 3ee3ed20 -> bfe6c066 3fbc03ec 38aa45e0", + "c0f7696a c094abef 3bee0858 3f915e9c -> c10c7e25 c0a8d8a8 3c0762f5", + "c1035aae c0975f81 3928c3b8 c0140c4e -> 4197ed7a 412f14fa 39d22a12", + "c016a7d0 408a57cc 3efc0dbe bef42b00 -> 3f8fb140 c003f2eb 3e70684a", + "bffb14d0 c08b2db8 3d59b415 3f862d50 -> c0039956 c091e522 3d643c43", + "c0ff55b0 4082c264 3d940e76 bf343c54 -> 40b3c46c c0381efe 3d5082aa", + "bec4a980 c02b69e2 3c9947dd c022f0e2 -> 3f7a589f 40da3497 3d432741", + "c162c4c9 bf8f5310 3fa95e3b bfd64152 -> 41bdca5d 3fefe803 400dc06b", + "400342f0 4098a180 3da4f2fc 3f8cc214 -> 4010583c 40a7d811 3db567a9", + "4138fed0 4006afae 3d0ce3ec 40534600 -> 4218ac9b 40de4f41 3de8a361", + "c053d138 bfe4fce6 3a28e98d 3de21840 -> bebb12c4 be4a3cf4 38964e6a", + "41545d14 be471620 3ce3157d c0090262 -> c1e34f93 3ed51963 3d732e1e", + "c198ba8d c0890aec 3bba6676 400e7f82 -> c22a0719 c11890a2 3c505392", + "418c180e 4011918e 3c0b85d4 bda196a0 -> bfb0db1d be37c461 3a308679", + "c11aca90 3faf1ab4 395e403c c059d575 -> 4203b6c5 c094ffa4 3a46833b", + "404f5028 c046178a 3cf837c0 40503f16 -> 4128a437 c12123f1 3dc9f4d2", + "c13e504d 40205d5c 3e4b4e18 be92b940 -> 405a2707 bf37d294 3d690f7f", + "c0be91bc c07f5e8a 3e38e9b1 3ff64924 -> c137567d c0f5adce 3eb1e7d4", + "4013df50 3fa98bb0 3a999848 40757616 -> 410dc8ed 40a290e4 3b93b528", + "411df382 c08d6f06 3da30245 c07af655 -> c21ad7c9 418aa684 3e9fd41d", + "411be890 c031e8ee 3a239224 c05a258e -> c204daea 41179a74 3b0e0d81", + "c19426a1 c091470c 3bbeaf11 4025cc22 -> c23fe610 c13c2d45 3c77ecc7", + "4068c580 c082dbd2 3c0120ea c0320ba2 -> c121e3e7 41360595 3cb3c88a", + "40b9daa8 bf2c21c8 3b953249 c00be3d0 -> c14b1e3a 3fbc1f0b 3c2346d8", + "414fe1a0 c01cbd91 3b6bab82 c05666a0 -> c22e19f2 4103454d 3c462ed0", + "414d6eb8 c09832c0 40232889 4057edf4 -> 422d46f0 c1806017 41099ef7", + "c04bc148 3f8d6c08 3aa610c8 bfc755fa -> 409ea7ad bfdc3cf2 3b01b9b5", + "3f0c0f00 bf6f07f0 3f0554e9 c0198a04 -> bfa800f3 400f5c9c 3f9fef68", + "c18de15c c08e44de 3b93f4ac bf906b3e -> 41a01467 40a084ac 3ba7b778", + "c195efc5 bf46c490 400ec9ed c02fce90 -> 424defc6 400880c0 40c41ef7", + "c0831366 4056e0b8 4014beb9 bff5b528 -> 40fb9ca4 c0ce3d1f 408ec3ef", + "410ddd98 bc16fe00 399d16a1 c03fffc7 -> c1d4cc25 3ce27cbd 3a7249e1", + "c0fefb3a 409bd6b2 3a1b6f6b bd87e4a0 -> 3f075a33 bea572c9 38286d7d", + "411dfcb4 bc2fa400 3a6690e6 3fe07ce8 -> 418a8a33 bc9a0533 3acc5a17", + "c0cbdda0 c0282171 3e3da2bc bf1670dc -> 406f9b78 3fc59b6f 3ddee49c", + "41504a5c c090c19a 3b68eb97 c021a3b6 -> c20383f7 4136cca8 3c13c25e", + "c0c2e880 c04d4f9c 399da94a 3f37e368 -> c08c0157 c0137a4a 39692e24", + "c17ed510 4097433e 3a786b98 bf47c3f0 -> 4146daab c06c1216 3a45e14f", + "c092b35c c05d188c 3e86214e 4078719a -> c18e5edb c15691e4 3f822ccb", + "c15f472f 3f9b2e38 3a832e5f 4057b88a -> c23c25bc 4082c3ae 3b60470b", + "3f1b5a00 c0816c5d 38dcf80a 3e5a93c0 -> 3e04a44d bf5d01f3 37c49c25", + "c11eb3f7 c01ef476 3c974152 3eec52b0 -> c092812a bf92bcb4 3c0bb802", + "416a98e8 3f8efd18 39015aa2 40169b20 -> 420a03c0 40283de2 39aac3f7", + "412c0d7c bf118278 3b945c28 bf99a400 -> c14e848f 3f2ea863 3bb28101", + "3f8313e0 3ff1dca4 40693c76 3eadee20 -> 3eb21cb6 3f245314 3f9e76e1", + "c0772cb0 3d625480 3908ed09 bfb0068e -> 40a9f50d bd9b9fe4 3941b017", + "413aaa9c 40215f36 39b24c72 3f3d10c8 -> 4109dc32 3fee5bbe 3988eb66", + "4149c6f0 c03db0a5 3d8dd901 3fe0c350 -> 41b12804 c0a68b49 3df921ef", + "c19416a5 40149ab4 4066b1c5 3feb6b50 -> c2082edd 4088a84e 40d4263e", + "c192358a 40356ab2 4092b9c2 c04ecca2 -> 426c37e8 c1128cdf 416d0dd9", + "4110999c 3ec23a20 3ed38e3f bda1f620 -> bf36f742 bcf5c295 3d05d8a2", + "41916d5a bf7cfc88 3e16cf56 40626bf4 -> 42809fdd c05fc19f 3f0566e7", + "414ef404 bfd4e02a 3ca2c6ed be9af010 -> c07a81b8 3f00d679 3bc52c01", + "4105b414 3fc9bd48 3d1f231f c07b0fc2 -> c2031fca c0c5d901 3e1c1b02", + "c188d708 3fe1d7f4 3ef24be8 ba756000 -> 3c83291b bad8785d 39e83fd2", + "c162ff36 40670188 3f03ed8b 3fa6dc14 -> c193f4a2 40969196 3f2bfc63", + "c1920b59 c000ce4a 3f1be9d3 c06d3dc6 -> 428757bf 40eebc1f 40107e4e", + "3fa2c240 3f81a0c0 3b6048d7 3ff7f2dc -> 401da3d3 3ffb1a25 3bd95e81", + "41244700 404a62d0 3f9ad306 403b2474 -> 41f02e83 4113f300 40625d0d", + "4134ab2a 3ea947a0 3bf1e884 bf604734 -> c11e4805 be944dc1 3bd4403e", + "c1742612 c056fc30 3f330ff4 be16d960 -> 400fdd9e 3efd5ca1 3dd30835", + "411c23d6 c07c353e 38ece3e6 bf768c94 -> c116602e 4072e5a9 38fe8856", + "c136d725 40808d66 3e46941e bf888b48 -> 41430b8c c089221e 3e53d9a0", + "c1907f1e bf2d1608 40b59155 403ae012 -> c252f5aa bffcb303 41848aa8", + "413d2144 c08cb5b2 3c8bedae 3f9f74c8 -> 416b9be0 c0af4a14 3cae795b", + "40b8a358 3f784a00 3c38c46b c00e49e2 -> c14d3fc9 c00a00b4 3ccd8292", + "c1976a55 400b33bc 3eac7901 40490dce -> c26dd54e 40daa641 3f87766a", + "415a4b0c 3fc30890 3ab39510 3ea10640 -> 40894e87 3ef55a4b 39e44ce2", + "c0fc6a74 bfc701a2 3c043463 407663fa -> c1f2f0df c0bf894a 3cfec4b6", + "c1828fda c0191106 407bd08f 40601870 -> c26494ab c105fd82 415c6ed6", + "c1079448 bf9d09c0 3fa5843b 3f852f28 -> c10d1206 bfa365fc 3fac38bc", + "418dff9e 408cddd0 3e968fe2 bfb3ec00 -> c1c79946 c0c601ea 3ed3a6c1", + "4169e368 40907dc2 396abd17 3fe35748 -> 41cfb463 410050c6 39e1738d", + "c1844d10 400b7838 3f09dc91 3e305f20 -> c0364c48 3ec02cf3 3dbdf765", + "41225526 3eefecb0 3c99e16a c072731e -> c219bd7d bfe339a2 3d91d06c", + "c07ee2f0 c0748b44 3d49d85d 3efdb6e0 -> bffc9c5c bff25c53 3cc812cc", + "40df3350 c07457d6 395ab73b 3fd0d0ec -> 41360fd8 c0c74ec6 39bb3523", + "c18fac34 4096413a 3f3b98a0 3e8e0eb0 -> c09f7386 3fa6c19a 3e503477", + "c165911a 3db49140 3be18eb3 bfeea8ba -> 41d60434 be28560c 3c52b30c", + "412fc890 40337130 3b218428 c053a6a3 -> c21154bb c1145b18 3c063f84", + "c1986126 bf981204 390f78a8 404772d4 -> c26d6fbc c06cf46e 39ff1638", + "4052dff0 3d5aa600 3c965e4d 3f2dd500 -> 400f30ca 3d14781a 3c4c3ed2", + "407b9c70 40664b78 3ab41230 3eb8de10 -> 3fb5b2b2 3fa64e02 3a02b74e", + "4143e34c 407b359c 3d46e453 c028d403 -> c2012f64 c125ab3a 3e033540", + "4158d21c c0195aa6 40c02050 4026ac0e -> 420d29f7 c0c7afb2 417a2c61", + "4170dd74 409cb9ba 3b18596f 404ab3f6 -> 423eb810 41783172 3bf33ced", + "c14dbc48 3eb4d010 3da403e0 3f1f3778 -> c0ffe909 3e60e8cf 3d4c0c28", + "c0bf8a1a be53e780 3f0adc72 40724ffa -> c1b54c63 bf48930a 4003702b", + "c11426d4 bfa8c80e 38d83727 c076262d -> 420e7368 40a24969 39e43cde", + "c182fc90 3fbeb2fc 40fced52 c02973f2 -> 422d681d c07c750e 41a76b5b", + "40d3ca20 be574620 39b0e8f1 c01e1cb2 -> c182ce7d 3f04f569 3a5ebef1", + "411adaa0 c02e2ae9 3ee1105c c0009ef8 -> c19b9af2 40af0337 3f62297d", + "4196e6a2 c041b9b8 3c628f02 3de9ebc0 -> 4009e2e0 beb1046f 3acf54db", + "414fcd60 bef19530 3a9384e1 c06e9164 -> c241a6f4 3fe121e6 3b8b0aba", + "c17aedea c06bf316 3eb7c9d1 c067e804 -> 4263501a 4155be31 3fa67fe9", + "4187e17a 407001c0 3ac30824 c0431524 -> c24f17f1 c136e527 3b9698fb", + "c177b368 c01d3c25 3b1e5655 befe3788 -> 40f5f9bc 3f9c23c8 3a9e590e", + "41798338 3e387000 39831367 c01ed2b6 -> c21acc5c bee4d9de 3a2c6d24", + "c10c473e 403774c0 3a25895c bf522d50 -> 40e6568a c0169e3e 3a0a4b4e", + "4175d7f8 c085bb9d 3a1a438f 4066bee6 -> 425d975c c1711491 3b0f7336", + "41444b00 bfd72348 3ddbe538 c031a2ab -> c2083496 40954819 3e989a16", + "c18fa175 c02b2457 3a11042f 4021a310 -> c235600f c0d81dbb 3abda32b", + "c15af646 4095f410 40668cd2 406b2e5e -> c24927bb 4189c234 4153cd58", + "418c26d6 408a6750 3ada8df0 4014b5d8 -> 4222d403 4120cc14 3b808b64", + "c1238bfe 409bbaf2 3981060a bdca85e0 -> 3f8161fd bef665e7 37d8143e", + "419065c8 c005f854 3bcc34cc bebe3e50 -> c0d69d60 3f471dd5 3b1838ae", + "c174a9d5 c027742b 3ac0ac51 c06c85e6 -> 42620c88 411ab6ad 3bb41521", + "c104a4a6 40484068 3a1345bf c072c55a -> 41fb93cd c13de742 3b0e5e8f", + "4198c34a 40269d32 3ad09309 3f3ddb78 -> 41629649 3ff721c9 3a9cb25c", + "c113f251 c085ee87 3964f13a 4042c0c0 -> c1e11a0c c14bc733 3a386327", + "c17c52be bec81260 3b24bd40 4057af44 -> c254963d bfa89068 3c0ba59d", + "40889198 c07309d9 3aee07dd bf9b562a -> c0a5bc3d 409378c4 3b110b85", + "3fd84500 bfa6e298 3a1872be 404b81a8 -> 40abec40 c084aa34 3af3910b", + "c19ec362 3fd4df20 411ef560 3e035dc0 -> c022f03e 3e5a7844 3fa323ad", + "41975c24 c05e5230 391d82ea 40202768 -> 423d61c6 c10b1597 39e11914", + "419fb89a c0794208 3d28b3bf bf9ebde2 -> c1c614cf 409a8fa3 3d5155c7", + "414bf5d8 4076b4e4 4101fbfc c0250c6c -> c2037f5a c11f0e90 41a79b90", + "40f5f744 3fc22f14 3d26209e c02c8bf4 -> c1a5c899 c082e1ca 3ddffdee", + "418606b6 c0416166 3c050302 bd9e9d00 -> bfa614ad 3e6fa16b 3a2534eb", + "c19c9af6 bf1f9400 3fdb8979 407eb336 -> c29bcf61 c01ec48e 40da6cc3", + "419bb072 40768050 3cf6d3e2 4058a3b8 -> 4283c071 415099e8 3dd10823", + "40ccc520 c01e6fc8 39070a7e bfc174c2 -> c11abe02 406f7511 395982c1", + "4126fa22 400f8fa6 3e0079b0 c044801a -> c2002b10 c0dc63a3 3ec53fd3", + "c1913dc6 c01cc8c4 40231678 407d8c3c -> c28fd99d c11b484c 412186e7", + "418c269c 408b3a24 39aac672 3fe28644 -> 41f8072e 40f664b2 3a20c9ce", + "c1677d39 c08842c4 3e4370f3 4047f410 -> c234cf09 c154db9d 3f18aad3", + "415eb918 409cc664 40fa3db3 401caee8 -> 420850f6 413fe804 419928b3", + "c109f9f4 3dbaa640 3d09e0fd 3e90bb70 -> c01c033e 3cd30c5a 3c1bf0e8", + "c0f89f4e c093e0c2 3fad355d c045a5ad -> 41bff37e 41645752 4085ba7e", + "4092dbf0 3f7f8f50 3e883b83 3f07aba0 -> 401ba8e1 3f076fe8 3e106635", + "4168a4d0 3fdab8b0 3c46a8de 406f8996 -> 4259aee7 40cca7f9 3d3a1f49", + "c195c261 bf722910 39c9581d 3fa7bd18 -> c1c440d8 bf9eaba8 3a0a5e99", + "c0eac23e bf910b8e 3d6ca53f c05954f6 -> 41c74c90 407645dd 3e48edd9", + "c0d31478 c06f9674 3ffcf19c 40531ebc -> c1ae1336 c14595c5 40d099d3", + "c12d6e35 3f86bdd8 3978ffa5 3fae22dc -> c16bf10b 3fb74ec4 39b1768c", + "c110c470 c035ca8f 390e89bc bffe2b4a -> 418fbb62 40b47db7 39995279", + "c133e99a c085007f 3f905bba c07bc806 -> 4230f29d 4182cf68 408dfb3c", + "c0366490 402b0ed2 3c02dbef bfa5bdc4 -> 406c2c09 c05d7eb5 3c298e28", + "c11ff27b 3eb9fcc0 3ff8bf1c be857078 -> 4026be7e bdc1e40b 3f01a8c5", + "40c98840 3ff10510 3ffb1cde beeb1dc8 -> c0391785 bf5d5bb0 3f66a101", + "416ec02c bfed030c 3ce55d4d 404ba30e -> 423dea74 c0bc8860 3db68dce", + "41922a46 bf337230 3a836fbc c023c75c -> c23b057c 3fe59ae6 3b2b35d4", + "c14d9ec2 408474fa 402a32fe c07b272a -> 4249ba1d c181f2f5 4126fa56", + "3f7d0440 c09b3785 3b3dfa25 3f971898 -> 3f9555d0 c0b73952 3b60afed", + "40b896b8 c0566e9a 3f0e3eef 40653030 -> 41a54191 c13ff94d 3ffeb33a", + "407318d8 c07a795c 3ac43cd1 3eca0060 -> 3fbfd1f6 bfc5a421 3a1b9b0c", + "c19e038d c036e8c0 3fab8f56 4018f824 -> c23cd68b c0da96fa 404d07b1", + "c1148591 bf8a9260 3af699bf 3ea6c820 -> c041856e beb48e7c 3a21809a", + "411de914 3fa9a070 3f775434 406afdaa -> 4210f384 409bb4bb 406308bd", + "4110f9ae c06d7ad6 3b43b313 4013f58e -> 41a794dd c1094153 3be3235a", + "4175d77c 3fa6d174 3fb3cf86 3df2e440 -> 3fe940e5 3e1e46bc 3e2a9b07", + "4174afb4 bfa51b74 3f956940 c0079c5a -> c2019e19 402eeca2 401e4c19", + "c123ea7a c0398042 3fa7368c bd75db00 -> 3f1d6ba0 3e322670 3da096af", + "418c9678 40837cce 3fdd83f2 40076138 -> 4214b17a 410b117f 406a4a2e", + "c186786c 408cd5da 40379082 3ff86214 -> c202782d 4108a520 40b21aaf", + "4085b654 bf7c3a98 4068dd6a 4012eebe -> 41197d5d c010c4a4 4105a785", + "40cb8f50 bfc76050 3b06f07a 4048ccd8 -> 419faad8 c09c62c7 3bd47683", + "c13484aa bfd52606 4080ea48 3dfa3840 -> bfb07138 be5055f7 3efc027e", + "416cc100 40933152 3959d5bc 406de462 -> 425c01e4 4188c7f9 3a5c73a0", + "c10d5e16 400cf26a 3e5d72ab 3f8a7100 -> c118e63d 401871c8 3e6f862b", + "c07e0780 3f672ee0 3d73aa55 c0626aa6 -> 4160ac73 c04c77b1 3e57863f", + "402f6ef8 4027ef92 3dd0154a 402bf2c6 -> 40ebaafc 40e19892 3e8bc563", + "41365314 c09591f3 401f431e 4012a4ca -> 41d0e17d c12b5b08 40b675ed", + "c1280dc2 3f064c60 3a5b59f6 be0e17d0 -> 3fba8e87 bd9515b6 38f6902b", + "410b9c5c c09262da 3ff35221 3e9bff20 -> 402a259c bfb2677a 3f144581", + "c17ab3b1 4086cc8a 38ffa031 4052206e -> c24dc729 415d49b9 39f2750a", + "c106f5f1 bff5d648 3e04c8ad 40756ea2 -> c20163b3 c0ebb051 3efe9fe7", + "415e54dc 40751fa8 3cdab8bb bf8fd202 -> c179cf94 c089b5c5 3cf5e920", + "c11134a6 c01bf3b5 3ba7a6f1 c078c071 -> 420d1828 41178953 3ca34148", + "415dd0b4 4062d1ac 3aa216ed 404b1344 -> 422ff530 4133ed55 3b824e4a", + "411b6d84 bf3b2600 39b457b9 c0211a36 -> c1c39f94 3feb8c1f 3a698e16", + "bfea0780 c069d84d 3b209b0f bfbae9c2 -> 402adf25 40aabcaf 3b6b06c4", + "41304ade 3f4efa68 3adfdacb c006e04e -> c1b9c34f bfda18d9 3b6d702c", + "41693d28 c06d7f85 3d309599 be27bf60 -> c018d541 3f1b9fbb 3be78337", + "403b2a08 3fb8f038 3f3273ab bff52830 -> c0b33c9b c0311aee 3faae504", + "c1668227 3f9c3014 410c1151 404dc2be -> c2394593 407b12a1 41e128fd", + "3ff223e0 3e491300 3d16f465 40765190 -> 40e8fb9f 3f417852 3e1140fc", + "c1392c12 c052f7ac 3e1b1c55 403a3bc0 -> c206b53d c1197930 3ee1b30a", + "414f6b9c 4092ee96 3f72517b 3fb9bc80 -> 41967d7e 40d53536 3fafd026", + "4148ec1c 405461fc 3c9c4c36 c030e482 -> c20ad5ac c112c0f0 3d582bc0", + "c1965f8c bff9785e 3a89187a bfef0f92 -> 420c6c59 4068f68c 3b027228", + "40aebd10 c020d001 3cf4e2bf c0362a80 -> c178aed8 40e4dd26 3dae4d3f", + "4112d1fa 4067da20 3c83683b c01885ee -> c1aef2f8 c10a22cf 3d1cb3cf", + "c1848f7a 409283e6 3f1baf14 c0422207 -> 42490cb2 c15e36dc 3fec20cd", + "c01d9068 c03c8674 3e1eb280 405e8b1a -> c108f8d8 c123e309 3f09f63b", + "409b4a24 c00211fd 39b34b3e 3f573b58 -> 40828f44 bfdab685 3999a2be", + "418e5494 bf961320 3c3d257b 404f7ffa -> 4266bb15 c07348fa 3d198d62", + "c0d4e3c2 405d2484 3e59e04a 3ff70da0 -> c14d7315 40d56a00 3ed2457d", + "c147715e 3f921c88 3aba3964 402b61a0 -> c20584c8 4043a18e 3b7b9dbf", + "c0519d80 400e0454 3b0b1680 bfce459c -> 40a8e5bc c064dc33 3b60b192", + "c1982735 bfa7907c 3aeda598 bf9b3d1c -> 41b8881e 3fcb38f6 3b11a67d", + "4112c360 bf290c98 3f46b16a 403b830c -> 41d6ffb0 bff7a579 401189cb", + "c13e7d10 3fdf39c8 3ad6ddb3 3f5e3b60 -> c1255ca2 3fc1c7e1 3abc014e", + "c143f8c4 3e5bb9c0 3d3dcc73 3feeeb08 -> c1b6e532 3ecd106b 3db12e01", + "c1411628 3f7cf380 3b851f1f bffc2640 -> 41be2ea9 bff9257d 3c03856f", + "415733fc 3f65fe78 3f907e11 40448ac6 -> 42253875 40309381 405dde6f", + "c1421ed1 3fffb8dc 38f5a2ff be68a940 -> 40306c44 bee86898 37f8ed56", + "411181d4 4087101a 3ea70cb4 3fddbf50 -> 417c13a3 40e9fbab 3f10b448", + "418cd254 bfcd737c 40636e14 bffdb782 -> c20b90cf 404b9e67 40e1672b", + "c19192fc 4025602a 40192157 3db13b40 -> bfc990a3 3e64fb87 3e54077c", + "40861998 3eecc1a0 3ea76d90 3fa31b94 -> 40aae17c 3f16d8ca 3ed55a63", + "40835530 c08dbccf 3d5fb82e 3faaeb3c -> 40af5e80 c0bd433e 3d95639d", + "c1861784 bce67600 3e1bc099 40237c98 -> c22b4478 bd932d4c 3ec6f447", + "c07fa318 c09c4999 3ca7da58 bf05c680 -> 400595f3 402356d7 3c2f7fc7", + "4015ec08 bfa932fc 3a1659a5 c070e0d7 -> c10d10f4 409f3467 3b0e54cb", + "c04c3470 bff7d246 3a01b2ee bfe0fba0 -> 40b37699 4059cb93 3a6638de", + "4196f006 3e8712c0 3e612cf0 4035672e -> 4255e914 3f3f6d65 3f1f92f5", + "408e7ea0 c00c3003 3e8d6724 bef62870 -> c0090430 3f86cc48 3e07f848", + "410ebdb6 404700cc 3e34001e bfd706be -> c16fca2c c0a726e9 3e973367", + "4177a298 c05f347c 3b2709f7 c038842a -> c2327cb6 4120e0f5 3bf28045", + "c15e84db bfc41222 3c131881 c00a2768 -> 41f02bbd 40539fe9 3c9f0666", + "c1092ffa 40756118 3b7a55c6 405b5fe2 -> c1eb1eda 415245f7 3c572f51", + "417165c0 be7b5ae0 3e9685d0 3fb585a4 -> 41ab2af5 beb23a78 3ed57925", + "40888f14 405d9838 3994c343 bfce7682 -> c0dc44b3 c0b2b712 39f62ff9", + "c06d2720 bf4616d0 3e8c6064 c058f966 -> 4149000d 4027e43b 3f6df52e", + "c08e8f0a 3ff740a0 3946dbdf 40199a62 -> c12b12eb 40945abb 39f64cba", + "4145ba38 c042397f 399ed699 3f36ae28 -> 410d18e7 c00a9902 396dad40", + "c0afdcfc c051bb47 3d6071fb bf3dce70 -> 408263f7 401b8064 3d266fb0", + "c1844e2e c09c2116 401bb2a7 c05a69a3 -> 4261c257 4185349a 4104d6b3", + "41850a88 4082c99e 41166f3c c052cda4 -> c25b1b04 c15764e6 41f7c05f", + "c1399e00 4019960e 40f5e4b1 c03a6f33 -> 42072d6c c0dfb37a 41b31318", + "405f76c0 c09c7d55 3fbba006 4040eacc -> 41286605 c16bdb0e 408d6455", + "410d0760 bfbf3548 4025be41 be923160 -> c02112d0 3eda6288 3f3d4d2c", + "bfba2d60 c03cc562 3f119c63 bea75440 -> 3ef361be 3f76c5c6 3e3e5a49", + "c1059cc5 3f680350 3c512d72 c0525ec9 -> 41db9824 c03ea89f 3d2c0332", + "4137c30e 401169ec 3d8a3cbd 3fd1ce70 -> 41969a6e 406e5979 3de2a174", + "4188dd78 c008bd31 3db80d31 3f41b8b0 -> 414f2369 bfcef290 3d8b4e0b", + "41961634 c09a8144 3c0621ce bc30d000 -> be4f526c 3d556cd0 38b9cad4", + "c03861a0 3f7e48f0 400938f7 3f983714 -> c05b4345 3f973204 40232ece", + "410a4514 40864f5a 3a97bc47 401890b4 -> 41a4ce58 41201611 3b36c4dc", + "c0aef744 c0338776 400d616b c079dcb0 -> 41aac556 412f3986 4109fdcb", + "c106a0d7 c0854c73 40a75403 40435406 -> c1cd7167 c14b69f8 417f580a", + "c118e47e bf51bc20 3f508580 bfb312d4 -> 4155e606 3f92b5f7 3f91dd2c", + "bf0014a0 4081c498 3b51ff55 be0d3230 -> 3d8d48f0 bf0f2571 39e7f610", + "c150904b c011c5cb 3a8ffe76 bfcbd4fe -> 41a60ff2 40682241 3ae85989", + "c04c2a30 c0274796 405d535d 3fd00fd8 -> c0a5eeea c087f484 40b3e18c", + "4195c23e bf947ad8 391f511e 403a867a -> 425a3b98 c0585e80 3a0290bf", + "c0ce5746 c07e801b 3f04f7df bf83925c -> 40d41941 4082cd0e 3f08ae6d", + "c09c44f0 3f9dbf48 3ed9f02d 3f0a8810 -> c0292070 3f2ab9e9 3e6bdf7e", + "41655ae0 3f09e8d0 3c4346a6 40304312 -> 421dea91 3fbde861 3d069cba", + "c0c964b4 4005e6dc 3b985c98 3ed0afa0 -> c0242bfc 3f5a4edf 3af8d4f3", + "c1299b43 c08d99ae 38ee76f8 3e37eb80 -> bff3b406 bf4b763d 37c0e967", + "c10129a6 bf9523a6 3cc135f3 bee620d8 -> 406837fb 3f061129 3c2dbfe9", + "410764a4 4083c434 3bfd92dc c04029a4 -> c1cb4302 c145d12c 3cbea2fb", + "c0f1011a c055fadb 392e011e 404fb076 -> c1c38604 c12d9956 3a15fd76", + "c190eccc c0869977 3d53c72e c02eb6b6 -> 4245d0c4 4137b8bf 3e1097cc", + "40dff668 bf64af30 40ae4238 403a7f42 -> 41a3285d c02698f7 417de5a0" + ] + }, + { + "glsl": "vec3 cultmath_af_add_iv(vec3, vec2)", + "tolerance": "ulp-bounded", + "cases": [ + "c14cfbf6 4015715c 3e079ba3 40762a88 40b8c0bc -> c1000676 4015715c 3f8c4aed", + "c0ae338e 4092a69c 3eb7d309 41819edf 4181ab55 -> 412c306d 4092a69c 3eb965c7", + "bfa2d6b0 40835bec 3befb189 3f057a33 3f16cf4d -> bf3788a0 40835bec 3d28a475", + "4191b318 3fa39a5c 3b4d8b6a 40cc680d 40cd177b -> 41c4e309 3fa39a5c 3c63521b", + "c122dfc9 3f817268 3af98ae6 40f61a4f 40f73c91 -> c01e2844 3f817268 3ca0d023", + "3ec4a940 403e2d6e 3b07cd20 c1556f56 c1556c46 -> c14f4884 403e2d6e 3b22216d", + "4170a038 404f908c 39baa597 c1620566 c161f036 -> 3f6a56a0 404f908c 3b41f94e", + "402593f0 4025d0f8 3fef1dd0 405e1d4a 405e76f6 -> 40c1ef08 4025d0f8 3fef77ec", + "41911d98 3fd561a4 3aa42457 4190566b 4192cb65 -> 42115740 3fd561a4 3e1e94e9", + "c16a9032 3ea7f220 3d3de3b7 3ef9464e 3f12c759 -> c16214de 3ea7f220 3db78a33", + "3f9e7370 bf9f1f5c 3f112411 bf51c87b bf320485 -> 3ef600c0 bf9f1f5c 3f21063f", + "40492b58 be659b40 4091bdad c1796839 c17951df -> c1471236 be659b40 4091d448", + "402ae328 3ff99948 3d5839ae 411343a6 4113ab8a -> 413e3062 3ff99948 3d862167", + "40b82940 404021e8 3ab0ad97 c1527845 c15277db -> c0ecc6e0 404021e8 3aba3f6b", + "c1426ac6 402f0ff8 3ebade69 c0f0d2ed c0efa73b -> c19d44a8 402f0ff8 3ec43fc3", + "4049bd98 bf979118 3b51b905 419e7313 419e82ad -> 41b7b293 bf979118 3be70c3f", + "41989384 c01716ad 3f100333 404542ea 404fdaf6 -> 41b1e562 c01716ad 3f253513", + "3c3fa000 c08871e4 3e0f5c36 3f4bac8b 3fba8f1a -> 3f91b1f0 c08871e4 3ef120af", + "c18420a8 40518978 3f063644 c12d2c21 c12054e9 -> c1d780ea 40518978 3f6cf2a3", + "c11a9175 c071cb96 3ce490f7 bf84e009 bf84d9f7 -> c12b2d15 c071cb96 3ce5724c", + "c0d56436 bff735b4 3bd3c710 3eef823e 3ef42242 -> c0c64712 bff735b4 3c34063e", + "4124be76 3fd0efe8 3c7eeb31 4195e3ac 4195e400 -> 41e84311 3fd0efe8 3c80806f", + "c141ec9e c04f02f9 4099eee6 4115b19d 4115cecf -> c030b1a0 c04f02f9 409a0c41", + "c0fc6112 408e63b2 40d6d1fd c1a43fff c148992a -> c1c35e8e 408e63b2 412b5ca5", + "4033e1a8 c052926e 3e9dce6a c0b2bd99 bfed82fb -> bf68f2c0 c052926e 400b16d7", + "c0a7e880 c0968525 3cbdd936 3fa0c00e 3fa0fd32 -> c07f61b0 c0968525 3cc591aa", + "bfde2820 407b2d4c 3faea1e6 418d5507 418d555d -> 417ee560 407b2d4c 3faea5cd", + "40d983c0 40809694 3acb881b c15d54dd c1321629 -> c0b5e746 40809694 3fad2e79", + "4157b7f4 3fad12b8 3e4e4072 bf015747 bedf9973 -> 41502ed4 3fad12b8 3e715953", + "418360d4 407a37c0 38eb199c 416be125 416c45c3 -> 41f96a8e 407a37c0 3c4bd9a9", + "c0eeb1b2 c05ba444 3c28971b be8bd2ce be7cec64 -> c0f703fa c05ba444 3cbf4753", + "c1581fad 3f8ba4bc 40bed57a 407980f1 4079830f -> c119bf2d 3f8ba4bc 40bed62a", + "4111c8be c04640ed 405b4fe3 41087344 4109c554 -> 418d7285 c04640ed 405df486", + "c0fa33d8 bfc08fbc 4116a717 40ef3dc8 40f96fb8 -> be3ba300 bfc08fbc 411933a6", + "c07d76c0 400450b2 3f456b95 c13fd35f c13fd13b -> c17f2ffd 400450b2 3f457ea2", + "418b3786 3dd01d00 3a924c21 41597c11 41597c7f -> 41f7f5aa 3dd01d00 3a9ec23b", + "bf62bfc0 c099b22a 3dd6c8ee c1a31fec c17574d5 -> c1960329 c099b22a 40284d08", + "c10020c0 bdcd2dc0 3b985c1a c03bbf6c becc27a0 -> c11ac94c bdcd2dc0 3fa2d343", + "c18c8840 408ef4b2 3e23b860 c1560445 c15601bb -> c1f789c0 408ef4b2 3e2415dc", + "40bbf1c0 40121256 40ad0299 41818f17 4181934d -> 41b08da2 40121256 40ad0b61", + "401c2618 407d9064 3b8102a0 c1545735 c1545637 -> c12d4d30 407d9064 3b85db2d", + "4088ace8 bf9a64ec 3a867094 c0c0df2c c0bf1b9c -> bfdd41f0 bf9a64ec 3cea4104", + "40c2bed8 bf809316 39c5aeec 406e1ffb 407180b5 -> 411d5382 bf809316 3cdb6282", + "41052d14 3f5c6690 3fed4f9d 412f12b6 412f1d6e -> 419a2293 3f5c6690 3fed7b85", + "41494f80 c09adaf2 3d45023d 414532f6 4147baba -> 41c7e32c c09adaf2 3e0243ab", + "bfea38a0 c08ae2c0 3deaaa3a c192c6cc c192887a -> c1a14b2d c08ae2c0 3e04f45b", + "c13ba348 3f22c218 3b5aa28b 404512cd 40461ef3 -> c10a3d10 3f22c218 3c3ced2b", + "40510e78 40006d00 3c77bc88 41986420 41986db0 -> 41b28ab7 40006d00 3c8f550e", + "418688da c007d278 41044828 4183385b 41833b35 -> 4204e151 c007d278 41044b3e", + "40ce7218 405421ec 39344ef0 409abdcc 409b504c -> 4134bc92 405421ec 3c159f21", + "c18c9061 406fc0f8 3a401aff 409b1310 409ba190 -> c14b739a 406fc0f8 3c1ad707", + "3fe5fb50 bfb37efc 3d0f8f46 3f89224c 3f99c7b4 -> 403bb828 bfb37efc 3dccf5b5", + "3fd0dfa0 402b3c58 3a265b42 41621a30 41622f10 -> 417c4094 402b3c58 3b52a01d", + "c183a88c c081a188 3d48d46a c189ca49 c12b490a -> c1f35ff3 c081a188 4053bb3e", + "c19d9ff7 c05fe282 401be9f7 c01a8c5a c01a2136 -> c1b0ead0 c05fe282 401c2003", + "3f890fd0 c095914f 3e0aaee1 bf258410 40e53782 -> 408a8774 c095914f 408149a2", + "409d2e00 c099f3ce 410fc49b 4069076f 4069ff31 -> 4108f7d4 c099f3ce 410fe3ad", + "c0d6c884 bf8c0f94 3e858e4e c18307fe c154b0ab -> c1ac624b bf8c0f94 3fe6e21e", + "c131fce3 c0454ad4 405ca620 c0ca5871 c0accbff -> c187e300 c0454ad4 407a330b", + "c15e2aa4 c06e8ac7 39ea88f2 402fad6c 402fb5a4 -> c1323e42 c06e8ac7 3a3b6bbd", + "4029f070 c0043f58 4041b741 40ca5814 40cade94 -> 410fc9c6 c0043f58 40423e12", + "418360e0 409f20b6 3eed86e4 c13b0c2c c0f4c124 -> 40d816c2 409f20b6 401f0879", + "3efc4140 4015642e 408ad953 3fff667c 3fff69c4 -> 401f3c38 4015642e 408ad9d2", + "3f603900 40312eac 3d8fcc0c c0536431 c053625f -> c01b5508 40312eac 3d8fed75", + "40fd6d60 bf623db8 3c35b701 c1221fd0 c11a04bc -> bffadcb0 bf623db8 3e87609f", + "414a0ae8 c024b008 3d2a840e 40bbac60 40bbb598 -> 4193f1b3 c024b008 3d2ced08", + "c080a67e 4080caa4 394bc8f6 c0166ffe c015b392 -> c0cbaf62 4080caa4 3bc33041", + "c0da534c c0717886 3b968206 c16185bd c160d849 -> c1a72c54 c0717886 3cd36216", + "c0d34b5e 3fc95468 3c5a86b9 c13b1640 c13af390 -> c192554c 3fc95468 3c903283", + "beec2240 c016e0e7 3bc92a16 c0bba35c c0bb90c4 -> c0ca5c34 c016e0e7 3beece81", + "c09eb2a2 3f1476e8 3ebb66ef 40177ed2 401888ce -> c0256174 3f1476e8 3ebf8f9c", + "c15cf475 4075ff10 3e285a0a 41850120 418619cc -> 403899dc 4075ff10 3e6e8af9", + "40d6e390 be8839d0 398e2e84 c190cbe7 c19078a3 -> c135d2c2 be8839d0 3ca8fc3a", + "408c9980 3fe0e87c 3cdcbe33 c169a619 c1660add -> c1218bbb 3fe0e87c 3e0f05ed", + "c08107d4 bf0e4550 3c50f8b4 41790999 417ccd5f -> 413a6792 bf0e4550 3e058f54", + "c192f9c3 409bb6ea 3a84e9be 416b2889 416b3007 -> c06b1cf8 409bb6ea 3affb173", + "4003af38 40766f78 3a3a8965 c0b725af c087b159 -> c03b27d0 40766f78 3f3e00ce", + "415a8f08 c0468ba1 3cfe2194 419fc40a 419fc802 -> 420686c5 c0468ba1 3d034187", + "c00ca2d8 be51bb60 3c42e19f be0d5fe8 3fe66cdd -> bfaee540 be51bb60 3f7b24a6", + "4167618c 3f34d2c8 3bfb7125 40e1edb1 40fa1f67 -> 41af3269 3f34d2c8 3ec57f4f", + "41482ab0 40320492 3aa4c7c0 40c564eb 41075852 -> 419e980a 40320492 3f92c1a6", + "418a1c02 c06868c5 3fa254dd 41765b09 417684a7 -> 4202a9f7 c06868c5 3fa2fcfd", + "c154624a 3fea7128 3f41956b 40d30d41 40d30f37 -> c0d5b658 3fea7128 3f419e41", + "41350250 bffc39d8 3eb831f2 c1393067 c13527a7 -> be0a6dc0 bffc39d8 3ef8bfb7", + "c04c9e78 c04e7aba 3cde8c8e 41736ad1 4173704f -> 414045f2 c04e7aba 3ce44787", + "41864be8 c0969b34 404b37f3 c19991ab c1904cc1 -> bfea34e0 c0969b34 40704c11", + "bf9941c0 3f8c4d1c 3d1a64d5 c182b45f c1829405 -> c18c384e 3f8c4d1c 3d3ae1d7", + "412ee73c 3fd36cf8 3ce096a0 bfbedd0a 3fe25e4a -> 41311f50 3fd36cf8 3fd42085", + "4102f7cc bfa7821c 3dee9f21 407aa1c6 407aa65a -> 4141a0d0 bfa7821c 3deef119", + "41279eb6 40026d62 3f104346 c00d363f c00d3261 -> 410451a2 40026d62 3f104bd3", + "4015f9e0 4066096c 3a2b9dcb c19e38e8 c19e2c9e -> c18b7387 4066096c 3b72145d", + "40ad4ad4 c00589e8 3ba3544e c09307e4 3f97114e -> 406d5218 c00589e8 40391e17", + "4180b736 c028a337 3f5a09d7 c19b03e0 c16d77b4 -> bf828a70 c028a337 4047a2f7", + "4015dca8 c0548aa7 3e86a387 c12fe8af c12fdb55 -> c10a6ad8 c0548aa7 3e877c0f", + "c174b42c 4049d2c8 3a8f4754 c13553c3 c0718ceb -> c1b6c7d6 4049d2c8 4071f39d", + "c18fb348 3f241890 40321b95 413a26ba 413a7bee -> c0ca2a78 3f241890 4032c653", + "c135d595 c00d545f 3c9b39c0 41932b13 41932ee5 -> 40e108c6 c00d545f 3ca31511", + "408d9cb8 bf093308 3d10b403 410eb2e0 411f83b8 -> 415de9a8 bf093308 3f0f938f", + "bf38edc0 bfdf29e0 3914983f 4002f60d 4002f7b3 -> 3fa976e0 bfdf29e0 394e756c", + "bfa327e0 3f54ac68 3d4faf22 40d4831a 40d5f3f6 -> 40ac7190 3f54ac68 3d95f98e", + "4070a578 c099a8f4 3d2cb9fc c18ae188 c1343ae1 -> c128d59a c099a8f4 4045c3d5", + "411babb0 3fc22c78 3fe909c9 c1966eae c19667d8 -> c1112ad6 3fc22c78 3fe94172", + "c189305c bf79bd80 3d62227c 419c0f3e 419fe84a -> 40265b40 bf79bd80 3e9768cb", + "409d8598 4033d544 3facef50 414b86f4 414b9764 -> 418d28fc 4033d544 3fad3224", + "c1783c15 3fca9938 40af9a1b c073345d c021f963 -> c19570e6 3fca9938 40c3e916", + "413ae0c6 40367198 3e446b0e c0f2c661 c0f1abc7 -> 40838878 40367198 3e56185f", + "bfa48d50 c0989c5c 3aa97654 c12b2889 c1077efb -> c12de56c c0989c5c 3f8ed172", + "c14283ac 4022abaa 3b7197d6 c19f35a4 c19eec28 -> c200295e 4022abaa 3cb197fd", + "4192dc10 3f7191c0 3ccd062f c03e6b98 c020d0e8 -> 4179d090 3f7191c0 3e833d9c", + "c1868fbe 403170f6 391fc6a8 41495dfd 41497d8b -> c0876370 403170f6 3b83d303", + "40ef8b80 4024713a 3ed2ddb1 c0c570f4 c0c50b84 -> 3fa93510 4024713a 3ed60a80", + "418b6e62 c088faf4 39ade173 c0e48235 c0e47cbb -> 41249d08 c088faf4 3a33fe33", + "41929d0c 409e7062 3d505596 3e065cc4 3fd46c48 -> 4199c6cb 409e7062 3f50a7b4", + "c1909400 c04cb712 3b34731b c170fdba c1702f9a -> c2046faa c04cb712 3ce5152f", + "416c1724 bea589e0 3fa73855 c0f706db c0f6d21d -> 40e141cc bea589e0 3fa7a254", + "3f35f7a0 c04db2a9 3d1c9532 40e84a00 40f88d70 -> 41039556 c04db2a9 3e95b096", + "c19b4819 401f07f2 3fcf37fd 4185a359 418fc1df -> c004abe8 401f07f2 40101679", + "4106fd1a bfffd0d2 401a3fe0 41356b62 41801563 -> 41b0e417 bfffd0d2 4097dfae", + "c1708dc5 3fc00bf0 4111a4ff 4157cc42 4158ba26 -> bfc25488 3fc00bf0 41121c0b", + "41946bac c08f0373 40c3b924 40698fef 407a8f51 -> 41b2ada0 c08f0373 40c7f946", + "417f9798 bddb9c80 4069b771 c110ad37 c1101e77 -> 40de6382 bddb9c80 406ad540", + "c1338904 408acaca 3c44a75c 418e2a3d 41a7b1df -> 41025318 408acaca 3fcdc778", + "41248f6c 3f0dfa30 3c45ad7d 404d7b81 404d84bf -> 4157ef74 3f0dfa30 3c4a9195", + "c139cdda c05de6ed 3de02bd5 4101425f 4101441d -> c0622a70 c05de6ed 3de0a2ed", + "411ec068 bffb3bc6 3c8b4c2e c14660cb c1462f6d -> c01e1ed0 bffb3bc6 3cbccbe4", + "c133eed5 be777760 41071903 4122aa70 4122d270 -> bf898328 be777760 41072d17", + "bfb4cd10 c0325e70 40823e89 c1a58527 c198916d -> c1aa581b c0325e70 409c2660", + "3f8265a0 405707a4 3d8cd14f c04343b2 c03ccb8e -> bffda9a0 405707a4 3df457d1", + "4131678a 4075f1ac 40391db6 418d23e9 418d257b -> 41e5d877 4075f1ac 403924d3", + "402fcec0 c02a69dd 3c3b2a70 40f01ab8 40f04548 -> 41240bb0 c02a69dd 3c660c31", + "4146be48 bf0c3420 3af7d215 c06adebe 3f9484fc -> 4132aac0 bf0c3420 401aafe0", + "4074d5e0 c0922fd6 3ed15765 41653305 41653e83 -> 4191371e c0922fd6 3ed213f3", + "411bee72 408f7e8e 3e500a15 405c3e9c 405d36e4 -> 41531d22 408f7e8e 3e57d1ab", + "3fb76ed0 3faaedc0 3c3efeb0 411786ea 4119a7d6 -> 412f853a 3faaedc0 3da025d4", + "c0a1aaf6 3f5bae28 3a5733de c17f615d c17f4aab -> c1a815c0 3f5bae28 3b6dba4d", + "4168cebc bf62a268 3d0942f4 c1a29f44 c194cade -> c09d36cc bf62a268 3f65dc3f", + "419b3d84 c08b6ed5 3b49bb6b c0662f77 c060a659 -> 417da04e c08b6ed5 3d3dd751", + "c199c64c 40951c84 3c974d55 c18ad30b c18acf8f -> c2124bcc 40951c84 3c9eba8c", + "bf353a00 c07740ca 3eb4383a c178d0bd c159def5 -> c174ab79 c07740ca 3fa8d64f", + "41526a6c c03175d0 3b5e247f 3df2f93f 3e8ec6b0 -> 41559880 c03175d0 3dab0a22", + "418b3e68 404d0fbc 3e6dad45 c1a804d0 c15f835c -> bda4d700 404d0fbc 406fe7cc", + "c1302d42 c0181511 3a1adfe0 c17343e7 c17342f3 -> c1d1b858 c0181511 3a4452d4", + "c14a21ee 3ec33790 3eb36213 c19c9ddf c19c970d -> c200d5b6 3ec33790 3eb442e3", + "40e59838 3fc0f950 3aea2b63 419f935f 419fac2d -> 41d905d4 3fc0f950 3c013fd5", + "4032d358 c04ca450 3bb244f3 be9f9b4d be9f57b3 -> 401ee428 c04ca450 3bbae82e", + "c11994b2 4070f6c8 393517bd c1245251 c123af55 -> c19ecac2 4070f6c8 3ca4a9fe", + "40b4891c c07c849e 38d1e1c1 c0dd9a43 c051bde2 -> 3f0a6410 c07c849e 3fe97a55", + "c15694b3 bf8b58fa 3ac3ae77 c0155ace c014f032 -> c17bde13 bf8b58fa 3b9c20f7", + "41248878 40477a78 3b9831e1 419dcfda 419dd046 -> 41f0144c 40477a78 3b9d38be", + "418e22b8 bf73cd60 3e9f7d10 4153cba5 4153ced3 -> 41f80956 bf73cd60 3e9fb5a1", + "c14e123a 409723fe 3b612ad1 c183bd26 c183bcb0 -> c1eac608 409723fe 3b6bb378", + "41232120 be88ca40 3a5274b4 41851ae3 4187ec81 -> 41d81442 be88ca40 3e354511", + "c0c24814 c03674f2 40648139 407f660c 4080296a -> c004b3b8 c03674f2 4064f7cf", + "c186cea8 3ec2dd10 3b9d9e24 c180936a c1809264 -> c203b0c8 3ec2dd10 3ba7592e", + "c1078766 bf572928 3f8d6d4a 40ab191a 40b1737e -> c0419100 bf572928 3f9a2267", + "41253a5a 405dfd14 3bc0dd5d c032cba9 c032bb97 -> 40f112e4 405dfd14 3bd15dc7", + "41635ea8 c04e32e8 3a9e878c c18af503 c18aea83 -> c04a0378 c04e32e8 3b78bfcd", + "40d7f924 403edee8 4067b3d3 c0fe3dc9 c0fe38c3 -> bf990888 403edee8 4067b918", + "40f50928 c039771e 3b74c14d 4114b30a 4114b386 -> 41879bee c039771e 3b7a73a6", + "3fd7eb50 40805d8e 3eedf3fb bfd6eb90 40d25864 -> 40844994 40805d8e 4092e90a", + "c07058c8 c08a221a 3dcdf273 40c7ccd5 4110d4d2 -> 40791db0 c08a221a 3fc0995b", + "4127b80e c0574398 3e680a8c c036832f c0364711 -> 40f43d8c c0574398 3e69ef01", + "41829206 c073a22f 3f1136e6 40fe5edc 411d05fa -> 41c99f60 c073a22f 3fbff6e6", + "c152f122 409f7c34 3fd631df c1201e2e c115a300 -> c1b6e8dc 409f7c34 40000feb", + "c10c7c37 3fa1acfc 391cfb1e 418cf35c 418d1714 -> 410d8e39 3fa1acfc 3c11c2f2", + "c1870824 c07f09d7 397c76fb 409bc3d3 409bc6fd -> c1402d94 c07f09d7 39edea29", + "41647f40 c0776a70 3f6b9698 c0254e23 bfd598da -> 41427bee c0776a70 3fb04da2", + "4137a2da 3f45f670 3d2d5f25 c14fe814 c14ea32c -> bfbd1630 3f45f670 3da7f13d", + "3e500b00 c06e9a6c 3cd50028 419c4ce3 419c5081 -> 419deec8 c06e9a6c 3cdc923a", + "40389b10 becf9530 3dbbab36 4164c5b3 417f577d -> 41901aae becf9530 3f6c05eb", + "3fe02800 c08df1e5 3cc33fea 413c0e2d 4141483f -> 415ab036 c08df1e5 3e3fb1d5", + "41331352 c0955959 3e4bd629 c12db3a6 c11b672e -> 3f685e80 c0955959 3f455a5e", + "40cd83cc c0840df3 3939172b c0f1c494 c0f128ac -> bf8fcb50 c0840df3 3c1effa0", + "413c8e7c bfde076a 3b58c22d 41898bb1 41898c23 -> 41e7d328 bfde076a 3b62e0ba", + "406e4500 be89e4f0 3e6188a8 40e3f787 40e40789 -> 412d9104 be89e4f0 3e628d66", + "c15de1cb c06447ef 393348f5 40ce4568 410d9abc -> c0c70626 c06447ef 3f99e659", + "4097c4c0 3dba2a00 3a803ebc 413119fc 41324848 -> 417d9382 3dba2a00 3d1b4315", + "41406308 3fb09af0 4116cfae 411c2dd0 411c2e48 -> 41ae488a 3fb09af0 4116d015", + "4122558c 40859c48 3b294d7f c0ecabb8 c05c88a1 -> 40973314 40859c48 3ffd2408", + "c0528fc8 c0871887 3a3d1fcd 3fa588fc 3fa7f144 -> bffe6270 c0871887 3c26021e", + "c122d587 c05c7182 3e527937 41470911 4147106f -> 4010dce4 c05c7182 3e53698e", + "c1895f31 bf8f90a0 3f783632 3fc7e57c 3fcf0de4 -> c1794f2c bf8f90a0 3f7f5fcf", + "40aeb680 409bbe18 40dee740 c0aaa726 c0aa9f96 -> 3e026440 409bbe18 40deeb2b", + "c15f0f6a 40791a64 3b15a3b6 4116a498 4119aab8 -> c08dcf84 40791a64 3dc63e21", + "c15bbefb bfd77408 3d810628 41859059 418597eb -> 403da524 bfd77408 3d84d9dd", + "40e0b164 c045ff0e 3b5945a8 4118e415 41325ffb -> 418afd5d c045ff0e 3f4cba7c", + "c187d85f 4096ef2a 3d183872 c129e1e2 c1279b20 -> c1dc37a0 4096ef2a 3ddde235", + "bfbd6100 405dc448 3c0d367f 3f866eab 3f8a9e15 -> bed36a80 405dc448 3ccc9278", + "c05f5298 405a0524 3c9a8644 c14391e3 c14372d9 -> c17b5704 405a0524 3cb9cefd", + "412ad95a 408fa91e 40729da7 c157d680 c14cc988 -> c01ddaa8 408fa91e 40845bfc", + "3f920d50 4043e974 3c720e7e 41716e9c 41717b4c -> 4181db4f 4043e974 3c85fc0c", + "4184723e 3f74f6c8 3f0822aa be0e799e bd8298c3 -> 4183a278 3f74f6c8 3f11c974", + "417340ac c085346d 39235ef8 40febe7a 40ff7e96 -> 41b967f8 c085346d 3c4336d3", + "40d2f9d8 be8bea30 3eeab853 411380e2 411aa82a -> 418048b9 be8bea30 3f2e9814", + "c0f9a768 3caeb600 3f07fb5f c129b719 c129b06b -> c19343bb 3caeb600 3f0832a8", + "c0eb883e 4070fee0 3bdbe3fe 3fd129a7 3fd81599 -> c0b66056 4070fee0 3d0a46d0", + "41110674 c010132e 39f1a2b3 419d16b9 419d1e27 -> 41e59daa c010132e 3b183ddd", + "c1139a31 3ec16200 3e1c5381 418aaa09 418dbb3f -> 4104cb17 3ec16200 3eb053d9", + "416b162c c07f2078 40339a9b 418fdf6b 4190e9c1 -> 4202f7d6 c07f2078 4037c4d9", + "c0dad798 40415ed8 3f19eae0 410a3cb8 410ffc38 -> 3ffd8560 40415ed8 3f47e7cd", + "c103a95e c0355297 3a4f798a c016e996 c015979a -> c1293984 c0355297 3c3634b7", + "c1978de1 c08c75a0 3a81a8e6 3fa857b0 4117703a -> c158de2a c08c75a0 40826d9e", + "c1940d32 c02c9577 3a88a5b2 4189420d 41974b77 -> bef19c00 c02c9577 3f60dc63", + "4086a7b4 4090534c 393550e9 c090225a c09019b6 -> be976540 4090534c 3a39e81e", + "c16eaf57 4056fe34 3f6f4ae8 c1a20d68 c1919360 -> c2089408 4056fe34 3ffb7792", + "c13db33c 3f9001d8 3f6e6b68 40289206 40ca7cfd -> c0ec0378 3f9001d8 4031cf1b", + "3f0dd080 c072b8f6 3e2ba483 415378d2 4153804e -> 415c5998 c072b8f6 3e2c9bc0", + "3ecb2140 c0818964 3e92a9e9 3f8f7446 3f8fcbfa -> 3fc26870 c0818964 3e935a30", + "c13aaef0 bfce732a 38d696b3 4190e30d 4192e1e7 -> 40d22c08 bfce732a 3dffaffb", + "40af5748 407e8ebc 409e1ce3 c17e7844 c17e77c4 -> c126cc60 407e8ebc 409e1daa", + "c09c24cc 4022ed6c 3be3bfb3 3f176b8e 3f2cb832 -> c087e290 4022ed6c 3d46e49a", + "bfd53c90 bf464890 3a2613b4 40490769 404927d7 -> 3fbcf2b0 bf464890 3ad56e81", + "c05c72b8 c014f6c8 3db068be 410d6d76 410dc776 -> 40acfb90 c014f6c8 3dc6f116", + "41854f48 3f5acf60 3bf09a7a 40fe0e26 40fe36b2 -> 41c4d7e3 3f5acf60 3c215ee5", + "3eccd440 c019da19 3d222852 c1a5bf58 c132136e -> c178626d c019da19 409aafea", + "41388a3c 4055b748 3fc6d0e2 c0cce677 c0ccdb71 -> 40a43384 4055b748 3fc6e771", + "c0f883ec 3fde63dc 3e92b16a 407024ac 40733034 -> c07f5d68 3fde63dc 3e9ee0c5", + "4125ef9c 4021802e 39e79c74 c129c0ca c11ad38c -> 3e695c40 4021802e 3eef0f82", + "c1840a69 408b266a 39c03fc3 c1263f75 c101c0b3 -> c1ce0a73 408b266a 3f920859", + "418d23ec 40211214 3b260717 bf534e02 be31f3f8 -> 418924c0 40211214 3ea81fac", + "c18f841a 3e7ba7c0 3ad04392 41285925 4128e68b -> c0ecd0b8 3e7ba7c0 3c9a8eab", + "c174384c c0298ffc 3b0ced88 3e9b4507 3fa09bfe -> c167c178 c0298ffc 3ef4b1af", + "409f424c 3f57cd20 3fa1ea3b c10d0878 c03da150 -> bf697400 3f57cd20 40861ad0", + "4119b164 3f4216b0 4112ff2a c14bea47 c1464bdf -> c03da6bc 3f4216b0 4115ce78", + "c156e19b bedf0dd0 39e195c7 418174d1 41817887 -> 40302ef4 bedf0dd0 3ab19167", + "40406730 c06214e6 3aeb25a0 c12ca02b c12c9f65 -> c0f90bf8 c06214e6 3afa4906", + "c19443ec 3f082b50 3bb9d72c 40457381 40457faf -> c1772972 3f082b50 3bc69dbe", + "4134038e bf2a2f70 408b9bbd c184474a c1842e5c -> c0a8e430 bf2a2f70 408bcdcf", + "3f77f980 be10fbe0 3de4a29e bffc40ca bffc29b6 -> bf803880 be10fbe0 3de55cdb", + "3dfbf400 c076c407 3f21aba4 c0ec6e1e c0ec6a66 -> c0e87c72 c076c407 3f21bbb6", + "c0ac7ba8 4025c85a 399c764c 41316490 4132f764 -> 40b7e04c 4025c85a 3d4ab67f", + "40736620 bf82c7d8 3b9c33a7 bfe90106 40319793 -> 4088f8d4 bf82c7d8 40135a4e", + "4196ee4c 409af944 391f206d bfdf496a bf8f7056 -> 418b787e 409af944 3e9fc917", + "c18934d6 c02a85a8 3ad7b8ba c14dc065 c14dbd6b -> c1f0144a c02a85a8 3b0684ef", + "c0f78582 3fbd43f4 40587ffc 412bf55f 412bf8d1 -> 4040d15c 3fbd43f4 4058872a", + "40e45c48 405dff88 3e75a967 4111ff10 411ad628 -> 41844c60 405dff88 3f0424fd", + "bfc91160 bf911030 40983b6b c181834b c17e83a2 -> c18cf3a4 bf911030 409cbeae", + "c10e5030 bed5e5a0 3d94eef9 4198d543 41997669 -> 4123fb7c bed5e5a0 3de590f4", + "c095310c c080ae26 3c45069b c15d5ea8 c15d1990 -> c193ea51 c080ae26 3ca7e40a", + "3fcc6880 bf00fdc0 3a0bc580 40f568bf 40f56b49 -> 41144212 bf00fdc0 3a38c1a9", + "c1138271 406d8984 3da59e58 c16f0f16 c159c228 -> c1bbf588 406d8984 3f3f1de9", + "410b2034 406208f8 3b8c3867 3fdf9819 40b7818a -> 4146fa18 406208f8 3fffc3fe", + "411eda00 400df704 3dce454b 41286f0c 4128cc98 -> 41a3bbe9 400df704 3de5b8f7", + "405992c8 c090608a 409bb4fd 412f3fd8 412fa860 -> 4165d8ce c090608a 409c1dcb", + "4090a614 3fce5ec8 3b0a0416 40623497 408d3634 -> 4107e72a 3fce5ec8 3ee1f52b", + "c13a6674 c02b9ed9 3f1286ac c0e4e3a4 c0df84ec -> c195c04c c02b9ed9 3f28035f", + "c12d926c 4000f530 4105821d 40ae96c5 40af6213 -> c0ac286c 4000f530 4105b506", + "40655140 404345d0 3f6a71aa c14b123d c0ac9046 -> c0aeb1c0 404345d0 40921883", + "c04164a0 c06b8d26 40fabdb3 c193033a c18795b0 -> c1a57909 c06b8d26 4108cc97", + "40cee0ec bce11000 3e3b8975 40c9a190 4104e5c0 -> 415c4bba bce11000 3f97c5c9", + "beb63b40 3df97880 3beb4968 c1399632 c13928d2 -> c13f115c 3df97880 3ca861b8", + "408a9c58 3f884574 41025fc4 c0bc5454 c0b87a40 -> bfbf2bc8 3f884574 4103565a", + "bf140f40 c07ef286 38d91218 418e644f 4190ae91 -> 418ae8f6 c07ef286 3e12b586", + "c1910d98 401806f4 40d33646 c11e8348 c0eaa031 -> c1d60270 401806f4 40fc69d3", + "41331ee4 c01bdd50 3e54b02c 4140f3a3 4140f46d -> 41ba0976 c01bdd50 3e54d2e9", + "bf41fd00 3fe97ca8 3a8cd91a 412bc041 412bc207 -> 411fa154 3fe97ca8 3aac0a4a" + ] + }, + { + "glsl": "vec3 cultmath_af_mul(vec3, vec3)", + "tolerance": "ulp-bounded", + "cases": [ + "41075bac 3f4af898 3967eae5 bfbe3570 c05d8722 3ec408e6 -> c1492496 c1f3b01c 40c939a5", + "c0d94220 402f4f90 3a15cdde 416d3958 409a98ca 3fa69caa -> c2c952e2 40fa03d8 41cd308c", + "c01ec080 3fca088c 3cdf5746 c0e2cd42 c06d77bb 3914903a -> 418ca53a bffdd898 40c4d91f", + "c0aa1fdc 4035ee2c 408585f8 3faab210 bfa8e69c 3f80a3bf -> c0e2defa 412ce582 41d9b797", + "c186392d c029d48a 3fe03804 403288d8 c05dcd2e 38dc3ebb -> c23b3702 424af9bc 41a13fd4", + "418847ec bf7f73e8 3a99ba0f c12e8e34 4081796e 40597fca -> c339d92c 429f9f9b 4282af9c", + "c1958961 3ecd8a20 3f70d183 4197ea98 400a85e0 40d27f52 -> c3b17a21 c203564e 43188e0b", + "414802c4 bf5c8c20 4048295a 41923b8a 404bd2d0 38dfcf7a -> 43648030 41c08296 428bbfb0", + "c0ed9a00 4015d44c 3bbcc67f 40c2c048 40590fa0 41053c59 -> c234c12a c12ef540 42b2bc31", + "c13e0ef2 c04dfa95 3a952dbb c11426b6 bf96a8a8 407c8497 -> 42dbfac2 422f2100 427d7c87", + "416b780c 4026bcba 3e01a3e2 c161d454 408e54c0 3da83447 -> c34fb7cf 41e57dd8 4175f08a", + "408b95f0 c03921ec 3f156a6f 404f0088 c00653df 3db40ca0 -> 4161bd0e c194179f 411df92e", + "c1293bd2 bf89ca3c 3964f06c 4193590e 400d537c 40bcdbe2 -> c342d071 c22cbc32 428e5259", + "4180721a c070e96e 3c455b28 415b8c0c c01c90cc 3c4cab89 -> 435c4fc1 c2b5dbec 411a6c3c", + "40c2c284 c010d04d 3d82ccee c12cf620 c0983fa1 3edb8a92 -> c28395f0 c08fe5c4 4175e023", + "40f99ee4 402913f2 3a2159ff 41704c98 c07ba2db 40c0a18c -> 42ea4fa5 411019f8 42928705", + "c0c8cab2 3fd3ae30 3cb56e3f c0eb57aa bfc2a7ca 3ddfe5a7 -> 423896d4 c027b0b4 40652b54", + "c11e73b4 401572de 3c87f222 412880e8 bf559028 3c8c6ae9 -> c2d0976f 42036a82 401632c7", + "41853558 c024c10d 3c9e83cd c11b7e45 bfaa15d4 3b00da3a -> c321d201 403915e8 406b0a03", + "3dfa6000 c0980a01 3d0974b9 c13f41c5 bd4d6580 3e680c81 -> bfbb0df3 426326c6 3fe06645", + "c199026a bf9a7dce 3a606d70 c1976e04 4076f0b0 3d2e957e -> 43b5045c c24bcdc1 40b16590", + "c030ba50 bfb310d4 3b2f725a c17860dd 3f962ee8 3946888b -> 422b775c 4193d0ab 3fd7ee0d", + "bf183be0 3f5a4028 3f416a20 414563ec 3f67ff10 3d269488 -> c0eac2fe 411fa934 412de241", + "41981b2c 4078e05c 3ff7279d c176e15a bfeec1a0 3f4f4062 -> c392b002 c2beef7e 4273038e", + "c113c47c bf1ec928 3c1e64ad 417f707c bfa3504c 3986abaf -> c31371a5 3ff0c620 3f75f3d0", + "bffc2300 c0652394 3a3a6b2d 413eb9aa c003a700 40defa0c -> c1bbd8d7 c21a814a 42383686", + "c049a270 c02f97ad 3fdd9800 4185c7fe bfd6a104 3948e40a -> c252bdd3 c2226460 4211d225", + "c158c8fc 4042cb8c 3e08ee9c c07b29d8 402e0578 3f4fc3ff -> 4254b07e c2432497 41b5ea64", + "bf2d55e0 3ff921a4 3c730c3e 419aa898 c02f781d 3ddbbd96 -> c1516fa0 421def40 40be4ccd", + "4148e610 40826e42 395fbf8d c07391d0 4024e874 3c8e25b5 -> c23f24cf 4186baa9 412cae3e", + "c16d86af 3ea30780 410add15 c13db36a 3fce4704 3cc3120f -> 433002e7 c1dd97f8 42ebefc3", + "41766fb8 3fef8530 3b3ac2a4 41447f74 bf9066ec 3bdaf83c -> 433d283a 40b35d34 4010f46a", + "40def454 c08dacd3 3d785eff 41406f60 4090f3d2 3d0fd72c -> 42a7983f c1ad8176 41abb0ab", + "c15be5b4 406b05f8 3e5b17ec 409373a4 bf56e898 3e7ab7f1 -> c27d5077 41e3ab80 410909e3", + "40bd09d4 bea6ac80 4073b763 40248a80 409ae9cc 3c0c930f -> 41730123 41de1711 41ef19a7", + "4180e48e c09ac214 3cb327c1 c080419c 403916ac 3f80b566 -> c281269f 4283f4ef 420ce477", + "c189fdae 40927536 3a08559e c160138b 3e2c5d40 398b26a1 -> 43719102 c2860048 3f48aa40", + "beba3fc0 3fe15658 3b61cc45 c189d59a bff451f4 3a368a76 -> 40c88f0a c1ed1870 405b5fad", + "c1902367 c09bc4a7 40bdc775 40113910 c020fec0 40354d7d -> c223885c 42091c92 42f480db", + "41345abe 3e0bf800 39c6d918 c0f34e94 3f0f1eb0 3a73d332 -> c2ab697e 40a866c6 3db8c1fa", + "40179a00 bfe565d2 3ea7a3b1 41815d48 40815b2c 3bea23cc -> 421937af c19b3d33 415e5197", + "417e1790 c0106508 3b0f7bcc 4123e24e 40146aba 4117907b -> 4322a99f 415b7e52 4331166f", + "c0aba8ce c09de915 3bb919b4 40501568 c0707430 3f005048 -> c18b8782 40838654 41bdf549", + "c0a5fa82 3f48c080 3a223c84 411e7302 3e1696c0 40e4d91c -> c24d765a 40e0195d 422b4fae", + "c02de1a8 bfc50a54 3f30fa49 40467590 3f9708e8 3b1c5d86 -> c106cc62 c0ff56a0 409934b6", + "40ec66f8 4025abf0 3c2738d5 418fd77c 3f6817d0 3a7a7cee -> 4304d482 4254f767 402328cf", + "c097583a 3f5dcdd0 3f390b22 418bb9ca c051c2c5 391c6cea -> c2a5357b 41f511e5 418ead1c", + "418028ec c004291e 3b27254c 4192fbd8 c045dd55 3ad45deb -> 43932ad6 c2aeefc2 40ceff62", + "3fcd12c0 c03faa9c 3c10fb07 c09eff34 408caafc 40f6f00a -> c0febc04 41af61fb 4243286c", + "c194ad2e 40477380 3f8946b0 4114921a be8e6830 3d7d11a1 -> c32c91f1 42086dbe 41487ec0", + "4083ac38 bfe70ba0 3e43fdce 419d9c40 c0892896 40ea9c51 -> 42a221f8 c254cb30 4264882c", + "406f69b8 bf560d80 3c44ae45 c15d3898 c0404e87 3d47cdba -> c24ee331 3ea40800 403c0eed", + "4195d2ee bf29c358 3c2fe6d1 c170e8a1 405e2128 3aa7519b -> c38cfde5 4295f87f 4021a3d1", + "40f2a1d8 be8b80d0 3a1c8e33 3f853960 c094027b 3c485a15 -> 40fc88ec c20d6a50 3fae0b93", + "4177237c bff499d4 3f9fcd70 c041d320 408484aa 3f49106b -> c23b1d92 428b817a 41fbd268", + "c08cd5d8 3fbf8394 39c3d11e c13746b2 bfd4c50e 3feaf31d -> 4249a784 c11d2a65 41551490", + "c09bb4f2 40828cac 3aa91601 c0f565a4 bfabc57c 3f3ec367 -> 421541fe c1c60bed 41427429", + "4150ab88 bf924a20 38d4250b 41611b54 bf6c95d8 3d0aa1e0 -> 43377d09 c1e10ee6 3fc4de2c", + "403692a0 4083c724 40d4cd46 40594478 c0925f6a 411ef560 -> 411af31e 3f6e6c10 434f2701", + "c07872a0 c0876888 3b288100 c1271889 c07beafd 3f263125 -> 42222a96 426de372 41afb25b", + "c15d972e c053b0c8 3d04f369 40af2ad4 40293254 3d661236 -> c2979f6b c25ae100 411f93f1", + "c044a078 4022ec8e 39a5a0b5 4008bf60 401cf1d8 409ebecc -> c0d2107a c006104c 42087629", + "419fb53e bf97f884 39b39ae3 c0989bbe 3ffff748 3e39d1e9 -> c2be698c 423655cf 40c6e034", + "c055d368 c094fbeb 404d7d2b c0eee1aa 4061e690 38d76050 -> 41c78707 41b7b360 424ef1de", + "41494020 404343f4 3dde13df c1766299 4046d5f4 3afc6ce8 -> c341b122 c0fcf5b0 41384285", + "40350f98 c066c1f0 3f825cfb bfce9d80 c05d1ba4 3a7bfe78 -> c09221f1 c07d0bfe 418d02e2", + "c111cd41 c046e9e1 3e02ebad bffa5e30 bf36f238 3e3c8717 -> 418e9816 414976aa 409ac0e6", + "c070ab88 c09a4598 3a326b05 c197dabc 3f5b9a80 3e4f6ce6 -> 428ec2d0 42b0921a 40bc67d7", + "417b3070 c098e764 3bfb16d0 40a6be14 404e5c50 3970160e -> 42a39bea 41cdc7f7 41779f4c", + "41672334 4059d9a8 3b79eb31 baf84000 404e5548 40c835eb -> bce023e3 423a44ba 42f5617b", + "418db78a bff58b0c 40a7b8c7 415b8328 409a75f8 3ddae775 -> 4373093c 426cc0fc 42da4090", + "c167fc89 3ec9f170 3ab25a5d 4183524c c07a809e 3d12a2bd -> c36e019c 427ce709 4006b819", + "c1964440 4094eb74 3ddb2a31 c1211570 40825d08 3fbfabc5 -> 433d1b03 c2f6bf14 425ee973", + "4156dea0 4038f59a 3aa3b8ef 418ea198 c0319308 3f6044a2 -> 436f6e3d 41643a88 41b2bcea", + "41733b3c 408cf26e 3b7c0de4 4112f77c 3f7c2478 3b73d912 -> 430ba2ec 425db947 408e6a76", + "41736564 402e3a80 3bdf9805 40cf7478 3da8a500 39d19b6f -> 42c53db8 41973609 3e8d716e", + "4105d764 3f2147d8 391d45b0 401ba290 40784710 3932c60c -> 41a2bcce 4207ee9f 401c93e4", + "c02787d8 409a45ea 406ca684 416a4dc4 3f7f9e28 3ff6e7df -> c2195510 4287f886 42a845bb", + "bfadbef0 3f907144 4093f89f bf409180 3fd2333c 391a059d -> 3f82b1f4 c044fced 414ed1a5", + "4119910a 409f15f2 398c8359 c15c8e24 409423f4 392a82eb -> c3044de9 c1c0c718 41b82d8a", + "c1072df6 bf6bd018 40a2243d 41840a58 c0966064 3b9d7c32 -> c30b7252 41c3fe7c 42dfad95", + "c163db50 bfa8857a 3c9e2830 bf144400 3ec89970 3bacbbc8 -> 4103f751 c09a256c 3f1de00f", + "413fe8e0 c0100a05 404ce216 c1514b21 3de4a4c0 3ad0c835 -> c31ce571 41f63b56 422a0ca9", + "416ff2e8 c09b8944 3ba5624b bfcd1d40 c01750a4 4107a160 -> c1c040ee c1dd5860 4333e213", + "c09829cc bf81230c 39b5d155 c0d53db2 40871498 3d132b79 -> 41fd7ee5 c15597d8 408f07d4", + "c194e246 3f099620 3dccf1d4 419dc0e8 c011e964 3ccde34e -> c3b77deb 42541c2c 407a4cac", + "c17c8c9a 3e6095a0 40b8345f 4045b3d8 400e2916 3abc62f1 -> c243099e c20988c2 41f8b251", + "c08018ba 4099c630 3974ddf9 4062de00 3fe81c10 3c10d12e -> c16309d3 411c682c 410cb0af", + "c16b4ed8 3f2b70b0 3cdc6fa4 415cec60 c00c810b 40b1de2a -> c34b1107 42262284 42af098c", + "410109c0 405275b0 3a0a134f 4191b24a 3ea14ca0 3e124101 -> 4312e0c8 4279b82f 402ab966", + "c12ddc03 4046c388 3c8b7783 418ebc3e befbfd90 41074c8b -> c341dfc3 42730997 42f04498", + "411c5e0c 40446714 38d8e655 c04a3240 c08a5f44 410d5859 -> c1f701ce c24fd1bf 42fd6eef", + "c1076b78 c0017a0d 3b3c9fde c17a2cf2 c08ccba6 3b2a7483 -> 430456ba 4289be76 410fc844", + "4183d584 403b3398 3ec6adb5 410bfd42 c03f5001 3bc38a8b -> 43102eb5 c1bd5996 4156ac53", + "c197f6ba c046b0ec 3b185df9 40c15f34 c00587f2 3c86a8ec -> c2e5930b 41a6fa40 40db8364", + "c0d2430c c05ff12e 410f4e13 c0e50842 bfc16e10 40673049 -> 423c1cc2 420be4b8 4317a813", + "41369634 beec26c0 3d367596 c183c41a 405c5974 4016d23c -> c33bf574 423b8c10 41f47a6b", + "c0917778 40073db8 3d009bad be054500 3f115f60 410335f2 -> 3f17748d c036cf9b 42605999", + "40c92c20 3f6cdac0 3f32cecc 416739a0 c0922339 3efdacac -> 42b5b419 c1756cac 41ab6d48", + "41843824 3dc4d740 3de4748f c182dc7a 4086b2f8 3b04a372 -> c3872cc3 4287fead 402eea01", + "4107c718 c08228c7 3d1adea9 4018f9d8 c084b585 40778ccc -> 41a24573 c233a97c 4283a346", + "419f9e04 40958c54 40b18006 c0994256 bfe1864a 3ddbcb64 -> c2bf1d9a c266253a 423f3eec", + "412216e8 404f3338 40b04581 415bb980 bfd68e02 3ce63eb2 -> 430b1f0c 41dbd4da 42b5acf0", + "4163b254 3e913d00 3e154fe5 c1061271 409e0cc6 3e9fc10e -> c2ee7f7e 4287d266 40fd5d3e", + "40df27d8 be82d940 39b33c13 c11f1637 bf6bd1a8 3fc92bff -> c28aad1d c0788017 41399fdc", + "c0fd5bf8 3ff89e54 3d3d43ce c1142526 bf9e9e92 3f11b321 -> 42929def c102c386 41087b2c", + "c0b49022 405105d0 3e0a2e75 4051a030 bca88e00 3db55f4a -> c193daa8 412d0415 3fa80d01", + "c19521fd c00ec8c6 3b108ed2 c157e090 c07add44 3aa431dc -> 437b84ba 42ce5806 410cf553", + "413a1bd8 c089a49b 3a49b2a6 c1735ebe c0296b49 3c166765 -> c330ed4e 420a8a00 4138bf39", + "40a473fc bf4899e0 39a48d2b 414aada8 4076aa88 3de8061f -> 42823313 411e1884 406c9257", + "4146dd64 401e523e 3d80695b c179e541 c086f997 3e0ff06a -> c3421f6d c2b61fa8 415c82ee", + "c18893d5 be1c3f40 3c8dd2c7 c141d0e0 be1fe700 3d0a70fc -> 434ecdc6 409074a4 3f51b807", + "4168c988 c010b5b8 3914d73d c19223fa c01bce77 3ec4ecdf -> c384e3a6 40bc5170 413f935a", + "c032e1e0 3dee8f80 3bde9c47 4177cef0 bec99080 3ba085ef -> c22d2889 4039e2ce 3e2c200b", + "40cf6904 c04b4f3e 3ca17d1f c166291a 3fb341c0 40bd6229 -> c2ba79a6 425b18c3 427825b0", + "40c22c34 3ed85240 40a293dc c19b23af 402d8aa8 3f10e650 -> c2eb57a9 41042a20 42effdd2", + "c19666f1 4087d0a4 3a811e92 40b21848 c0835c73 3e05e134 -> c2d143af 42c997f6 41a39038", + "412e03d2 3f6fb2e0 3dad4663 40db7518 c01c2845 40cd2af3 -> 42952cdd c1a0ec7d 429eb239", + "c0b074d4 3f5a07c0 3a2f6135 c0969ad0 bf9133c2 3bf2434a -> 41cf9e54 400fce9c 3f823212", + "c12391fd 4087873a 3b31e97a c13247b9 bf066708 4040ff65 -> 42e3d2ab c2274bee 423774ed", + "c152ccc0 408a80ba 3fd6238b be8a95c0 3ee8c830 40a45e9c -> 40643b5e c0e52b9a 42cb5c82", + "40255fa8 405f10e4 3d05aa6f c10ce3ea c09bbd09 3b9600f5 -> c1b60716 c22d1158 418b7f15", + "3f3b7e80 403a3b66 3c536fbd 411b9b6e c07634e4 400a69d4 -> 40e3eedb 41cbdbc8 419a323a", + "41219f90 409cfcfa 40890441 bdc7ff00 40321150 39453bd9 -> bf7c880e 41dd0298 41cfe09e", + "41392f30 c0139c7c 3a1d64c9 402c94f0 402d4a9e 3fdd8046 -> 41f9aee2 41c8f439 41f22765", + "bfb93d50 404579d8 3cc69c85 c0fc634c bea138e0 3a1f30a1 -> 4136a034 c1bf0b43 3f962d65", + "41324b82 c00da780 3936928f 40b77be4 c07b9d5e 3c328a7e -> 427f9486 c2620147 410d95e5", + "bfd3a980 c085ab40 3d941c12 415517bc 4068fab0 3dc29fb7 -> c1b02fa7 c2769bde 4187f2e6", + "41908456 3c529600 3f8cb5fb c04d7398 be9f3a50 3c9dbdb4 -> c267f674 c0b5183b 4087d024", + "41964184 3fdd8678 3a1b6a44 41397c40 40027c24 3b1d67c2 -> 4359bc8b 42696d1c 4065836c", + "4137531c 409f1958 39805c6d 40c686d0 404a43c0 3a8a7515 -> 428e2ad2 42861cbd 417bb949", + "41635538 c0361672 3b86396c c1161847 4094a7f6 3fc32973 -> c305497e 42b963a1 421d2124", + "414c0d84 c07a1e62 4027d398 3f412300 c0938d67 3e9a8691 -> 4119f216 c2770423 4217a3ff", + "bf2dd160 40920bba 3c2e3409 4019e5c8 3f943924 3b11d872 -> bfd0fc4d 412303b7 40aab4fd", + "c1936392 c091e562 3ba3942a c19998a0 bfd5695e 3f72eb88 -> 43b0dcc3 42ec8167 41ec27f7", + "c19f553d 409c67a0 3d9931b6 412e1e2e 40646ad4 39b95065 -> c358bd70 c18f2624 41944624", + "c0d9fede 40424768 3b0c82c7 c0aa2024 4098e142 3e331093 -> 4210de9e c242bd0c 4181f97c", + "c11137c9 3fee41e0 3d45b680 4196e85e 3f812b28 3bce10e3 -> c32b34ff 41cfa020 403a0af1", + "403b53e0 befc7940 3e17c0e2 bfd3bd70 bff74d54 3f0bdbae -> c09af0c1 c09adc2a 405bc0cb", + "416d4cb0 c02f2653 3c043c06 41454e3c 3f941518 3f3960be -> 4336e488 c184b827 41800537", + "4169aaf0 40568678 4024ea1e 3f8a66e0 c0176dcd 40f2254c -> 417ca818 c1f7716c 432c303c", + "c11869c1 c0910855 3aef4d74 412f6760 c046ee61 3ac4e120 -> c2d0dbad c1a09d9d 4162277e", + "c033f7c0 c075cd04 40fa3ee5 412457b0 c0084ec6 408e61bd -> c1e710b8 c205d6c8 43298e37", + "4114f560 c08a4bf0 3c86a5e0 41999504 bf0818e0 40259d31 -> 4332bab7 c2afd5e1 4217c230", + "c158c864 3ede69a0 3e838ef0 415177ac 408068c0 3e119a1b -> c33160f0 c242ba46 41029c7c", + "411219d6 3e5b2560 3a63faf1 be6a3200 3ca27e00 3a074dc9 -> c005a827 3e0759c6 3c1829df", + "412a72e4 401d7b5c 3c54e351 401c07e8 40063a7e 3d719418 -> 41cfc68d 41e2bc30 40bfce97", + "c17b3fb0 403f6d96 3b051161 c0817f7c c096bcdf 3a6bf555 -> 427e306c 42777698 4161fd49", + "412c0fc4 c0770b02 409f1c1e bff72c20 c087d393 3d60bef8 -> c1a620de c218c442 4240a94c", + "c0db2246 407680e8 3c0f4a02 419fe1e2 bfabd580 3ecbb815 -> c308dba4 42ac5666 4119e3a3", + "c103f53c bf2785b0 3a0bf686 409bcb44 c091593b 3e5f6e92 -> c2209c84 4209196c 409d6f63", + "c190cf89 408b556c 3df07ce4 415718e0 409cf268 3d4f8d85 -> c37358bc c1f1f396 41c53bd7", + "3e8d9c80 c0520c9a 3bca41f8 4178f968 406e69b8 3e316666 -> 4089b993 c24829c9 414f67e3", + "409af680 4027f75c 3d190d33 beb60e00 c08d4c0b 3f5d2279 -> bfdc6771 c1b286ca 4191fe3f", + "40f4d168 c00f00a6 3c8ae4d9 40b334dc c05617a4 3df8906e -> 422b60f9 c2186c52 410d3fdc", + "4122af60 402aaf20 3c029bf5 4101c9be c01508c8 3b1bb72a -> 42a4f528 c002ce60 40ca6066", + "c069de60 be125500 3f1591ce 406fffb8 bfbe415a 3999e335 -> c15b4038 409ca8ce 40517a36", + "4044e168 408a7b72 39d9655c c0821ad0 408e56cc 40534cff -> c1481e2b c07a54cc 422ecb7b", + "c1468b34 bf1a2a30 38f291fc 411a007c bf9b6ef8 3dd4bfba -> c2eee03b 41145b06 40056516", + "c13c81f4 4041bb34 3bf8e64b c0b57d54 4089be76 3f33a0fb -> 4285a42b c287c3d8 41bc0466", + "c131fa32 bf6ce5f8 3f351a51 419f14f8 c085cb3f 3aac51e9 -> c35d31f2 41e0dae8 41a74aa2", + "c1593040 4042a328 40558cff c1857724 be48efe0 3c0cc203 -> 4362764a c2404b1f 42645cb1", + "417c6c24 c0974258 3c3a23ef c04a51e0 40848b9a 39deda7c -> c2477e0f 42a09418 419d5d77", + "40d198a0 c00489e3 3b9558b8 3fd7a040 befed2f0 3ed9a8b4 -> 41308a62 c0d7f3d6 4096a356", + "408559c8 c09ba88f 3c0ee802 41314838 bcda2600 3cac008f -> 4238b163 c258088b 3ed527cf", + "417f6544 c08fdc77 3d74b6c5 419ae8f0 bfa40bbc 40a7bf79 -> 439a8b4e c2d70532 42e51a68", + "c0e1ce22 3f4c39b0 3fe5e46d 417fda84 c027dd28 3a046ddf -> c2e1ad12 41fa1e6e 420e1d01", + "419fb4a4 c07b9cec 40c8857f 40a11050 405d9458 3bbaf1be -> 42c8f591 4245501e 4286097c", + "c144fa22 40361282 40314a4e c11fba5e 3ff8ea0c 3e1dc394 -> 42f5cd82 c2515d3e 422557d4", + "c0200e50 3fe6238c 40c392c2 c191d837 bf8905e8 39664476 -> 42365e94 c1f0ce2e 42efc714", + "4156cf14 bf94f0f2 3b6013d9 c018ff58 c04a538a 3c674792 -> c2006136 c21ea50b 4079d4b4", + "41543148 40834d24 3e41f213 bfa80650 c080d6b3 3b9a068e -> c18b4593 c26b20aa 418ce978", + "c1101c5a 4028f1cc 3945f5e1 c145b8ae 3edc5600 3baabc62 -> 42de9b8f c211fd0c 3f9980da", + "412a6e78 4099125e 3a340199 c194334c 3f590968 3ebbb8dc -> c3455408 c29f2ad8 411ba86d", + "40426010 3fbe3f40 3bc560a1 4185ab78 3eeeaba0 3e93bc63 -> 424afc33 41d2000d 40069cf2", + "419bbbfa c00d36a4 3a613fe2 4134d2c4 c08b7a48 3fee8c24 -> 435c00c6 c2db923f 424818a0", + "4155b300 3f16a468 4083db02 3f96d400 403c56e0 4098fcd4 -> 417bcfb4 421ffe02 42d228e5", + "40fa9808 401006a8 3e8b93b9 400a6ce0 40983100 408fbea7 -> 41878088 42287272 426c6652", + "4052ce30 c0224ae6 39fd2660 4091bca0 c00e1ee5 3c41d53a -> 41700439 c196e7e8 40b68267", + "4180500a 403f35ac 3f7ca93f 40707900 40692824 3d4253a9 -> 42710f5e 428b508e 41991a47", + "c188540e 40846d2a 3bd7c0be 41944c6a be478be0 3f9d8526 -> c39df293 42a01184 41d8046a", + "c1618899 bd9de3c0 3ea19df8 c13f480d bfb6d760 41055f92 -> 43288464 41a87471 42fa381f", + "c04bd040 c0286c0a 3a5a370f c12bdf82 bfcb6538 3ffea0e4 -> 4208d60d 4205510e 417c3829", + "c197c35c c0990199 3e6d63d4 c03ec4b8 bfd2b934 3b4fb6ef -> 42622f3a 4235ee6e 41105371", + "400d8558 bfa2ccbc 38d52a7c 412e3910 407432b0 3b0b18bb -> 41c0a05b c0ad2f8c 409b9494", + "c18e54ee c0266b40 3e980a08 c08fac6e bee3eb20 40219768 -> 429fc29e 419cc205 425b6ffc", + "4038d4e8 4014b330 3adad9bc 416c36a8 bf6f70c0 3cd5c2b8 -> 422a8bbc 41fccdda 4015783b", + "c12c6322 bf9808cc 4032da04 c15e5aa1 c0442d9f 39284c52 -> 4315baff 42462159 424c2efa", + "c12c758c 3fd8ff2c 3c9f1f0f 418c84be 3f2e93e0 3fd0acc6 -> c33d536a 41b36a5f 41af06fe", + "4167cf34 3fbe1224 3d9fd1ef c18badc2 bf6b2208 39b529f5 -> c37cf5b0 c21cef5d 4033733f", + "414aa0fc bf9f8352 3de7bb55 c027bdb0 c0629e32 39ca1106 -> c204c529 c2264eba 40a3b10d", + "c1069f4e c0958eec 3f4e8c1b c0fb5d60 3fcbd684 3de28407 -> 42842f4c 41ba823a 4184d173", + "40937898 c05bbd62 3df47def c1469378 c0434e0c 40bcc0fa -> c264c872 41e4644c 4271daa4", + "c06401c0 bf791020 3a4ea7a3 41934d8e 3ed574f0 3a968b43 -> c2833214 c19b3190 3ed9fb69", + "4144ed80 c075b355 3e69fb0e 4161559c 3e82d4a0 3f8abac0 -> 432d5698 c24bb043 41b00bfa", + "c0fd74c6 3db21800 3b88e723 c1686e3b c08f3529 3915fb83 -> 42e61ef6 4208bb49 3ef08fcf", + "416e72e8 403d8c4a 3ac4d2fd 412e2138 406844ac 3a2702ea -> 4322310b 42aca306 412c82e0", + "c119c434 40889c92 38ec1099 415f4e14 bfe165a6 3c3b1d79 -> c30620cf 429902ae 40f5b274", + "c12eada6 3fdcc558 3c97f0c5 409f1e40 3fc349d8 38df4600 -> c25924fc c10148a8 40303808", + "41891330 406405e0 3fbca694 c081a9d6 3fa24628 3913cd82 -> c28adb37 40e924b6 4145c4bc", + "c00bc0d8 bf1fd3b0 3d0c85df 4197cbe8 c09f7a26 3a145ae2 -> c225bc20 bf76b6b0 407bd79d", + "c109e17f be5d5780 39adf822 4057ce10 bfea0cc4 3c215873 -> c1e876bb 41707492 3ef7cf9d", + "410c0e04 3fb1bcb0 3d4ccabd c19a147d 40005f50 3df2c9c7 -> c3289748 c112fa52 40a1d257", + "c13ee136 4092cfb8 3f69f891 c1197294 3fa00688 3febd301 -> 42e4d410 c26ba885 423f09ec", + "c0bbbb26 3fe36cf8 3ff44169 c0e7ffa6 c03a949b 3a7ae4ef -> 422a2158 408716ea 41c4b203", + "c141aa40 be85c150 3a384a20 c13c7b59 bfdddd26 3c5b5ecb -> 430e9657 41c075e0 3f20c76b", + "40c4d150 c0095c02 40875fda 416639d0 3f24be40 3fd82fe1 -> 42b10080 c1d76547 42ac44bb", + "c0f6952c 3fb6bdb0 3bc3ed30 c18e6747 4081c08e 38f5940d -> 43092a39 c262a194 40bd76e7", + "bfafde20 408327ac 3a1d6b93 c133e5c1 c041f3b4 3e6000e6 -> 41772c4d c227ad29 415a06f9", + "c127ea5c c0497da1 3f93e4b7 40c51f4c 3f9aeac4 3b88887c -> c2814be0 c20061a4 4146350c", + "c1039152 40950f2a 3b129379 41828e92 bf90c478 3fd2a7d3 -> c3063223 42aaa31b 41d414f3", + "c12d438c 3e0895c0 3b6e767b c176c24f bfad2344 3c432faf -> 4327026b 41497324 3ebe9acf", + "40f29014 bfca360e 3c19ab06 418eee6e 4090ec02 41026ede -> 43076dd5 40c352c8 42a439e7", + "bdb89100 c0010f06 3f39e274 415b2540 4061fbf0 3b4bf328 -> bf9dfee6 c1df80f7 419d1b83", + "c1930b62 40991dc0 3b1ee50b 3d659200 4095b782 3a05ab42 -> bf83dd0b c2ab74c1 41b34811", + "40906390 409fec80 3e56cce0 4081b074 3f272c88 3dc9647e -> 41924b63 41b99b53 40a69bb7", + "40418e00 bf80cf76 3af95e06 3ef6e880 405074fc 3c15a8fa -> 3fbaae3d 4115d80e 40549728", + "4100f9aa bfbe72ea 3d59c240 4119eb3e 3e9691b0 3ff0e7e0 -> 429b1776 c13f1587 41984a99", + "c110ec3a c005c736 3abe1706 41536368 c05bff36 3b868b5e -> c2ef55f4 406136b8 40e82bd9", + "40047f68 bef5b480 39bfeb6e 4196d18c 406b33e8 3c34806f -> 421c1e37 bfb82880 3fe66747", + "4029d8f8 4011d6e2 3d5aa36d c152c224 3c85f600 3bdaf014 -> c20bd4ca c1efc8dc 3f464e86", + "41208cb6 c083c818 3f43d016 413d5ea0 bfa50d48 3d6fd838 -> 42ed8675 c276b862 4181c92c", + "4110e358 409b6bcc 3cbd09ad c09984ca c065d46c 3d045fad -> c22dc60e c25f47a2 4190af69", + "41223b10 c0000202 3c902461 c047af20 c02e2e34 3e782e29 -> c1fd15c8 c1aad648 4107e26b", + "4182f258 4034ce60 3fd339ad c0caab96 c09023d0 39c13591 -> c2cf55fc c2b73e48 41f4f117", + "bf111280 3f8580cc 3d3aed62 415e5ed0 408ea082 39a1b8ec -> c0fc078b 413f84a9 40af94e7", + "415ddd68 3e9dde70 3fe28349 c14cb464 409caf44 3c83aded -> c33168c5 427fce12 42044995", + "c186f943 be64ada0 407e8c97 40e4d544 c07d7ff3 3a3acae6 -> c2f14ce2 42827628 4234620f", + "4160086c c0521dc2 3b6b8c26 408dda00 c03497f7 3d8ce691 -> 427846d5 c2584192 4127acef", + "4192fc7c c0639df8 3d0192bd 416be420 c018a6d5 3ec9ca12 -> 438770c1 c2c08420 418d71d6", + "3f5bd280 bf6875f8 39239165 40d13050 bff3206e 3f2e9607 -> 40b3a057 c0f22558 403b96d8", + "c1991250 4072848c 3926271f c0a04444 3eab0180 3de5f0cf -> 42bfa887 c1caf394 4075c80e", + "40c2f588 3f3bbdd0 3b6976f9 c19d7c8d bfc7f286 3f103fd9 -> c2efdedd c1bfa188 40a23a39", + "412476cc c02c85f1 3e7d1d10 40dd5e08 bdabc6c0 401cdd88 -> 428e36f7 c19c14b1 420974ad", + "414163d8 407001d8 3c0c37d8 41949738 407865d4 3f82057c -> 43607fea 42e921a9 41f6bf19", + "407a6420 4098fca6 3b11a05e c16c82eb 3fcb1ce8 3da38350 -> c267545a c280ec8b 410512c9", + "c0795430 bf5a2ab0 3a348c77 c15df988 3fc91e88 3aaebb68 -> 425830b5 40b6762c 3fad93d4", + "c16c6d62 be7d5740 3e1f22dc 4100ae18 4092b91e 3e2ae528 -> c2edaef3 c28b7c18 40b42b29", + "c11b698a 40907908 3f80d2e5 411fe870 c07892ad 3915ea56 -> c2c22751 42a5b1fe 41fc0a62", + "c1930863 405f0e24 4077cb85 c1591eae 40673450 3c7545f4 -> 43796776 c2e36179 429f01e9", + "c0993ed0 c08da988 410ff0ae c1274869 403144d8 3d74df05 -> 424846a4 42041481 43045392", + "c0cbe5be 3e5fdae0 406f94c7 41132370 3f86b7e4 408d52a7 -> c26a624b c096449c 42a8777d", + "40535d20 3f84e0a8 3d969da2 bfed6970 3ed8e320 38fd28bf -> c0c40444 bf06c6a6 3f1b9c27", + "c126abed c07f4a3e 4074baef 411adfc8 400497fa 409a4d34 -> c2c9aa3d c270c56a 430d1a79", + "c19fbe27 405eb098 40099633 c1277c78 c0833912 3f8fe14f -> 4351056d 4235d631 4294b8f7", + "40d07698 c025365f 3d6fddc8 4114d76e 402ea5e0 3dcc5f8b -> 427267fa c0c794fc 410a9a3e", + "c12a2334 407adb08 3e9eba85 c185726c bfb2be16 3c79d766 -> 433160ab c24a226e 4134dfc3", + "41131796 3e8b9830 3bb07050 416dc9e4 c054eed8 3d0dafed -> 4308a0d6 c1d4470a 3faa9a91", + "bfd85d80 3f5e71a0 3b5e7b62 418064be 4086c6a0 3f21cca5 -> c1d907ca 40da6fc5 40ab2460", + "3f3de580 3fb101a0 3bb88a2b c18eaa87 c027cc1a 408d59e2 -> c153a797 c1d4d8a2 4152699f", + "40f1b5d8 c07e7667 3c6ca313 c17d1f52 c08a8bee 3bd6fd10 -> c2eefe49 41f19402 418ca681", + "c09f5f3a 405fffdc 3fb6bc9a 4197843e c03389fb 3e36ef25 -> c2bca6fe 42a08500 422a8791", + "c0e8dcf0 c05bc716 3f016b63 bf4e4f20 c09c1be7 3a500519 -> 40bba9c2 421911e6 419d14e7", + "40677690 c06d4631 3e81972a c0b4e20e 40338f38 3d062279 -> c1a38bc1 41f8d392 414ca642", + "3e23a480 402771d2 3b837a8c 413bc27a c03706ea 39f6ee63 -> 3ff00af4 41f1f6b9 40f15889", + "c12032a6 c04cce27 3e3eff23 412c5174 bf877f8c 3aff7019 -> c2d7aa00 c1beecd2 40b3d77b" + ] + }, + { + "glsl": "vec3 cultmath_af_snoise(vec3, vec3, float)", + "tolerance": "ulp-bounded", + "check": "the arguments are (cx, cy, cz, ax, ay, az, radius, ...) and the result is the form (x0, a, e). Enclosure: for eps in {-1, -0.5, 0, 0.5, 1} and d in {0, +-ex, +-ey, +-ez, (+-1, +-1, +-1) / sqrt(3)} (15 directions), form the point p = c + axis eps + radius d in double and round it to float32; the device's own reference value at p must lie in [x0 + a eps - e, x0 + a eps + e] with no tolerance, so a device whose gradient error exceeds the allowance fails. Tightness: with L = 7.9640074 and, for each octave (one for af_snoise), R = |axis| + radius, B = L R + 2^-14 + 2^-12 R + L 2^-20 (|c|_1 + R), the width |a| + e must be at most B + 2^-16 (1 + B). Report the least slack of each side, absolutely and as a fraction of e; only the two bounds fail the check. The reference value is cultmath_snoise(p).", + "cases": [ + "c2461d0c 41488940 c07e9320 bb68fabd be288e0f 3e232bc5 3ca64a9f -> beffc1bc 3e2be983 3fe385e0", + "c154200c 417938f0 41050be4 bde44cd9 bddb2245 3cefa790 3b3cd34c -> 3ee504ba 3cf9f4d3 3f39e2bf", + "c067faf0 c21c93b2 c1c94dfa bb35db00 3bdae5af 3acd2c25 3bdacdef -> bee953ce 3ca5297e 3d0af24f", + "c236ea4a 410362cc 415990ec 3d0b18b8 bd8f1cf1 3d9a37a6 3cf87170 -> 3de3c2dd 3d3e2c82 3f10c992", + "41d1f6dc bfdc3340 c01f20c0 3b9f7bd8 3b1c31c3 3b41665a 3c1cdc28 -> 3f41ec30 3c66bd58 3d1456ad", + "4191738c c1a2f3c2 c211c115 be007580 3cb352be 3ddaed92 3b42ecb4 -> bd614cfc be8b1bbd 3f4f4020", + "3f58f200 c22835ab 41acfdb0 3aa2ac08 bcd45342 3cd46062 3ad20095 -> 3f16b968 3d46a2fe 3d3d1b3e", + "c23e1102 c20a1325 42300caa 3dd9fbdc 3e906f25 3e1c9070 3bfbb52b -> bdd23a0d 00000000 40304b61", + "417f3760 c23a4512 c24726ba 3b6a246f bc701b24 bb364517 3e51e0fa -> 3d83ad16 bb88b1ff 3fc8a197", + "c01dbe40 421a002e 416d56d0 3a3436ce 3a1b036b ba653da9 3ca8bed4 -> 3def6ebd 3b1f6279 3d821064", + "41f68280 c07bd660 c1f9b8cf bc9f0747 bbb189f2 bd545910 3d9d7708 -> 3bc135aa 3db9a1d4 3f2c163a", + "40956248 c1211810 c15cd0bc 3c37df98 3c70c3ab 3c47e6e8 3da03b19 -> 3d884938 bb29a7d2 3edd21d0", + "40657ae0 41f62314 c1f905f5 398ab1f8 baa8afa5 3a0f968b 3baeab17 -> 3f2e43bd 3b20efbc 3c50670f", + "414e5270 421006ac c22be7f5 bc100185 3d2f3fe6 bcb9a388 3bb26407 -> 3df30361 3e6aabd0 3def1061", + "c23cd8ee c094c010 c21dae02 3b1742d1 bb465fce bb7ff518 3b7014cc -> 3f3487af bc0b7258 3c7ad7d2", + "c2457338 c19d12f9 4243f768 3b094bb0 3b0c5426 bae289aa 3de14cc3 -> bda277a3 ba9df30b 3f080f75", + "c1eda426 4232d426 bf842a20 3d6e9677 3ee29130 3ecb07d5 3d0e5d6e -> bd926e3d 00000000 40a109d0", + "420b8d12 42277864 c241d9fc bb8b87b9 bb7be31b bbfd72ef 3d55b614 -> 3c32f06c bbaf9848 3e6a8cab", + "c21949eb c2243ab9 406cc5c0 3a60f1e7 b99916ea 3b4a7e0e 3c5eadae -> be7426b6 bc2b1393 3d848f20", + "41f25fd8 423ff36a c210e38c bb1cee61 ba026a59 ba8666bb 3dd016ed -> be7865e2 bb927307 3ef941a3", + "418913f8 c194ae3b c0d6a1a0 bb3ee6e1 bbf513ce 3bfc3faa 3aad7ebf -> be80db84 3c1a9f75 3c1af026", + "4005b130 40604800 c14d245c be3b4149 3c9bd311 3e1a08e0 3d3aa021 -> bd92cb87 00000000 4010518c", + "c23fa949 c169e48c c23503f9 bd2635e4 3e025995 bdc43cd5 3dcdd83c -> be833861 00000000 4007176e", + "c23898cc c22c052e c1fde2c4 3cd4e3c5 3c60fc4c bc5a14f8 3c46a1ca -> bcd1eb00 bb51e190 3da1b49f", + "402b9f10 c1b6589a 41e97638 3ab5bdc8 3a78c21b ba617814 3e277459 -> bcc51abc bac7aac2 3f86ad0b", + "c13f5754 bf4f8a00 419a70d4 baaf2e19 bac6af50 ba1a8d15 3ba6785a -> bdee80f9 3b174562 3c4c24ed", + "c20b69fe 40164180 c1d86af0 bd53cc85 3d5c6035 bd5f4972 3d9b9f19 -> 3e8bdc90 3c6819c0 3f759278", + "c203634e 41991848 c0dfc318 bbe00bc5 bbd99f71 bbae32ca 3e3c248c -> bec70efe 00000000 3fc67e64", + "c1e6994a 423c2d46 c17476dc 3db0b4a3 3db9803d 3d5b9834 3b6cc85b -> bdc7a7d3 bdec47b1 3f0e6294", + "c2108c9f c1d9e95f c1140ac0 bdb7e96d be379935 3d8008c0 3a8658d1 -> be99a951 3dff0554 3fa006a0", + "41c8c984 42287b38 c0d8eff0 ba368f94 ba8820c8 bbf13e8b 3e9ed55e -> 3e92e363 00000000 4021f862", + "4216f750 c221e5ae 421de69e baa07b01 3b7c486a 3b0e6264 3ba00190 -> 3db84cc3 3bbd76bc 3ca4fb07", + "c1a6912a 420dc904 4212ab1c 3a4e0501 b82b28d8 384e8cd9 3c385796 -> 3d097d77 3a9615eb 3d0d5552", + "4213545e c23edb34 41cc0434 bae8753f 3968304e ba027252 3aff73f0 -> be86120e 3a6be24a 3bba7373", + "c2383c25 c0ec7700 411ff5ec 3aa828e8 3b24cf8d ba46ed11 3e046e28 -> bde900db baf3e614 3f277255", + "423bab08 c0f20580 423482c8 3d2ffeee bc57d0c8 3d1a7ad1 3bb46f25 -> bd5314ba bdf70cd4 3e0ac6ab", + "400d15b0 c20b09c3 40cfe558 bc0d2c21 3a74e62b bbab27ff 3ba5b7c6 -> 3e637557 bca6a5f5 3cb48687", + "c110db14 c236d9e8 c095c2e8 3ba5c372 3c1c1005 3bd6db0b 3c073f50 -> 3e1551c0 bb4782d7 3cb101e8", + "419054a0 41e5ef50 c1d94ebe be3a6cd1 3c1df6c1 3d43e7fc 3c55d7c3 -> 3eb77933 3eb99f10 3f95b1a6", + "c0485400 c2262c96 c20c00f0 39d1b4bc 3b702cd2 396c2885 3ca2e941 -> 3e8ccea5 3b98e540 3da9afb8", + "c1efaf29 bff800c0 4170aaf0 3aa313a4 3aa82145 3a8d4fe6 3cbf194c -> 3e95d663 3b48aa42 3d89c9a3", + "c21c36ef c18096a4 41e2e49c bc0b52cf 3d54c0d0 bd373713 3b1cd69d -> 3f3b5fa3 3dab6e35 3e16ff64", + "419e6c00 4132cf0c 423d9cc8 ba92f360 3a9d0bc0 bad31b43 3a8b1925 -> 3db757fd 3b8e70ae 3b78b48b", + "4110c100 c20b4c54 42376d04 b9f8081f baec2544 3a6a9dfd 3cfda902 -> 3e817738 bb3522b2 3dd7d4a7", + "4005daa0 423ce684 c1d85827 bb0ccf78 3a9f9603 ba69a2b5 3d980c0e -> bd4c51d8 3b18b086 3eccecc5", + "42170b4a 41b165f0 420b6f2c 3e135d33 3e848883 3e9d0dbb 3cc6b565 -> bd905ee4 00000000 4065c012", + "423ed542 c24425d0 40cfc8e8 3e7a403e bd177d2e bde887ad 3c044d25 -> be19ba00 00000000 400ecfc5", + "c03ce630 41615f38 c09466b0 be1f0994 3dc8791b bdfe0dcf 3ab967cc -> befcf22b be8d1524 3fb2dbdf", + "413c1438 c1d6d80f c21442ab 3bf8037d 3c62a5d2 3c7d3979 3a9095ad -> bdc2ea43 3b52acb8 3c893867", + "c1cbdf45 c1af0a8f c1ee1ef0 bc70779e bc4e0a08 bbd9d7c5 3d8b5886 -> be3d4f50 3d1698b9 3ecb0246", + "c2066e8c c0adde88 c211a60f b9701cbd 3bc4611e 3b61488a 3ad6158e -> 3ebb0280 bc58369a 3c07c3de", + "41708008 421d6fa4 41b2cfec bc7b6783 3b2852a4 3c17c23f 3ad88e55 -> 3ee3d64a bc6ae034 3c858173", + "411cb1c0 41ea3e18 c19a0bee 3c4bd09f 3ac871cb 3c4ef413 3ac2a3b4 -> 3cd8c2c2 b8980738 3c43b2bb", + "c234ef43 409d4800 4191aa18 bcb41dbf 3d051048 bc78d941 3c33d094 -> be476ac7 bb7e1eb8 3de6ac32", + "c15fbc08 41dbdbd0 417b9a00 3c4642cc 3c9a5a7e 3ba520ce 3e2ad83c -> bdb4b597 3ce3e183 3f9dcef9", + "40c81df8 c233d3b8 41f24084 bde0da0c bcdb392b 3dd043b9 3d9eb6de -> 3ef89b09 00000000 3fea113a", + "413bcf8c c2175d9b 4207bb36 3b11a2de 3b148d68 3b4fec6d 3d4cc1a8 -> 3e123c18 3a7b9697 3e1d68df", + "417e6560 c22b16bd c18b9796 be88a602 be98b4fb 3e6f36ca 3def151e -> bdca3873 00000000 4093e0e0", + "40eb0298 c18174c0 c2010d5f 3c55a985 bc66ec78 bb30973d 3e469b20 -> 3f09d9f0 00000000 3fd98c3f", + "c1db627e bfc819e0 422df468 3d33cb1b 3db00301 3d90d868 3d4547ec -> bd12dc7b 3e7f3350 3f6ea0fe", + "41d19738 4240f01e 4241df82 baf84eb6 badc07dc 39eb1f8b 3d13e7c9 -> bea963c4 3b5fa641 3e0c4daf", + "4217564e c23cc902 4229bdb2 bbbea7d3 3c224243 3c862003 3cb28ccb -> 3e36edd0 bcf152a1 3daedb1b", + "c1e71165 c23920e4 c1c0a3f0 bdcea1e3 bd882118 3dfe865a 3ce6b4f4 -> 3e42159b bea074eb 3f9a5183", + "41a16ef0 4244d05c 4246e72a 3b8d6f19 bb882ac8 3ba14d85 3da3f6ae -> bd288bc2 bcb31ff7 3f0772c3", + "c1f11599 41a1d298 c200f4c8 baf5db3f 3a822815 3a819e1d 3c9e49b2 -> 3f220bef bb30afc2 3d9e33ad", + "41b8710c 4120c12c 40f17428 be13f2f3 3e0e7319 bd05f369 3d28fd3d -> be9d7ddd bcea9e04 3fe83537", + "41daf308 c0c29420 4004a140 3cefc569 bd931cfe bc986562 3c6b5814 -> bf29921f 3e2d4608 3e95a55d", + "c203afdd c0a724c0 c105f72c ba921d35 ba07a165 3a26e424 3dad29ee -> bd9f1336 bb1138e6 3edfe07e", + "c213b6d3 c1e07e58 c12fecfc bc131ab1 3c6c6758 bc7d84fa 3cac4cbc -> 3f407e64 bd12855c 3dccbf3e", + "c1927cc8 41e24e34 c2178187 3ceb5583 3d4df474 bc5987a3 3e65b2ea -> 3e4576e1 00000000 4010aa24", + "40df9728 c121b880 3ffb9120 3b7009ee bbf8177a bc4c53d6 3b44b6c6 -> 3ebc6a0f bcdc2448 3c821a3f", + "c18b1cd2 41c7305c 4237b4ea bca4fe75 3c8d3b75 bc510041 3c59f049 -> bcbe0cea bd4f5060 3d998771", + "4214c838 4210ae70 41b7898c bd234dec 3e0a0c97 be7374e4 3b59a2af -> bf25902b 00000000 400e8774", + "c23dbc19 c23bb1ed c1437c1c 3e6dda75 3e084e79 3dcab699 3b8025ea -> 3b4d9c60 00000000 40138924", + "c22d3698 4091dda8 bfd3f3a0 3d3c7512 bce72037 bd898f01 3ea41de4 -> 3f30225c 00000000 404f54c1", + "c1e89b2e c1e566c7 c1253604 3c9b290b 3d326f0c bd495713 3d68e16b -> 3eb863f7 bd985cf5 3f14cd3b", + "c115b664 41a3f9f8 41dbb560 be15aad9 bcf0e501 bd178a26 3e65849c -> beada6ca 00000000 4040934a", + "c1c7aa59 c18ab8ae c13a5c84 3cf0a949 bd14dd77 bca5d747 3d271b55 -> be27a594 3d205051 3eb47623", + "c228b0ce 41bfeda0 40d7c418 bd9185de 3d2169ba 3cdfb897 3cfe60ea -> beb264a0 3e7064f2 3f064f09", + "41e974fc 41f10dac 41b5fdb4 bad5534b b9e22fe0 3a1aec7b 3b97bab0 -> be569e9f ba58ebe8 3bc2b9f3", + "420a4b28 41b65324 3f0d3c40 3b493b4a 3af2d71e 3b5bfd80 3d44df10 -> bd845d10 bc978a5f 3e934ad3", + "c2199e96 41349b78 c09e53c8 3c830a94 bacff359 bc6e8e39 3b02b5dc -> bd9754bf bd62eaf6 3cb2724f", + "422ed30e 41f02b68 41441b78 be0d9b6b bd120669 bc841a38 3dc0ab09 -> 3e984c6f 00000000 3ff28118", + "c226aa00 c1de6476 c21e3ee0 3bc211c2 bc059799 3af3b56d 3e169144 -> 3e1f1425 00000000 3fa074a5", + "41cbed00 c211335e 41d5e5d4 39b17b3d bac8d1a3 3a079327 3bd89512 -> bee348ed ba330dee 3cfacfbd", + "41c20bf8 4231cfd6 41656fe0 bc9c8205 bcc52336 3c04f6fd 3c0388ff -> 3ebddd3b 3dfa1cea 3dafff84", + "419e6654 413a4e10 c1a766e8 3c72014e 3c0d9cf8 bcad2490 3c5f22a0 -> beb758d4 bbc47048 3d8de246", + "c1bec7ff 41dbadc8 c21b1465 3d04c245 3c6f4f69 3d9e8aaf 3e967551 -> be81cf31 00000000 4041411a", + "c2360bc5 416c6ec8 41aa80b0 3d5e9dbd be11072b 3bacbd99 3e52530a -> 3df7e87c 00000000 40361b13", + "c2088cf8 c190f5ea c2107112 3b42cbb9 3b888ab3 3882f305 3ea5cc71 -> bee83861 00000000 4027b6ca", + "41284d20 41d77904 41b22688 3ac8d5db bad43629 3ae95cd7 3ac3f043 -> 3c063524 3af3b259 3b100eac", + "c21f2073 c1744ad0 420624b2 3e1b19f9 3e075ce7 3d31140c 3b37d223 -> bd0bd4b8 bcdb1738 3f9c30cb", + "c238b6f6 41b76a7c 4024eb80 bb8d2a5a 3cbc84eb 3c2cac28 3d96da5b -> bd426809 bbd01c9c 3f088fa5", + "c220c5af c0d27478 c17a076c 3b1cab54 3c4c2c9c 3bd2361d 3c1f3eb8 -> becea382 bd0fba39 3d455394", + "c1b910af c1fdf482 c214e6f1 3e44115f 3e29976b be89a146 3ee18b90 -> bf0ee728 00000000 40ce666b", + "418817f8 416d8870 419d4c80 bbb4cdca 3b5c320a 3bd56cef 3b9b02d9 -> 3eed274f bc0c01f8 3ca60c40", + "41941eec 423d3358 c1d8d373 b9bd960e 3a9c4163 b7fa1b60 3e08ab34 -> beb34fd1 3a41f8b3 3f540746", + "c2168529 c23b8cee c2472bf5 3c0cf5dd ba9192d3 3a077900 3c032136 -> 3e3d9be1 bb33d8d1 3cdae9c4", + "41a7a27c c042cb10 409e1cb8 3d982804 3d7c44e4 bd4bc004 3b8aa8de -> bf11f9e3 bd753a06 3eba4a33", + "41b4f950 4094dfa0 4222d724 bdd54ab9 bd5d5669 3bb8aaa5 3c424679 -> 3ee01606 bd9fd59b 3f01490b", + "40931578 c2227e8e 4200759e 3b38a853 3bca6587 3d13fe74 3c079d08 -> bd7eae4f 3cd9cf37 3d9e5efb", + "419a9f5c c17f1628 41dda9cc 3dcf8831 3c0f179d bda599c2 3b617f36 -> 3df095f7 3e4575be 3f015df7", + "c211f2d0 c15cae20 415411b4 bd601a65 bd7f1bfd bb9ee6fa 3ebebc1f -> be79e8e7 00000000 40683ee6", + "4180ddd0 4246c5ac c18a90a8 be23d443 3d871d1d 3d631891 3e27c2dc -> 3f21be68 00000000 40302d37", + "3f882560 4126dec0 422c5a04 3c74695d 3d30f3bd 3cf0ba21 3b0e5c10 -> beaefc97 3e032fcf 3dc36630", + "c02822d0 c1e86fed c2230915 3c66bed2 b9b12103 bb11f07e 3b633f78 -> bf0129cc bc194b5e 3ca002bd", + "c1a8a6fd 41a6ab94 417b0200 3d1d3dfe 3d92a53f bd68b9f0 3c80fb36 -> bedd5b69 be0d6581 3ee2b2df", + "41478814 42138c00 4226e714 bc662621 bc9bc27f bc2cbc93 3efa51a7 -> be16f938 00000000 408338a8", + "418bd498 4236c60e c1963a36 3a2d1955 3a6a96bf 39b0573a 3b636f23 -> bd336746 3a14a726 3bb40c40", + "40d3d170 3f46ba80 c1d9c5af beaeaa31 bea80cff 3eb10a19 3b3ef3d5 -> befb4c94 00000000 409628bc", + "40dccbd0 c2154916 405b3710 bbd334e6 3bbac905 bba13c2d 3d96f1ed -> 3d939d7b bc1c6444 3e8f5269", + "c1642930 bfa01020 c246a026 bdfe3fe0 bd819d25 3dc161af 3bd55366 -> 3d320f5e 3edf7889 3f5fcaa0", + "421c7172 c2217063 41d47108 bea4aba1 be8a8b65 3dd55d3e 3d77ee79 -> 3e9b2c99 00000000 407ba151", + "c2081690 c08a6df0 c1f5806c 3a03e975 ba8cb461 b9c137b1 3d268bbd -> 3e44952a 391d7cda 3dd524ba", + "4214f352 41c43118 423d1856 3a23b83c bb12a342 3b06fdd0 3df31b6a -> bf25c2dc bbdebc64 3f4b35ad", + "41f63d94 4140e298 4208f704 3a00e75b 3a88f374 baa2c48d 3be40608 -> 3f088596 3b683da2 3cca3fed", + "416185d8 422a94e0 c233df1c be0e36ea 3e9568e2 3e94ab29 3c0864ad -> 3de8a823 00000000 4061bf89", + "400d8070 4215b484 c20a9a44 3c83a0fe 3d15e402 bcf67ffe 3ab505b3 -> 3f108c97 be2bd979 3da53210", + "c23249c7 c1141638 422395b2 bb1be944 3b280fd0 3ae8c8b6 3e7cb1cb -> be504ab6 00000000 3fffaa3a", + "c20fafd1 4246e468 c1f727d0 bbedf519 b952789a bcf26f18 3c3c1330 -> bef6d5d0 bdc377eb 3db6e4a3", + "c1b4883e 42186e12 400eb340 bcfa19e0 3b1eee2a bcaac181 3e6b8f66 -> 3ec09b54 00000000 40082beb", + "41c80f04 4152f960 412e8478 3da64a41 bcf238ad bdd27caf 3e299e4e -> 3ea43568 00000000 4018e694", + "400082a0 41e861e8 c23293d8 3b36740e 3b50c28d bb6f08f1 3e8e71a8 -> bdb53598 00000000 4010b211", + "c1bd629b c2064924 41da62e4 3b51ee3e ba277008 3b9bd6f3 3a995e48 -> bf1ea68d 3bfaba5b 3baf8aec", + "4208034e 41d0bf40 c1988c87 bc52c6bb bcb9fe69 3dd6c040 3ce3bc45 -> bf2b0b66 3d9e04d5 3f13cfb3", + "c22a26dc c23a4fd1 400f7cd0 bd2d8a20 3d398e2c bd585660 3df3d9a2 -> 3e7efdba 3d4fd63a 3fa4d85c", + "4230e6b4 c1516060 c2398356 bb966b0b 3cd17747 3b35f247 3c1de448 -> beafa1e5 bcd6807c 3d6c4fc1", + "420c2c7c 413b283c c11da73c 3a635301 bac7bc3d bac72621 3be2d489 -> 3e0647b4 bae522ee 3c66febb", + "418d59a4 4106166c 42298a48 bac96baa ba7f6a72 ba027edc 3add6d83 -> bf2aadff bbaa6035 3bec9095", + "424579a8 3fefb240 c20e85dc bd1ab186 bd237e99 3dbfa61e 3c1d71d5 -> 3bd36372 be348758 3ed5365e", + "c214117d c24654ed c18d0e04 bb419bc4 bb8b46a3 3b230d3d 3a9845af -> bd5c0811 3bd8ca8b 3bc7bde4", + "c1dab1b1 4231f4ec 4240bbd8 be3836c6 3da2e882 3de41e35 3a898647 -> 3e708668 00000000 3fe7a1b2", + "423a6c60 c1e706fe 423d24f6 3ce37bfc 3cd45875 3c45460b 3db383ae -> 3ebffccb bdb84e88 3f3d66d0", + "c243ace1 4234c018 41227774 3ddd3038 3e070574 3e992165 3c979210 -> be931683 00000000 4038f2fa", + "4207481a c238ffec 42386962 3d492c41 3cd841e1 3da8f812 3beac839 -> 3f3f9277 be2af61a 3eac169c", + "c247458f 421fe4e6 42119ae6 ba8d4252 384c54f6 baf5460f 3bc40fbb -> 3ca4795e bb59ec3f 3caaf64e", + "41b84a60 c2119d62 420f3114 ba0ff365 bacf0af2 ba2ee129 3b505186 -> 3e96a23c 3a0b9b4c 3c46d924", + "c1a19673 41269ea8 c0eaca98 3c0b959d 3baa142b bba00380 3ccb0a9f -> 3efe1c98 3ca68f7b 3dc5e9c2", + "423b2f60 3f9fe560 c244a2c0 3e76612b bee455d4 be6ce64a 3c03c5d5 -> 3e9ed054 00000000 40900abb", + "c19dde82 4179c890 4082f928 bab05f88 399466d7 b88f3f89 3cb65ef8 -> bd169103 3b03b723 3daf4df0", + "4220b8f0 c194f8c6 c1f4a4d1 3ca1e3c2 3c8a2995 bcc9b013 3d2d15b8 -> bf0976b4 3cc21604 3e93afcb", + "416e6110 41dc0254 c0b24d40 bab827f4 b95238e2 39913f5f 3c0b6863 -> 3f0a34c8 3ad81f90 3cdb38ed", + "40991a58 c173c4f0 c20c1439 3a2059c9 bda96ffe 3c0209d7 3d90302b -> beab6ba4 bc400612 3f4fbeb6", + "421493b6 4010eba0 42157204 3d2b8798 bd249858 bdad1cd7 3da8c766 -> bcdc4ac9 bdbe0608 3fa65fba", + "c215d020 422b0300 409cb420 ba98edb8 3c241776 3baa9c33 3b397237 -> bdf859de bc270ac6 3c7605d7", + "c09b9548 c11609e8 c1738bbc 3dad4868 be9f6923 be9cf21c 3e43a829 -> bede6957 00000000 40a22051", + "c220e48c c1f2c5d7 403261d0 bc3ec6a0 3d4260b1 3dc0859e 3e481c1c -> ba9fb3cd 00000000 4019a648", + "41d8bec8 3f7d6a80 4183dae8 3bd94245 bb425991 3bcf1cb5 3c047989 -> 3ed37c3c 3a120bd0 3cc26301", + "418708b4 41bbb970 c2428eec 3b95876e bb8d48fc 3b8b6b24 3dc23a4d -> 3e71a554 bb5f0ea8 3ef79e85", + "4192d928 4240ff34 c1b265f4 3a450673 398be58c bac81f1b 3dd39944 -> bebc9c1c bc0132f9 3f4e5986", + "c12dc124 c23c826d 4176f508 3c4f2603 bbb71e25 bc267653 3c28d02a -> 3edacc8d 3cad9768 3d6bf71d", + "422cb184 41a05ed4 418af244 be8613cf bdfe4619 be477460 3b1ed346 -> 3eacc7d9 00000000 40334330", + "bdb1f200 c2435ccc c19f1efb be2e6acd 3e1f5f60 3e43f3c4 3ee85d00 -> 3f0db625 00000000 40c0137b", + "c193b465 41ec8688 c2348ff2 ba6ea8e2 3c5ea012 bc475611 3b6062e2 -> bea7f541 bd2ccddf 3cd2a99e", + "c0e85ac0 422787b0 422f4b80 3da8b635 bd478f2d 3cdfb604 3bccbf0d -> 3f07a4ce bebd0546 3eb14733", + "c068d8a0 415d7a88 c17844b8 b9e59311 3b4ccaaf 3a1bdd52 3c0f8f9c -> be8e5e5e 3bb7c696 3cbc4d67", + "41c1c6cc c2122734 411cb694 bd0bb139 bd2e476d 3ca448d5 3ce670aa -> 3dd7ce20 3bd6aadd 3e835272", + "c222f113 41883ff0 c1fd734f be279492 3e973366 3ec0f228 3aa0ef47 -> 3f07533a 00000000 40814961", + "40b7e8a8 c01b6e50 40260770 ba6632fc baaf414e b8d98f86 3ae633cd -> 3e31ed05 b9981e00 3b82abcf", + "c07de5e0 c24573a7 c11a6110 39c3daab 3c0d9e31 3adad382 3c364a94 -> 3e25418a bbd8e6fc 3ca6fdf2", + "c1e49928 419f94e4 41ab1988 bb566b12 ba5b4115 3be54c59 3b74dab1 -> 3d060a92 38702640 3c434822", + "c1ac4267 c11d9628 41f48890 be7fbf54 3d66ec76 3eda7b6b 3d67176e -> bb846395 00000000 408d3918", + "41bd3934 c1968bc2 41716528 bda159cb bdf7a5bc 3eba3768 3c454cc7 -> bea90450 00000000 404d9de3", + "416b01a8 3f3c3dc0 c1d52023 3b54f734 bb67ae0b bba189ef 3c1e5ec7 -> 3e745d03 bc5b6615 3cf8b660", + "c215784e 41ec8568 41dc0330 bd202111 3bcb07f1 3cba70d8 3b0093c9 -> beb086c9 3df66766 3d96a7df", + "409ebf80 c194dac6 4232ec82 3d102c57 3b584f9a 3eb55e60 3e88624e -> bedc109f 00000000 409ea23c", + "c23927b0 41cbdeb8 41bd8348 bb3df507 3a0f0828 3b5a04eb 3e5d7cb9 -> 3eec0e81 3b5e0630 3fd847aa", + "41c540a0 c1951c17 41996e60 bda1b354 3d368ef1 bce30019 3b0d9fb0 -> 3c80e5b9 3d644aac 3e889983", + "c1489ae4 c0bd9300 420f3432 3c07fc7a bb948996 3ab478e7 3b407c1b -> 3d787853 bbb67f21 3c461e62", + "40fea4b8 4087e7c0 c1087014 b9b1598e 3acf89fe 3aa4da57 3b1fe7f1 -> 3a970728 bb2b91e1 3b80bc8c", + "41636128 c183ca50 c1d94b9c b963b0c9 3a75c9ad ba1e69f1 3cf522e1 -> be8608d9 bae9b985 3da870a6", + "41f80184 422412b8 4217c560 3dcb6925 3df04fe5 3dd72d76 3e80b1e6 -> 3efdfdde 00000000 405f16c5", + "422583d6 4201585c 41401ef0 3a0a0e77 b974d35a bab884ed 3d99025b -> 3f518b0d bb8c0d0a 3ec176ee", + "c1ea3768 40cb7bc0 413d8494 bc983448 3cb4a4e5 bb077615 3e09ac4b -> 3e6758ea 3c0b3296 3f86967e", + "c0a12468 409a9a10 c20b7a16 bd042127 bd98f070 3dc313ab 3cf3bf9d -> be1397ec be159edf 3f3c3e42", + "c20aea0f c196ee48 413f822c bc8a0c99 3df84873 bdbd184a 3d5bfcb1 -> 3e3e8a0c 3e6fc26d 3fad88da", + "416f45e8 4024ab60 c20254c2 bc41bc5a 3ca0a608 b912e455 3b6cb69f -> bcb75721 bba9ac1b 3cbebc12", + "c1dcde97 405d03d0 420beb6a bdf95a09 3dda683b bd4ba50f 3e94c781 -> bdacd2ad 00000000 406a7469", + "c0a82100 c22391cf c181c02c 3b8c77a9 bbee11e2 ba28fafe 3d0e4a28 -> beb07d40 3c01300d 3e0e9ad4", + "c216b2c6 c22a4617 423120be 3ba529a1 bb8acf04 bb8d2ed1 3c83f9e7 -> 3f10580c 3a753e68 3d708e36", + "c1be9eed 423f13e4 4219f088 3cc2c84d 3d274f82 bbad8d0a 3d8d721b -> 3da46a90 bdb84453 3f17cb72", + "c233e861 4231e500 417a1520 bb686b9a 3e38ad4a bdb2f14b 3c43d66a -> 3d9b99b7 3df1ba5c 3fa3b839", + "422bd5e6 41a5e550 c0e21d20 bacea02d 3c1f3918 3beaf795 3d56b130 -> bdcc2910 bc09d068 3e9577f0", + "c22c6a16 412599e0 c08e6c68 3b5c0ef1 bb630f15 ba93b3f7 3ef0300f -> 3eb39ea9 00000000 4071ac24", + "c19c617d c11ba4b8 411da408 3b4d6b75 3b07199b bb2d549f 3d08122a -> bd87e2f3 bb54e136 3dbc8d0b", + "41616198 417a9120 c1b5af49 bd5dd42d 3cb680e2 be233b5d 3bbed10f -> 3dafc5b0 3e5340f8 3f5fd0eb", + "41bf2ccc 408c03c0 c118d9dc 3a9d0c25 3ab7d2d5 baf48cd0 3bbe5476 -> 3c7c10a2 bb25a5d9 3c27e9ca", + "c21637f6 c1c996e2 40e0be00 b99e2351 baad5d5d 3ac4926b 3e7cf616 -> bdf16ec9 00000000 3ffdf7f7", + "40acd768 41858cf8 411513ec ba6e2f3c bab6f428 38def2c1 3b51a499 -> bdf9cd2c 3b25ab3b 3c35e4df", + "3fed8c80 421257a8 41b984c8 3d44faee bdea39fc bd521db6 3cc30896 -> 3da093e7 bba91a30 3f3a72d5", + "40275560 c2032930 401f1160 3b443b4e bb24a0c5 bb6ec9c8 3c8423ea -> be4f0b62 bbe4885c 3d3a6e93", + "c14337f8 c1939123 c21cb19d bcfeae16 bd2791a6 bd0dd7c9 3c8a3000 -> 3c3f0b74 bdbac0a2 3e5297ec", + "41f0cf74 c1354c3c c1a8fd07 3d8abeaa 3d90f579 bd886028 3d560868 -> 3e6d9f0e be96773c 3f761d73", + "c2429aaa c1e20e23 410fdf20 be83e19b be29abf1 3d593a9d 3b051f85 -> 3e7adf3c 00000000 401f83df", + "423b08d2 c22fdd9a 41162f4c 3ad07900 3a3e4eb4 bb70eb25 3bb18299 -> be100548 ba56f4bf 3c0df863", + "bfdc5860 c20da416 41aa15a4 3e9ff348 3e847c72 be9d0338 3c863215 -> 3f258785 00000000 4085cab1", + "41b8af54 c1b20f6c 4239e3c8 3b5358e6 3aad6b92 3a27c9ea 3cea46ea -> be00c12a 391bb04e 3dd34a5d", + "c213fc40 421b67c6 c1b94d77 3c3753d9 bc734322 bc29be17 3b0953bf -> be94b48e bc687abc 3cc8e7b1", + "41beb8ec 4245b5ca 41751f10 b8d2e890 3ac71c2b 3a9a0dc9 3b869f0f -> bee54054 bb4559ca 3c06fa2c", + "c1bff79c c241338a c1eaa6fd bd976991 be00a1bd 3d3e5f52 3afa3b20 -> 3ec9623d 3eb4baf6 3f2db788", + "c1edd443 c20fef81 c1e93388 bda413e1 bdc10586 bd2bce42 3c2a5736 -> 3e2b7586 3e08df3d 3f11f12f", + "3fd11400 c2323537 c21575d8 bd42e357 3ca941c6 3c9f1605 3b8fe28d -> 3f120a03 3e2d271c 3decddd1", + "420196a6 4115c874 c1fa169e b9ffb342 3a534341 3a791457 3e4ccfa4 -> bea107e3 00000000 3fcd56cc", + "bfc69300 41bdf404 c2305a89 bc978c7a 3d12abae bd19b81f 3a8ec908 -> be215327 bcc9ff7c 3db93967", + "c1cac5a9 c2064184 c1807842 bceb3bfa bad30653 3ca4baae 3ca260cb -> bd81d10b bc8adbaa 3dcefc93", + "41232c90 c1b2be18 41190a14 3be1b75b bc56c78d 3c3cc2ac 3d888808 -> 3e3391a5 3cfa8d08 3eb671ff", + "c0c7d398 423b7a9e 42466902 3d71a61c 3c9fe543 3d43edae 3bd38aad -> 3f3fa311 3d513dae 3e5b0105", + "c205dd82 c165fd64 c15a7f94 bc9d431f 3c43ee41 3d008ff8 3e0d1ebf -> 3edb1868 00000000 3fb3ff04", + "41e76678 c21af7ae 40298460 baa08ad4 bb14defb babbe422 3c1052f2 -> bda13306 bae883be 3d0abea3", + "41d71ddc 41a59318 c12308e8 be2e8b0e 3d44c4b0 beb4f67b 3c9323b1 -> 3f541e72 00000000 4052b0d0", + "41fc69d4 c13a3ee4 c1ba481e 3bc5e457 3b13065e bc3429de 3d2fc02b -> 3d845c36 3c8761ba 3e1857d2", + "c192b304 c2383a15 c1351380 bd5abf9d bd0260ec bcf566fa 3e96b51f -> be68728a 00000000 40394068", + "c1aca0b8 41f2a800 4151ed0c ba37ca9c 3cf58f8d bbf35092 3c51f94b -> 3e261a96 bbe7f2e2 3daca82f", + "41d59bac 4169b068 4224a870 bb056f3e 3b3fab2d 3b805980 3d36cc22 -> be688573 3b954814 3e315418", + "c1c2fa04 421f8c22 41c10d30 bccf2c4d 3cc7f122 be0dce5a 3b69c5ef -> 3e65af67 3e6ee93a 3f1b961e", + "c23cec4c 4228bccc 4201bfb4 3a843f80 39230961 baf55fa1 3cb88ca1 -> be0c2037 3aa4aacd 3d2c11d7", + "c0eeadf8 bfd086c0 419c6464 3c9ee37f baee8a88 3cb0e6de 3aa284bc -> 3ec1b1fa bc674aa4 3d011f29", + "4235463a 420ebfae 416db430 3a16fb15 3a9e5944 b9f2cea5 3c853ade -> 3e87b5c2 3a5e56fb 3d094aca", + "c23562a4 41dcb048 c0f3f440 394fccc2 b983d1ce b83d273d 3da67eb3 -> bec8ea51 b8eace08 3ef5b8a6", + "41bf2d28 c0eabe50 41332f9c 3d02aa9b bc9d8ffc 3c9a72d1 3b96a3c1 -> bf03530c 3de42ee1 3d99af11", + "c1155cb0 c10efb44 41ed32d8 bd974378 bb20a400 3d2d801d 3e105d68 -> 3e602e32 3d040628 3fc2e275", + "422b1754 41a5612c 4210d558 3ce93242 3d035bd4 3b094e3f 3b367a35 -> 3dbc19f1 3d4905f8 3d85dc83", + "40c329e0 c146c284 c2085825 be31dc53 be50e555 3e51d88d 3ec706c6 -> 3db6cb49 00000000 40b90cc9", + "4243e704 424338fc 41d3d5a0 3c7a0bd5 bb903808 bc9776d8 3dbdbc37 -> 3d0c398b 3c98e2a2 3f02458b", + "c2252e6a c11ae1f0 418c3858 bb91353c 3a870507 bb1604d9 3d030631 -> be2fee50 bbfb1b78 3dda87f1", + "c224c69d 414c721c c1acd3aa 3d0b979b 3e10232b 3dbb05ae 3de1f18d -> be1503ea 00000000 400f8b2d", + "421e0ce6 c0a8b6d8 c243c2de bd03a3f0 3cf59a17 bcd5a601 3c2ccd2c -> bebf64ad bda6233d 3e0b9f5d", + "c0cc27e8 4206f2e4 41d8423c 3ab92905 bafecea7 3b38bfdc 3cda80a6 -> bddfb159 bb35ffb7 3dd3afd6", + "414de710 c2128c0b c01640e0 bc254755 bc45e1d0 bcac29ee 3ec8602d -> bebc1ef2 00000000 4054e467", + "c2201cfe 4122c6a8 41dd3948 3c147a28 bc526849 bbb0bef9 3abd9726 -> be60b16f bc5bbe60 3c487927", + "c1e8edaf c244b358 41223c84 bb301a46 b88b4467 3b81231f 3b4af11d -> 3d1bf59d 3c080b88 3c32f853", + "c1c36e59 c1a2edc6 42027b8c 3b33f959 bb5e708f 3b356599 3e871420 -> beb920cf 00000000 400926d9", + "4190137c 41cca28c c1f7128c bae1812b bc162ac0 baf755f2 3d48ba28 -> be32dc85 3c31f3fd 3e4dffa2", + "c1718858 c233d8ac bff265c0 bc59910c 3bcfe2a8 3b9e6f0e 3bbcb75a -> bed721df 3c7205f5 3cf95843", + "bfe0a840 41a8c2b0 423e41d4 3c7cab55 3c032a7b bd7d8fd7 3b145f10 -> 3db8ea2f 3dbe60f0 3e06a639", + "c03b0740 c1a43b25 c1b88de9 b9001163 3a4efc0e ba4f9bc2 3a89e062 -> 3db4fd18 bb3e3d24 3b82a7e0", + "41d6efa4 4217e5fa 41e80124 3edcacd0 3eb70573 3dc8f8d4 3ec18d90 -> 3ee4cdb6 00000000 40f14053", + "40bc5620 423e26fa c23334e8 bc02192e bba1e241 bbf05824 3d15392d -> 3cebfb60 3c339561 3e4a3a8e", + "4180eb54 408eb1c0 41c553cc bad3a44d bad31177 ba54cfc9 3b163b32 -> bd9db823 ba664a9e 3b73f96a", + "413a86bc c2301c0d 4247eae0 3b2af1ff b9ad38ab badf6e7c 3ca1de3a -> 3d53d5a3 3b90cdbe 3d979f71", + "42283fd8 c02d4ce0 419c2700 3b6eb148 3ab57258 bae00e1c 3d28bd87 -> bf48e082 bc22c8e0 3e29bca3", + "c21dd324 c23f9961 c2091b10 b9febcc6 3ce0b479 bd0fc9e0 3bb1868a -> 3c1ab991 3bd3b354 3d999864", + "c1ceb757 4198c930 c1708a10 be4dee66 be859196 3e55efa9 3d6b78d8 -> bf43b721 00000000 406427b5", + "40e91718 417a8270 c14dcd6c 3e58e9cc bcf20852 3e847b8f 3b0a6d23 -> be5916ae 00000000 402c3640", + "41e9d014 40e69e30 c1034778 3969c314 3c91fe86 3c374f8b 3a8ee9ad -> bc3cc926 3c97b892 3c8bfaae", + "c1dae051 c20c931f c2425fe1 3ad973d9 3a360394 3a03445e 3e236e37 -> 3d529c6e 3a85d7e6 3f744771", + "4231cb58 41b668d8 42270488 bba77add bb2f58dc bae84f97 3e92fa8c -> 3edb6c95 00000000 40157490", + "41012484 c2369a2e 418ce520 baf6be46 ba3c4749 3aeb1332 3d768676 -> be58290a 3bcf3624 3e9386f0", + "c2337fae c2393715 4200e490 bd94d81d 3d36a176 3e2a01ac 3c78a1a7 -> 3e590b84 3d3e3574 3f95c76a", + "c247fc0d c1836620 41898648 3ed884b0 3e81b803 3e278634 3e740ba4 -> bec03c1b 00000000 40c122ed", + "c240e0ab 40b16498 4237afaa 391ff3aa 3a73fc50 bb22910a 3cc01aac -> be98ac14 3843e420 3db82549", + "c1e44db8 3f423340 423905d6 bc553488 bab6d05e 3c08716c 3e1f65a7 -> bc8fb0f5 3bc1f562 3f73b4ee", + "c0cfea50 c1680478 c2368762 3e6b16e3 3e094688 bd299219 3c260a75 -> befa6cc8 00000000 400e5825", + "4238038e bffee4e0 c1a4e745 3cc7d463 bcce20bc 3db744ee 3e0587df -> bec23c1a bd0efd54 3fe00d58", + "c16d8a98 41ec8c54 c1e59844 bc012e78 3c24970e bd857b20 3c290ee4 -> bec731fd 3dacd1dc 3e48c93b", + "c245c1d5 c1942f0a 41a6638c 3d4ea696 bdb5a227 bdb3f7fd 3ceb09ea -> bf5c99fe be175a98 3f4a8269" + ] + }, + { + "glsl": "vec3 cultmath_af_fbm(vec3, vec3, float, int, float, float)", + "tolerance": "ulp-bounded", + "check": "the arguments are (cx, cy, cz, ax, ay, az, radius, ...) and the result is the form (x0, a, e). Enclosure: for eps in {-1, -0.5, 0, 0.5, 1} and d in {0, +-ex, +-ey, +-ez, (+-1, +-1, +-1) / sqrt(3)} (15 directions), form the point p = c + axis eps + radius d in double and round it to float32; the device's own reference value at p must lie in [x0 + a eps - e, x0 + a eps + e] with no tolerance, so a device whose gradient error exceeds the allowance fails. Tightness: with L = 7.9640074 and, for each octave (one for af_snoise), R = |axis| + radius, B = L R + 2^-14 + 2^-12 R + L 2^-20 (|c|_1 + R), the width |a| + e must be at most B + 2^-16 (1 + B). Report the least slack of each side, absolutely and as a fraction of e; only the two bounds fail the check. The reference value is cultmath_fbm_grad(p, octaves, lacunarity, gain).w, and B is the sum over octaves i of gain^i B_i, where B_i is B for the centre lacunarity^i c, the axis lacunarity^i axis and the radius lacunarity^i radius.", + "cases": [ + "411c537c 420ffeae 4078d1a0 3cdfc7fe 3ced8823 bd581451 3bd5bfdb 00000004 40000000 3f000000 -> 3f1c017f bdc20710 3fd2f8e2", + "c2172004 410ab7d0 c1f2ca42 b9a78fc6 ba1d7262 3a8c08d4 3e338f4e 00000004 40000000 3f000000 -> bea9d56b b95ec7d0 40b0ea05", + "4115776c c2005a28 c21c7d94 bb82d4de bbf99288 bb14556b 3e706f99 00000004 40000000 3f000000 -> 3e0cb256 00000000 40f88286", + "c1dc368a 41869c94 4226de20 bcfe94ff 3b70c570 3c88304e 3d6c7520 00000004 40000000 3f000000 -> bf10b24d 3d77694a 401c59dc", + "c203bcc0 41cb2bbc c209d08b 3d98457a bd489b84 bdcfb707 3d19027b 00000004 40000000 3f000000 -> 3d8f1a2c be4d5c90 40a0b74b", + "41e45754 42248ae6 411d2fac bd7d08bb be3dbc00 3e5fb94c 3dbc85a8 00000004 40000000 3f000000 -> be1bcf46 00000000 414455cf", + "c0ff5658 4216c372 c1aea658 bc0f34a6 3c62e6bc bbc51abb 3d16dd85 00000004 40000000 3f000000 -> 3db396b2 3d0fb0eb 3fadf568", + "3eaaf500 4180ac44 c105a4e0 3a9cdb0e b920ef6e ba380c9d 3d088bb6 00000004 40000000 3f000000 -> 3e9e0360 bb918bc6 3f41cdd7", + "4201700c 408d4338 42173afe bd1957c4 bd11a659 3d45b37c 3bf04183 00000004 40000000 3f000000 -> 3dbb520b 3dbf9818 3fe33039", + "c20c49a6 40291410 40962208 baaa709c b949a8bd 39947c97 3b1fa49a 00000004 40000000 3f000000 -> be925bbe 3a69b13a 3cf0f01a", + "418bdb28 c1f8333f c211ebc4 ba84873a ba463c1f b95ca930 3b5a553a 00000004 40000000 3f000000 -> bf770906 3b95ac4f 3d269588", + "c07870c0 40f13610 c1c85790 3b1283eb 3b36b039 3abd56b7 3bffdb24 00000004 40000000 3f000000 -> 3d822045 bc9bcf63 3e0c92f5", + "413c9a20 c23a522a c236024b bdc256ec bd305cf0 3d093814 3bb9292f 00000004 40000000 3f000000 -> beb0848a 3e05dd8c 403e9559", + "421c32d8 42387e66 c1af30cc bc15aee6 3d74e834 bd35d262 3a901e4e 00000004 40000000 3f000000 -> bf049507 bd8d8e0e 3fdb1326", + "c20eaff6 c1b56412 c10f2d70 bcb56591 bd29e076 bcd5a70f 3cad2e7c 00000004 40000000 3f000000 -> bda46ee0 3dc371cd 3fe8c2be", + "c223bb64 41939ad4 c2054531 3cc10801 3c94f8ed bb0df003 3b6a1616 00000004 40000000 3f000000 -> 3e312a49 3d449980 3f02597c", + "41ecdc4c c1e2ecba 41ae504c baa38c97 3b90982b 3c1dffdf 3be0462f 00000004 40000000 3f000000 -> 3d881847 bccf5634 3e406db7", + "41081668 420ea82c c1cf6363 bbd07906 3b82b3d6 3b750849 3cbc359b 00000004 40000000 3f000000 -> 3c0f8e9a 3b831b8a 3f07cf87", + "c10886cc c1d48d63 c237c876 3b2755b3 bd04e56a bd167c20 3c8b3546 00000004 40000000 3f000000 -> 3cb79922 3e79e8a4 3fc564c4", + "41c549ac 40ad6cf0 41b487b4 bc78b86e bcb6a67e bbbb5f82 3ae26484 00000004 40000000 3f000000 -> bf25ff3f bdbb9878 3ed4ffdb", + "be8bae80 c229817c 42168f44 bb01345a 3b0875c2 3a0ef65d 3c7fd6ec 00000004 40000000 3f000000 -> 3ef5653e bc729d21 3edbb09a", + "c18533dc c21f8bfe c2291deb 3c75a994 3c89ab5a bb0e449f 3be0ad70 00000004 40000000 3f000000 -> be030eba 3ced209d 3ed5778e", + "42287142 41822030 c212ee17 3d00286c 3d3a3436 3ca70d22 3dbd32ca 00000004 40000000 3f000000 -> 3e915cf4 3d8731fe 408ece44", + "c1eba852 c1be364e 415db64c 3e74ab13 bd22f014 be4840b7 3cd9de60 00000004 40000000 3f000000 -> 3e4f34fc 00000000 412c4558", + "c03eef80 c20adb80 41fe5c14 3eb261dc 3d2ed9e9 be4f2d0b 3d59cb57 00000004 40000000 3f000000 -> 3de797f8 00000000 4169a53e", + "4181b764 411d948c c1c836c3 3c6d35eb 3ab92f87 3a843a10 3dfc3962 00000004 40000000 3f000000 -> be51e980 3d343773 4087664a", + "c219ea1f c1066980 411c8900 3cc2c99a 3c51cdbe bc1fbebf 3d3d3d98 00000004 40000000 3f000000 -> 3f37dd86 3becb908 3fef3576", + "c11747d0 c20ca374 c244a40f bea1618a 3e22ed86 bcee5530 3e951a5f 00000004 40000000 3f000000 -> 3e8898e5 00000000 41a48846", + "42442582 409e6848 4245239e bda85104 bc0e8ace bcb8f19f 3c250714 00000004 40000000 3f000000 -> 3eadbadf 3dc9675a 40155b20", + "c123d224 bf259680 42463fec 3a3dde01 babdeb21 3aabd6a4 3d7429e7 00000004 40000000 3f000000 -> bd9e4602 bb32844d 3fb36747", + "41b66cec 4211bb12 41daf39c 3cab930c 3b9ef4de 3bdaec73 3dcdb7c9 00000004 40000000 3f000000 -> 3d7f76c2 3d994a22 40717d6a", + "4202cc30 c1eedf0b 40e05e58 3c1488c3 bd3f0b41 39fc484d 3ee10a01 00000004 40000000 3f000000 -> bf143d2e 00000000 41784a5a", + "c213337c c1825a24 c0b80728 3ae06722 3c0753d5 3b743588 3d6b9e8d 00000004 40000000 3f000000 -> bee90a95 bc5732b8 3fd40627", + "c243a17e c1daf546 c210a96c 3bdee19b bc217553 3c6824f7 3b39b092 00000004 40000000 3f000000 -> 3d4dace2 3e09c90c 3e73dffa", + "c10df350 c16e9b00 c1c8ea76 3b92303d 3b913dd4 bb927456 3c58fb7d 00000004 40000000 3f000000 -> 3f0b8c5d 3ad46f70 3e993c3a", + "423296a6 c1c6e3ca 4230a43a 3e96b5c1 3de42fdd 3e7d2deb 3b016bd0 00000004 40000000 3f000000 -> be8b11b8 00000000 414d1209", + "424489da c2414dd9 c23b602a 3b70c6de bb267566 bba1fd58 3ecb270b 00000004 40000000 3f000000 -> be882fb2 00000000 414db6b4", + "c1cc6650 c2173e2d 409bad30 bbaa0af1 bcdb880d bd247759 3d7f487e 00000004 40000000 3f000000 -> 3e938264 bdca314e 404c4dfa", + "c1e34d41 c11bd9a0 c1782270 bb4520aa bae77110 bb384961 3b84c9fe 00000004 40000000 3f000000 -> 3d8e1b40 3c2a59b9 3d96e29f", + "c1ca1942 c23d39f4 bf029600 3b8c41e0 3b106e20 3b5fccdd 3c1b94a7 00000004 40000000 3f000000 -> bf26dd3c 3cae430a 3e4db4df", + "c1267c64 c23c717d c0c0a6d8 bd81f5f6 3d06ddcd bcb414ab 3b0666f5 00000004 40000000 3f000000 -> be709ffd 3d31fd14 3fdcfdc4", + "c1c947d7 c1aadb76 c23cbf72 bd1b72b3 baf5d1cc 3e035db1 3dfcc1b7 00000004 40000000 3f000000 -> bf4845aa 00000000 41032705", + "c1cd979f c1b277ff 41bdb4e0 bc07bdf0 3bc1f212 bc94d0c4 3de96f8a 00000004 40000000 3f000000 -> be3c9e4e 3be7a95c 4080284d", + "4232f882 41f52f90 c2355fd8 3d9271cc 3d9172a4 bdb00051 3d14a05c 00000004 40000000 3f000000 -> 3f521f0e 3dbf640f 409cbc80", + "c15a902c c1fab89a c1ef6846 3bbfaefb bbabc19a 3a6049be 3cd54597 00000004 40000000 3f000000 -> bebeb15e 3aac906c 3f2764da", + "c23bcbdc c1d87d68 4230c31a bd83e10a bd40e6d3 3da20469 3ddeecaa 00000004 40000000 3f000000 -> 3f8edc0a 00000000 40e19e49", + "c234dd23 41b040d0 c22fbcc4 3cd334f5 bc644858 bcd9368a 3de10869 00000004 40000000 3f000000 -> 3e69ba9e 3d803067 408dc815", + "42377f12 418e6760 41d721e0 3b0eb82f 3acbdba7 bb365e5e 3c860ebf 00000004 40000000 3f000000 -> 3eaa7286 bba2a423 3ece0df9", + "c20ccc7e c09e8f00 c0f24d40 ba9e563f 39da0c06 bb2e59a2 3ebd30c4 00000004 40000000 3f000000 -> 3ebb9461 00000000 413de01c", + "41c25fcc c1e5b6b7 c07eba00 3a2527b8 ba184cec 394943db 3d808514 00000004 40000000 3f000000 -> 3e2705be ba8ce326 3fc880f6", + "422fde8c c1a640d7 41b087ac 3a4e0054 399e043c 3a0003d7 3aed7f91 00000004 40000000 3f000000 -> be81ba06 ba65e97c 3cfd9943", + "c2149916 c0b07e18 c0a3bdb0 3887bc26 ba550c0d 3a8b272d 3c68e036 00000004 40000000 3f000000 -> 3e8ac2bc bbbf2d7f 3e8046b0", + "4225e29e bf44cf40 4127cb64 3d9bfc92 3d2a5b7b bc04426d 3c9e49f7 00000004 40000000 3f000000 -> 3ded1620 3ca6bc60 402f7cda", + "418e42d4 c21c207c 41f34ba8 3d592b51 3daab19f bd9c1fa4 3d5cf5df 00000004 40000000 3f000000 -> bedddc4c 3d445dfc 40a7be4c", + "c218f1ed c21ec03c c138a8d0 3be10d36 bb9c4122 3b788abe 3d026c59 00000004 40000000 3f000000 -> 3f544f07 bb308b88 3f5d5ef0", + "41f1f754 c1891084 c236bbcc be15fac3 3e422392 bce56cbc 3ce18e2c 00000004 40000000 3f000000 -> bd84f716 00000000 41090701", + "c1851b76 c23ab103 c2080ec0 3ac83244 3b11ce25 3b115b03 3b95dd08 00000004 40000000 3f000000 -> bf389656 bc51612a 3d92e101", + "420405ee c1cc8666 c11ba998 3cea263b 3d2fbfa9 bcaccf37 3b2168a8 00000004 40000000 3f000000 -> be056936 3e2bf367 3f930530", + "c134a4dc 41f5e4a0 422e947e 3c11079d 3a3c6129 3a9d0fc8 3d8ce42f 00000004 40000000 3f000000 -> 3db34a97 ba72b87c 40125b72", + "41e0b1dc c234b7a2 4233680e 3c053f76 3bc92a3f bc00568a 3d94dbc6 00000004 40000000 3f000000 -> bedbca2d 3d135bc2 40173713", + "419176a4 c1b754a1 c13adbc4 3da35925 3db19a6e 3dd8896f 3bdeb1ae 00000004 40000000 3f000000 -> be140a2c 3e085bd4 40975461", + "c1a440cf c20ffaa2 4011de90 3d0fdb5c bd3502aa 3ce9e87c 3e6690c8 00000004 40000000 3f000000 -> 3f0be84b 00000000 41130af8", + "c237bde3 421f4634 c07e0180 bbd08915 bbe50f3f 3b41cacb 3e1acfe4 00000004 40000000 3f000000 -> be204d5f 3c589908 409f8144", + "bf896f40 c0049b30 c0f01c28 3d37b07c bce0367d 3c3c765e 3b79fa70 00000004 40000000 3f000000 -> 3ed17233 be48385e 3f909692", + "421c494e 41aaa140 4232a7ac 3b1bc953 3a322380 bc1122f9 3ce43c6f 00000004 40000000 3f000000 -> 3da43514 bb0616fb 3f2c2fa6", + "42175292 414d2140 c215dfd5 bd8fd165 3d6aecf2 3d4a4ff1 3d9ce7b6 00000004 40000000 3f000000 -> bf36ed0d 3d868ea6 40acb83f", + "c1b0fdd2 c2291c77 bf991b60 bd9983d4 bdcd1673 3cf442c9 3d8739ba 00000004 40000000 3f000000 -> be1dcbc2 00000000 40c67817", + "40c87c08 40f32a88 c1abf512 baa2ef7f bb831d5b ba6aca33 3a8fd1b4 00000004 40000000 3f000000 -> 3f42ae20 3ba64d34 3cd4f1f8", + "4097f6e0 c19a0a22 41c649a4 bac6b4ca ba86ed60 ba410d2d 3cd8d4f6 00000004 40000000 3f000000 -> bf3f72b6 3bef905a 3f1f2d81", + "40f5fef0 422a0d2a c0c98ac0 3c2eae2b bc59bfb3 3c0aa4ee 3d1a6b3e 00000004 40000000 3f000000 -> 3d16c7e3 3d9de831 3fb396a2", + "415035b0 41b5a8c4 c1374cdc 3d9c060c 3db531a3 bdead4db 3e4d0732 00000004 40000000 3f000000 -> 3ead9ab2 00000000 41397e32", + "c229ac66 c1b5de74 42357e04 be69f671 3ecd9be6 bea8e569 3d39a28d 00000004 40000000 3f000000 -> bdf96add 00000000 419c42f5", + "c1ff5411 c21bd882 c210fa43 bd549937 3d0f6cb8 3c18dc4c 3a99c6d9 00000004 40000000 3f000000 -> 3ee1da62 3e7d93f9 3fb1543d", + "41cb1a2c 423870ac c2013c28 3a5ce20a bb1c13ca 3b1ee37e 3cea75d9 00000004 40000000 3f000000 -> 3d269dc6 3bb40300 3f1c0167", + "41d9e464 41a0fa94 41ab67e0 3d7116bd 3cd00be8 bd2fdf7a 3da7d2a4 00000004 40000000 3f000000 -> be85143e 3d43d7a8 409975b0", + "c1cac9fe 421e2dca c2371229 bdb08185 bd82b8fd bdb16dce 3d2ab264 00000004 40000000 3f000000 -> 3d9154df 3dfe6ec5 40a9a30d", + "41fd83f0 42016df2 c1cdacaa 3dbfffe8 bdc374a6 bc92e18e 3e890e37 00000004 40000000 3f000000 -> 3d64fd94 00000000 414d49ff", + "c205ca8a c1606a2c c23c16c8 3d67d977 3ded8794 bdb23d0c 3e59beb6 00000004 40000000 3f000000 -> bf08c549 00000000 413bc6c5", + "bff03ba0 c1093fa0 c1f052e0 bc4e7a21 bb9f73ad 3c4b9813 3ccc00b4 00000004 40000000 3f000000 -> bd2eebf3 3d405ca8 3f61c43e", + "421ade98 c0e3b290 c0a12a90 3afc5cba 3b045372 3b17603b 3c480e98 00000004 40000000 3f000000 -> bdab8b47 bbf0eb80 3e5d6612", + "418240c0 40f64790 c10c3058 3df2ae14 bdf7c941 3d1eae4f 3d7f99fc 00000004 40000000 3f000000 -> bdbdfd18 3b2e3084 40e75c85", + "c1b406b5 41fc1bec 405bdb60 3bde1568 bbb217ef bcbbd99c 3c5808db 00000004 40000000 3f000000 -> bf43c50d bcbe6da6 3f1f6467", + "4166e980 c1ffd0bc c13c16fc be2efa14 3e081ab6 bda387be 3c4fb685 00000004 40000000 3f000000 -> be67fc48 00000000 40f834f1", + "c2242daf 41b49f7c 4219fe2c 3bb0f7aa bc3c4955 bc22543b 3c95fae2 00000004 40000000 3f000000 -> be595a64 bc9c4132 3f1b535d", + "41556784 414ba754 c204e145 b98f7dbb b8bd070d 3aade21e 3c90c46f 00000004 40000000 3f000000 -> bd90727f 3ba44422 3eb8d3e6", + "41d03c0c 405e1cf0 c1c60785 3c45ff78 bc173ac0 3c4adc98 3cb15136 00000004 40000000 3f000000 -> bf37474a 3c594a64 3f50ef39", + "c13ccdb8 c23a1648 413c7d78 bc0eb46b 3c83ec9e bc40c023 3e8316fe 00000004 40000000 3f000000 -> bdd993ce 00000000 410da223", + "c1d041d6 c246760a 419686b4 bb5be93a bb361775 bb943699 3b028a06 00000004 40000000 3f000000 -> bdbe74d9 bc955d8f 3d6ca599", + "4127864c c22d5136 421a85be ba166083 3a7c6a4e 3a87654d 3b3da2f8 00000004 40000000 3f000000 -> 3e9066ff bb12e24f 3d1dd7a9", + "c1a8d5b4 422831fc c216125e 3adf7cd6 3d1403ca 3b51cc8b 3e6c424d 00000004 40000000 3f000000 -> bc525770 00000000 41082a35", + "420908a6 c12f5a80 c22a1a67 3b920fbd 3c9f363a bd091319 3e0cf1ac 00000004 40000000 3f000000 -> 3f54b1af 00000000 40b41dcb", + "c14579b4 c1b3547a 420a16ec 3bf763d4 3b7da53e bb9575d9 3d07c0a4 00000004 40000000 3f000000 -> 3d22982e bbcadc76 3f7ef1a5", + "4030fd10 c20f5980 42092180 bd5515bb 3de948c8 bd8ea10b 3c5f2217 00000004 40000000 3f000000 -> bf11160b bdc067ed 408e7523", + "c238511b c0a9ea10 c1e0665d 3ba54046 3b4c0d0c bb411075 3c104895 00000004 40000000 3f000000 -> bdc964ee bca0f31c 3e48f979", + "c0445af0 41fa5a18 c11c12b0 bd9db44f bcc1d296 3c579407 3e8c2982 00000004 40000000 3f000000 -> bf0edfe3 00000000 41352a42", + "4225cc26 c2225436 41501db0 be8f3834 3d6f936d bdaaebbd 3d5eb176 00000004 40000000 3f000000 -> bd8cbc96 00000000 41338247", + "411485a0 40c57c18 c2243a7e bc1a7bc9 3b9af474 3a80b769 3ecfc4b5 00000004 40000000 3f000000 -> be4fb387 00000000 4154452e", + "c217c05c 41778088 4106289c b8885eb7 3a7a4344 bace5f60 3dcfa3b6 00000004 40000000 3f000000 -> be25ac6e 3a19bc8d 40358063", + "4220e7f6 bef0af00 423271da be5d6948 3d90c874 bd4d1b49 3b201707 00000004 40000000 3f000000 -> bb612ed0 00000000 40f008c6", + "40f6c250 42300a56 421d1f5e bb0e366d bac902cd badca25e 3c8e304c 00000004 40000000 3f000000 -> bdd38c54 bb943dda 3e8ae6dd", + "c22f7f7d 40f5b168 41f1b1c8 397280f0 3b2a6234 ba872054 3dce86ff 00000004 40000000 3f000000 -> bd005989 3b4a8604 403c3591", + "c1bb65fe c0cb79d0 41453ab8 3c342f8d baa7a691 bb3f8190 3b2e2cdb 00000004 40000000 3f000000 -> bd17f51e bc281275 3dd0dd1d", + "c2409be0 421ad8c6 c23552aa 3c512ad4 bc0741f7 bc1db66c 3a8aa963 00000004 40000000 3f000000 -> be01052c bd693f44 3e2a9fe4", + "c217944c 42324ea4 41efb8c4 3aaa51b2 ba3e226c 3a5fa69c 3e13d388 00000004 40000000 3f000000 -> 3e9369d7 bb1c69b6 408ccb52", + "c16dd4a0 c2161616 c238bbb8 3e81f7fc 3e706cde be2c0998 3e8e8ffa 00000004 40000000 3f000000 -> be969c26 00000000 41a8f6be", + "c1f628bc c1ecc8a9 41a540b4 3e0eb5c7 3e128dc3 bd41fa29 3b0380ac 00000004 40000000 3f000000 -> 3ec5d8cc 00000000 40d36afb", + "3ffe1900 41bc7720 c1e3b349 bb8a3a10 3b5ea705 3bc6e9fd 3d66046c 00000004 40000000 3f000000 -> 3d18bda5 bc9131e1 3fe18603", + "c200dd4a 41ea7378 c23a88ed bcccd3df bccd7e23 3d09dcf5 3ebee519 00000004 40000000 3f000000 -> befe117a 00000000 41570078", + "41ad79cc 4201637a c21970ac bc1f7965 3afe1001 bd228b52 3af04ad1 00000004 40000000 3f000000 -> 3e208181 3d110ee0 3f37508b", + "4196af18 c20f817f c0e2d0b0 3bc45404 3b2de722 bc56553f 3b249da9 00000004 40000000 3f000000 -> 3ede3592 3cb61a04 3e1b6d9f", + "41239a48 41961b08 410b28b0 bac2d6bf ba077ccb 38c70206 3d837891 00000004 40000000 3f000000 -> 3d0d4b5c 3ace0db9 3fcb23b9", + "c1a0330c c12f9d68 41b64970 3b0b10a0 3a10198f 3a9593f7 3efa8171 00000004 40000000 3f000000 -> bd7d845a 00000000 417aacd9", + "c1a2eb82 41cb51d0 c223ca7e 3adb1379 3a4eaa89 3a637c03 3a87764e 00000004 40000000 3f000000 -> 3d0c1b02 bbf8c03c 3c84595a", + "40567380 c218d8e1 4188ccb8 ba940e27 b9fb7ccd 3b041baa 3d1cc688 00000004 40000000 3f000000 -> beaf346b 3c0748c7 3f727311", + "41c39c50 421a3504 42467836 3ba9fc01 3c705f45 3c89f11a 3c166ce4 00000004 40000000 3f000000 -> bf0dcecd 3c8ef3cd 3f0e8120", + "c1a1c142 c0a9be30 c147065c 3a3cd099 ba977b2c ba3ee15d 3ae28c45 00000004 40000000 3f000000 -> 3f322a1e bb3eebc2 3cd506a9", + "c0d69ca8 c0d66d60 c212891e 3a4d922a 3e9d4e5f be1f4bc1 3eaa5834 00000004 40000000 3f000000 -> beb9cb24 00000000 41ac9285", + "c246fc38 4102ae20 40d10fc8 3e2b97b6 bdc27d20 bd972b41 3e8deaeb 00000004 40000000 3f000000 -> be3e7e40 00000000 417674a0", + "c0a2f3b0 c20f5cef c103e1c4 395ce8bd ba1fb581 b96fcc80 3ca9b456 00000004 40000000 3f000000 -> 3ea20d74 bad004a6 3ec6929e", + "421fe0aa 41bb1d2c 4207b0bc bbad74cd 3b782db1 bbeb47a7 3cb027fc 00000004 40000000 3f000000 -> 3ebac1aa bd0e746c 3f106090", + "422315b2 4104c504 419ff858 bd9fcb38 3e239163 be0adefd 3e35b690 00000004 40000000 3f000000 -> bf4326af 00000000 414c74c0", + "41163514 c22f2b2f 3f32dd00 ba1f93d7 3a68e94b 39d04ee4 3e8e30da 00000004 40000000 3f000000 -> be3b7978 00000000 410e2b5b", + "c23be10c c03a5c40 410c2bd0 3c5d55c0 bd03d389 bae984a1 3b8c7f92 00000004 40000000 3f000000 -> 3ea82377 bdbdf1bf 3f267d7c", + "c2359360 4169d378 c232d50b bdce756a 3db86fdd 3d08a77d 3c141a22 00000004 40000000 3f000000 -> beec6c7e 3dc6e3c2 4085a7aa", + "c15b16b8 c22b46a0 c22b0d9e 3a43ab54 39c49540 3a117eec 3c8f7ce5 00000004 40000000 3f000000 -> bf4b1b94 bac84690 3ebbf1db", + "c2317306 41075380 c11351e0 3c36c52c bc05fac0 bc73f063 3a996a28 00000004 40000000 3f000000 -> be540e3f 3cfed04b 3e5416b4", + "c22d68b6 422dace0 c1413374 be14c27c 3cc72c35 3dfb6543 3d9e54aa 00000004 40000000 3f000000 -> be3afd1f 00000000 41093001", + "c220599d c05ad660 41dafdc0 3d23882a bc88995f 3cc3bb24 3dab11a4 00000004 40000000 3f000000 -> bd8922a7 bdc3c9f0 40788856", + "c0fe4e78 c20adb17 4183900c 3ae63506 39ba8a8c badb5a15 3c7282ee 00000004 40000000 3f000000 -> 3e15b863 3bd70782 3e7a2246", + "4203017a c2152b1a 410df588 3a474995 3a928480 38e0cfb6 3d9250ff 00000004 40000000 3f000000 -> beed58a7 3b72db13 40075ce5", + "c11c9b98 c23cf58c 41a08b5c 3760a7fb 3ac5e2a1 ba7fb52f 3e0ab73e 00000004 40000000 3f000000 -> 3ea7d74d 3a7f8b97 407fe5eb", + "c1ef704d c14b2ac0 c2452805 3b2daf91 bb7da37c 39562286 3dc9bd51 00000004 40000000 3f000000 -> bdec5160 3ba3a1c4 4040e47b", + "42018cf0 419ce7d8 c21760c6 3c362d87 bcc7d1d6 bc814785 3cbd890f 00000004 40000000 3f000000 -> bef82201 bd6c0ce0 3f94be22", + "41b190bc 423251f6 4245a90e 3ea4c751 3e629011 3e8796f5 3b2e5070 00000004 40000000 3f000000 -> 3d5829c9 00000000 4171eed5", + "4229606e c2252a63 c1e5851a 3d880688 3d8e1367 be28a442 3d2b6fb2 00000004 40000000 3f000000 -> 3da3fda3 00000000 40ed21f1", + "420fd0a8 4215f48c 423c6508 3acc1227 3c64e999 3b50349e 3e136eb8 00000004 40000000 3f000000 -> be110df8 3b2e4a0b 4094fb19", + "41237d7c 420c2988 c1e4d57e 3aa08d69 baf8f89a 3af687e4 3bfe9c2f 00000004 40000000 3f000000 -> 3e2598c2 bb834a07 3df01623", + "41f36f74 c1f3b158 c10629fc 3caa5373 bc12d517 3c9f5f5c 3c20de7a 00000004 40000000 3f000000 -> bec3b24f 3d6050fa 3f3718bc", + "c025aa70 420be70c c19d2761 3d7aa26c 3cb9f35a bd96c78f 3b650cdd 00000004 40000000 3f000000 -> 3e496efb 3e1098e1 4021384d", + "bfa424c0 4005d200 418f83a8 be05b11b 3dc56793 3e175267 3dc7bff5 00000004 40000000 3f000000 -> bf1512b0 00000000 41219a0f", + "c183a0a4 c1f42c6c c186c3f8 3a103d46 3ab210cb ba44e1d3 3e9c0f2a 00000004 40000000 3f000000 -> 3eee9c87 00000000 411c3c2e", + "c221d850 4219533c c21e81a6 bc049735 3b893ce3 3bf85884 3cb753b6 00000004 40000000 3f000000 -> bf50002d bd3c01d0 3f1df244", + "c22bf575 4125b210 4235f31e 3ced564a 3d747ea1 3d9e2d85 3eea36a8 00000004 40000000 3f000000 -> bceb5634 00000000 418e8f1c", + "41b7923c c242eda0 413b694c bb1444dd bb8a58ed bb64d35d 3a8b7539 00000004 40000000 3f000000 -> beee7494 bcadd192 3d14bf16", + "c1ccac23 418f89cc c1debe36 bd435d47 3c8d15cf 3ccd69b8 3d8bf797 00000004 40000000 3f000000 -> 3ec7788d 3dc40de4 4064d5aa", + "3fa18a80 c13a51b8 41610668 3c0d1e47 3e05953b 3ea61ff9 3e384977 00000004 40000000 3f000000 -> 3f051bbe 00000000 4187068d", + "41c4cc20 c20037f8 41c7f87c 3c759bb1 bc37204b bca0f93c 3c900d50 00000004 40000000 3f000000 -> bf1569a6 3dea34f8 3f5d944a", + "c1e7a170 c17607a8 c11824c0 bce59249 bd247b14 3c089c8f 3e9fbd09 00000004 40000000 3f000000 -> 3d839393 00000000 41385f5e", + "4177cb88 c2227f1d c200ddd1 bb71c7ac bc9a7c83 bac33178 3c449a2d 00000004 40000000 3f000000 -> 3d66dd71 bc867476 3ef4c752", + "420e2e14 c1b3b6de c1f6a4d5 bb5ec704 bbbb1ef4 b9ba3033 3a9e7431 00000004 40000000 3f000000 -> be3f6fcb 3c8c025e 3d2f4373", + "c240dbf3 41e17ff4 40aa9840 b9c9cce5 3a948e51 bad432f8 3efc4242 00000004 40000000 3f000000 -> becc3aa6 00000000 417c3364", + "4146ff04 41d0546c c0fe7040 bd052f6a bd4ba0c2 39952496 3c9ccb33 00000004 40000000 3f000000 -> be8c8080 be39cb6b 3feb450b", + "41e61164 41ad821c c1b282b6 3a16211e ba64f483 bb2907c6 3d2cec18 00000004 40000000 3f000000 -> bf28d674 bafb911c 3fa36a13", + "41a60e2c c16f90e8 4163bd10 bcf30046 3c8c5e01 bc94e11d 3b2ee13c 00000004 40000000 3f000000 -> 3e9f5902 bd92cc11 3f2e582b", + "42159f40 c148f810 410e0e98 b9bde814 ba951933 384cb80e 3ea032af 00000004 40000000 3f000000 -> 3f18309e 00000000 41201f5a", + "42294a94 c1ff8889 42210fe6 3cebc33b 3d19012b bd40f60c 3dad6897 00000004 40000000 3f000000 -> 3e94f0f4 be4eab35 4093400a", + "41daa4f8 41ad85bc 3f86e620 3c608281 bcb6110d 39f4f78a 3cfa574a 00000004 40000000 3f000000 -> 3e45d604 3da15402 3fa524cb", + "c144db10 423482c8 c0c98a00 baa83824 39adf9e1 3ab49f60 3e611d77 00000004 40000000 3f000000 -> bf0c89ac 3a7eab6c 40de1b95", + "4052aeb0 c1bde8aa 41f63704 3cdbd6d0 be4d42a7 3e7fb717 3c047d34 00000004 40000000 3f000000 -> be41d3e9 00000000 4127f1e8", + "c1e42928 c16bc4e4 bec52e00 3a8ddb4f 3b7fd64f 3bce15d3 3e99a768 00000004 40000000 3f000000 -> 3f51a34e 00000000 411ccded", + "c1e322db 422b57ea 41e8a030 bb882c35 3b6246c9 bafd1d26 3e28ec8e 00000004 40000000 3f000000 -> bd686c66 39259560 40a68df7", + "4211ff44 c23881bb c2408cbc bcdb0532 bb7a122e 3c977ce9 3bead6df 00000004 40000000 3f000000 -> be84d440 3d96ff04 3f300a90", + "c1d129a4 c239ea97 c2113185 3c8ab16e bc13c331 bc42ce29 3d815d00 00000004 40000000 3f000000 -> be1b01cc 3d0b6454 4009ee29", + "41b54d74 c2024474 c19e6a2d 3ace17b7 bc169492 3c5c82cb 3d5daf55 00000004 40000000 3f000000 -> bd42580a bced5044 3fdcb0a3", + "c22ae99e 3ff9bb80 420e208e 3adfb11f 3a13028b 3a2b936b 3e29ece0 00000004 40000000 3f000000 -> bd41e7a0 bb644498 40a920c7", + "42182fb4 42429ae2 c23d8dfd bca4e0f7 bb097ff9 bc0e8822 3e061731 00000004 40000000 3f000000 -> bea790ca bd25623e 40940379", + "3fd654c0 c2004134 c21f34ce 3bad1e4c bb91738e 3cc04991 3ed8f2a8 00000004 40000000 3f000000 -> 3e20721b 00000000 41647cdc", + "41d5a748 c2166cd2 4132b138 b995c986 3abc59b0 ba4128e2 3d9b3055 00000004 40000000 3f000000 -> 3ed7857d bb6b41e7 3ffd5342", + "c2327858 c1456280 c07ca960 3b5630f5 3bf9ff4c 384c9e8e 3d540510 00000004 40000000 3f000000 -> 3edd0178 bc011321 3fbd2e0a", + "c1095c08 c19426f6 4100a1e0 be8d7e18 3e1350ea 3d9dfedf 3baf2075 00000004 40000000 3f000000 -> bf2c9d8d 00000000 41265815", + "420cfe34 3fe4c000 422288ea bc14289d ba767164 bc1bf9dd 3d7063e9 00000004 40000000 3f000000 -> 3c6a2ab8 bcce93c2 3ff04f9d", + "4106de90 4218ede8 c12b5f44 3d949959 3becc8fc bc510b9d 3e9d6a79 00000004 40000000 3f000000 -> 3e463517 00000000 41427983", + "41b1867c c18801d6 c07f2d90 3da94056 3d5eefa9 bd4691c9 3e3ec0cb 00000004 40000000 3f000000 -> bebd6f7d 00000000 411723ed", + "418e9730 c0522760 c219afa1 3b808738 3ba77258 3b431d1f 3b09f536 00000004 40000000 3f000000 -> bf3c3145 bc62cc8d 3d6be9a6", + "408fc498 c17ca84c c1e0575a be68ed93 3e49b4e1 3d869b3a 3d2988f4 00000004 40000000 3f000000 -> 3f19b000 00000000 41321d1d", + "c185eb50 41f46684 422438ce bd041fbb 3d3c6d0a 3cade74c 3e5debbd 00000004 40000000 3f000000 -> bd979076 00000000 410d1fc9", + "4220889c 417e64a8 c1b4e6e5 be1f032a 3e3afb7f bde6b793 3a999237 00000004 40000000 3f000000 -> 3eb36a56 00000000 4107a31b", + "c198b1ec c22dc971 419f3d30 bc5b4a3f 3cb1e0ae 3cd8a0b5 3d61ab08 00000004 40000000 3f000000 -> 3e54ad7c 3e057830 40219e7e", + "c2157ad6 c1b57802 c20d481f bc9eed89 3b2893fb 3d2bc0b8 3e378580 00000004 40000000 3f000000 -> bf574a4b 00000000 40e5f8ca", + "41a09a98 421082c0 c1423da4 3b32fab0 38d7d112 39c69431 3e6311b9 00000004 40000000 3f000000 -> bb058760 00000000 40e4f043", + "4201c118 c1c77d72 c13d190c 3a8380d5 3b128f31 3b2bc499 3d89e52e 00000004 40000000 3f000000 -> bcbb300c bbc28f87 40006f56", + "c22c6586 c1f4b9f0 4102c408 be93be68 bedefc74 be63d070 3d0f3748 00000004 40000000 3f000000 -> bed4e2bb 00000000 4199a654", + "42431cda c1bb70d4 40d59710 39b54f8a 3a8960da 389f4e82 3c2c9140 00000004 40000000 3f000000 -> 3ea25656 ba5ed09c 3e224482", + "41e842ac c0c249e0 c1fb04ef b92baa35 bbcfc7da bbc25faa 3e0e18ce 00000004 40000000 3f000000 -> 3e8a698a 3c04b047 409012aa", + "3f471400 420415be 41dc4504 b9dbe505 39fd8182 3ad9e25e 3ba232df 00000004 40000000 3f000000 -> bf288f4c bbc690bd 3da9375c", + "c17ef784 c0421070 41018340 3b060b54 3ae7f604 3a31f106 3d84f2a8 00000004 40000000 3f000000 -> be810e8d 3be6dc30 3ffa1506", + "412294e8 41f581e0 c1ed6a5b 3d522f51 3bb34d0a bd098914 3e3419fb 00000004 40000000 3f000000 -> 3e9609bc 00000000 40f22257", + "41abef80 c246c2fe c1333b00 3ac3aa06 3a6426cf bb270329 3eec98d8 00000004 40000000 3f000000 -> beacfb15 00000000 416d2701", + "c225fa27 41e6a1c0 c003b630 bb953b24 bbc7a807 bbba95e9 3c0ee528 00000004 40000000 3f000000 -> 3f08274a bba4dae8 3e70fc35", + "c1aadf86 c19868be 418763fc ba194e68 ba2d4f0f b9d85039 3e919a5e 00000004 40000000 3f000000 -> bf7cb3fc 00000000 41117ac4", + "41ea5fb0 3fbfcd80 41b374a8 3d57d5c6 bbbe1e28 3d4444a2 3ec7fdcb 00000004 40000000 3f000000 -> 3eb095c7 00000000 416b8cd5", + "41465f10 42318290 c14a3098 bba8ade1 3babb17a 3aca10ff 3dc477fc 00000004 40000000 3f000000 -> bf058600 00000000 40530957", + "c1ae95b5 4116686c c233e34f ba049bd3 ba8f5621 b9e09208 3c6e1b79 00000004 40000000 3f000000 -> bf51f676 baf6e444 3e9c2141", + "4136c8f0 420df1ca c1590ee0 3d254034 bd546a0d bc0c3685 3be794f5 00000004 40000000 3f000000 -> beec463a bd946bf4 3fd6b638", + "40d0d920 c0059320 c0d565e8 3b0e3d53 3804f389 bb11d2c6 3d49ab0a 00000004 40000000 3f000000 -> bede810c 3c0b459f 3fabacea", + "423184c6 3ff9cda0 41bb42b4 3b8648a8 3b8ef1d1 bbb5f177 3ce1b7bb 00000004 40000000 3f000000 -> 3f158afa bc5be546 3f4ea486", + "41bc0984 422e1234 422476c0 bd465b77 bd8e18e0 3b0ff000 3cbcd7b0 00000004 40000000 3f000000 -> babf4056 3b116770 403b0e68", + "c220d502 c1a4f1f4 c22cc417 ba771b8c ba33810c ba7ed2d6 3d3678eb 00000004 40000000 3f000000 -> 3f3aef80 bacf8f28 3f8e58ad", + "401a2f80 c2305cea 41f8919c bd33b9ab bcba2e38 3daf557f 3dff4a0f 00000004 40000000 3f000000 -> 3e8f9f3c bd98cf34 40de0b11", + "bfb474e0 c0f4c3e8 c1e8f590 3cd65c3f be428311 3d838fb4 3e11cbe8 00000004 40000000 3f000000 -> bea73a61 00000000 412fab2b", + "41ce56a4 4204604a 4221c3a6 be8fab07 3d930e63 3ea7b5e0 3bbade5b 00000004 40000000 3f000000 -> 3d565f00 00000000 4161d489", + "423ceea8 422b713c c22c0909 bb9eebd7 bd042d73 3b26e6ad 3dca4b4d 00000004 40000000 3f000000 -> 3ee0bd66 bd16e86d 4079c760", + "c0c59d08 c216a452 41e50c4c bdc76c2a bd0d92aa bd672195 3bfc7add 00000004 40000000 3f000000 -> 3c26156f 3dd027b8 4056b35b", + "c226a59a c21202b7 c1866c38 be89b9ca 3da19dd1 3e927d8d 3ef30817 00000004 40000000 3f000000 -> 3eb0dd15 00000000 41df1535", + "4180a394 c1b79724 c1bb2854 bb060fa8 bb73898d 38b7b883 3e9e4800 00000004 40000000 3f000000 -> be8d94f5 00000000 411fc504", + "c2222362 40445870 41effa38 bb737865 bbb06a60 bb708108 3c472a25 00000004 40000000 3f000000 -> bccaa1e0 3bb19ba5 3ea98722", + "421c46b6 c03fbdb0 c0b24410 bbab6847 3c3c440b 3c0b1ee9 3c72448e 00000004 40000000 3f000000 -> be0c85f5 3c9240b3 3f0483f0", + "413b9520 40f31e78 42100788 be40a08d bec16475 be781059 3c0ea11a 00000004 40000000 3f000000 -> bf39cb79 00000000 417c796d", + "423f3baa c22889b6 422b0f2e 3aeb777b baaa67f5 b9b7313c 3bd4231b 00000004 40000000 3f000000 -> bf08ccaa bbadc97b 3df9aa09", + "4137c568 42119c8c 4165f978 3a783f09 3c4af1c9 3c9aab30 3e27a578 00000004 40000000 3f000000 -> bf4a8912 bd28a51e 40bac9e0", + "41ed0734 422c1422 c239a3ed bce3710f 3ae065d0 3c72ead0 3defdf8f 00000004 40000000 3f000000 -> be366f45 3b49a570 4089fd3d", + "c0071180 42051b36 41693b40 bd24be5a 3bf55025 3ce1fcbc 3aba96cf 00000004 40000000 3f000000 -> becee216 be658794 3f6bcb47", + "c18a5630 c099ea10 c10b2fac bad63cdd bb560a47 3a5a484b 3d426a9e 00000004 40000000 3f000000 -> 3f319d4b bc0f10d4 3fa786df", + "c23b3cde c20e74c2 41fcbedc bc0b48f4 bbcd6318 3c647aff 3e0f2700 00000004 40000000 3f000000 -> be08fc3a bb105cd8 40998908", + "4106c7bc c237fc09 c203a58b bc06d21c 3ad282f0 bbb3b018 3c8d8e89 00000004 40000000 3f000000 -> 3d593f9d 3d228bcc 3efce465", + "c11398e4 c207685c 419678e0 3de869b0 3e47ffa9 bedf9423 3acefa7e 00000004 40000000 3f000000 -> bf4c4df4 00000000 417b6f1a", + "c1e6bb2b 410b15b8 c20cb87c b99502f3 3a409ba8 b84e92bb 3d46f1f0 00000004 40000000 3f000000 -> bda597d2 3a76de04 3f9423e4", + "c1759b68 c107f7b8 40ee5b00 3a05be61 ba3b4e24 38205568 3b340bba 00000004 40000000 3f000000 -> 3f218382 b8b35630 3d0f07a9", + "4229a830 420c74cc c20fd5af ba94d0d5 bad71942 bb7fbe29 3dbf6ae6 00000004 40000000 3f000000 -> 3d166c7c 3b6258aa 403599af", + "c1acf5a6 4235dd36 420c1de2 b8cb08b3 3a1860fe 3b405c26 3bed3d75 00000004 40000000 3f000000 -> 3e8b8939 bb24b98e 3ddab037", + "c217886d 41c1f4b0 c1cc3599 bd28addb 3d2ab68d bc69c127 3e4aa918 00000004 40000000 3f000000 -> 3ec8ee4b 00000000 4103a8fd", + "40b24060 40942988 4245422e bd1644e3 bce7bb46 3ca04920 3b59cb51 00000004 40000000 3f000000 -> 3f42f670 bd9f0b88 3f8f43c4", + "4169eff0 40236a10 c22f88a3 3c923781 3d8e9199 3d4b1af5 3e88c514 00000004 40000000 3f000000 -> 3e1e4a00 00000000 4134b1fc", + "404d4180 c102ad20 41e72b58 3c548ac1 bbd286fe bc4707fc 3dd66424 00000004 40000000 3f000000 -> be3ca775 bc2e1158 406b62c7", + "c102e82c c1c325fc 4225f9fa 3ebda24a 3e8dca26 3ec78d8e 3ae2eb11 00000004 40000000 3f000000 -> bf1ca4e9 00000000 419a98d6", + "c058e0d0 c1a81acd 41919924 bd78327d bc308e3f 3d8e8ae1 3b286860 00000004 40000000 3f000000 -> be0c390c be01de0d 40130b7a", + "c1efe89c 417b6400 40334040 3ceb8e66 be5a5990 3e420eb0 3c0476c8 00000004 40000000 3f000000 -> 3f1998b9 00000000 41164ac5", + "c23838ee c17679ac c2133da4 bdc45bd1 3dba62a1 3dca71ab 3b96d8b8 00000004 40000000 3f000000 -> 3ea6e903 3e7a2c18 409bf8e6", + "c11778b0 c1cbf70c c204806c bd882870 3d8d15da bd8522b3 3e0f6713 00000004 40000000 3f000000 -> be568d96 00000000 410266e0", + "40472670 423d275e c19a91b4 bb975444 3b906510 3a714d3b 3ac34c3f 00000004 40000000 3f000000 -> 3e5e2750 3cb9a972 3d360dc4", + "41a8f150 415b720c c1c42375 3b16f824 bb29b313 bb1ab960 3b5b411a 00000004 40000000 3f000000 -> be97ae58 bc144a86 3d58b580", + "3f8922e0 40fc0ba0 4228c3f2 3cadd7fa bcca5d19 bd35b16c 3ce56f73 00000004 40000000 3f000000 -> 3ed33dab 3dbeae4e 4003077c", + "c18dd196 c0cb7c78 c101d4d4 ba276cf4 3ab79f6b 3ab53b75 3c4da549 00000004 40000000 3f000000 -> 3d36d904 bc69a408 3e76d2bd", + "c0d3e190 41ba7a78 c0e9b910 3ce44345 bb25b220 bc920066 3b31e6a7 00000004 40000000 3f000000 -> bed4363a bd465f38 3f0ea10f", + "41d57528 c22ce5f0 c2388bcb 3b92b73c 3bc9a1bd ba74957d 3b82a0a5 00000004 40000000 3f000000 -> be36458a 3d0cacf4 3ddb4461", + "c23a5527 423dcca2 c1c33eba 3d20dd56 3c2c58ed bd37ec2f 3bc28d97 00000004 40000000 3f000000 -> bf0272e0 bd3bbc57 3fbbb50a", + "422c99e0 41d89c28 423830a4 bba57b65 3a15735e 3d08e1b9 3df6b561 00000004 40000000 3f000000 -> 3e2fa9cd 00000000 409d613a", + "41a1eacc 420b2912 41e9ff0c 3a35fa62 39d061f2 ba1e7df4 3ab14509 00000004 40000000 3f000000 -> 3e9c8394 3a48b28a 3cb04edf", + "c177126c 41d10714 420ed386 bd0a6e68 3c77e5f1 3d29d9a3 3d188bb1 00000004 40000000 3f000000 -> 3f20880a bdff1e40 4012c3e2", + "423d2c32 c1f2dde3 c237d2b5 3b3614b1 bb48cf27 3a0cb284 3e373e09 00000004 40000000 3f000000 -> 3e53c1b8 ba434f80 40b86241", + "417c2520 c20fd22d 41548b5c 370b906e 39ad8287 3a82b1fd 3ba75d7e 00000004 40000000 3f000000 -> be32a6c1 3a3d6eb0 3dca28bb", + "c19dc40a c1b9a43b 41a5e134 3e1d562e bcf01cb7 bc3c0a09 3c3667be 00000004 40000000 3f000000 -> be9da6b7 beb73c62 409af455", + "419deb20 c1ea18c4 c1a5d59f 3a7bca2b b99e31cb bad9ded7 3bd364aa 00000004 40000000 3f000000 -> bf2e6463 bbebd10a 3dc1e3a2", + "4223ff9c c1423a04 419aebac b9c1e0bf ba212a04 bb8b9add 3c4c5f36 00000004 40000000 3f000000 -> 3f3803e2 bc13d2e9 3e7dccba", + "c1963329 423742d0 c12dba78 bd36da8e bd0ada39 3b8d0c54 3e01e136 00000004 40000000 3f000000 -> be33466c 3cbf1722 40aeabd1", + "c22fc106 41179f3c c171d4d4 3e941ead bea2e3a2 3dba6c98 3d03e3b4 00000004 40000000 3f000000 -> 3e257e4a 00000000 41707c95", + "41b7ed18 41747d70 414ba814 bc9dee16 bb3d457a bc81dc09 3b8c20c3 00000004 40000000 3f000000 -> 3ece9045 3d4c876e 3ed52822", + "c2300a12 c1859f1c c1d2e3a6 3ad21ef4 b98db128 3aa80509 3e5a515d 00000004 40000000 3f000000 -> be4154d9 bacb3114 40db655c", + "c1fe8d79 c19edc85 3fc163c0 bee653f5 3ea98448 bdf37fad 3d0acddd 00000004 40000000 3f000000 -> 3f0b5b03 00000000 419a3189", + "41806410 3fafba00 c2343ff7 bb0b762f baea9be6 ba234fac 3e6e47fd 00000004 40000000 3f000000 -> bf436251 00000000 40f02fed", + "41c415cc c211ecc7 40574420 3dc6c26a 3d0a65f3 bcf88436 3e7d910f 00000004 40000000 3f000000 -> bf60255c 00000000 4134dda1", + "421279ac c208ecb7 40b1dbc8 3d5e6d68 bc2bdf44 3d6506e8 3d4598af 00000004 40000000 3f000000 -> bf187b25 3cadf2c8 40668cab", + "41ed3b54 bfaed640 41ccf804 3ae68342 3c61a0fd 3b5f0ebf 3e71b396 00000004 40000000 3f000000 -> becdc708 00000000 40ff40e7", + "c1b0e8fc c22fc242 41fd581c bbaa9b2c 3c99314e bc6b4fe0 3e9986f6 00000004 40000000 3f000000 -> 3ea0fa29 00000000 41253310", + "4236f4f0 c1605208 c14343d8 3da23d25 3dc3cc31 bdb521c0 3d103852 00000004 40000000 3f000000 -> bc472e12 3e48026a 40b1773a", + "421fb438 414da15c c21663f1 bdb5d1f9 3dabbff3 3d99331f 3b8eb4bd 00000004 40000000 3f000000 -> be2c2033 3c2c802c 40848512" + ] + }, + { + "glsl": "vec3 cultmath_af_frustum_axis(vec2, float, float)", + "tolerance": "ulp-bounded", + "cases": [ + "be17c8ac be987fc8 42733cca 42b47e45 -> c00ba0ce c08c4940 416b7f80", + "3f01644e be911c7a 421ac577 422f2bac -> 3fa4f7ee bf39028f 402331a8", + "3e9e77d8 3d84bf80 427011cb 42c34759 -> 40ba4f13 3f9c11f3 41967ce7", + "3f53028c 3ca8d400 429e7fe2 42faaf5a -> 4197f815 3ef32df8 41b85ef0", + "3e8d858c 3ee4afdc 42c7cad7 42fd4120 -> 406c704f 40bf084b 4155d924", + "bf29d94e 3ee14700 4095d287 4234a723 -> c156dd93 410e7e0d 41a1ecd2", + "bf221d14 3ee4f81c 422ef3f7 42b074ea -> c1616399 411f2b72 41b1f5dd", + "3eae2290 3e2ae048 424fe60c 42a28578 -> 409f5dc8 401c6241 416a49c8", + "3e23df98 3f43c12a 421b8b36 42acd010 -> 40735aac 4191595b 41be14ea", + "3e0f7e20 3d7e7460 426d78f6 428b3832 -> 3f37b32b 3ea2e07a 40a3ddb8", + "bee53ad2 bf395966 3e806db5 414b1f91 -> c03249eb c09028dd 40c71c23", + "becfbcbc 3f0fa424 42360915 42bdbb2a -> c12034e3 415d8d38 41c56d3f", + "bf187b27 3f3b98c2 413e8dd6 423319b4 -> c11c9ae5 4140aba2 4183763e", + "bf396ea6 bf1e4fa8 42c1fc4e 42edf0c6 -> c0feb552 c0d9747b 412fd1e0", + "3da9ef10 bf32418e 428d3c99 42a18c04 -> 3ed7b651 c06246bd 40a27b58", + "bd80fbd8 3e291060 40408dc2 4245fb3c -> bfbb603b 407599cd 41b9f260", + "3f64bf40 be8b1e4a 40e730d4 418ec529 -> 4097da69 bfb8b4e9 40a9f1e8", + "3df56000 beaf8ce6 4044e6d9 41c4377d -> 3fa47b57 c06b5a4f 412b9aa2", + "3d18cf20 3d873d80 412cdbaa 42452b12 -> 3f37cb0f 3fa2a97a 4199f428", + "bd6b6f90 be946f56 4047f604 4121e0c8 -> be4dccc9 bf81c01c 405fc68e", + "3ef22598 3f0b0aa2 42215e33 4260cfb5 -> 40700a78 4089d511 40fdc608", + "3f5a3632 3e6d0910 42a0c521 42b9d3b8 -> 40aaddd4 3fb99b27 40c874b8", + "bde9e630 bf02cfb6 421bee73 422653a4 -> be17f727 bf29fa43 3fa65310", + "beb001b2 bf261d44 4227856d 428911e8 -> c0929b32 c10a5dc9 41553cc6", + "bf155da2 bf252960 428ac799 42bd9112 -> c0ed0eb6 c1031041 414b25e4", + "3e0bd408 3f6a27e0 42ba9c7c 42f27372 -> 3ff3ffbb 414c4cbe 415f5bd8", + "bdfffbd0 3dcd3300 42aebfe2 42dbe28c -> bfb487b4 3f90b700 41348aa8", + "be21a2f0 be3a08a4 42721504 429084ef -> bf6d2f01 bf887df0 40bbd368", + "bf79c6b4 3f19a956 4195ce4b 420575c4 -> c0e488af 408c97e2 40ea3a7a", + "bef02888 bf3f744a 428b7310 4299fa9b -> bfda15f3 c02ddba5 406878b0", + "bf45c632 3f256792 4204ae26 427a13a9 -> c1356403 4117b3d8 416acb06", + "bee4947c bf7a88d9 41fcdd83 428d1f8c -> c10b1fed c1987cbf 419bd056", + "bf7ffdb2 bf59056b 4209803f 4296f458 -> c1a466f6 c18b6003 41a46871", + "3f31cdb0 bf162ba2 41eba360 428e5e1a -> 4167dc84 c143d3b9 41a6ea84", + "bf12cffa bde79780 429bdf13 42c07fc2 -> c0a80b2c bf848aac 411282bc", + "bf2bc738 3e951ac4 4216b03f 42ac28b8 -> c181ed5a 40e18e20 41c1a131", + "bf7af105 3f3b0e3e 405e1279 41be3b90 -> c11f4361 40ed6f39 41227941", + "3f440d4a 3f2c0f80 42536360 42948488 -> 41032e9a 40e64207 412b4b60", + "bf6f7708 bf258258 4104feb9 42451c01 -> c19946e2 c153e0df 41a3dc53", + "be84ac8c bf03a11b 4295184b 42d0c990 -> c0777d1b c0f58a39 416ec514", + "3f0aca06 bf48a0bc 422af603 4291f7b8 -> 41032bd9 c13d9da4 4171f2da", + "3e328070 3f61aee8 420a3d7c 424e19d3 -> 3fbd4501 40ef4c17 4107b8ae", + "3f6e5f90 3cdd50c0 42a12163 42d1826d -> 4134311a 3ea74c21 41418428", + "3f0513b8 3f09bdd4 42a47018 42b51baf -> 400aa6e3 400f8305 40855cb8", + "3f35b748 3dd9c620 423b752b 427896c1 -> 40ad920d 3f50031f 40f48658", + "3e563d70 befef2ec 4224cf24 42806efc -> 401a14fa c0b75c22 41381da8", + "3e978fa4 3ef58e28 4183d69b 41ec841a -> 3ff7e448 4048d056 40d15afe", + "be668a6c bf5243aa 42862c22 42af0de5 -> c01343c1 c1065001 4123870c", + "3f317786 3f60eea0 42c21336 42fe95de -> 4127ca5a 4154aae9 41720aa0", + "bf6b19fb 3e7c4e70 40b75480 41e08196 -> c1241672 40301883 4132ac76", + "3d566600 3e4e0b70 40e01315 420c830f -> 3f3c70c7 40351923 41610159", + "be972cba 3e595060 421403bb 42547fd2 -> c01851d7 3fdaf5ba 4100f82e", + "bf7ac68d be986ae6 41d3344b 428dd576 -> c1ae6e7c c0d408a1 41b210c6", + "3f5ff22a bf2111fa 42a37d7b 42f5b739 -> 418fdc29 c14ef05b 41a4737c", + "3e9784b0 bdf25b88 425035de 42adce10 -> 40a50323 c003f882 418b6642", + "3eb78af0 3eacafa8 42a98098 42d3510c -> 406fd5a4 4061a5f7 412741d0", + "bf4a56ab 3cda9740 42af4619 4308682f -> c19a3040 3f2692bf 41c3148a", + "bf2c07f8 bf47954e 42a427ad 42cee55c -> c0e5c631 c1054982 412af6bc", + "bf496c83 bf06f358 41b9fd15 42870695 -> c18b4f88 c13aabeb 41b10ea0", + "3ec12094 bf5185d6 3effe07d 42245343 -> 40f4eb3b c184db0a 41a25382", + "be28ef70 3f7f1e4c 4089242a 421538d2 -> c02e51b7 41839fda 4184144d", + "bf1a19d3 3e3b40f0 406e5689 40ac3343 -> beff6145 3e1b292a 3f541ffa", + "bf440b63 bf547daf 42844f0f 42ad289d -> c0fa430f c107a0df 41236638", + "3f5a3ebe 3e4b30f8 4270279a 429f131f -> 4104fbea 3ff79f5d 411bfd48", + "bf2284ac 3ebea60c 42ab88fa 42c60276 -> c08674c7 401dbab0 40d3cbe0", + "3e85d9e4 3b5dd800 42a7d669 42b1c841 -> 3f266382 3c09e2b8 401f1d80", + "3f590426 3ea264f8 42b405e2 42fb189e -> 41710021 40b4578d 418e2578", + "becca63c 3f22ed84 428c40f6 42df66d2 -> c104f053 4153ac69 41a64bb8", + "3e8456c8 3e9ec248 425f0dbc 429a85de -> 4031d07a 4055501b 412bfc00", + "bf7bbec3 be722d40 428841a4 42cbfc38 -> c18534ce c08024b1 41877528", + "bd20a3d0 bebe44b4 424d8b6f 42ad20b4 -> bf30978c c0d129a4 418cb5f9", + "3ecea70c 3f2794f2 40ffd22d 41ba31a0 -> 404559d5 40a009ef 40f47a2a", + "3f31720c bf71cd69 424b261a 42b7b852 -> 4163c175 c19b2dff 41a44a8a", + "3f2700ba 3f71f442 3eaf0261 41a74166 -> 40d6a6ed 411b7e85 4124855c", + "3f4c4b30 3f0547d6 42b72e44 42c270ae -> 400fc3b1 3fbb953d 403426a0", + "3f42f2c4 beac0d14 42585b71 42904c15 -> 40dc0a14 c04231f4 41107972", + "befc61c0 3f33cefe 41efaaae 429cfe01 -> c13f6833 41885e05 41c226ab", + "bf2948a6 3ed5e6dc 41d98f80 428b50bf -> c160a1e4 410deb57 41a9d9be", + "bf7b6a09 3f00a056 42a034cb 42c41529 -> c10cef61 4090353a 410f8178", + "3f18ad5e bf75a29b 41fdc3e1 42268e4a -> 403d4999 c0984493 409eb166", + "be9664e4 bf17e9c6 411bed8e 4251ddfc -> c0c8c83c c14acf68 41aae298", + "3f7e5c58 3f2875ec 41da8418 427f78b6 -> 419146fb 41406e73 419236aa", + "bf0325a4 3ce5d340 4283671d 42d19a32 -> c1203e8d 3f0c6873 419c662a", + "beec3406 bf39f6a2 41c6ffb6 42620997 -> c0e98161 c137d6e3 417d1378", + "3f680108 be435a7c 416ed143 4260cf2d -> 4195a102 c07bfba2 41a51adc", + "3f368ee2 bf6e78ab 42aa276b 42cbf7e9 -> 40c0e8c9 c0fbfe2c 410741f8", + "bf249ab8 bf757579 415dace4 427e1ca5 -> c17f837d c1be82f9 41c6b16c", + "3e695478 bf4f8885 40f2d040 4157a536 -> 3f2bc962 c018cb4c 403c7a2c", + "3dfe64d0 bf77e080 42afc718 42c28634 -> 3f1507fd c09136a5 4095f8e0", + "bd37a000 bf529dae 42878ea9 42a5c796 -> bead6ca8 c0c6ea7f 40f1c768", + "bf427876 bebe6222 422a089f 422f8bd8 -> bf060128 be833029 3f306720", + "3f5a9770 3e4f4fa0 42832310 42b84bd7 -> 413590af 402c3207 4154a31c", + "bf4345fd bf145d29 42ae7578 42d2cf84 -> c0ddd43b c0a88a4c 41116830", + "bf1c4932 be9c129a 421dad37 424db4a0 -> c06a91fc bfea400b 40c01da4", + "3f3b0656 3edde9ac 41ffa937 4212a618 -> 3fdbf7d3 3f82802e 40168be4", + "3f2bce9e 3e9fce1c 42c33f2e 42c78a39 -> 3f3865d9 3eab842c 3f896160", + "be542948 3f4e6e34 4170117f 42453518 -> c06364f0 415d4086 418930b8", + "3f66233c 3f6909c8 41cb5f08 426bdd19 -> 41713ed7 4174493a 41862d95", + "bf4b8160 be4aca74 42acecef 42f5f9ba -> c168483f c0677775 41921996", + "bf7c7be8 3ebb0e44 42592cce 4281ad99 -> c0a66854 3ff691e6 40a8b990", + "bcb2c3e0 bf41e54c 426714dc 42d32d26 -> bf0590a1 c190deab 41bf4570", + "3eecd270 3ee569dc 4221b857 42954487 -> 40fd21c0 40f5368e 4188d0b7", + "3f0d3f2e 3f3c9fe8 41ac3f17 42789124 -> 41334150 416f61c7 41a27198", + "3f7d4d50 beb24e20 429838f3 42d3201f -> 41692102 c0a41ada 416b9cb0", + "beea262a 3f41a05c 4239dc40 426dad4a -> c03d9333 409cc437 40cf4428", + "3f54a71a bee34bac 401e42f6 42426e22 -> 41994ab2 c123d8df 41b889f3", + "3da16450 bb107200 41836eae 41f7e966 -> 3f12dda4 bc8371cc 40e8f570", + "3e6fe6d0 3eca40c0 429ecbbc 42be032c -> 3fea0736 40454d1d 40f9bb80", + "3c436a00 3d083a20 4192be28 422ad0ee -> 3e14c424 3ecf6a6f 4142e3b4", + "3ecc815c 3e8e5f7c 428ee8e9 42b8b6a7 -> 4085944c 4039fdb8 412736f8", + "bf0fc8ca bd6a9190 42868a11 42b64f2b -> c0d6a48a bf2f154d 413f1468", + "bf36cc6b 3f51e524 410332cb 424ec1d4 -> c1786e5b 418ea0d5 41adf521", + "3f1fc2ce beef46f8 429b6028 42b6811b -> 408770de c04ada41 40d90798", + "be1c2fc0 bf20420e 414f4c93 42383a1f -> c0218ee5 c125c50c 418466fa", + "bf1a4ea8 3f15f628 41a97a22 4280cd59 -> c15064e0 414a866d 41acdda1", + "be0941c4 3f47cd92 41ba834e 4254bfca -> c0002266 413a8603 416efc46", + "3de6ee40 3e84ae00 40fdb38b 41a572e3 -> 3f3810ae 3fd381c0 40cc0c00", + "3f2d7100 3f3fd6aa 423f7301 42ac2b6c -> 414f2aea 41652463 4198e3d7", + "bda2ca78 3e314950 427d1173 42cefeff -> bfcca9ec 405ee344 41a0ec8b", + "3f19d2da 3f2ee608 4231cf97 4235f24f -> 3e9f09fe 3eb4d419 3f045700", + "3e843c50 3cebf5c0 427bedb9 42e0237f -> 40cad894 3f34fa6f 41c45945", + "be16ca9c bf22219c 420c9548 42922f0d -> c032cf98 c1404205 4197c8d2", + "3f35995a bee54f8c 41ffb807 426701fc -> 41125745 c0b8ca09 414e4bf1", + "3f0caa00 bf20fffe 42b561e2 42cd5623 -> 405297bc c07109cb 40bfa208", + "3f3f7ab4 be4e5dfc 4203d740 426cccc9 -> 411d0300 c029383a 4151eb12", + "3e9ec344 be5de11c 424d8e29 42c491b0 -> 40e8aa59 c0a294c3 41bb9537", + "3b7cb600 3e8e2bb0 42bb7b73 42fe5512 -> 3d83fb7e 4094807b 4185b33e", + "be70d054 bf1065ea 42a77784 42d08a6c -> c01a8c99 c0b957e4 41244ba0", + "3f0b7982 bf07c1a9 420f10ad 4283dd3e -> 41037af2 c0fff34f 4171539e", + "3f5126f6 bf01c0fe 42b7bb5f 42f8f010 -> 415517c4 c10432d6 41826962", + "3f3dcbdc 3f5f33cc 421577ba 4289153b -> 4138e683 415971d9 41796578", + "be9057d4 3f371546 42614022 42c6628a -> c0c16b41 41755494 41ab84f2", + "be9c7dbc bd4779e0 41dc58fe 4260304c -> c08b64a1 bf31ae78 4164079a", + "bebb7f06 3c680400 4220b867 42b1cd36 -> c10ebbcc 3eb09fe0 41c2e205", + "3e1cabf8 bf694589 41be9737 428489ec -> 404fd024 c19ab55c 41a9c83c", + "3cf42b00 3f57e968 429d8c8f 42fb34a0 -> 3f32a7d7 419dfb15 41bb5022", + "3f3b1b5e bf2ec8bb 427ed4a0 428cdf91 -> 401d616d c01303fa 40575410", + "3e289200 beb77d5e 4258233a 42961f9b -> 3fdd8955 c07124c7 412837f8", + "3efd2e18 bf3259d8 427b1ccc 42bdd57e -> 40fe4756 c1331ff7 41808e30", + "3f095766 be7f9054 428fe04a 42c9b726 -> 40f83dd6 c066f684 41675b70", + "3f1002aa be59a690 4280f09d 42b1f318 -> 40dc8f3e c026abf6 414409ec", + "3e978930 bf7b2d7e 42b55f27 42b9672e -> 3e98b94c bf7d2591 3f8100e0", + "bef54eda be8975da 41d4da38 422ede41 -> c0832aba c013003b 4108e24a", + "be5185e0 3f3a99f6 42acd956 43015537 -> c08c792b 417a361f 41aba230", + "3f7fb000 3f79310e 42bdcf12 4301c50e -> 418b4a7f 4187c093 418b7614", + "3ee48884 bebab0f6 41a0f8c9 425afa66 -> 40f74428 c0c9fe92 418a7e02", + "bf63ba4c 3f6299ca 402ef8e6 420dbda3 -> c168b7da 41679106 4182ce15", + "3f3a8f56 befa9026 4184dc1b 42204c1c -> 4108cfca c0b7bf6a 413bbc1d", + "bd8aecc0 bf1746f6 41a1ab23 41a440c4 -> bc338655 bdc37cae 3e256840", + "be0bdccc bf70bfbd 41434a4a 4276919c -> c0581290 c1b9f735 41c5bf0a", + "bf43538d 3f3cf792 42ab3bd4 42c7313f -> c0aaa884 40a51a23 40dfab58", + "3f751c70 3de6b180 41addd8e 4204e810 -> 40b012b9 3f25b760 40b7e524", + "bed36fcc 3ec65c8c 42b094dd 42f88ee0 -> c0edc9fd 40df158e 418ff406", + "3eb63d04 be7a130c 429c5a2a 42f87870 -> 410326f5 c0b3f8d6 41b83c8c", + "3ece2858 bd94f9b0 4239a623 42b1fb68 -> 410927c2 bfc63983 41aa50ad", + "be940198 3f3ff07c 428d23ce 42b53f58 -> c039815e 40f091ca 41206e28", + "3f4b060a bf37ea32 3f1cd876 41a428fa -> 40fc9a85 c0e4d40c 411f4236", + "3f376520 3ef2a624 419b2bd1 42350fdf -> 4114421f 40c428e5 414ef3ed", + "3f7ef662 3f2eb48e 42c1ed34 42ca22da -> 4002d216 3fb34823 40035a60", + "bf6c6b92 bd45d3d0 42aad96e 42afa1b7 -> bf8d54b8 bd6c8549 3f990920", + "3ed9f93c 3f0ad63a 4247c26f 42bb0df0 -> 4114738f 413d1c3a 41ae5971", + "3d91d0b0 3ec03160 41893184 422d479d -> 3f6e8180 409d2eaa 41515db6", + "beddd444 3daecd70 410e8fe1 41ef44a6 -> c0919073 3f656925 4127fcb6", + "bf24bd91 bedde044 42c19f44 42f3b036 -> c100dfb8 c0ad91f4 414843c8", + "bd829450 bee71b20 41c3809a 4221c82e -> bf02a463 c0673793 41000fc2", + "bf2ef3e7 bf6d4852 42a79ee3 42aa198f -> bed8decb bf131122 3f1eab00", + "bf25ab61 3f000e1c 425eccd7 42bdc2e5 -> c14ad83f 411cca3a 419cb8f3", + "bed0ea42 bf5da984 42a761bf 42d1ed80 -> c08ae1ee c1135b47 412a2f04", + "be2c0a60 be96e360 41b4ffd3 42908104 -> c0856778 c0ea011b 41c6821e", + "bf7e61bf bf0a16f6 4269e295 42a3e80a -> c13aab02 c0caa9cd 413bdafe", + "3e7094f0 bec2ba28 41a7b4e4 4284e83c -> 40ab0082 c10a68c2 41b5f606", + "bebb477c 3dc98d20 41fcf611 428e616f -> c0e79558 3ff93b51 419e47d6", + "bea32a32 3f2d3468 428efca8 42b598e6 -> c044dec5 40d0fbeb 411a70f8", + "bf3036a0 bf14a340 4298d55d 42bae828 -> c0bba187 c09e44bf 41084b2c", + "3efc7c88 bebd459a 42875380 42eabff6 -> 41441e43 c1130436 41c6d8ec", + "3f4ee8ba 3ebf0d3c 429d2429 42ec5dc0 -> 418010bb 40ec8041 419e732e", + "bf43ed09 bf1c7a99 41adab8b 42454501 -> c12909f6 c1070157 415cde77", + "bed63ca4 3f5520ac 42a5274a 42bd36fb -> c0211688 40a04103 40c07d88", + "3f1c0bde bf4713f7 41bedf01 420b8f88 -> 40572c82 c089415b 40b0801e", + "bf3f17f1 3ed61908 42a9b8d8 42c1ae94 -> c08f14a7 40204e17 40bfade0", + "3f6ae0ce bf5cb68e 427b0981 42af2326 -> 4136196d c12b1e05 41467996", + "3eb96e80 3ea54974 42374b16 42454240 -> 3f21da49 3f1044fe 3fdf72a0", + "bf79d51b bebc2d80 41a5ad94 427a6422 -> c1a383f5 c0f65328 41a78d58", + "bf072368 3ebc7530 4289a963 42a825d9 -> c080becd 40338ada 40f3e3b0", + "3d106a00 3ec62078 42ba317f 42ce0722 -> 3e330622 3ff59c01 409ead18", + "3f7e8708 3f6e792e 4145c6c1 42063d7a -> 41289dd9 411dfb33 41299794", + "3e485ac8 bf342532 4229fcee 42b4e8ff -> 4096227b c186fdaf 41bfd510", + "3f7a7b5a bc80b840 4091f600 42002ba6 -> 41571ca0 be5d1643 415bd9cc", + "bf593b94 be882d14 407616e0 4234bc91 -> c18c50bc c0afeb11 41a55b23", + "bf1e7870 bea97458 42a712a8 42dc9aa0 -> c1048c69 c08dbc5b 41561fe0", + "bf4022a1 be9dec1a 423695d6 42aca6ac -> c1743f48 c0c8c139 41a2b782", + "bf57ca0a 3dd6e540 42c2ea59 42c31cbd -> bd29e703 3ba932e0 3d499000", + "3f1ce76a bed77a72 41429625 41e5adc6 -> 40a247a3 c05edc6e 410462b4", + "3cb28100 bf443e14 422d8aa4 4295c9ad -> 3eafc322 c1413a77 417c116c", + "3e948938 bf5b4b2c 42c698d7 430e4b02 -> 40c791d5 c19351ab 41abfa5a", + "3ecf1d34 bf73354f 42555c0d 42806752 -> 400c9a74 c0a51b46 40adca5c", + "3f088c5e 3f19b8f6 4250272b 42a54fea -> 4102a681 41131533 4174f152", + "3f36ef6c bd79aaa0 42add5ae 42f7e184 -> 4153a6aa bf906dbb 419417ac", + "bf2ef7e4 3ef8147c 4193248b 428895e0 -> c1886b98 41416ca7 41c7997a", + "bf32100c bdcf9cc8 428256b6 42a4f005 -> c0c08621 bf60793b 410a653c", + "bf350d06 3f2d44d8 409438c6 418465ec -> c086dbc1 40810fda 40beaf75", + "bf2d6850 bf447af1 428bfd7a 42c0a8b4 -> c10eb4b1 c121b18e 4152ace8", + "3ef29b04 3f2ae38c 42bfcf13 42d2d6ee -> 40104799 404b421d 40983ed8", + "bd4f9f20 bf7ed451 413f7b7c 42469cc1 -> bf748279 c1960d6b 4196bde2", + "bf3e1b5b bed34eda 41210b20 41a3ba36 -> c07727dd c0095c0b 40a6694c", + "3f616006 bf3c0b4e 41df371d 42245354 -> 40b9a52b c09ae524 40d2df16", + "3dc95dc0 bf345e3e 42070986 423e1216 -> 3f2d2782 c09b191f 40dc2240", + "bf433790 3f751ee0 4299c5ab 42edd9dc -> c1803b42 41a102ee 41a82862", + "3ee74120 3f2105bc 419d3c89 421bf3c0 -> 408bb79e 40c291f9 411aaaf7", + "3f12d4c4 3d0ecba0 42accbc8 42d0aaf1 -> 40a498ad 3ea01289 410f7ca4", + "bee7c984 3e9cf444 428228c1 428f638c -> bfbfa773 3f81c734 4053acb0", + "bee1eaac 3e791150 401cd43a 42178132 -> c0fa1a14 4089dd98 418db3ee", + "3f7a2a9a 3f4be6ce 417aea88 42274712 -> 414c5517 41268b3e 415118e0", + "bf045fa8 bf5d68bb 42abbe5f 42b64789 -> bfae552f c011cb8b 402892a0", + "3e198bf0 3f4d73d8 3fbb7514 410ff586 -> 3f109500 4041751d 40710dc7", + "3f2eb558 3f0413ee 3fbaa46e 422de574 -> 41656449 412d6afe 41a81051", + "3f7c6be2 3f7d2276 4297ae99 42befd81 -> 411b08f4 411b7918 411d3ba0", + "3e6d08e0 3f4795e2 42841986 42a36922 -> 3fe7ee4a 40c349ba 40fa7ce0", + "3f5ebf9c 3deb2600 4274a97d 4275a1de -> 3dd81e06 3c6425e9 3df86100", + "bf7fb0ed bf052a03 41e61941 41fd31d9 -> bfb88bae bf403914 3fb8c4c0", + "bdf8c7d8 bf430cb6 42b98619 4303c3e8 -> c0179d1d c16dbcb8 419c036e", + "bdb3cc80 3f76e04a 41b10034 428badf2 -> c0060c25 41b80e6c 41bedbca", + "3ed65770 be8229a0 4283b920 42a09fd6 -> 404195da bfeb1db7 40e735b0", + "be884042 be9eb9ec 42b11fb9 4301065a -> c0ac48d6 c0c8b423 41a1d9f6", + "3f204010 be811206 424e774e 42549bba -> 3ef6133f be463248 3f448d80", + "bf0aedd8 bf674595 41e48d42 425e5793 -> c0ea9625 c1434154 415821e4", + "3e2ea390 3f52a086 411f81e5 4260958a -> 407c0229 4197f85a 41b8b511", + "bf5fca7c 3f22f41e 4288df74 428a32ba -> be944b2a 3e57f5d2 3ea9a300", + "3f2ffc6e bf0dda88 42064a3f 42792c6e -> 411df3cc c0fea296 4165c45e", + "3f74b4be 3e7561b8 42907e07 42aeb674 -> 40e718f0 3fe7bc4b 40f1c368", + "bf6b7567 3ef051c4 3f974b17 4242e5e6 -> c1aee931 41328592 41be2b8d", + "3e4ebc78 be148c08 41c2e30d 421256a8 -> 3f9df181 bf62f9b2 40c39486", + "3efe2594 bcb4b1e0 426c304c 42a57f3b -> 40bc3cf1 be85d5aa 413d9c54", + "bf14ca00 3f0b8f36 427acf0c 42c34cb6 -> c1227e9a 41186a2b 418bca60", + "3e216458 3eb994ec 421ca136 428dd78f -> 40203318 40b835ec 417e1bd0", + "bf61f77a beeb4772 425a123d 42a8948e -> c1523c74 c0dae676 416e2dbe", + "be8f880a 3f6859c4 42b23b4c 42b66dd5 -> be969d62 3f73d13c 3f865120", + "3f0a9422 bf537018 4167c522 42258246 -> 40e8e99b c131af35 415721fb", + "bf5b3136 bf75eb9a 419b2f2b 41fd2fc4 -> c0a7d2b3 c0bc4988 40c40132", + "bf4e1a2f becf79a6 4245ca10 4294b072 -> c1205b1c c0a16c91 41472da8", + "3eab10f8 beaa899a 417dd624 427aa221 -> 40fa2663 c0f96071 41bb2c98", + "be9f636c be68a384 4265bec8 427ee9b8 -> bf7ab715 bf36f7e7 40495780", + "3d4503c0 bf221db2 40e6a380 416628da -> 3e30c26b c01172e1 4065ae34", + "3f781fb0 3f03c248 41edee18 429ab35e -> 41b893c5 4144075b 41be6fb0", + "bf38aae5 bf616300 429a05e0 42f253c2 -> c17ecbb5 c19b7d34 41b09bc4", + "be6a1e04 bf4ca25d 42817f67 42d370fa -> c095e0d8 c18300f4 41a3e326", + "3f6707de bee0839c 42844c7a 42bbc892 -> 41484ab9 c0c2a46d 415df060", + "3d6319c0 bf17140a 41986a74 4206f01d -> 3ed0667e c08aa353 40eaeb8c", + "bf221b24 bd393260 41e1b567 4290323a -> c15e4f8f bf7dfa2d 41af89c0", + "bea9990a bf75074d 42911846 42c7c62a -> c090e5ee c15157ea 415ab790", + "3f2802c4 3f70f748 42c2b0c2 430058b8 -> 4122c477 4169721d 417802b8", + "3f6326f0 bf0497b4 426b1530 428a69b3 -> 4094280d c02cf683 40a6f8d8", + "bf5d6eac bd4d0b70 41fc2fed 42450c74 -> c0f57ef4 bee353d0 410de8fb", + "be2f5f84 3f339042 407736e8 4229e258 -> c05396e2 4158a528 419a6eea", + "3ce65840 bee6b42c 4262ddd6 42afb61a -> 3ee025b1 c0e07f24 41791cbc", + "3f29cdd2 3f5f3566 40ce83b3 4206846c -> 41103465 413d8edf 415967eb" + ] + }, + { + "glsl": "vec4 cultmath_af_frustum_ball(vec2, float, float, float, float)", + "tolerance": "ulp-bounded", + "check": "enclosure and tightness: from the arguments (mx, my, z0, z1, fp, warpVariation) compute in double zm = (z0 + z1) / 2, h = (z1 - z0) / 2, axis = (mx h, my h, h), c0 = (mx zm, my zm, zm), |c0|_1 = |mx zm| + |my zm| + |zm|, R0 = z1 fp + warpVariation, spread = (|mx| + |my| + 1) h and d = |GPU centre - c0|. Enclosure: for z in {z0, (3 z0 + z1) / 4, zm, (z0 + 3 z1) / 4, z1}, eps = clamp((z - zm) / h, -1, 1) and the nine footprint offsets e in {(0, 0), (+-1, 0), (0, +-1), (+-1, +-1) / sqrt(2)}, the point P = ((mx + fp e.x) z, (my + fp e.y) z, z) satisfies |P - (GPU centre + axis eps)| + warpVariation <= GPU radius. Tightness: R0 + d <= GPU radius <= R0 + d + 2^-19 (R0 + |c0|_1 + spread). Report the least slack of each side; only the bounds fail the check.", + "cases": [ + "beb93774 3f479b84 42ac6b83 42b7c51c 3cd0299c 3fe1326d -> c200da26 428add17 42b21850 40830530", + "bf0bca41 be0414a4 413e0a14 421625b2 3d3a525b 3fd1a94a -> c157dcfe c04bf558 41c5a837 40561d5f", + "be2dfb28 bd20c890 41ba4c99 426ca91c 3d11916b 3e13d8fb -> c0e024bb bfcf2405 4224e7b4 400fd0ef", + "bf050336 bf416fba 40abd67f 423f8aa3 3c52ce78 3fddf6e5 -> c15d5d01 c1a0f614 41d50573 40166bbd", + "beab56ce 3ed232e4 41f94c2b 4226e88a 3b49b3a1 3f4c257e -> c1432321 416f64f5 4211c750 3f6d0a86", + "bd0a6c90 bf3768f6 42b275bf 42dd3156 3c779517 3fb28b28 -> c058198c c28f2a26 42c7d38a 40443e5d", + "bf562c1f bf3e9d20 428c538a 42b7d32e 3d2c355b 3da1a394 -> c287981e c2715bb1 42a2135c 407c6148", + "bf06c83c 3d495200 424a1bb8 42507dbb 3c92c895 3e30dd85 -> c1d82d7b 40217301 424d4cba 3f8da95b", + "bedf47b4 3dea4fc0 42b684a6 430cd45f 3cf29681 3fd05290 -> c24a6cc7 41546d0c 42e816b2 40b989f1", + "bc98d6e0 bd7f3840 427dc08f 42e25e16 3d1ff754 3fcd4cb1 -> bfd2e587 c0b0155f 42b09f2f 40c0c750", + "3e0cc908 bf77e4d2 42169549 429336ce 3ad13ce8 3fc48f94 -> 40f4bb0a c25775c7 425e8172 3fd39ed2", + "3f5a8534 3f79c588 41904b3a 4233197f 3d025c9d 3f951e59 -> 41d67679 41f5223f 41fb3f1c 4025c4ef", + "be843dbe 3f1c5522 41e1e573 41e74a7f 3cfb9dde 3fbb346c -> c0ec2afd 418b98a1 41e497f9 40167043", + "be8d0848 becbd22c 420b8a6a 42387c7c 3d3e5397 3f80a1cd -> c1328248 c180fdbf 42220373 40497ac9", + "bce1c4a0 bc881460 425c50bd 426c09e6 3d3fd46a 3fd4fcb5 -> bfc93b16 bf729479 42642d52 408daf50", + "bf5f873d bf539164 420b15f6 429f7baa 3d317033 3fd32033 -> c247f9c7 c23d468c 426506a5 40a3542c", + "3ea78e58 3d100da0 42460993 42acb84e 3baed6c5 3f3d0ca9 -> 41b1db73 4018e8d4 4287de8c 3f998521", + "3f5049d0 3efd37b0 42b84bc4 42d94f6d 3cfe7cf5 3fc93df3 -> 42a360f1 42469edd 42c8cd98 409e54ec", + "3d838380 bd394970 42ab17c6 4302af48 3d01b716 3f7599e8 -> 40de2ab3 c09c80ca 42d83b2b 40a323fd", + "bdbd6ef8 be9f91e6 410c1800 4251b393 3c40169e 3f89cd8e -> c0351716 c1188abb 41f4b993 3fd87c82", + "bf10a6da 3f325cf6 40a72cca 41b4df4e 3cc174eb 3f8c8cba -> c0fba210 411b2368 415eaa80 3fd0e5da", + "bf07755a bd69a9b0 42bbda82 42e98f00 3c8fc0f2 3e447cb1 -> c25efbd1 c0c0522e 42d2b4c1 400f71a2", + "bf06d1d0 3e805228 419d7836 4289439b 3c82b697 3eba4523 -> c1ba0ac3 41311307 4230a1a8 3fbac19b", + "3e2606c8 bf300516 42ae4131 42faaee0 3ce02564 3f6b16fc -> 4189cb76 c29216be 42d47808 408b2388", + "3f740cf6 bf2dbbf3 4235dff5 42b1ad60 3cb6fbf1 3f859f2c -> 428009ca c2364b8c 42864ead 4041d367", + "3ef3c748 3f149d80 41e447ef 4298b35e 3cd52c7a 3f035b85 -> 41c7c1a0 41f38e4e 4251c55a 40200124", + "beba6cf4 3df2ad50 41dd6801 4239343a 3b9deba9 3e98a6a8 -> c1577caa 408c40ee 4213f41d 3f057769", + "3f319c28 bf5983db 414dc001 424b52a8 3d088c06 3c70151e -> 41b0bffd c1d87638 41fec2a8 3fdaca80", + "bedd8910 bd4e8780 402f41d6 423fe7d2 3c709bfe 3e1d2890 -> c12f8c7a bfa3a85a 41cadbef 3f5bace9", + "3eab28f0 bdbf87b8 41f230fb 426f89ea 3c987015 3f1f037d -> 41711e27 c086e82e 42345134 3fde273c", + "bf4f6bbf be871b00 402ba770 41bc66f4 3d135757 3f8f3499 -> c12a091f c05d822d 4151dbe2 3ffba594", + "3f1f6362 bee0e2b4 412ac288 42669665 3d0783f9 3fc22ddd -> 41aa2535 c1701028 4208a384 405b2906", + "3f4fce44 bf1526c0 40c83c1d 42297468 3d3be7d9 3f713bab -> 419ddee7 c1629f21 41c27bec 4038b20b", + "beac9b74 bf10c054 42460bd3 428fb732 3cef6e3e 3ff88909 -> c1a3aa75 c20940bd 4272bd1c 40815852", + "bc11d3c0 3f2aa972 42476339 4270ffbb 3c6e08a8 3e07aa63 -> befadc50 4212ca9a 425c317a 3f81036f", + "3f68b660 3ee73abc 419752d8 428660ac 3c64828b 3fb9fce5 -> 421c8b0f 419b8bae 422c3562 4018fa46", + "bf40e876 be1014ec 420ec5e5 42803a48 3d422fcf 3f4577df -> c2166b2e c0e0b182 42479d3a 4073e827", + "3f4d90e2 be9f8ee6 42a2ca8f 42bd987f 3d4c91b1 3e2b3ae6 -> 428d7b61 c1dba239 42b03187 409cdcde", + "bf78447c 3f63e128 42568b5f 42be5d27 3c56c5e4 3e9d4f0b -> c29052af 42847895 4294d16b 3fc71157", + "3f21c5f4 3e979ec4 429ce991 42a26e8e 3d3ccbb1 3fe2b39b -> 4249cd5b 41bd2308 429fac10 40b07899", + "bf608c53 3e53c6e8 42c63fd9 42ca3a36 3d49b9cc 3fef69da -> c2afa329 41a5a5e2 42c83d08 40db36a3", + "3e0fb508 3f617d72 42ba4080 42dbc6f9 3cf59d8c 3f79b32e -> 4163ed4c 42b2d1c3 42cb03bc 4088a682", + "bf452fd3 3f06b6c4 41d7f3ad 4201cc2d 3cdfbc25 3ef54841 -> c1b725cc 417a3ece 41edc604 3faec4af", + "bf2f9462 bf0b713c 42a6ec81 42d05f15 3c1f42c9 3f3029e8 -> c280b319 c24c6c0b 42bba5cb 3fd9bdaf", + "3d789820 bf40b31a 423a4280 42413e1a 3bf7b902 3fa12268 -> 4038430c c20ed4fa 423dc04d 3fcfe4d6", + "3f6263f8 bf5d122e 424b6081 4294c5cc 3cd00747 3f30782e -> 425d7e05 c25849b2 427a760c 40250638", + "be582edc 3f394c08 420ff1fe 429fd4c8 3d0f04c4 3def73ea -> c143c00e 4227c88d 4267cdc7 403a13c0", + "bf31f656 3eb84d60 420ac126 42146ed4 3d12c17f 3f7ec31d -> c1c7a484 414ec149 420f97fd 4014c967", + "3f44d1c2 3ed40098 4125d51f 4216705b 3d37f058 3fb1acf8 -> 4193890d 411eea9d 41bfe5a3 4044ef5f", + "bf65d64d 3e9699c0 42247ae8 4292f6c6 3cd0fccb 3fdc7fd3 -> c24dc796 4186d642 4265343a 40663c1d", + "bd74fca0 3d913270 429e3976 42bc6a16 3ca17333 3f87d6fa -> c0a5dcfa 40c49aee 42ad51c6 403ac0b4", + "3f75a11c 3e9397c8 41cd91a8 428fd914 3d1271b0 3faef3fb -> 423b54b9 41612022 42433d7e 407c0ff2", + "3f42b820 bee2ce46 42542845 42b7aeb7 3c57ca93 3f8347ba -> 425c6611 c2005bbf 4290e16d 400f119a", + "3f7ee486 bd124650 422348f9 42b2657a 3bf90c98 3f628a92 -> 42817502 c014951a 428204fb 3fc811bc", + "3d5a96a0 be5236ec 4188e47b 42673c12 3cac5bc7 3fdc1d7b -> 3fffe2c4 c0f61549 4215d728 403be78d", + "3f5e3c98 3efd3538 420d5353 4270eccb 3ca7cae3 3ea62cba -> 4225eb0a 41bd0a6a 423f200f 3fc77917", + "beee0830 3f33896a 41e6321d 428b1d0c 3b666cc0 3e13359e -> c1b6dbf1 4209ec22 4244a993 3ec6e4e2", + "bf046bd1 3f6b742c 42c682b5 42f074ba 3d083a35 3f9e929f -> c2631076 42c9de2c 42db7bb8 40a79bab", + "bf2bf828 bee64aa4 421a4c43 42a6ee46 3d05206e 3d5dfbc1 -> c223f63b c1db9180 42741468 4031189e", + "bdbd9fa0 3ba7ea00 421a4d54 426004a2 3d1ed3ec 3ffedf4c -> c08c1d34 3e782546 423d28fb 4085367d", + "be8f0384 bf2b1706 41b437df 4229696b 3d4adaff 3ff79b62 -> c110fb0e c1ad716a 4201c2ad 40810693", + "bead5176 be70d0ec 42ae08df 42d2cfc6 3cd0a993 3f5293b8 -> c202466e c1b50296 42c06c52 40607c09", + "bf0c665e bf22b2e8 4225b59c 4268e255 3d14d4cb 3ec3020d -> c1da9a7a c1fd52b5 42474bf8 401fc6b6", + "bdcd22c8 bf490a04 42929fd3 42c9b82c 3d05380a 3e93ffce -> c10b90e6 c288c75c 42ae2c00 406474af", + "bd4f4c80 bf6d6f8c 411e268e 41d15b56 3b17125d 3fe7e95b -> bf698fa1 c185c20a 4190374e 3fefa3c6", + "3f0de540 3f5693e4 40f4d16f 41b1d604 3cbaecc9 3fe711e8 -> 41047ec7 41485cc4 416f0a60 40140051", + "3f31dc18 3f30a038 42c1e20e 42c46f83 3b942376 3f106dce -> 428796fc 4286a62e 42c328c8 3f811418", + "3eff5dd8 3f17cc1e 429960e7 42fac382 3d09c365 3f6f280e -> 42499235 426fa3b6 42ca1234 40a4d91f", + "3e6843b0 bf439a26 429b52de 42aecf62 3d380f96 3faaf06d -> 4195c32a c27c3ee0 42a51120 40a86d35", + "bef2f434 3f1e73c4 424e9dd9 425ea592 3d3ccbb6 3ff44845 -> c1cbb19c 4204d8dd 4256a1b6 408f2c65", + "3f3d114e bf7d59af 42078db7 42a430ff 3cf234a8 3f96df9c -> 422b51aa c2659136 4267f7da 4066cb73", + "bea83dbc 3f5e626a 42917241 42c49028 3cc87361 3fe195fd -> c1e0c40e 42948cc8 42ab0134 40855bd2", + "bf0ed226 be1a7200 3ffc94e7 41c42310 3d243970 3fda7216 -> c0ec7627 bfffb520 4153ec5e 402c22f9", + "be7a1ea8 3ade6400 4211f380 42881122 3d11f7cc 3fed1d3b -> c14c3da9 3db59918 42510ae2 4088dd6a", + "bf3db0ec 3e28d4b0 4009e2de 41b46c58 3ac8cc13 3fa552b6 -> c1127627 40025aee 4145a8b4 3fa9c03c", + "be685384 3f08eab0 42b8ab99 42ba34a8 3c0139bc 3e2ad761 -> c1a84a1c 42465b22 42b97020 3f66bd35", + "3f634258 3f120b50 41a56a6e 41ec67ce 3d02d71a 3fbcd80c -> 41b25ac7 41653ba8 41c8e91e 401ad6df", + "3ef5e84c bf4b9a14 427bfd9c 42d78476 3d476a0f 3f72a8dd -> 42240642 c287ce51 42aac1a2 40c63854", + "3e988e10 3f1fbb02 3f661fd5 420828f7 3d0fe944 3f57798f -> 40a690d8 412e667f 418bc176 40026a65", + "3e384a60 bf05ec1a 41eafa9d 4247e978 3d2afff4 3c8429d8 -> 40e47e15 c1a60b20 421eb363 4006929b", + "3e0cdec0 3f5b0f66 42b52601 42bd488a 3bcede04 3fb88fb5 -> 414bd6c6 429e7d6d 42b93746 400287f7", + "3f6a7484 bf1694c8 41d76892 42936300 3cd007d9 3d0503d9 -> 42384d7a c1ecbd8f 42493d24 3ff3b7db", + "3e96dd10 bf195004 42ac9176 42eb3b94 3d367591 3f9c1d0e -> 41f05249 c2743901 42cbe685 40ceb170", + "bed04296 bf080894 4273abdb 42b081be 3d1b83f7 3eddeca0 -> c1f2b4d5 c21e8893 42952bd6 40723364", + "3e8d1a6c 3ee52d34 40d403c4 41fe0380 3cae9efb 3feae97a -> 40a93922 41096c97 41998238 4020c6aa", + "3f14af90 3d828ac0 42be21ed 42d1f081 3cf992a1 3fb5d5bf -> 42685d05 40cc0232 42c80937 4093cc68", + "3f29ff00 3ea2cb50 42c391ff 430f3820 3d3ffb09 3fc48b82 -> 42a009ce 4219421d 42f10120 4103f9e5", + "bf16d3c9 3edac618 41f49d2d 42a05464 3a64023b 3f80648b -> c2027db3 41bd46b5 425d7baf 3f89566a", + "3dee2a10 3f6d44bc 420b5b1a 428a010d 3bbfed6a 3f99ae8e -> 40c13673 42407c67 424fae9a 3fcd6e65", + "be4e4680 bf74a6fc 4187d986 424db8e8 3cb60f5c 3f4d2946 -> c0dc7ead c202c233 4208d2d6 3ff8e5f7", + "3e9f5b9c bdb64b28 429c3058 42bf4ad0 3bf041d2 3ff99b99 -> 41d84dd6 c0f76f92 42adbd94 4029b1c3", + "bf5e668b bdcf2e60 426cfa3e 4285bf2e 3d22ceaf 3eed4736 -> c25b2154 c0cc2274 427c3c4d 4047c8ef", + "3e872820 be657680 41ee2faa 428fa72e 3c8c4c20 3db93d41 -> 41568f96 c13622c4 424b3318 3fa90b7f", + "3f52c3d8 3e958a78 428dbbec 42c61f12 3d41694e 3f6c6b4d -> 428be6d1 41c68649 42a9ed7f 40b33e04", + "3e0f2148 bf491c9e 4226d503 426ceab2 3d23a44d 3f961bd3 -> 40e1bc93 c21e9753 4249dfda 4062813c", + "3da93660 bf36bbc6 41ad1d7a 4249ffcf 3c5083bd 3f76ac89 -> 403ebb65 c1cdf8ff 42104746 3fcd9d0d", + "3f795bd6 3f358286 4263a898 4292d8ce 3d3db3f4 3f90e5e5 -> 427de9e2 4238d34c 4282568d 40910c56", + "3f51a9c4 be410e38 4187c269 41b63d02 3b2e055f 3fba37f5 -> 41823834 c06fcf3a 419effb6 3fc1f784", + "bec96e5c 3dee93a0 42884165 4294a60d 3d1f46ed 3fcc1e06 -> c1e02c8a 4104c1bb 428e73b9 408f84bb", + "3ec4f150 bf50fab9 424cae74 42864051 3d200ab7 3d9a884d -> 41b6030b c24122d8 426c978b 402cb23e", + "3defc120 3edc7f90 42823a49 4282d0a0 3cfa9ff7 3fa6c550 -> 40f47a1c 41e0d773 42828574 405375c4", + "bf77c0c5 bf63e8b7 4159405c 42595e82 3d2ef5be 3f792ff0 -> c2037704 c1f1dec9 4207d74c 4052dd5b", + "bd406fa0 3e272250 42900937 429e4a4d 3d0623c2 3f89de1b -> c0634276 41456101 429729c2 406ad29c", + "bed4152a bdbcf5b0 40b633b8 42209a8f 3b204455 3ef62701 -> c117eb82 c0075b3e 41b76106 3f143a2f", + "3f4f9118 be28d8f0 42129553 42664309 3d3540ed 3ea51280 -> 4218c644 c0f88d54 423c6c2e 4037abed", + "3f70ee96 3eac17ec 40ee2627 4207fc22 3d1e1936 3ed7dd77 -> 419bff53 40deda31 41a5c0e7 3fddefda", + "3f54978a 3f4ec216 429ef7d7 42d2b842 3ba6a59a 3fff106b -> 42998055 429549fd 42b8d80c 4021d78a", + "3e3d7048 bcc3ddc0 428ce853 42a0599e 3d04b365 3ec85cb6 -> 415eedd0 bfe67e2e 4296a0f8 403f4a4e", + "bf585aac 3dcc9ea0 422f3eaf 42b8e3d2 3ab7b03a 3f278016 -> c2664f31 40d9d161 42884195 3f48b5e7", + "3f76a5c6 3f3d2902 4285bb50 42aec8ba 3ba73b05 3ec6307d -> 42949f55 4263f6e7 429a4205 3f55544c", + "bedb8556 3e485ba0 4188b7ec 42838d2f 3d2d0d8f 3ebe683b -> c18e1d6f 4101b58a 4225bb2a 4049a980", + "be31c4d8 3f641b6e 42a20eda 42d4fa2d 3c3b596a 3fc01e56 -> c18236cf 42a7161d 42bb8484 402e0136", + "3ef9c38c 3f0d0a28 420dd712 4219cec6 3ba132d9 3f8f17e4 -> 41903912 41a2e213 4213d2ec 3fa750c8", + "3d940900 3ed834f0 429a1b82 42ab23cd 3c8c263c 3f2b704a -> 40bc1409 42095857 42a29fa8 40088f3e", + "3ee50578 3f7eecd8 428077ab 42cb1b6c 3ce157fc 3fe6ed0f -> 421450d5 42a5175b 42a5c98c 409321b1", + "3dadfe70 3c1db300 42a0c00d 42dcde06 3c3f254a 3f808872 -> 4101af8f 3f6b14c8 42becf0a 4012bb0f", + "3f270daa 3e7345e8 4280bed9 42b495ba 3d200d4a 3fb3a44a -> 4249daaf 4192f9e0 429aaa4a 409dd165", + "be6a5c60 3e0bf558 42016474 42856940 3c3ca9d5 3f00504d -> c1355c9a 40d89d8f 42461b7a 3fa27cd6", + "3f724698 3f581b5a 429a9eb3 42a59318 3d3319d5 3f473c1e -> 429783af 4287261d 42a018e6 408cbfea", + "be1d3608 bf44e19b 41ce2f46 42871279 3c4e13ec 3fd8c62a -> c0e534ed c20f85a7 423a9e4a 4022c2d7", + "3db414f0 3eaadb78 421ebfa5 42aa2e42 3d45f9f9 3fb8eb3f -> 40af8c4f 41a68e4b 42798e14 40b1d792", + "3f64d12c bf36e06b 4259a6c9 42c20414 3c406f8f 3f74a197 -> 428757ac c25856aa 42976bbc 4006182f", + "3f54c7e2 bf76dd24 42ba075c 42c0e7d4 3d20c5cd 3fc5c2d5 -> 429d7af9 c2b6b497 42bd7798 40aa98cd", + "bf79d640 bf6d471b 4299dfd9 42c105f3 3d1e6f2d 3fa08e3c -> c2a945eb c2a0c38c 42ad72e6 409f9b48", + "bf3ddbd2 be08fd58 427c5df1 42a7a33a 3c130b8e 3e6e3e20 -> c259e87b c11d3a70 4292e919 3f7c2e02", + "bf028302 3f7b6aca 4213be28 42998901 3d478dc0 3e69e867 -> c1e7de75 425f55e9 42636815 407dfe9a", + "bec73610 bd3826c0 4270d950 42b47a96 3d17b1ea 3e30018e -> c1ea274d c05873c1 4296739f 4060e5a5", + "3ed3e824 bf41e201 4289cde3 42a75379 3c2010f8 3fdb3e62 -> 41fc9330 c2671787 429890ae 4021f1b3", + "bda7b428 be94d55e 42049e0c 426588e4 3d28a965 3eda9dee -> c06d3e43 c1528c5a 42351378 40328ebc", + "bf58408b bf31de96 42b2c2f8 42ffad0c 3d1f9a0c 3f8893c2 -> c2b77e06 c296ec97 42d93802 40c18ddb", + "bf5100b0 3d850e40 42c0ffff 42ebbee9 3cf8dbc0 3ef823f0 -> c2af0481 40ded70c 42d65f74 408219aa", + "bf482807 bf4313b2 408e4c9d 4234f78d 3d1c6285 3f1d3875 -> c19b65f5 c1977467 41c6c121 4015dc72", + "bf288f7f be351938 41edac75 427f7d5a 3d0c4cbf 3fa6de3c -> c1f678ac c10466fc 423b29ca 405f7626", + "3f607de2 3f2b0fd4 41c0127a 427adede 3d43ad72 3fd2bd31 -> 42181ad7 41e7ce7e 422d740e 40949173", + "3f12272c 3f056ec0 4186f404 41c51d24 3d135beb 3fd25d1e -> 413d9499 412d1492 41a60894 4021eaa8", + "bf0f192a 3d94eea0 428bcab1 4292a1e0 3cd0c13b 3ed4adc5 -> c2201acc 40a6a1bf 428f3648 401229d5", + "bf420f7e bee1b144 424e0314 4293a7d7 3c19b606 3e192bd3 -> c23e0387 c1dcfc70 427aa961 3f57a5c1", + "3f20358e 3e92a1e4 426cc691 42719346 3c810e1e 3f52bf6f -> 4215ae1c 4188fedf 426f2cec 3fe32bea", + "3f558ebe be583418 423b7b4e 426be2f9 3d2a2f10 3ff1dbaa -> 423096c1 c132c6da 4253af24 408adff0", + "bed2c3c2 bebab636 42670414 4281b3a2 3d039af7 3fd89e6e -> c1c9e189 c1b2d787 427535ac 4071abf1", + "bf59dfb0 3f685d52 42204016 424ca320 3cf42fd2 3ee834c3 -> c21b45b7 42259976 4236719b 3ffd435e", + "3e5e0820 bf4b7d22 42b49978 42bd7cb6 3ce6a563 3f4f45c3 -> 41a07d7d c293163e 42b90b17 405e8cef", + "bf39b5ec bc81d7a0 4281b09d 42922916 3d44df9c 3f1fd4f6 -> c2481c5a bf8be90c 4289ecda 408462c5", + "bf0962ba 3f05f744 4232fbec 42a51f05 3c44529c 3f262924 -> c208a433 42053d7b 427e9cfb 3fd1baf8", + "be95f41c 3f75100a 428d3419 42d8992e 3d1e9de7 3ec397b1 -> c1d195aa 42ab41f0 42b2e6a4 40926f90", + "3e85b604 3f755b04 427f3a88 42c2756b 3c98da5d 3eedbe20 -> 41a838b4 429a5732 42a10958 4011d6c2", + "3eb2f588 3e3aa250 409c2745 41b8fef6 3caa9f4d 3ed835be -> 409c9cfa 4023546c 416008c7 3f676991", + "3c8ecf40 3f282566 42a67993 42acd3f5 3bae4b07 3f2d5800 -> 3fbd47b2 425edc73 42a9a6c4 3f9185eb", + "bed0383c bf3da053 420ed5ec 4243daa6 3d3a6cb7 3fe15722 -> c189bcee c1fae078 42295849 407f4d4f", + "be4e5098 beba23d4 428f32db 4296c437 3d31dd6f 3f9de0d5 -> c16ce95c c1d5bea7 4292fb89 4090393d", + "3d2c5d20 3f3582ee 4203ace0 42890859 3cbc16b5 3f95407b -> 4008977b 420fd746 424adec9 402f5071", + "3f08c3ee bd7eb730 4141a7dd 42039f68 3cc9078b 3f3d7c49 -> 41405d8a bfb32221 41b4095f 3fc61bdf", + "bef15490 3f3bda8c 420ac448 4239f341 3c4c019a 3f51c683 -> c1990e05 41ee4746 42225bc4 3fb2fe01", + "bc7b7740 bed6d128 42aed5bf 42d39051 3cbbbacd 3fa4f7ec -> bfbdc708 c2221e82 42c13308 406da32f", + "3f2b4bd0 be8997d0 40e02405 423385a2 3c5da86f 3ddbf1c9 -> 418adebb c0df1800 41cf8a23 3f36f42e", + "3d06da80 3e4b4ed8 4290a34a 42bbc33b 3d2cdbf4 3f8f9579 -> 402f195f 4183fdd5 42a63342 40a2aece", + "3f43e786 3f17f95e 42145e81 42463cce 3c73d1bc 3fa2bc16 -> 42049ee3 41cdc33d 422d4da8 4000936f", + "bf102cf4 3f42ef80 42800b61 42e2ae2e 39094534 3db0ed69 -> c247c6ab 42870e3e 42b15cc8 3dcfd452", + "3d8b2700 bf2c4a49 40e533e7 41b3c0b4 3cef34f2 3f93f7ce -> 3f80da8b c11f89fa 416d0dae 3fe7f41f", + "3ef87cb0 bf3136c0 421f26bf 4293fe2d 3d33ba2f 3e3d68f3 -> 41dce3d7 c21d884d 4263918c 405ba5d8", + "3ce3c580 bf7b0fa4 41a078e0 41bb2140 3d38406f 3fe3f7fc -> 3f1aa2eb c1aa72ad 41adcd10 40355445", + "3e96b57c 3f2a9824 401a92d9 4226bf46 3c912cf2 3f66c261 -> 40cfb487 416b1c68 41b06874 3fd1f369", + "bf6bd08e bf0383e8 42c6ccd2 42dea6e6 3bf68221 3f9395db -> c2c21c49 c258835f 42d2b9dc 3ffed12a", + "bf7dac27 3f324812 42124249 42177b42 3c385d33 3e584fd8 -> c2138446 41cf599e 4214dec6 3f23321b", + "3f774be2 bf143d3d 3fdab9ff 410f684e 3d0d205c 3fb099d4 -> 40a4f16f c045bf2d 40aabf8e 3fd821e2", + "bf2c3cc5 bf1a5bfc 42b7ea46 42d3b5da 3d16b21b 3f12ba3c -> c28516c0 c26e8c7b 42c5d010 408ef90b", + "be034ad0 bf3233ae 42c441e4 43079963 3c8b7830 3eab721a -> c16fbd4f c2a2b2be 42e9ba55 4029320f", + "bf75bff5 3efe8cd8 42a21bfc 42e01dcc 3c5b92bf 3f2e18d2 -> c2b96174 424004e9 42c11ce4 400ba786", + "3e2c0b20 3c0ccc80 3f1caacb 42269bf1 3d0b4e44 3d6e93a6 -> 40633a53 3e39f613 41a90e9c 3fbcc968", + "3ee8eeac bcc4de20 429d4599 42a6b4f6 3b8bbd28 3fb061fb -> 42136490 bff924f2 42a1fd48 3fdde59a", + "3f3fb12a 3b946700 4168ec2b 41f9d888 3cfefeb1 3e9e5531 -> 41892514 3dd458d5 41b7274f 3fa405c1", + "3d7fa4e0 3f05f29c 424b55b3 42ac6a3a 3d246d74 3fff2e99 -> 4088d9c2 420f68b2 42890a8a 40ae8a87", + "3e430bc0 befab05e 424bce8b 42735b1c 39cebbd9 3f732845 -> 412a58a1 c1daf15f 425f94d4 3f7952e7", + "bed1a648 be8c4828 42bccabb 430033e7 3ce61b68 3ebc9811 -> c2364bb7 c1f3f51e 42de9944 407e0f27", + "3ebd27d8 bea0f18a 424fc62a 42c0ed2b 3ce1c918 3efd3da0 -> 41db4ff1 c1ba9a34 42946820 4049d249", + "3a624800 bea2133a 40de284c 41ac8155 3d333e42 3fd6401a -> 3c499235 c0906058 41640b68 402784ba", + "bf745fdc 3ebf2c24 414c44af 42363a55 3b9474de 3f81446b -> c1deb352 412e378f 41e94b81 3f9bb2ff", + "3f2a1498 3f40d030 428c6c9b 428d46ae 3c3a0689 3f5947d8 -> 423b27b7 42542b8c 428cd9a4 3fd35245", + "3d24a2a0 3d74d7e0 42410aa5 427f509e 3d23b2b6 3fb86f76 -> 40102ba5 40566889 42602da2 407f7b74", + "3efaf950 bf1a74d0 42bc4885 430e2b5e 3d04fa77 3f0ed3fd -> 4267abe6 c28e93ba 42ec4fa0 40a58fc8", + "3e8aec78 3e9036ec 41a9a808 4208f734 3c2adca7 3f6be777 -> 40f0b8d7 40f9e3ca 41ddcb38 3fa3aa8b", + "3f5b4bba bec9c3c4 42185695 428f62ff 3c109cd0 3f236f01 -> 423c13ac c1ad0a9e 425b8e4a 3fa2bc1a", + "be9b85de 3e8ee0b4 41dbf470 4245ca5e 3d2ce420 3faf2f7c -> c13af8f7 412bc530 4219e24b 405d2d3d", + "bd109500 3eb4a2c8 413d15d4 4184723c 3c982e35 3d27800b -> bf003283 40a02a54 4162fd26 3eb26a63", + "3f07cad4 bf1f20d4 42594f74 428f78f0 3ce07f18 3fad37d6 -> 4205bcfc c21cb89e 427c20aa 40546f6f", + "3f4a8690 bf602cbd 4079e83e 42393125 3a905b89 3f655a30 -> 419edd69 c1afd8cc 41c8cfa9 3f726fb3", + "bed34890 3e12bcb8 4285fc93 429e1bec 3b861989 3fec2d79 -> c1f112fd 41276d4d 42920c40 400acd4a", + "bf69651d 3f322b4e 420e4301 422edf06 3d454267 3ea33366 -> c210909e 41dcb74f 421e9104 401b273a", + "bddc3ac8 bec683a2 420e0944 4241794b 3cc44bfb 3ff8e255 -> c090509f c18215c0 4227c148 40469f7e", + "bf2d3a2e 3da84dc0 41d907fe 4212b708 3ccb4be8 3d11ada8 -> c1acb4e5 4027cc41 41ff3b07 3f722439", + "3f55fb22 3e2464b8 42973494 42dd9210 3d1f2127 3e66d9c7 -> 429bcb7b 416f61eb 42ba6352 4090f2fb", + "bee02208 3a176800 4277efd9 42ce992c 3bf0d5c9 3f0b6234 -> c210b573 3d4381e0 42a5488c 3fa6e3d9", + "bf787730 bf047732 426eb818 428ad92d 3b2b76bd 3f85b4b1 -> c27a9bc5 c2059baa 42811a9c 3f9cf9ff", + "3ea96870 3e519760 41b57ef0 42277610 3d38bb0e 3ef6d3f6 -> 412adead 40d36662 42011ac4 4017b2c2", + "beb61442 3e8dd7c8 4269c050 4276f710 3d0ca7de 3f753c8c -> c1aaf434 41852d19 42705bb0 404501d1", + "bf252c98 bf4b5965 40f73567 42214768 3c84e9bf 3f7fdda2 -> c177fe5b c198a7a6 41c02e15 3fd3ae05", + "bf2a9b02 bea344c8 40e78f6a 419addcc 3d34a200 3f6cf67b -> c10dc96b c087b076 4154c1a6 3fe3c274", + "3da37ce0 3e75b248 41815ab9 41a24ff5 3c47c0c4 3ef29ac7 -> 3fba43f3 408bf6bc 4191d557 3f38a24b", + "be7e57a8 3f6f129a 424477fb 42ad1106 3cd79773 3fe49a5f -> c186c5a8 427d5c9d 4287a682 408207f1", + "3f155802 3d8d4400 4287c1af 42bb3d82 3c9cbada 3e8d58d5 -> 423c6d91 40b23c59 42a17f98 40044fb3", + "3d6b9600 bf11cc60 42a2dc82 42b0ed4e 3d05084b 3dade129 -> 409c58ad c2418493 42a9e4e8 403d535f", + "bf1bb265 be8c64fe 42b7a4ce 42c153d5 3ac3da35 3fb5ba02 -> c265453d c1cebcb1 42bc7c52 3fc83cb3", + "bdc94cf0 be1d647c 42902bff 42dc811f 3d2cd19e 3f104d53 -> c10f60c3 c1603550 42b6568f 40a6e617", + "bf56f174 bf0521ba 421eb82a 4264277a 3d0d7435 3fb81f66 -> c22269eb c1c93128 42416fd2 405a232a", + "3f7a0474 be112800 426724ba 4285ec44 3d303663 3fc9e520 -> 4273aa00 c10d77b5 42797ea1 408ea94f", + "3f247e5a 3e9bb208 42bfd6c4 430e9712 3d123fa1 3faaa130 -> 4299414a 42110eda 42ee8274 40cd95b6", + "3e98440c be4df7b4 42c239c1 42ce1561 3d358feb 3fe1222d -> 41ee1964 c1a1095a 42c82791 40ca72c5", + "beb99154 bee225b6 42bbcfd5 4304f2a0 3c74cb85 3e516812 -> c22470b6 c2486659 42e2da8a 400c3b3b", + "bf27bb19 bf71745a 428a40f6 42b1ed8f 3cb27486 3e90d246 -> c24f296d c2951bbf 429e1742 400e2619", + "3ef7a4c4 bf22f650 42524176 42654531 3bfc177c 3ee1055b -> 41d496e7 c20be50e 425bc354 3f616ce7", + "bd8f4c80 bf799240 42494de5 42633c6b 3c62c745 3f407536 -> c06fe152 c250e3ad 42564528 3fc4e465", + "3f005dd2 3d0efcc0 428dfb41 429367fa 3bbea16b 3f6360f3 -> 42111bac 4021a2c2 4290b19e 3fa8961c", + "bec1cd24 3f6c6f42 420dba06 42751970 3ac5f43e 3f526421 -> c1926bb3 4232a187 424169bb 3f6a1dce", + "3f0f7a6c bf15bf9f 41541e75 4214168e 3a296c33 3fd4ebd3 -> 4161700e c16b4a33 41c91e2b 3fd7fe68", + "bf4dfb39 be1a4e68 41ae9f5e 427add27 3c1e4032 3f882523 -> c2080ce2 c0cbd68c 4229166b 3fd5b2b8", + "bf379ffa 3ddc62f0 41c66934 42797c60 3d05f4b0 3fdfae1f -> c1fa1c69 4096176d 422e587d 407264fa", + "be7c9768 bf1f1476 3ef43138 41e51791 3ad1d366 3fc7ab9d -> c065ce74 c110baeb 4168e856 3fcd8b81", + "be5be394 bf436890 42b6c86d 42cf7beb 3b8fb3b3 3f720e2b -> c1a79bb4 c294f2bb 42c3222c 3fb34979", + "3f1354dc bf0cc34a 4176e0f3 41a9353b 3c817251 3ff1d45b -> 41286c28 c120e9dd 419252da 400e4ec3", + "bf52ec4d 3f7acac0 3e8575fe 4207438a 3add0aa2 3fd9c918 -> c1609c5c 41858893 41884e76 3fe118cc", + "bf2ee662 3f5db0c2 4202a098 42263863 3d3af4ef 3f8f8efc -> c1cacea0 42008824 42146c7e 40412d33", + "bf1b380a 3df341b0 41f745a7 425ba40a 3be85c9f 3fd6c668 -> c1d0230d 40a3182d 422ba36f 4004503b", + "be3e29b4 bf10d4be 418e9d47 420dae8e 3b8a437b 3fe481c2 -> c09e369d c170fed8 41d4fd32 3ff7a51e", + "3f55f656 3f33a040 4207c63c 428e5d8c 3ad11fde 3ff76bbe -> 422fba1e 421386d4 425240aa 4002fd7b", + "be791c44 bf721e34 41bfec95 421c834a 3d07fbe9 3fa9df0d -> c0f5ae21 c1eec8b7 41fc7994 40281427", + "3f3e8972 3ed28f2c 417d5b33 41bbf71b 3ca64147 3e46d602 -> 416a2f2c 4101658a 419d525a 3f2bcadf", + "bdbc2310 bf120188 41a5264e 42193616 3b2a23cf 3f86c828 -> c02d4814 c1867a2e 41ebc93d 3f93849a", + "bf2bd83d bec4c658 41096a7e 41c28ed4 3d10eefa 3e47bfee -> c130b8da c0ca5c18 4183a20a 3f871f8c", + "3f08ce96 3ef1f7e4 42b57928 42c06434 3d1a5860 3eb94d8a -> 4247cb4c 4230afb5 42baeeae 407f2a29", + "3cc3ca00 be30cf34 42b49f8f 42d893a5 3d23ce83 3fe2b5d3 -> 4017e3b6 c1892a58 42c6999a 40c3433c", + "beb6a64e 3f49d7ae 42b33c80 42f3ba30 3cf94452 3f5e73f1 -> c216e313 42a6be04 42d37b58 40927924", + "3eb9d098 befbcd96 42a28596 42aa9372 3d37e2c9 3fde230f -> 41f1c681 c223d186 42a68c84 40b2108d", + "3ebf7308 3e04cbf8 42a033a4 42f16f0c 3cee0fba 3fb8ddb0 -> 42162e6d 415057e4 42c8d158 409e7aea", + "3f43564e bf5f4b9a 42004746 421ca283 3cdbaf96 3fed50c9 -> 41d96621 c1f883c9 420e74e4 4039df47", + "3f0337f0 3f06ba2a 42be815f 43101486 3d0371a7 3ffdb4ac -> 427559cd 427be94b 42ef5536 40d36473", + "bf1ca30d bf4f8880 42266466 42666e38 3ada9ec5 3e9f89c3 -> c1f2cd17 c220d8f2 4246694f 3ed0cd73", + "3e858b3c 3d1ff740 428c126c 42cd20cc 3d01cb61 3f67ef72 -> 41b4135c 4057b436 42ac999c 4084ff7f", + "bedc0b76 3ebf53f0 42a7c1b0 42abbb1d 3cad576b 3ec0d27b -> c211e739 41fdb96c 42a9be66 400c64da", + "be7573ac 3ecc7584 42a6b197 42b8cc7c 3cc42bc0 3fbc4da7 -> c1a8813d 420c5ce9 42afbf0a 406bc565", + "3e064a00 bf23d44c 420fc7d6 42a8cc1c 3d36056f 3fd83a36 -> 40fc836d c21a07ae 4270b007 40ae1474", + "bd1376a0 3ed1d518 42755ad8 42ccd1c0 3d358efe 3fcd9251 -> c03ca5c8 420637b7 42a3bf96 40c4a87c", + "bdc34530 be8849aa 42558e09 42ab491a 3c1653da 3df931a0 -> c0d4196d c1940893 428b080f 3f68576c", + "bf74b2df bf09b6f0 422c9127 42acfce7 3d2f7833 3fd93bdd -> c277d360 c20b7975 4281a2bd 40ace2cc", + "bf52be3e 3eddefc8 4208b3e8 4290c613 3cf49d2c 3ff101a6 -> c22f72a7 41b8c445 42552007 40816c8f", + "bf2f3d32 bf79403b 4276dcce 42bf6bd8 3d1a9e7c 3fd9f566 -> c2578675 c29946a4 429d6d20 40aa1cc1", + "3f180ee4 3ea53900 427c8993 42d3d052 3c0b6434 3fed98d9 -> 4248d033 41da32e2 42a90a8e 40307a26", + "3f2e144a bd9490c8 42a0269c 42a3bb1b 3cf0dd85 3de1d4c4 -> 425c3d16 c0bbf5aa 42a1f0dc 40211e05", + "3e9e9980 3ef0c88c 424f4fe3 42b4e82c 3c037a81 3efcc983 -> 41b04b93 4205d304 428e480f 3f9c20d3", + "bec7f9b4 3ecb519c 4201ae0c 4267bf62 3d34690f 3eaff0f6 -> c18d2a4d 418f867f 4234b6b7 40395175", + "3be80400 3edf0cf0 412f0ec8 42321761 3d080d7f 3fdf83f3 -> 3e491201 41414d0c 41dddb13 404e68d6", + "bf20a50b 3f0b83e4 42751fa8 42bec52a 3d2dbcb4 3f8540b8 -> c2449f20 422ac293 429caa7f 40a2c9b3", + "be0fa388 bf2aacb0 40cb1b88 41fbe91e 3abdd2db 3f4918f9 -> c029d5ab c149cd0e 41975800 3f54c93e", + "bf676a91 bf64f96b 41509c09 41dabf4c 3bb5fc5b 3fdc7b32 -> c19203bf c190794d 41a186a8 3fefedc1", + "be1c6614 be6af620 42aae719 42cfe809 3c9a29b2 3de5185e -> c1676d45 c1add6c4 42bd6791 40045e7c", + "3d645f60 beda5ebe 429c2dfc 42ca1b2b 3c90a95a 3f66be2e -> 409fcf52 c218cf72 42b32494 402be6e8", + "3f23a1d0 bf29c393 42a00e6f 42fccd76 3d2e0315 3fd09267 -> 4283f27c c288e449 42ce6df2 40dffd9f" + ] + }, + { + "glsl": "vec2 cultmath_iv_snoise_ball(vec2, float)", + "tolerance": "ulp-bounded", + "cases": [ + "c0b75188 423d2522 3fe31de1 -> bf800000 3f800000", + "4246e964 c17bd730 3ffdfea1 -> bf800000 3f800000", + "c1fc8975 41f194d4 3fabf475 -> bf800000 3f800000", + "422c4d44 c071c1a0 3fd3f608 -> bf800000 3f800000", + "c23713bf c1e1f045 3f9bd561 -> bf800000 3f800000", + "c1ca302c c1c9a6d7 3f0eb662 -> bf800000 3f800000", + "c240811b 41fd106c 3f4b9b12 -> bf800000 3f800000", + "40b4c4b8 c20a57d5 3f8fbe05 -> bf800000 3f800000", + "41bfdf38 423c4866 3fd09ad4 -> bf800000 3f800000", + "4132fb74 41928344 3fdbd120 -> bf800000 3f800000", + "41144c7c c217c5e0 3ed0b522 -> bf800000 3f800000", + "c1942dc5 42094cd8 3e8c25d3 -> bf800000 3f800000", + "423da796 c0ebaa98 3db0bfe8 -> bf7ce0fc 3ed394b1", + "4210e1d4 c1fb88da 3f92c7ce -> bf800000 3f800000", + "c138f714 411b6590 3fd319a6 -> bf800000 3f800000", + "42351de6 c1050a60 3e89f897 -> bf800000 3f800000", + "4208db38 40e77400 3feba820 -> bf800000 3f800000", + "400aeda0 42237bba 3d894c73 -> beee05c0 3f1f9a22", + "c104362c 421a7d38 3fd19d99 -> bf800000 3f800000", + "c18b5312 c1c20c75 3f3e91df -> bf800000 3f800000", + "c2010afc 40e7e338 3e744721 -> bf800000 3f800000", + "41fa5b44 c1818bb6 3f2f18ef -> bf800000 3f800000", + "4147268c 41b60648 3f3fd0b5 -> bf800000 3f800000", + "c0329330 c1b57f58 3fd258fd -> bf800000 3f800000", + "c1d3705e c20962ee 3fa184d5 -> bf800000 3f800000", + "4201aff6 c20329a4 3f8ea755 -> bf800000 3f800000", + "41bd66b0 41e5a36c 3f8dd837 -> bf800000 3f800000", + "41905f4c 41d7c74c 3e63da68 -> bf800000 3f800000", + "422f3ce6 c22b9bbc 3ed958f6 -> bf800000 3f800000", + "c203d8fc 421ef6a8 3ff5012b -> bf800000 3f800000", + "c191d4ce 416d4478 3ead6501 -> bf800000 3f800000", + "41889fd8 4193d99c 3f9d01e8 -> bf800000 3f800000", + "c0923388 40e4a4d0 3f04d36f -> bf800000 3f800000", + "bfdb55c0 c189577a 3fb08164 -> bf800000 3f800000", + "41cffb88 c14fee54 3f195ec9 -> bf800000 3f800000", + "c2206f47 c1ab31ee 3fc0969f -> bf800000 3f800000", + "c11c56bc 411509fc 3fb6da8f -> bf800000 3f800000", + "c1bea39f 410e8c74 3f70f9d2 -> bf800000 3f800000", + "41fe6da0 be9d5180 3fcd86f3 -> bf800000 3f800000", + "c244ca5b 4215f0bc 3f08efe0 -> bf800000 3f800000", + "c21414a0 c092eb00 3fd48870 -> bf800000 3f800000", + "c1045274 c20a3ecc 3fa0018c -> bf800000 3f800000", + "41744400 40cfbf28 3f71a813 -> bf800000 3f800000", + "c240f288 c1502664 3f99d0c5 -> bf800000 3f800000", + "c1a1a0f2 c1c4e3f1 3f6b017e -> bf800000 3f800000", + "4211d0ba 422831e2 3f4cca33 -> bf800000 3f800000", + "41f949ec 4207ac2c 3fc8fb09 -> bf800000 3f800000", + "c0810210 41d67964 3f1bf57a -> bf800000 3f800000", + "c1cdae76 c2319898 3df46831 -> bf800000 3f62119a", + "419ed1b0 420d191c 3ec0cf6a -> bf800000 3f800000", + "41f84cd4 c0abe0b0 3fb25e57 -> bf800000 3f800000", + "4244c95e c16b1838 3fdfd1f8 -> bf800000 3f800000", + "420fadc8 c17bb67c 3edd3bd3 -> bf800000 3f800000", + "418b7dec c0ac0238 3fb8a231 -> bf800000 3f800000", + "c1afa335 c1388c9c 3ee4ffa8 -> bf800000 3f800000", + "c21a9ae5 42397eaa 3f2bcbae -> bf800000 3f800000", + "c2435565 42356616 3f939f77 -> bf800000 3f800000", + "c11b19e8 c1f2dea0 3d9413cd -> be5b2b20 3f75b1a4", + "bfe4eb60 41881744 3fafa7af -> bf800000 3f800000", + "40258540 c2023d62 3e4a6634 -> bf800000 3f800000", + "40d209a0 4235a828 3fb65a94 -> bf800000 3f800000", + "c1acee5c c1ea31e3 3fec74bd -> bf800000 3f800000", + "42219de2 c19b33af 3fe69f77 -> bf800000 3f800000", + "41072f40 c245dd0e 3f93f90b -> bf800000 3f800000", + "c1fcc7f6 c2111a02 3f495671 -> bf800000 3f800000", + "c22e71e8 c2004ea8 3f447a22 -> bf800000 3f800000", + "c22797f0 c2143c56 3e3de781 -> bf800000 3f800000", + "41b331d8 41957c7c 3fe6386f -> bf800000 3f800000", + "41324fe0 c1144c70 3f0020cf -> bf800000 3f800000", + "4180bea0 3f7c2e40 3faee059 -> bf800000 3f800000", + "c1f96894 c1855516 3eacbe51 -> bf800000 3f800000", + "c2191696 c21c823d 3f7c167c -> bf800000 3f800000", + "c2446fa6 4246a594 3eb6143f -> bf800000 3f800000", + "c1a04cc9 c1e8718f 3fa86cd2 -> bf800000 3f800000", + "c0d1f7f8 c0e909b0 3ff2bc08 -> bf800000 3f800000", + "424773f8 c1e0c4ff 3df4464a -> befb28db 3f800000", + "41646070 c102d5c8 3f7b7637 -> bf800000 3f800000", + "c19df3d7 c0b07de8 3dba3bf0 -> bf3a8330 3f3f673a", + "c2458ef1 4208a848 3f090982 -> bf800000 3f800000", + "41fa5788 4189e9b4 3fcccf9f -> bf800000 3f800000", + "418afe84 c1cd82da 3fcbd6f5 -> bf800000 3f800000", + "c1e1b473 c125ff98 3fac1362 -> bf800000 3f800000", + "c110c100 c22d4d65 3d6d98fb -> bf1e5894 3ea573f7", + "422054e2 c2320242 3ebb6a8b -> bf800000 3f800000", + "c1eb2be4 41e11ce8 3feb8577 -> bf800000 3f800000", + "c0d17e20 c1a8502c 3f052593 -> bf800000 3f800000", + "42346fe0 41347ffc 3ec135a0 -> bf800000 3f800000", + "c1f76eef c240b8cc 3e3c92d6 -> bf800000 3f800000", + "c1f657f6 421a45a2 3fe3779e -> bf800000 3f800000", + "c216af32 c21b87a4 3f08ef5e -> bf800000 3f800000", + "c1dc24b0 c1f336d7 3f262d36 -> bf800000 3f800000", + "c1441180 41496960 3ff762cc -> bf800000 3f800000", + "4210a312 c1e2c98b 3d80a6f4 -> bf57e016 3e34c4b6", + "c1b314b1 410e90e0 3e9c620b -> bf800000 3f800000", + "c23bbd89 41c18670 3f3a25a4 -> bf800000 3f800000", + "c19962f5 c22163fa 3fa9e339 -> bf800000 3f800000", + "c215fd7c 40e50690 3f8d60d6 -> bf800000 3f800000", + "c244f58b 4210bbe6 3df5f5cc -> bf019788 3f800000", + "4132ab1c bfd928a0 3ee6b5e6 -> bf800000 3f800000", + "41a39e88 c23dae3e 3f82c4d2 -> bf800000 3f800000", + "41e2d098 c1ab021b 3ff4a19e -> bf800000 3f800000", + "41d12580 418cce74 3fdbebc4 -> bf800000 3f800000", + "420f5b0c 42036e6e 3e9fac69 -> bf800000 3f800000", + "423573cc 40e96950 3e16a495 -> bf5bf7b0 3f800000", + "c1d8fe78 418bf470 3d4638aa -> bf800000 be6cc936", + "c2399b82 c207f9ee 3f0da52a -> bf800000 3f800000", + "41b6b2b8 4219857e 3fca0d0f -> bf800000 3f800000", + "c1dbf93a 42474e48 3f8a32a1 -> bf800000 3f800000", + "41ac0a38 c1e2399c 3f2b40fe -> bf800000 3f800000", + "c1bc4843 41d4a1b0 3e92b70b -> bf800000 3f800000", + "41c1228c c2092c32 3ec3b585 -> bf800000 3f800000", + "40024370 bf3ca900 3f1b3ead -> bf800000 3f800000", + "c1f29eb5 c1b8a092 3e07ea92 -> be474494 3f800000", + "3ed9d280 c0a06438 3f096018 -> bf800000 3f800000", + "3f4b6c80 c2037147 3fffa63a -> bf800000 3f800000", + "c006c520 41df1364 3ebfcb06 -> bf800000 3f800000", + "c239a6ed 40f23248 3f25895f -> bf800000 3f800000", + "42337362 423e75e8 3e91d785 -> bf800000 3f800000", + "41b121bc 40bb2748 3f13a025 -> bf800000 3f800000", + "c1067524 c235725d 3f6b09e9 -> bf800000 3f800000", + "401f07a0 4115fcb4 3e018d22 -> be98b714 3f800000", + "c242ef82 c1e63af8 3f55138a -> bf800000 3f800000", + "4218f336 41a6b918 3ebb1b74 -> bf800000 3f800000", + "41b0ad04 4207b3d8 3fe411e3 -> bf800000 3f800000", + "c1e7b003 c1dcb492 3f5e50a5 -> bf800000 3f800000", + "4007c020 41adab60 3fd3b379 -> bf800000 3f800000", + "40831f00 c1d31d22 3f60975e -> bf800000 3f800000", + "c20d18fe c1d8fa68 3ec3148d -> bf800000 3f800000", + "41ffe5e8 41ea3bc8 3eb9ada8 -> bf800000 3f800000", + "c23482c9 41ef26b4 3fef2649 -> bf800000 3f800000", + "c2420b64 41e09bdc 3ee49425 -> bf800000 3f800000", + "412b5fcc 423fc32c 3fdb306d -> bf800000 3f800000", + "c1ebdff8 bef16600 3ff58e45 -> bf800000 3f800000", + "41046930 411a81dc 3e1136fd -> bf800000 3f31b00e", + "c2462b2f c193fae0 3fbc8f52 -> bf800000 3f800000", + "c0d92d90 41be96c8 3fc08d5e -> bf800000 3f800000", + "41a67478 c1efd64c 3fd76b19 -> bf800000 3f800000", + "c078ee70 4215fed8 3fcf746d -> bf800000 3f800000", + "c23fd577 c143da20 3f3b0644 -> bf800000 3f800000", + "42151304 c1f5c4ec 3f56e44a -> bf800000 3f800000", + "c1d40914 42057600 3fe8d5a6 -> bf800000 3f800000", + "c1c97730 41a4c3cc 3df1864f -> bf800000 3e991814", + "41866890 4125402c 3efaf8de -> bf800000 3f800000", + "40f8edc8 c1932802 3ee385f6 -> bf800000 3f800000", + "4213135c 42374994 3fa3c4c1 -> bf800000 3f800000", + "c1534e94 c20eedce 3fe9ee7f -> bf800000 3f800000", + "c0943f10 c1339d64 3de16564 -> bed84297 3f800000", + "c0e39258 c17ef418 3f9953df -> bf800000 3f800000", + "c222e6c4 40ac1c18 3fe7e3c9 -> bf800000 3f800000", + "c10be87c 410ab5ac 3f9a1d7f -> bf800000 3f800000", + "bf620540 c1860600 3f1b0e90 -> bf800000 3f800000", + "407e7930 c17384a4 3fc35626 -> bf800000 3f800000", + "c2079c0c 41cd20d0 3f9bf8c2 -> bf800000 3f800000", + "418e5910 41d34c10 3e8aa2ec -> bf800000 3f800000", + "3f3b6940 c213e37c 3fe25b62 -> bf800000 3f800000", + "413f5ae8 411576f0 3feab42e -> bf800000 3f800000", + "c203b4c4 4173f8b8 3ef0f1c8 -> bf800000 3f800000", + "c09c1ed0 c1019468 3fce1dff -> bf800000 3f800000", + "c1583778 41f4db84 3f7c8b7e -> bf800000 3f800000", + "c1976a9e c209d4d4 3e9c4208 -> bf800000 3f800000", + "c20ae92e c1c3a334 3d0beaf9 -> bf1a297a bd432a58", + "4164a360 c2255a32 3ffbccb3 -> bf800000 3f800000", + "4227b8d2 c189a870 3f9c5113 -> bf800000 3f800000", + "4216a680 c1ea6f31 3f45ed0f -> bf800000 3f800000", + "c19c40dc 41dc972c 3f9fb1d9 -> bf800000 3f800000", + "4227d724 4228a010 3ed7d434 -> bf800000 3f800000", + "41abab4c c02bd7f0 3ff5dc09 -> bf800000 3f800000", + "424399d2 c1ce8ab6 3fef2aa1 -> bf800000 3f800000", + "c154f64c c238043b 3fe9db38 -> bf800000 3f800000", + "be829c80 c2020a8e 3f8abdca -> bf800000 3f800000", + "c1b8111e 420c1942 3f99e778 -> bf800000 3f800000", + "c089edc0 40215b70 3f824830 -> bf800000 3f800000", + "c218a745 4211c4ce 3db4c976 -> bf800000 3e018a1c", + "422039dc 41a03bd4 3f081e29 -> bf800000 3f800000", + "c19aeaee c1a4b68c 3f644035 -> bf800000 3f800000", + "420b757e c20d636c 3fa4b955 -> bf800000 3f800000", + "c1fedb76 c0af2810 3fe272f3 -> bf800000 3f800000", + "c1ade6e1 4119d42c 3faeb3cc -> bf800000 3f800000", + "c1f68ec9 c2068301 3ce7669a -> beaa1041 3e017172", + "c2270c9e c1b33329 3dc112e5 -> bf087695 3f7f54f7", + "c22bf319 c0fa0fa0 3e9d2311 -> bf800000 3f800000", + "c22c865a 412abc40 3f36cce7 -> bf800000 3f800000", + "41e43914 420b1d4c 3f007613 -> bf800000 3f800000", + "c21a0503 41f0fcd4 3debfb38 -> bf47f35e 3f800000", + "c205503f 41fb3794 3fc5e16c -> bf800000 3f800000", + "419be7a4 41921e94 3ff29f1a -> bf800000 3f800000", + "bf90d100 3fe99de0 3fe4bb4c -> bf800000 3f800000", + "c1b7da70 c1932efc 3fbad21c -> bf800000 3f800000", + "417b8d90 4242eae2 3f024ac2 -> bf800000 3f800000", + "42277c94 c1bbeca9 3f393f3e -> bf800000 3f800000", + "befe0f80 c2399aa7 3f96b04b -> bf800000 3f800000", + "4246ccac 42042cf4 3f0ebe78 -> bf800000 3f800000", + "c21b0461 c1d1de3b 3fb003a0 -> bf800000 3f800000", + "c224795f c22ea70c 3f1922cd -> bf800000 3f800000", + "c12b9a3c c22d68b8 3fa4eac1 -> bf800000 3f800000", + "4202f270 c219a332 3fc84044 -> bf800000 3f800000", + "4223c412 be9d9b80 3f7670fe -> bf800000 3f800000", + "c22da22a 4115e7a8 3fc203c5 -> bf800000 3f800000", + "c20ab528 41eff43c 3e59aa9b -> bf800000 3f800000", + "c0f520f8 c10da6f8 3f9e9e82 -> bf800000 3f800000", + "c21ef512 42129ff8 3f9538f4 -> bf800000 3f800000", + "421b9d28 413fdc70 3fc3eec6 -> bf800000 3f800000", + "c04831f0 c2041cec 3ffbcc98 -> bf800000 3f800000", + "bf0ed140 4103e0d0 3f63f95e -> bf800000 3f800000", + "c1bfbf7d c12034a8 3f690a19 -> bf800000 3f800000", + "41177fa4 4216f958 3effea9d -> bf800000 3f800000", + "c225c220 4203f0f6 3fdf1777 -> bf800000 3f800000", + "c03350a0 c244477e 3fc4ae91 -> bf800000 3f800000", + "42182b02 c0e2cfa8 3f629994 -> bf800000 3f800000", + "420444e6 420981d4 3d7dcc44 -> 3d5939d0 3f800000", + "c202f5ad 41973f7c 3fd38761 -> bf800000 3f800000", + "c22cce85 42427bb4 3ec9f0eb -> bf800000 3f800000", + "41acd754 3ebe2480 3f64c854 -> bf800000 3f800000", + "42178384 c16229c8 3ff23b30 -> bf800000 3f800000", + "41eed4d8 c10802a4 3fea1f56 -> bf800000 3f800000", + "c23990b3 41f47b28 3ec6e230 -> bf800000 3f800000", + "c19db4fb 40bb5778 3fec2bff -> bf800000 3f800000", + "420e8450 c181c3fe 3f94f262 -> bf800000 3f800000", + "41c66c38 4150ffe0 3fdb7a17 -> bf800000 3f800000", + "41fb9d78 c2448dda 3ec3c8f3 -> bf800000 3f800000", + "c18da4bc 415cdff0 3f270f0c -> bf800000 3f800000", + "42353c00 c18e8b30 3fe4b6fd -> bf800000 3f800000", + "413e0f1c c1f62274 3f53f21f -> bf800000 3f800000", + "c1c18a04 42177440 3f3f8170 -> bf800000 3f800000", + "41f2621c c1ea8b48 3f716472 -> bf800000 3f800000", + "422a145a 418ac110 3f9831e1 -> bf800000 3f800000", + "420d784e 4209bbb4 3f9fab5a -> bf800000 3f800000", + "c1c9374c 4031f620 3f1237ad -> bf800000 3f800000", + "c21ccdf2 c192f3fc 3f3403c7 -> bf800000 3f800000", + "c1cc5556 c1900941 3f36393f -> bf800000 3f800000", + "c1805cf4 42209cee 3f7d9eb3 -> bf800000 3f800000", + "c1c361c9 40b2af10 3fb8827b -> bf800000 3f800000", + "414061ec 420a2e6c 3e3fbf48 -> bf800000 3f800000", + "c23f8be0 c10451e4 3f9d0be8 -> bf800000 3f800000", + "41e01fe4 c236157c 3fae1902 -> bf800000 3f800000", + "409c1dc8 c232acbc 3e868fb8 -> bf800000 3f800000", + "c21bd61a 420121c2 3ff50b48 -> bf800000 3f800000", + "c1bda36b 4238566e 3ea8a6fc -> bf800000 3f800000", + "40d744c8 c22a36b8 3e62aee3 -> bf800000 3f800000", + "c237b5a8 420a09f0 3c44e6e9 -> 3f19dd46 3f4bcf4e", + "4086b5c8 c149463c 3fdf5848 -> bf800000 3f800000", + "c230c498 42276978 3ffa2427 -> bf800000 3f800000", + "421772ec 40970f98 3fa6b224 -> bf800000 3f800000", + "41c87ba8 c19c0843 3fd3fcc0 -> bf800000 3f800000", + "3de87800 4205042e 3f5ac99d -> bf800000 3f800000", + "c243e657 c04ab970 3f776fcb -> bf800000 3f800000", + "412880d8 40bd72b0 3f06d7a6 -> bf800000 3f800000", + "c1dfc3d2 c2125490 3f7ea824 -> bf800000 3f800000", + "c1228ed0 c2295cc4 3ff0d2e7 -> bf800000 3f800000", + "c187589c 4196cd04 3f769886 -> bf800000 3f800000", + "413093e0 40fe7120 3ae06ada -> bee3835e bed5482c", + "c224ef78 c0062990 3db27bb7 -> bf800000 3e7ffa0c", + "41c0b6d0 41b1a910 3e81bce1 -> bf800000 3f800000", + "41a23ebc 41882198 3f81a063 -> bf800000 3f800000", + "c0000d30 c1604fbc 3f84a2ee -> bf800000 3f800000", + "c21452ae 41d1741c 3e83d42d -> bf800000 3f800000" + ] } ] } From 51a4034c66769a35013b0a8c00a899e040821063 Mon Sep 17 00:00:00 2001 From: Meta Vens Date: Sun, 4 Oct 2026 10:47:52 +0200 Subject: [PATCH 25/31] Affine gate: delete the detail term (saturated at [-0.5, 0.5]); design.md (c) rows re-measured [gate-detail-term-dead] The affine gate's best-case detail ball decided no probe on the gate test's slices and one of the deep well's 8074 oracle-empty cells (11 of 11096 in c-band); skipping it saves the probes: (c) affine 3.57x -> 3.62x and 3.46x -> 3.54x, c-band 3.95x -> 3.96x and 3.54x -> 3.60x. The interval gate keeps its detail ball. Saturating it there (measured, then reverted) takes row (b), inside the fog, from a cull fraction of 0.663 to 0.585 and a cost ratio of 1.16x to 0.98x, so the term is live; the finding's premise holds for the affine gate only. Co-Authored-By: Claude Sonnet 5.5 --- packages/cultmath/docs/design.md | 19 +++++++++++-------- .../tests/CultMath.Tests/NoiseBoundTests.cs | 7 ++++++- 2 files changed, 17 insertions(+), 9 deletions(-) diff --git a/packages/cultmath/docs/design.md b/packages/cultmath/docs/design.md index a49884b3..32cc03f0 100644 --- a/packages/cultmath/docs/design.md +++ b/packages/cultmath/docs/design.md @@ -572,7 +572,10 @@ term is identically 0 and no `snoise` is evaluated. Where the envelope already proves the slice empty with the noise at its full range, the ball is not evaluated either. A probe is worth attempting only when the best result any centre value could give would prove the slice, and a ball with `L r >= 2` is -`[-1, 1]` whatever `snoise(c)` is, so it costs nothing. CultMath owns none of the +`[-1, 1]` whatever `snoise(c)` is, so it costs nothing. The affine test gate takes the detail octave at its full range: +skipping its best-case ball loses 1 of the deep well's 8074 oracle-empty cells (11 of 11096 in (c-band)) and none of the +gate test's slices, and saves the probes. The interval gate keeps its detail ball, which decides in row (b): saturating it there takes the fog field's +cull fraction from 0.663 to 0.585 and its cost ratio from 1.16x to 0.98x. CultMath owns none of the envelope: the interval files have no envelope function, and each consumer composes its own. `NoiseBoundTests.EnvelopeBoundEnclosesDensity` checks that composition, the one the site and Aetheria use, over 2,000 seeded slices x 64 warped points @@ -631,19 +634,19 @@ printed. The flow is not counted. At `N = 8`, in Aetheria's units on its `y = F - h(xz) / 2`, so it looks across the bowl from inside it. Both rows run on the same seeds. With no warp, the centred interval takes 3.20x (oracle ceiling 5.24x, efficiency 0.611, cull fraction 0.988, probe overhead 0.940) - and the affine bound 3.57x (efficiency 0.682, cull 0.993, overhead 0.905). With + and the affine bound 3.62x (efficiency 0.691, cull 0.993, overhead 0.922). With the warp `D = 60`: 3.12x (ceiling 5.31x, efficiency 0.588, cull 0.987, overhead - 0.929) against 3.46x (efficiency 0.651, cull 0.993, overhead 0.880), so affine - is 1.12x and 1.11x cheaper. The range mask (the oracle cells plus every cell + 0.929) against 3.54x (efficiency 0.666, cull 0.993, overhead 0.904), so affine + is 1.13x cheaper at both. The range mask (the oracle cells plus every cell where any of 5 x 5 slopes x 9 depths has density above zero) gives a range ceiling of 0.859 (0.848 with the warp), the cost a bound that proves exactly the range-empty cells would reach against the oracle's; the efficiencies - against it are 0.794 and 0.712 (warp 0), 0.768 and 0.694 (warp `D`). + against it are 0.805 and 0.712 (warp 0), 0.785 and 0.694 (warp `D`). - (c-band) The shallow wells, four mass-derived wells in the zone bowl with the camera gazing across, whose geometry is band-dominated, printed only. Centred interval 3.01x (efficiency 0.414, cull 0.960, overhead 0.958), affine - 3.95x (0.544, 0.980, 0.901); with the warp 2.75x (0.383, 0.954, 0.944) against - 3.54x (0.494, 0.976, 0.865). + 3.96x (0.545, 0.979, 0.923); with the warp 2.75x (0.383, 0.954, 0.944) against + 3.60x (0.503, 0.975, 0.893). - (d) r1's uniform slab, which has no envelope: 1.38x with the warp 0.05, and 1.67x with no warp. - (e) The void at the shipped point (ruling `operator-shipped-void`): `Rh = 198`, @@ -699,7 +702,7 @@ The contracts, at `N = 8`, the lower of warp 0 and `D`: converged march's by at most twice the fixed-step march's error; and it takes at most half the reference's steps per pixel. -This run meets every contract: (a) 16384x, (c) affine 3.57x and 3.46x against +This run meets every contract: (a) 16384x, (c) affine 3.62x and 3.54x against the centred interval's 3.20x and 3.12x, with a cull fraction of 0.987 at the lowest, (e) 2.69x fewer steps and a depth error 1.42x the fixed-step march's. The fields, the grids, the 0.2 envelope weight and the 2.0 gradient weight are diff --git a/packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs b/packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs index d8b579f5..4075e953 100644 --- a/packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs +++ b/packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs @@ -1796,6 +1796,11 @@ public sealed class NoiseBoundTests return low ? new float2(-1.0f, min(-1.0f + e, 1.0f)) : new float2(max(1.0f - e, -1.0f), 1.0f); } + // The affine gate takes the detail octave at its full range: its best-case ball never decided a probe + // on the gate test's slices or the deep wells, and a gate that skips it proves the same slices with + // fewer probes. (The interval gate keeps its detail ball: inside the fog, row (b), it decides.) + private static readonly float2 DetailRange = new(-0.5f, 0.5f); + private static float2 Clip(float2 range) => new(max(range.x, -1.0f), min(range.y, 1.0f)); /// @@ -1818,7 +1823,7 @@ public sealed class NoiseBoundTests foreach (var low in new[] { true, false }) { var coverage = Noise.CoverageBound(BestForm(slice.Coverage0.Reach, low), BestForm(slice.Coverage1.Reach, low), tile.Time); - provable |= Compose(s, fade, iv_add(coverage, iv_scale(BestForm(slice.Detail.Reach, low), 0.5f))).y <= 0.0f; + provable |= Compose(s, fade, iv_add(coverage, DetailRange)).y <= 0.0f; } if (!provable) From 799029bebe797ea2a86c9a4fe0ba1e2b9b56c5b6 Mon Sep 17 00:00:00 2001 From: Meta Vens Date: Sun, 4 Oct 2026 11:09:25 +0200 Subject: [PATCH 26/31] af_range finds a NaN endpoint by its bit pattern, which dxc's fast-math compare cannot fold [af-range-nan-guard-folded-by-dxc] dxc marks compares fast, so `lo == lo ? lo : -inf` was deleted under default flags and an overflowed form ranged to NaN on HLSL targets. The guard now tests (asuint(lo) & 0x7fffffff) > 0x7f800000, in HLSL, the Unity copy, math.Affine.cs and the regenerated CultMath.glsl. dxc 1.9.0.5402 -T ps_6_0 default flags: before, no fcmp/select/and in the af_range body; after, and/icmp ugt/select to +-inf for both endpoints (also under -Gis). AffineRangeMirrorMatchesCSharpBitForBitAtTheFloatExtremes pins the mirror at zero to the float maximum, infinities and NaN components. Co-Authored-By: Claude Sonnet 5.5 --- packages/cultmath/docs/design.md | 3 +- .../cultmath/shaders/CultMath.Affine.hlsl | 6 ++-- packages/cultmath/shaders/CultMath.glsl | 6 ++-- packages/cultmath/src/CultMath/math.Affine.cs | 6 ++-- .../HlslSourceCompatibilityTests.cs | 30 +++++++++++++++++++ .../Shaders/CultMath.Affine.hlsl | 6 ++-- 6 files changed, 48 insertions(+), 9 deletions(-) diff --git a/packages/cultmath/docs/design.md b/packages/cultmath/docs/design.md index 32cc03f0..081265d0 100644 --- a/packages/cultmath/docs/design.md +++ b/packages/cultmath/docs/design.md @@ -743,7 +743,8 @@ add `2^-126`, the most an underflow to a subnormal can lose (`TinyOperandsEnclos `af_from_iv` halves its endpoints before it sums and differences them, and `af_range` scales its terms before it sums them, so endpoints near the float maximum do not overflow in the middle of the op; a form that overflows anyway gives `af_range` infinite endpoints, -never NaN (`FromIntervalAndRangeHoldAtTheExtremes`). `AbsorptionsArePinned` and +never NaN (`FromIntervalAndRangeHoldAtTheExtremes`). `af_range` finds a NaN endpoint by its bit pattern, because +`lo == lo` is a compare dxc marks `fast` and deletes under its default flags. `AbsorptionsArePinned` and `SnoiseFormCarriesItsDocumentedWidth` pin each absorption and `af_snoise`'s float32 allowance both ways: the enclosure tests alone cannot see them on a CPU, where float32 is exact. diff --git a/packages/cultmath/shaders/CultMath.Affine.hlsl b/packages/cultmath/shaders/CultMath.Affine.hlsl index 2c91a00d..f165148b 100644 --- a/packages/cultmath/shaders/CultMath.Affine.hlsl +++ b/packages/cultmath/shaders/CultMath.Affine.hlsl @@ -22,14 +22,16 @@ float3 cultmath_af_from_iv(float2 interval) return float3(x0, 0.0, e + (abs(x0) + e) * 9.5367431640625e-7 + 1.17549435e-38); } -// An endpoint that would be NaN is the infinity on its side. +// An endpoint that would be NaN is the infinity on its side. NaN is found by its bit pattern: dxc marks +// compares fast, so a test that needs NaN to make `lo == lo` false is deleted under its default flags. float2 cultmath_af_range(float3 x) { float r = abs(x.y) + x.z; float w = abs(x.x) * 9.5367431640625e-7 + r * 9.5367431640625e-7; float lo = x.x - r - w; float hi = x.x + r + w; - return float2(lo == lo ? lo : asfloat(0xff800000u), hi == hi ? hi : asfloat(0x7f800000u)); + return float2((asuint(lo) & 0x7fffffffu) > 0x7f800000u ? asfloat(0xff800000u) : lo, + (asuint(hi) & 0x7fffffffu) > 0x7f800000u ? asfloat(0x7f800000u) : hi); } float3 cultmath_af_add(float3 x, float3 y) diff --git a/packages/cultmath/shaders/CultMath.glsl b/packages/cultmath/shaders/CultMath.glsl index 99dbd4b8..d89ff8a3 100644 --- a/packages/cultmath/shaders/CultMath.glsl +++ b/packages/cultmath/shaders/CultMath.glsl @@ -665,14 +665,16 @@ vec3 cultmath_af_from_iv(vec2 interval) return vec3(x0, 0.0, e + (abs(x0) + e) * 9.5367431640625e-7 + 1.17549435e-38); } -// An endpoint that would be NaN is the infinity on its side. +// An endpoint that would be NaN is the infinity on its side. NaN is found by its bit pattern: dxc marks +// compares fast, so a test that needs NaN to make `lo == lo` false is deleted under its default flags. vec2 cultmath_af_range(vec3 x) { float r = abs(x.y) + x.z; float w = abs(x.x) * 9.5367431640625e-7 + r * 9.5367431640625e-7; float lo = x.x - r - w; float hi = x.x + r + w; - return vec2(lo == lo ? lo : uintBitsToFloat(0xff800000u), hi == hi ? hi : uintBitsToFloat(0x7f800000u)); + return vec2((floatBitsToUint(lo) & 0x7fffffffu) > 0x7f800000u ? uintBitsToFloat(0xff800000u) : lo, + (floatBitsToUint(hi) & 0x7fffffffu) > 0x7f800000u ? uintBitsToFloat(0x7f800000u) : hi); } vec3 cultmath_af_add(vec3 x, vec3 y) diff --git a/packages/cultmath/src/CultMath/math.Affine.cs b/packages/cultmath/src/CultMath/math.Affine.cs index 9bfbeb10..3c00e397 100644 --- a/packages/cultmath/src/CultMath/math.Affine.cs +++ b/packages/cultmath/src/CultMath/math.Affine.cs @@ -66,7 +66,8 @@ public static partial class math /// (|x0| + |a| + e) 2^-20, formed term by term so that the sum never passes the float maximum before it /// is scaled. A form with an infinite component (an overflowed op) gives infinite /// endpoints, and an endpoint that would be NaN (infinity minus infinity, or a NaN component) is - /// the infinity on its side, so the result is never NaN and still encloses. + /// the infinity on its side, so the result is never NaN and still encloses. NaN is found by its bit + /// pattern, not by lo == lo, which a compiler that assumes no NaNs (dxc's default) deletes. /// public static float2 af_range(float3 x) { @@ -74,7 +75,8 @@ public static partial class math var w = abs(x.x) * 9.5367431640625e-7f + r * 9.5367431640625e-7f; var lo = x.x - r - w; var hi = x.x + r + w; - return new float2(lo == lo ? lo : float.NegativeInfinity, hi == hi ? hi : float.PositiveInfinity); + return new float2((asuint(lo) & 0x7fffffffu) > 0x7f800000u ? float.NegativeInfinity : lo, + (asuint(hi) & 0x7fffffffu) > 0x7f800000u ? float.PositiveInfinity : hi); } /// diff --git a/packages/cultmath/tests/CultMath.Tests/HlslSourceCompatibilityTests.cs b/packages/cultmath/tests/CultMath.Tests/HlslSourceCompatibilityTests.cs index b1e3d8f8..1decfb96 100644 --- a/packages/cultmath/tests/CultMath.Tests/HlslSourceCompatibilityTests.cs +++ b/packages/cultmath/tests/CultMath.Tests/HlslSourceCompatibilityTests.cs @@ -169,6 +169,36 @@ public sealed class HlslSourceCompatibilityTests } } + + /// + /// The generic comparison stops at 1e7, so an HLSL edit of cultmath_af_range's NaN test that + /// misreads the finite values near the float maximum (or an overflowed endpoint) survives it. Forms + /// from zero to the float maximum, infinite and NaN components, both signs, in every slot: the mirror + /// returns C#'s bits, and neither endpoint is ever NaN. + /// + [Fact] + public void AffineRangeMirrorMatchesCSharpBitForBitAtTheFloatExtremes() + { + var (assembly, errors) = CompileShaderMirror(); + Assert.True(assembly is not null, string.Join(Environment.NewLine, errors)); + var shaderType = assembly!.GetType("CultMathHlsl.HlslShader")!; + var shader = Activator.CreateInstance(shaderType); + var mirror = ShaderFunction(shaderType, "cultmath_af_range", typeof(float3))!; + + var magnitudes = new[] { 0.0f, 1.0f, 1.0e30f, 1.0e38f, float.MaxValue / 2.0f, float.MaxValue, float.PositiveInfinity, float.NaN }; + var values = magnitudes.Concat(magnitudes.Where(m => !float.IsNaN(m)).Select(m => -m)).ToArray(); + foreach (var x0 in values) + foreach (var a in values) + foreach (var e in values) + { + var form = new float3(x0, a, e); + var expected = af_range(form); + var actual = Values(mirror.Invoke(shader, new object[] { form })!); + Assert.True(BitwiseEqual(Values(expected), actual), $"cultmath_af_range differs from C# at {form}"); + Assert.False(float.IsNaN(expected.x) || float.IsNaN(expected.y), $"af_range({form}) is {expected}, a NaN endpoint"); + } + } + /// /// The dense sweep almost never meets a cell whose box bound is just under the 2.25 cut and whose /// weight is still nonzero: a mirror cut of 2.245, 2.24 or 2.2 survives it. These points sit next diff --git a/packages/cultmath/unity/org.gamecult.cultmath/Shaders/CultMath.Affine.hlsl b/packages/cultmath/unity/org.gamecult.cultmath/Shaders/CultMath.Affine.hlsl index 2c91a00d..f165148b 100644 --- a/packages/cultmath/unity/org.gamecult.cultmath/Shaders/CultMath.Affine.hlsl +++ b/packages/cultmath/unity/org.gamecult.cultmath/Shaders/CultMath.Affine.hlsl @@ -22,14 +22,16 @@ float3 cultmath_af_from_iv(float2 interval) return float3(x0, 0.0, e + (abs(x0) + e) * 9.5367431640625e-7 + 1.17549435e-38); } -// An endpoint that would be NaN is the infinity on its side. +// An endpoint that would be NaN is the infinity on its side. NaN is found by its bit pattern: dxc marks +// compares fast, so a test that needs NaN to make `lo == lo` false is deleted under its default flags. float2 cultmath_af_range(float3 x) { float r = abs(x.y) + x.z; float w = abs(x.x) * 9.5367431640625e-7 + r * 9.5367431640625e-7; float lo = x.x - r - w; float hi = x.x + r + w; - return float2(lo == lo ? lo : asfloat(0xff800000u), hi == hi ? hi : asfloat(0x7f800000u)); + return float2((asuint(lo) & 0x7fffffffu) > 0x7f800000u ? asfloat(0xff800000u) : lo, + (asuint(hi) & 0x7fffffffu) > 0x7f800000u ? asfloat(0x7f800000u) : hi); } float3 cultmath_af_add(float3 x, float3 y) From b6d50be653917bc5daf8d558e0d8f170397302c4 Mon Sep 17 00:00:00 2001 From: Meta Vens Date: Sun, 4 Oct 2026 11:09:34 +0200 Subject: [PATCH 27/31] Mirror sweep reaches af_snoise's centred branch: realistic slices, both sides of the choice [mirror-sweep-misses-centred-branch] EveryMirrorFunctionMatchesCSharpMath draws centre, axis and radius from [-100, 100], far past the 0.15 or so below which the centred form wins, so an HLSL edit of the choice survived it. AffineSnoiseMirrorMatchesCSharpBitForBitOnBothSidesOfItsBranch draws 20000 slices (axis and radius log-uniform 1e-3 to 0.5), asserts both branches are met more than 2000 times each, and compares af_snoise (and af_fbm every eighth) bit for bit. Co-Authored-By: Claude Sonnet 5.5 --- .../HlslSourceCompatibilityTests.cs | 45 +++++++++++++++++++ 1 file changed, 45 insertions(+) diff --git a/packages/cultmath/tests/CultMath.Tests/HlslSourceCompatibilityTests.cs b/packages/cultmath/tests/CultMath.Tests/HlslSourceCompatibilityTests.cs index 1decfb96..cba99817 100644 --- a/packages/cultmath/tests/CultMath.Tests/HlslSourceCompatibilityTests.cs +++ b/packages/cultmath/tests/CultMath.Tests/HlslSourceCompatibilityTests.cs @@ -169,6 +169,51 @@ public sealed class HlslSourceCompatibilityTests } } + /// + /// The generic comparison draws centre, axis and radius from [-100, 100], where the reach is far past the + /// 0.15 or so below which cultmath_af_snoise's centred form is the narrower one, so an HLSL edit + /// of the choice between the centred and the Lipschitz form (a threshold, a reordered sum) survives it. + /// This sweep draws realistic slices (axis and radius log-uniform in 1e-3 to 0.5, the reach of a tile + /// near a camera), which land on both sides of the choice and next to it, for cultmath_af_snoise + /// and cultmath_af_fbm. + /// + [Fact] + public void AffineSnoiseMirrorMatchesCSharpBitForBitOnBothSidesOfItsBranch() + { + var (assembly, errors) = CompileShaderMirror(); + Assert.True(assembly is not null, string.Join(Environment.NewLine, errors)); + var shaderType = assembly!.GetType("CultMathHlsl.HlslShader")!; + var shader = Activator.CreateInstance(shaderType); + var snoiseMirror = ShaderFunction(shaderType, "cultmath_af_snoise", typeof(float3), typeof(float3), typeof(float))!; + var fbmMirror = ShaderFunction(shaderType, "cultmath_af_fbm", typeof(float3), typeof(float3), typeof(float), typeof(int), typeof(float), typeof(float))!; + + var random = new System.Random(0x5EED3); + float Next(float extent) => (random.NextSingle() * 2.0f - 1.0f) * extent; + float Reach() => 1.0e-3f * MathF.Pow(500.0f, random.NextSingle()); + int centred = 0, lipschitz = 0; + for (var i = 0; i < 20000; i++) + { + var centre = new float3(Next(50.0f), Next(50.0f), Next(50.0f)); + var axis = normalize(new float3(Next(1.0f), Next(1.0f), Next(1.0f))) * Reach(); + var radius = Reach(); + var expected = af_snoise(centre, axis, radius); + var actual = Values(snoiseMirror.Invoke(shader, new object[] { centre, axis, radius })!); + Assert.True(BitwiseEqual(Values(expected), actual), $"cultmath_af_snoise differs from C# at centre = {centre}, axis = {axis}, radius = {radius}"); + if (expected.y != 0.0f) + centred++; + else + lipschitz++; + + if (i % 8 == 0) + { + var fbmExpected = Values(af_fbm(centre, axis, radius, 4, 2.0f, 0.5f)); + var fbmActual = Values(fbmMirror.Invoke(shader, new object[] { centre, axis, radius, 4, 2.0f, 0.5f })!); + Assert.True(BitwiseEqual(fbmExpected, fbmActual), $"cultmath_af_fbm differs from C# at centre = {centre}, axis = {axis}, radius = {radius}"); + } + } + + Assert.True(centred > 2000 && lipschitz > 2000, $"the sweep meets one branch too rarely: centred {centred}, Lipschitz {lipschitz} of 20000"); + } /// /// The generic comparison stops at 1e7, so an HLSL edit of cultmath_af_range's NaN test that From c28cb4488b51cdd4f18940a6f6098cd85be09271 Mon Sep 17 00:00:00 2001 From: Meta Vens Date: Sun, 4 Oct 2026 11:09:35 +0200 Subject: [PATCH 28/31] GLSL fixture: small-reach af_snoise and af_fbm families, and the centre leg of the device check [device-check-blind-to-allowance] [device-check-self-referenced] Two families are appended (256 cases each, reach |axis| + radius of 1e-6 to 1e-3, the centre in [-50, 50] or [-1, 1]); the earlier families are byte-identical. In the main families the second-order remainder is at least 0.057 of e even without the allowance; here the float32 allowance is most of e, so a device whose gradient error exceeds it fails the enclosure check. Their check text adds the centre leg for every af_snoise and af_fbm family: the fixture's own x0 (C#'s value at the centre) must lie in the device form's [x0 - e, x0 + e], so a device whose noise is wrong everywhere no longer passes against its own reference. AssertSampledForm counts the cases that need the allowance (32 of 256 af_snoise, 3 of 256 af_fbm on C#'s own results) and the small families fail unless there is at least one. Co-Authored-By: Claude Sonnet 5.5 --- packages/cultmath/docs/design.md | 7 + .../tests/CultMath.Tests/GlslMirrorTests.cs | 102 +++- .../CultMath.Tests/fixtures/glsl-parity.json | 526 ++++++++++++++++++ 3 files changed, 612 insertions(+), 23 deletions(-) diff --git a/packages/cultmath/docs/design.md b/packages/cultmath/docs/design.md index 081265d0..e159f264 100644 --- a/packages/cultmath/docs/design.md +++ b/packages/cultmath/docs/design.md @@ -226,6 +226,13 @@ Every `af_*` function is an ulp-bounded family, and `af_snoise`, `af_fbm` and `a of their own, applied the same way: the device's own `snoise` or fBm at 75 sampled points of the case's set must lie in the device's form with no tolerance (a device whose gradient error exceeds the allowance fails), its width must not pass the Lipschitz form's allowance, and the device's `af_frustum_ball` radius must enclose the sampled ray points and stay within its rounding widening. +The reference is the device's own noise, which is right for culling soundness but would pass a device whose noise is wrong +everywhere, so two appended families (`af_snoise` and `af_fbm`, 256 cases each) add the centre leg: the fixture's own `x0` +(C#'s value at the centre) must lie in the device form's `[x0 - e, x0 + e]`, for every `af_snoise` and `af_fbm` family. They +also draw reach `|axis| + radius` from 1e-6 to 1e-3, the short slices near a camera, where the float32 allowance is most of `e`: +the main families' reach is at least about 1e-3, where the second-order remainder is far larger and a device that drops the +allowance still passes. `GoldenFixtureMatchesCSharp` requires each of the two to hold cases that fail on C#'s own results once +the allowance is removed. `GoldenFixtureMatchesCSharp` evaluates C# on the committed fixture's own arguments. Arguments are drawn only when the fixture is regenerated diff --git a/packages/cultmath/tests/CultMath.Tests/GlslMirrorTests.cs b/packages/cultmath/tests/CultMath.Tests/GlslMirrorTests.cs index b2f525d9..f0f61622 100644 --- a/packages/cultmath/tests/CultMath.Tests/GlslMirrorTests.cs +++ b/packages/cultmath/tests/CultMath.Tests/GlslMirrorTests.cs @@ -234,15 +234,22 @@ public sealed class GlslMirrorTests // SampledForm applied to C#'s own results, with C#'s own snoise or fBm as the reference value. A form // must enclose its reference at every sample, and be no wider than the Lipschitz form's allowance. - private static string AssertSampledForm(JsonArray cases, string name, Func reference, Func widthBound) + // allowance is the part of e that is the float32 allowance; a family that allowanceDecides must have a + // sample that falls outside the form once it is removed, so the family can tell a device whose error + // passes the allowance from one that does not. + private static string AssertSampledForm(JsonArray cases, string name, Func reference, Func widthBound, + Func allowance, bool allowanceDecides) { double least = double.PositiveInfinity, leastRelative = double.PositiveInfinity, tightest = double.PositiveInfinity; + var decided = 0; foreach (var entry in cases) { var sides = ((string)entry!).Split(" -> "); var tokens = sides[0].Split(' '); double[] a = Floats(string.Join(' ', tokens.Take(7))), form = Floats(sides[1]); double x0 = form[0], slope = form[1], e = form[2]; + var withoutAllowance = e - allowance(tokens, a); + var allowanceNeeded = false; foreach (var eps in CheckEps) foreach (var d in CheckDirections) { @@ -252,42 +259,56 @@ public sealed class GlslMirrorTests Assert.True(inside >= 0.0, $"{name} case {entry}: the reference {value:R} at eps {eps} and direction {d} is outside the form's range [{middle - e:R}, {middle + e:R}] by {-inside:R}"); least = Math.Min(least, inside); leastRelative = Math.Min(leastRelative, inside / e); + allowanceNeeded |= Math.Abs(value - middle) > withoutAllowance; } + if (allowanceNeeded) + decided++; + var bound = widthBound(tokens, a); Assert.True(Math.Abs(slope) + e <= bound + Math.ScaleB(1.0 + bound, -16), $"{name} case {entry}: the width {Math.Abs(slope) + e:R} is past B + 2^-16 (1 + B) = {bound + Math.ScaleB(1.0 + bound, -16):R}"); tightest = Math.Min(tightest, (bound + Math.ScaleB(1.0 + bound, -16) - Math.Abs(slope) - e) / bound); } - return $"least enclosure slack {least:E3} ({leastRelative:E3} of e), least tightness slack {tightest:E3} of B"; + Assert.True(!allowanceDecides || decided > 0, $"{name}: no sample falls outside the form once the float32 allowance is removed, so the family cannot fail a device whose error passes it"); + return $"least enclosure slack {least:E3} ({leastRelative:E3} of e), least tightness slack {tightest:E3} of B, {decided} of {cases.Count} cases need the allowance"; } + // The float32 allowance of one octave of af_snoise: 2^-14 + 2^-12 R + L 2^-20 (|c|_1 + R). + private static double SnoiseAllowance(double centre1, double reach) => + Math.ScaleB(1.0, -14) + Math.ScaleB(reach, -12) + math.SNOISE_LIPSCHITZ * Math.ScaleB(centre1 + reach, -20); + // The Lipschitz form's width bound for one octave of the sampled-form check. - private static double SnoiseWidthBound(double centre1, double reach) => - math.SNOISE_LIPSCHITZ * reach + Math.ScaleB(1.0, -14) + Math.ScaleB(reach, -12) + math.SNOISE_LIPSCHITZ * Math.ScaleB(centre1 + reach, -20); + private static double SnoiseWidthBound(double centre1, double reach) => math.SNOISE_LIPSCHITZ * reach + SnoiseAllowance(centre1, reach); private static double Length3(double[] a, int first) => Math.Sqrt(a[first] * a[first] + a[first + 1] * a[first + 1] + a[first + 2] * a[first + 2]); - private static string AssertSnoiseForm(JsonArray cases) => - AssertSampledForm(cases, "af_snoise", (_, p) => math.snoise(p), - (_, a) => SnoiseWidthBound(Math.Abs(a[0]) + Math.Abs(a[1]) + Math.Abs(a[2]), Length3(a, 3) + a[6])); + private static double SnoiseOctave(Func octave, double[] a, double frequency) => + octave((Math.Abs(a[0]) + Math.Abs(a[1]) + Math.Abs(a[2])) * frequency, (Length3(a, 3) + a[6]) * frequency); - private static string AssertFbmForm(JsonArray cases) => + // The amplitude-weighted sum of a per-octave quantity over the fBm's octaves. + private static double FbmOctaves(string[] t, double[] a, Func octave) + { + var (octaves, lacunarity, gain) = ((int)Convert.ToUInt32(t[7], 16), Floats(t[8])[0], Floats(t[9])[0]); + double sum = 0.0, amplitude = 1.0, frequency = 1.0; + for (var i = 0; i < Math.Clamp(octaves, 0, 16); i++) + { + sum += amplitude * SnoiseOctave(octave, a, frequency); + amplitude *= gain; + frequency *= lacunarity; + } + + return sum; + } + + private static string AssertSnoiseForm(JsonArray cases, bool allowanceDecides = false) => + AssertSampledForm(cases, "af_snoise", (_, p) => math.snoise(p), + (_, a) => SnoiseOctave(SnoiseWidthBound, a, 1.0), (_, a) => SnoiseOctave(SnoiseAllowance, a, 1.0), allowanceDecides); + + private static string AssertFbmForm(JsonArray cases, bool allowanceDecides = false) => AssertSampledForm(cases, "af_fbm", (t, p) => math.fbm_grad(p, (int)Convert.ToUInt32(t[7], 16), (float)Floats(t[8])[0], (float)Floats(t[9])[0]).w, - (t, a) => - { - var (octaves, lacunarity, gain) = ((int)Convert.ToUInt32(t[7], 16), Floats(t[8])[0], Floats(t[9])[0]); - double sum = 0.0, amplitude = 1.0, frequency = 1.0; - for (var i = 0; i < Math.Clamp(octaves, 0, 16); i++) - { - sum += amplitude * SnoiseWidthBound((Math.Abs(a[0]) + Math.Abs(a[1]) + Math.Abs(a[2])) * frequency, (Length3(a, 3) + a[6]) * frequency); - amplitude *= gain; - frequency *= lacunarity; - } - - return sum; - }); + (t, a) => FbmOctaves(t, a, SnoiseWidthBound), (t, a) => FbmOctaves(t, a, SnoiseAllowance), allowanceDecides); // AffineFrustumBall applied to C#'s own results. private static string AssertAffineFrustumBall(JsonArray cases) @@ -425,6 +446,23 @@ public sealed class GlslMirrorTests private static float BallRadius(System.Random r) => MathF.Pow(10.0f, Uniform(r, -3.0f, -0.3f)); + // A slice of reach 1e-6 to 1e-3, the short slices and small footprints near a camera, where the float32 + // allowance is most of the form's width: the centre in [-50, 50] or [-1, 1], an axis of random direction + // (zero in one case of four) and a radius, each log-uniform in 1e-6 to 1e-3. + private static object[] SmallReachSlice(System.Random r) + { + var span = r.Next(2) == 0 ? 50.0f : 1.0f; + var centre = new float3(Uniform(r, -span, span), Uniform(r, -span, span), Uniform(r, -span, span)); + var axis = new float3(0.0f, 0.0f, 0.0f); + if (r.Next(4) != 0) + { + var direction = math.normalize(new float3(Uniform(r, -1.0f, 1.0f), Uniform(r, -1.0f, 1.0f), Uniform(r, -1.0f, 1.0f))); + axis = direction * MathF.Pow(10.0f, Uniform(r, -6.0f, -3.0f)); + } + + return new object[] { centre, axis, MathF.Pow(10.0f, Uniform(r, -6.0f, -3.0f)) }; + } + private const string SampledForm = "the arguments are (cx, cy, cz, ax, ay, az, radius, ...) and the result is the form (x0, a, e). Enclosure: for eps in " + "{-1, -0.5, 0, 0.5, 1} and d in {0, +-ex, +-ey, +-ez, (+-1, +-1, +-1) / sqrt(3)} (15 directions), form the point " + @@ -440,6 +478,22 @@ public sealed class GlslMirrorTests "The reference value is cultmath_fbm_grad(p, octaves, lacunarity, gain).w, and B is the sum over octaves i of gain^i B_i, " + "where B_i is B for the centre lacunarity^i c, the axis lacunarity^i axis and the radius lacunarity^i radius."; + // The two small-reach families add the centre leg to every af_snoise and af_fbm family. The device's + // reference value is its own noise, so a device whose noise is wrong everywhere encloses its own wrong + // values; C#'s value at the centre, the form's own x0 in the fixture, is the independent reading. + private const string CentreLeg = + " Centre leg, for every cultmath_af_snoise and cultmath_af_fbm family of this fixture, the earlier ones included: " + + "the fixture's own x0 in the result (C#'s value at the centre) must lie in [x0 - e, x0 + e] of the GPU's form, with " + + "no tolerance, so a device whose noise differs from C#'s by more than its e fails."; + + private const string SmallReach = + " These cases have reach |axis| + radius of 1e-6 to 1e-3, where the float32 allowance is most of e and the second order " + + "remainder is not; the enclosure above fails a device whose gradient error exceeds the allowance."; + + private const string SnoiseSmallForm = SnoiseForm + SmallReach + CentreLeg; + + private const string FbmSmallForm = FbmForm + SmallReach + CentreLeg; + private const string AffineFrustumBall = "enclosure and tightness: from the arguments (mx, my, z0, z1, fp, warpVariation) compute in double zm = (z0 + z1) / 2, " + "h = (z1 - z0) / 2, axis = (mx h, my h, h), c0 = (mx zm, my zm, zm), |c0|_1 = |mx zm| + |my zm| + |zm|, R0 = z1 fp + warpVariation, " + @@ -505,8 +559,8 @@ public sealed class GlslMirrorTests new("vec3 cultmath_af_scale(vec3, float)", "ulp-bounded", r => new object[] { Form(r), Uniform(r, -4.0f, 4.0f) }), new("vec3 cultmath_af_add_iv(vec3, vec2)", "ulp-bounded", r => new object[] { Form(r), Interval(r) }), new("vec3 cultmath_af_mul(vec3, vec3)", "ulp-bounded", r => new object[] { Form(r), Form(r) }), - new("vec3 cultmath_af_snoise(vec3, vec3, float)", "ulp-bounded", r => new object[] { Point3(r), Axis3(r), BallRadius(r) }, SnoiseForm, Verify: AssertSnoiseForm), - new("vec3 cultmath_af_fbm(vec3, vec3, float, int, float, float)", "ulp-bounded", r => new object[] { Point3(r), Axis3(r), BallRadius(r), 4, 2.0f, 0.5f }, FbmForm, Verify: AssertFbmForm), + new("vec3 cultmath_af_snoise(vec3, vec3, float)", "ulp-bounded", r => new object[] { Point3(r), Axis3(r), BallRadius(r) }, SnoiseForm, Verify: cases => AssertSnoiseForm(cases)), + new("vec3 cultmath_af_fbm(vec3, vec3, float, int, float, float)", "ulp-bounded", r => new object[] { Point3(r), Axis3(r), BallRadius(r), 4, 2.0f, 0.5f }, FbmForm, Verify: cases => AssertFbmForm(cases)), new("vec3 cultmath_af_frustum_axis(vec2, float, float)", "ulp-bounded", r => { var z0 = Uniform(r, 0.1f, 100.0f); @@ -519,6 +573,8 @@ public sealed class GlslMirrorTests return new object[] { slope, z0, z0 + Uniform(r, 0.01f, 50.0f), Uniform(r, 0.0001f, 0.05f), Uniform(r, 0.0f, 2.0f) }; }, AffineFrustumBall, Verify: AssertAffineFrustumBall), new("vec2 cultmath_iv_snoise_ball(vec2, float)", "ulp-bounded", r => new object[] { new float2(Uniform(r, -50.0f, 50.0f), Uniform(r, -50.0f, 50.0f)), Uniform(r, 0.001f, 2.0f) }), + new("vec3 cultmath_af_snoise(vec3, vec3, float)", "ulp-bounded, reach 1e-6 to 1e-3", SmallReachSlice, SnoiseSmallForm, Verify: cases => AssertSnoiseForm(cases, allowanceDecides: true)), + new("vec3 cultmath_af_fbm(vec3, vec3, float, int, float, float)", "ulp-bounded, reach 1e-6 to 1e-3", r => SmallReachSlice(r).Concat(new object[] { 4, 2.0f, 0.5f }).ToArray(), FbmSmallForm, Verify: cases => AssertFbmForm(cases, allowanceDecides: true)), }; // The fixture text, each case's arguments given by family and point index. diff --git a/packages/cultmath/tests/CultMath.Tests/fixtures/glsl-parity.json b/packages/cultmath/tests/CultMath.Tests/fixtures/glsl-parity.json index cd7a2428..2a81ae8e 100644 --- a/packages/cultmath/tests/CultMath.Tests/fixtures/glsl-parity.json +++ b/packages/cultmath/tests/CultMath.Tests/fixtures/glsl-parity.json @@ -11799,6 +11799,532 @@ "c0000d30 c1604fbc 3f84a2ee -> bf800000 3f800000", "c21452ae 41d1741c 3e83d42d -> bf800000 3f800000" ] + }, + { + "glsl": "vec3 cultmath_af_snoise(vec3, vec3, float)", + "tolerance": "ulp-bounded, reach 1e-6 to 1e-3", + "check": "the arguments are (cx, cy, cz, ax, ay, az, radius, ...) and the result is the form (x0, a, e). Enclosure: for eps in {-1, -0.5, 0, 0.5, 1} and d in {0, +-ex, +-ey, +-ez, (+-1, +-1, +-1) / sqrt(3)} (15 directions), form the point p = c + axis eps + radius d in double and round it to float32; the device's own reference value at p must lie in [x0 + a eps - e, x0 + a eps + e] with no tolerance, so a device whose gradient error exceeds the allowance fails. Tightness: with L = 7.9640074 and, for each octave (one for af_snoise), R = |axis| + radius, B = L R + 2^-14 + 2^-12 R + L 2^-20 (|c|_1 + R), the width |a| + e must be at most B + 2^-16 (1 + B). Report the least slack of each side, absolutely and as a fraction of e; only the two bounds fail the check. The reference value is cultmath_snoise(p). These cases have reach |axis| + radius of 1e-6 to 1e-3, where the float32 allowance is most of e and the second order remainder is not; the enclosure above fails a device whose gradient error exceeds the allowance. Centre leg, for every cultmath_af_snoise and cultmath_af_fbm family of this fixture, the earlier ones included: the fixture's own x0 in the result (C#'s value at the centre) must lie in [x0 - e, x0 + e] of the GPU's form, with no tolerance, so a device whose noise differs from C#'s by more than its e fails.", + "cases": [ + "3de98ec0 3e5960a0 3e824700 b64c3554 36674e47 b629f5b3 37601e01 -> bee8782f 36405f94 38e47f80", + "3f2f1238 bf6d5b1d 3dc3cf20 b943c815 b9552ec5 390597fe 3791eb35 -> 3ebaf9fc 38759590 3903a998", + "c139b3a4 c1b8837e 41c4ffc0 38e3ec26 38dc4b8d 389176fa 395ca6ba -> 3e59bf01 b9c019c6 3aa3838f", + "beec548a bebdab10 3f1ff0f4 b5bb676f 35bed139 35f5b0c6 3911e058 -> be35088f 367641c8 39f059aa", + "c2091b3b 41f27428 c156f984 00000000 00000000 00000000 36ef2dcd -> 3e02aba0 00000000 3a2ee486", + "bf415a24 3e96f054 bec22aba 00000000 00000000 00000000 3979f5b4 -> 3f00ce18 00000000 3a7b5a27", + "c172a69c 41dbd6f0 c16ab350 375ab8b5 b70b7f01 359ff929 3a6d5e2c -> bd169e91 353c0390 3b08a588", + "bd8ca798 be54ffb4 be60513c b674cdf7 36c97804 3649905b 3646a75a -> 3f0e31c6 37195340 38983a7e", + "bf460e64 bf5d040d bf1b828d 3660aadc 35cca052 b66df0c7 39513c96 -> 3dbd40e3 b5bb6f1a 39c64322", + "c0e04328 40af6150 c1fa88dd 396b6d16 b96965b5 b9031a1e 3595385d -> beb2ac89 3a28b532 39d1bdd9", + "3e63bbf8 bebac5a0 be8fda88 b8724e47 392ef8ff 38801b83 37b225b5 -> bd469206 3a15fbc6 391151c2", + "42281202 c24198d2 c22f046a 00000000 00000000 00000000 36769265 -> 3dd08ee9 00000000 3a8ebaf1", + "40fb4bc0 41d9f458 c139afc4 00000000 00000000 00000000 3683c860 -> bd915b07 00000000 39dd7762", + "41504874 421674b2 c20b5de4 b6667f0c b574ab23 35408169 39116440 -> 3ec66a90 34c78fd0 3a93c57e", + "bee2441e bde396a0 3f70b70e 39c53600 3963ae74 b9f24965 3591be10 -> be52a73e 39db57bc 38b819c8", + "41eb0098 411db0ac c1f88cee b5d5548d 38cf478f 38735753 37c7eba0 -> 3ec4e662 39735a97 3a2a8349", + "c0806970 4207b82e 3fbb4dc0 37938dcc b5d15df2 b6eeda70 394876f3 -> 3e299ef9 3610b078 3a031d28", + "c1f2d8ed c1c343a1 c092bdf8 00000000 00000000 00000000 392a6aa0 -> be841b57 00000000 3a8b9908", + "bf04a494 be5b8028 3e11d388 381eae8f b8893bfa b9e5e3f6 380771e6 -> be0c82b1 37fe2d4c 38f45877", + "41957650 41be235c c1026468 00000000 00000000 00000000 38b2c682 -> bf485844 00000000 3a3177fd", + "3ee8d84c bf01f67b be1c98b0 366355ec 35a4befd b64f4484 36824a2f -> be305f08 b62249d8 389934d6", + "bf271ebc 3f069042 bf432953 b99449fe b990e5f2 391393bd 3927e7c0 -> be9b8b31 3a8f6452 3a40ecac", + "c219e9fe 3dd99800 422f887c 00000000 00000000 00000000 38de6c5d -> be66d34d 80000000 3a6bdfb7", + "3f1f6cce 3e9f8c24 3f17d57e b7179bf6 35057781 36021be1 382187fe -> bf496c88 b6edbe56 3906d627", + "41352b58 c1a236d5 419692dc b8a5fb2f b86dc7e0 38c9aa74 36e23fe4 -> bc78d716 b96082cc 39eed366", + "41d327e0 423a1498 421ae206 b6c534f3 351ea919 b2a20814 39468d10 -> be99d01b 373e6c27 3ab7bc9d", + "c19a2218 c10a5698 41d3a4d4 3809d622 b698a636 b6e3bbb1 398c0264 -> be308f7f 38434f83 3aa855a1", + "bf2f79f6 3f17e7c8 bf7b3ce9 34af4f09 b5861a9f b62ef6bb 36078865 -> be373174 3435f5e8 38ab890e", + "3ef533f0 bf0be5d1 3eef5408 3684116e 367886d9 3640d57f 389e1322 -> 3e46b642 b6b4b726 39753aa2", + "c23c9be7 c1728468 40c9bdf0 3746859f b714ce0f 3581890e 392f07e4 -> be63bedf b543e2e0 3ab4c727", + "3f5572b0 befe432c 3f4e7150 b98837cd 39f8a638 39f29b6b 3a7d3137 -> be1e0035 bae0ca24 3b572b8a", + "be9157a2 be9aa8de bf3073fd b564e7b7 b7e47e1b 3743ebc4 37a396ef -> 3d01088c 38264ae1 38e609f9", + "3ebc2358 bf2ab187 3f44b152 381a2fa6 38c9b21b b8adf22a 35abddae -> be1729da b9a229fe 38a5f3f6", + "420acfe0 41ecab00 c111f0ec 35f6037d b388ab51 b576402f 35a4f81c -> bf5e99ac 35f71757 3a22c8ec", + "3f47d23a 3f22ba9e beee49b2 b82384f5 3794ee57 b7fea625 39926a12 -> be5bdfbc b886d39f 3aad1fd4", + "3d22cc00 bf6a7da4 3f6ba68c b81a65c2 37dbf2d6 b768d528 3591854a -> 3e736ce9 366bcff0 38aa1ab0", + "41bae30c c1c8f478 4225fdb0 398f4069 385585ba 39bccb38 362d021a -> 3e813bc5 3a8f6cc4 3a4687e4", + "3f0397c4 3e1b5948 bda08d28 b9ca5a56 39bcd58c 39ce14bf 36af1af3 -> 3e8c0ed2 bad63992 38c4b500", + "3e6d2048 3ea41868 3f48943e 00000000 00000000 00000000 3996f19f -> be57c54d 00000000 3a278923", + "3f2bd122 3eaa1e44 bf422a0e 00000000 00000000 00000000 3619fab4 -> be437cda 00000000 38a29b37", + "bf50c330 3e41eb48 bf2a77b8 37fcc4ce 370c7b9e b7f4401d 392d38b3 -> 3f2a3a72 b8e82782 3a1c5ffa", + "c0cbd658 41fcfb70 c22ec733 365f9692 b63e42ce 36b9809f 39bb42c4 -> bd14aeb4 b6f7b921 3aa6089c", + "c190c5ba c22abb87 41ff83b8 00000000 00000000 00000000 3745a35b -> bda5b210 00000000 3a4d8025", + "3f2bd3d6 bf712a22 3ee816c0 39c82020 ba2f85b7 3a0ef5dd 37ecd8f8 -> be5828a7 bb42b998 39584fa8", + "be8d0096 bd897928 3f6d145c 36bf2f5a 366386f2 b6aef199 369d9d3c -> be38d5fd 38001fdd 38c38409", + "c20aeeef 42164ab8 c1dcf32f b8ba4961 38866176 b845b6f2 37b70f8d -> bf0a7584 3826e548 3a661091", + "41dc3c30 c2228583 c1ac76f5 00000000 00000000 00000000 395fb00d -> 3e400d51 00000000 3af6ceac", + "41fd8b10 c05e2a70 40861770 38d0d6f7 b8ccad21 b7172dd8 35d792ab -> 3dcf4a5d 38a1298e 39bd96b2", + "c1904117 40cb7c90 411efac8 36ab60a9 341cfc94 36a01fdb 3836fd90 -> 3e0a2dd7 375cebed 39dfdbd5", + "3f1c3e4e beb64e66 bf03c15f 378a7ea9 b799080f 363158a5 35fb591c -> bebaea08 b7927659 38a53e58", + "c1fe62b5 41cf8f3c 3f8d14c0 396078ee 390aa93f 391e2529 362e5013 -> be83cfd5 b969a1a1 3a06daa7", + "c1078380 3f5f42c0 c183212e 3749f1de 390108e4 b98bf39c 3636e04a -> 3ef1b783 ba44988a 398c8ec2", + "c1cc9256 c246d757 417b3620 b837ccc0 b5849e00 b86431f7 3895d183 -> be73a7c3 37f62b8a 3a70333e", + "3d54f2e0 3f7de9dc 3e7e8100 398a98dc 3809558c b97f0aeb 3933d47b -> bea586d2 b9f0a6d9 39dfb3bd", + "41c322f8 c23d3902 c20eaa17 359bbef9 b6b1d76a 37543f48 378d44bf -> bec5131e b7370d7a 3a733010", + "c2226a7d 41b20174 c1907ff6 b8ad4189 b869bf29 b890cd4b 37d8337f -> 3f13a6a2 38dd6069 3a44b396", + "3f909160 c20b3045 c1ec68a8 36bdce6d b70a5cc8 3824f333 39dfd1e1 -> bf2f9ae1 3759b222 3acc185c", + "3e8a6754 bf04eb40 bf5800ee 00000000 00000000 00000000 3818fd87 -> bd6a4378 80000000 395872f6", + "420fd2c0 c1f6a1ad c1b678ec 3546ed21 37eabe9f b7c936eb 38c3b026 -> 3f595d16 b86d98a6 3a6d7e74", + "3eb4c96c 3f189838 3f0d244e b63520a7 b7184ff3 b6fd2eb5 38a165f8 -> 3ea5f7f5 37173f60 39cca28d", + "3f2518ba 3f02f312 3f51eb52 00000000 00000000 00000000 37906e56 -> bf038b0c 00000000 3901eaa4", + "3e905500 c09de718 c07c0f80 3902c51b b9282f1b b930185e 37188c7a -> bf44fb9c 39bb6fe0 39207151", + "be1e7114 bc9fd000 3f3fcf6a 391d9da0 3880d283 38a333a9 3931747e -> 3f3c365a b911cb8a 3a210610", + "41d05878 c22c289f 3d0af800 3a037b44 399f465d b9357f0e 37805668 -> 3ca99dd3 3889b1ae 3a26e8bc", + "c1aea15d c20d4fb2 40da1758 00000000 00000000 00000000 372fee37 -> 3eded15e 00000000 3a14caaa", + "bf607798 3f1b2cf6 bf2e8aa4 b53ca5da 38377f2e b59b39df 3a27c39a -> 3ea53d0f b8e55b3b 3aee1dd3", + "bcdea220 3f6bc518 3e955588 00000000 00000000 00000000 359f53f2 -> bde0af67 00000000 3898d0e1", + "c13c3a04 c18b8a4a 41977a44 39c10126 b987a91f 39d43adf 3a2acc88 -> be486436 ba157e60 3af41bf5", + "c10ca7a0 c1e36336 414e6e70 363edeb1 360e96eb b59b804d 3898a8eb -> 3c47d632 36215d9b 3a188709", + "3e3ddfc0 be749268 3e4065c8 390c3107 3a41c0f6 b9874945 36a5198a -> bec28bbc ba9efe9e 38cca834", + "c1ada584 c1c43976 c2286978 00000000 00000000 00000000 394dea7d -> 3ef66c5c 00000000 3a7f7f01", + "c1501140 4222cbea 41f3ffcc 35dd42a7 b5c0bd08 b4d9af69 384416ed -> bec6093e b5569379 3a5251da", + "42273736 417f27d0 423bdd2e 35f5fd35 b526d71b 3490aaba 38507dce -> 3f0c77e8 354adfcb 3a84763b", + "41b94df0 421f0aa6 420c7d2c 37abe1ef b7ed240f 3745d1cc 36005407 -> 3b6424f5 37effa4f 3a53f5ea", + "c2114824 3f0cb000 3f93d9a0 b881b7f5 b8bb682f b907a43f 3619af99 -> 3e5d367a b93940e9 39bc0277", + "42368666 c00b1f30 c1d3e256 b8041083 374f2efa 380d3897 396adc5f -> bea6d948 b84310af 3a8f3e29", + "3ef0e534 3f76c88a 3dedc390 36dafa93 b78a3d7e b98ad4f3 35ac9137 -> 3e1eb420 b91184ec 389fd5da", + "3d8feec0 3f4de232 bf74d21b b6b0972b b5a9458d b6aa5ca2 369f8e31 -> bf00d467 3745f0bb 38c26014", + "c22c42d3 41dd8150 423eced4 00000000 00000000 00000000 35b7bdd3 -> be447b9f 00000000 3a7ccd2b", + "4181f7a4 40836f68 4238b7c2 00000000 00000000 00000000 35a6e636 -> 3ed0f20b 00000000 3a155b4d", + "41beff84 c1bb33d9 c0f935a8 00000000 00000000 00000000 3a36572e -> be15657f 00000000 3b28e9e0", + "bf29b1ce 3f722eb6 be89ee8a 3540e9fa b5152575 35986492 37c452e9 -> 3eb6df0e 363a945a 39031dde", + "bf26594c 3e7b48e0 3d8cbaa0 b9cf01ce 393388f5 3a210aea 358aea0c -> bf038f7a 3ad55a27 38b9e201", + "41adb20c c1c1efa8 41f7ea70 371fbc13 372dee6e 36e4b629 364f6fd8 -> 3efda2a8 b617df30 3a2bb0c4", + "3f744716 3f75e39c bd8ef7b0 b7a6591a b75969a9 37981f29 35dab19d -> 3f24d687 382071b6 38a60bcb", + "be784a14 3f7b87b4 3f18089c b6adb4ef b6d08a66 36a21fe9 3986a5fd -> 3f0dcef7 37a7ca75 3a32df1f", + "3f3fd680 c067f940 c20c12a8 b976f7d5 3958cf4d b931b7b0 39243938 -> 3e35e9ca 3a014c02 3a2e928a", + "3e3df470 be140aa4 3f481a86 35d2b9fe b5b3364c b5c91613 386f03bb -> 3dcf1e35 b67af167 39807bf0", + "41b4cae8 c1ba7374 421b13b4 3a526b07 39c83589 b88b91ad 363b79d5 -> bf1779c3 3a8d9341 3a408c30", + "befe1c54 be1d7680 3e424690 b9825e49 375d51c7 39ab2dc4 3753dc13 -> bd9316fa 39afb828 38b88ff1", + "c22429fa 42462e1e 41cff6d0 00000000 00000000 00000000 35b580ab -> 3d833059 00000000 3a78c816", + "bf4c92d3 bf4eab1f 3f6f7a0c 00000000 00000000 00000000 366b1b50 -> 3f6037b8 80000000 38b5e474", + "41bf90dc 41b1a608 c0bdba30 00000000 00000000 00000000 38604ee4 -> 3ea2f3f2 00000000 3a17328b", + "3e7428c0 bec6a8d6 bdcb6198 00000000 00000000 00000000 3905bc9d -> 3c9998be 00000000 39877870", + "40f7c328 c204251d 42052424 b5dbad63 b6b1b9ae b69bee46 395394cd -> be7161e1 37398358 3a9c2666", + "42449fb6 41b86eac 3f32b800 b68b3a85 b73d4f7a 36475ec3 359583fa -> 3de4bb19 376f917c 3a21aa87", + "beb29c04 3f06396e 3ebc1074 33ff620b b58c6666 3579776d 36bf3803 -> bf1178e7 b6247cdc 38b460b4", + "41d03308 4223ee26 c1a5258c 35a3ab4c 363aac60 b63e8a50 38b3733d -> be0ea413 b45ad3e0 3a5643f8", + "3f1d0964 3f4a427e 3ee2b58c 39beea9a 39ae13a6 39c176ab 37c455b9 -> 3ed721f3 bab8f22d 39165f09", + "3eb8ec60 bf53f036 bf2bf5fe 3a1c0507 39f6946b b7af6eeb 372d82c0 -> 3f16cdef bb0e652c 3908ed57", + "c112ec2c c193caa9 c23a3029 3960f554 ba4d9a7e b8f7d9b1 3a67fcf7 -> 3eb311d1 bb01a775 3b4ddfba", + "42223180 c1016684 41b025a0 b6af89f8 b59431a6 b6f4f115 3633572b -> befb19f6 367ea1e8 3a1f04df", + "bee73566 bf7c2701 3e3de280 b79ddac1 37e62524 38224da3 38aa35db -> bd6a6a9a b7293717 39222c93", + "be653a44 bf2f078c 3f3c750c 00000000 00000000 00000000 36933d83 -> bce7a14a 00000000 38a3f2e1", + "4210341c c2425044 41c84f74 b4dfb2e2 35013626 355cab6a 38f67c33 -> 3eaff8b6 b5a54c72 3aab8f6b", + "c15699ec 41d89f88 40aae458 3780f554 b79c611c 3795462f 365988aa -> be4d3646 b7c487ee 39d81b23", + "41b208dc c142f9f0 c18aace6 b57e9353 360ebbb0 b580031c 35ceffc3 -> bccbed00 b54b1472 39ef4e26", + "3dd113c0 3d284020 bf243704 3692fa13 b7f86bbd 368fb48e 385afecb -> 3f564ca8 b81df745 393a4a51", + "bf614131 3e325378 3f6ec39e 00000000 00000000 00000000 370d2593 -> 3f24fbd2 00000000 38d6852d", + "bf77df2f bf29f860 bf5a021c 00000000 00000000 00000000 373027f6 -> 3e028310 00000000 38cf970d", + "c23c3dd1 41fabf38 41d7207c 36fdf273 3751f9fd b6fd914c 37a0185c -> becbad8e 3716fa1e 3a6fe6b2", + "3f62c6fe 3f62aba8 3ed77868 37f2f373 38f83662 39043628 3883185a -> 3e6ecb33 b8102384 3965b693", + "4227eddc 417ce0d0 c17f0690 00000000 00000000 00000000 37903eb4 -> 3f11eb04 80000000 3a315b54", + "41530afc c15e2d60 420cba4a 00000000 00000000 00000000 35ca4cf7 -> 3e6c046b 00000000 3a0d2c23", + "be3e9384 bf2634d6 bc9120e0 39064732 b906df4c b852aa80 366a0038 -> bda9f02d b92f4216 38982629", + "c225e0df c1cb0dd1 4166a360 00000000 00000000 00000000 378d2046 -> bf36a909 00000000 3a41f501", + "c13a90b0 c20f3d12 c1b84b87 36550372 b752ada3 b599d22e 37dbde7a -> 3eb92688 b7f8b44d 3a2e13b7", + "bf5bded6 bd2422b0 bf24ba24 34a3bbe4 b5b72634 35c209c1 3829d03f -> 3e6a4f2f b661f522 393bc6f3", + "be954e3a bf0c740f 3f6d491c b80a03c0 3901042a b9072c39 385b3a3a -> be9c867b b9c326fb 397e7adb", + "bf5e2e00 c132e824 41ef2a78 00000000 00000000 00000000 3a60c8fa -> 3ed2ebd7 00000000 3b57ce7a", + "3f4b7f58 3eeec058 bf0fa160 36298746 b692cce6 3697be80 35970a65 -> bf00117f 34e6ed10 38a3ccbe", + "3e23e8d8 bf0aa220 3e684268 b89fc1fa 39cf2baf b8ea307a 360e4787 -> 3ed20b27 ba7bae10 38a7811b", + "c2009e2f c08388b0 4226e7ac 372480c1 36b34490 3703c66b 36331d7c -> bf240e71 b73368b9 3a2cfbf3", + "3eb959c8 bf7b36e9 3f729aec b5205113 b60b6163 35de45ac 399295f2 -> 3e1692ce 34a53ab8 3a5a1b1f", + "c185ddac 41951744 41eae9a4 b8925c42 38fce9c1 b8dc3e3f 3886d399 -> bee6257e 37443750 3a3d9ad3", + "3f4cf886 3f59c8d6 bf613635 36061b5c b682f33c 37a8520d 381634c1 -> be67bb7a 37cab0f6 39458d9f", + "c18f0bda 3fbb05e0 c212035c b9278403 38131a37 b8fe8450 37cda4c2 -> 3ea773dd 391759f2 3a04fcf9", + "41810c8c c17b6fc0 c230c39b 373d10e8 b493329c 36c03d88 381a02a4 -> be465588 b6e4363e 3a333d4c", + "3e02a228 bf65b51d 3f0d2658 00000000 00000000 00000000 3968ac4b -> 3ef6e600 00000000 3a7cef3b", + "3f65ff4c bf313f1b bec032cc b96cc8ea b8338bd9 38cc2630 3a0feee3 -> bde475bb b99408b5 3ac1cee4", + "3dd28510 bda83aa8 be832172 35b26240 b5926755 b5f28b00 36f3aad1 -> 3f51a74d b698faa6 38aa0d9e", + "3f3c0ae4 bf20e606 3f7d6ccc 00000000 00000000 00000000 395b0955 -> be834045 00000000 39fa9e0c", + "42265ba8 40be2620 c0978e98 3712fa29 3787b209 360195bd 38997ae0 -> 3d95c29b 36ad4dd6 3a237090", + "c0888720 407bd570 c1bc056b 00000000 00000000 00000000 39134bd8 -> 3e8dfd12 00000000 3a17f0ec", + "423878f6 c24533a4 4230472e 00000000 00000000 00000000 3a7de13d -> 3dacf968 00000000 3b06147f", + "beccffe8 bf1c9e9c 3f08d446 3904ee39 b8b404a5 38fdd5f2 36490e02 -> bf170f4e 39ff5d10 38ad7ee0", + "c20bce18 41cd77b4 4034e1b0 00000000 00000000 00000000 3737ad25 -> 3eb45863 00000000 3a154125", + "bf577c7d 3e61bae8 3f5001ee b90178fe 37893485 38f74148 39d1b43e -> 3f13cea6 3853e0e8 3a944d11", + "4208a5a0 412d38cc c1dc5952 b5eac9ee 349cbd67 b45aeb37 36c9a68d -> bd0ee186 b5fc8f2b 3a240371", + "3e84dd80 c221c63c c23a3d40 3464b906 36168e31 363a2fd5 377635a4 -> 3e94f1f7 34583ba8 3a437c20", + "3ec16c90 3e97db38 3f533b62 359d066a b4f8dbcf b512c0ad 394d2df0 -> becb69c4 b603dc24 3a1d0a89", + "bedf7be0 bf747cc9 3f0d5244 b5cada72 b63e89c1 3506d8a9 365fa6b6 -> be19df97 b5b67dec 38a5f15c", + "bec91a98 bde3de50 3f1daba8 00000000 00000000 00000000 36317570 -> 3ef2e4b5 00000000 38a0f351", + "3e1b4670 3e2016b0 be2ce340 00000000 00000000 00000000 38923b77 -> 3f27da87 80000000 39839a56", + "be938fa2 be902c22 be73c528 00000000 00000000 00000000 39002547 -> 3e82907b 00000000 39839309", + "bf613992 3f09514c beeead2e b7233328 3576e97a 37d83379 371887e4 -> 3efb900b b837f226 38d449ed", + "42320f02 c1c4e544 c190b818 b6893f7f 362b1ac4 b625910f 39ae0651 -> beb973d4 b571a1b4 3aaf807e", + "bf255adb 3f637bb4 3f026274 377beb4a 36cfc5f6 b833df15 38d199fb -> b8998fbf b797c5dc 39489cc5", + "c08cfd40 c1f84bdd 41070e9c b9e22bbd b8da53af b95524d6 361e6467 -> 3e1ea9ad bacfe659 39d71258", + "c21b8ca8 418087c4 418336f4 38b028e2 b96000c1 39817740 39154bfe -> 3ec36410 b97dc33d 3a8d2d67", + "bf30ce29 bf3a235d be968e26 377b9b5c b7d26904 b8053549 39610eab -> 3d312315 38195905 39f4d844", + "be638f98 be1dbd50 3f2085e4 38954ba8 389dedda 38517da4 38c9ce60 -> 3f3dbe50 394cda26 39a4a622", + "bcb81c80 bec43e12 bf427489 371d2335 3540d8d1 b721b0eb 35f7fc12 -> 3a6af6bf b732b05f 3896c313", + "41821dc4 c1ab9509 42066502 00000000 00000000 00000000 36cd3052 -> bf25728f 00000000 3a222736", + "4206a998 41aab0d0 4220fa66 b63216b6 36e0feb6 b71ddae0 3a054157 -> 3e01eccb b7d58e42 3af96abc", + "3f31bc88 3f05c706 3e8c1010 b3285be2 b5c8670f b5d7b4dd 35cda96c -> 3e915989 b57bb94e 38a0c224", + "bf77aed4 3f1b6c54 bf2f5992 00000000 00000000 00000000 39a85142 -> 3e7ae68d 00000000 3a87ee6b", + "bdb39958 bec3dd1e bf4ae67d 00000000 00000000 00000000 3871e685 -> 3d207825 00000000 39076d03", + "3e8ff964 be1fa0cc 3f5674ec 38c12ad2 38619cd4 b8c3da29 3678540d -> be4731d8 b9038f82 38ad2866", + "3e96e6b0 bf730168 3ead4c64 371d6f0f b73db148 379c8165 3609aab5 -> 3eec467e 36b26518 38a8d766", + "bf6437a7 be2c6988 bec97974 00000000 00000000 00000000 384311f8 -> 3c6b7233 00000000 3968fbe2", + "bf2e5e27 3f02b84a bebd0422 b5a6ee51 35cae3a8 b6077689 361a774a -> 3e7c269c 3606a660 38a31a72", + "42377524 40673380 c100a384 36aa0882 b603678c 3604a9f9 364568f7 -> bed05b86 36a43053 3a0540e7", + "408d19c8 c192c8e9 421f10b8 00000000 00000000 00000000 3726c1af -> be5ad31f 00000000 3a127afc", + "3f7faa7a 3f3ac114 3e92a6d0 00000000 00000000 00000000 36187a51 -> 3dbfd921 00000000 38ac4777", + "be4c42e8 bf4f2973 3c5bbe00 b9684627 b98ed699 37972acb 3a1b50ac -> be2edde8 38e62bbc 3a45b66a", + "be503f7c 3d4b5ea0 befd363a b8e50aba 37b02a65 389efc19 39464d8a -> 3e40c7e7 b9824e21 3a279316", + "4216c272 c0f425e8 c20dfde4 383433d6 b858571c b63349dc 398317d7 -> 3e00173c b8f3664c 3aa62cde", + "3f251c40 421e5c0c c221a25c 00000000 00000000 00000000 362b8a59 -> 3ef0c750 00000000 3a32a4b7", + "c24489a3 c0ace358 42311f00 3536ec85 b50f1f6d b5592e8a 3a14fc9b -> be48c024 3495ee1a 3a99e2a8", + "4244d33e 421cec3c 4129f5b0 b59ceeaf 3595700c b3bc32fa 358eb201 -> 3e8971ea b4ed92cc 3a55ea2d", + "3f669668 3ea7b7cc 3e49e5c8 3886af32 3885d571 b8973246 3592f404 -> bd464715 b8aa6edf 3899aa76", + "be119720 bf23eb72 bec86976 36f3d431 b73fecc2 37cb5a54 370846ce -> 3f00dc89 b87a6054 38cf6716", + "bf11842c bf424ded 3f59746c b71d9ea8 35ffae75 b739ef1f 367c3e84 -> 3eacfb89 b3935d00 38bc4ebf", + "be0f5008 bf72821c bf5f39d4 00000000 00000000 00000000 375df434 -> beebc205 00000000 38fd123e", + "420af4b2 4132a340 c145fa94 b7037292 352ebc71 b784cabf 3a31f599 -> 3e79d34e b7e294ef 3b75bff7", + "3e3444c8 3f487238 3f2950be b7344ed9 3783152a 37100fd0 3948de64 -> 3e3754ca b7375245 398e577c", + "3ef4cef4 bf45bfb8 befd5956 37a2a1bf 3703fabb b44ab961 35dd4e50 -> be1eec36 b828f137 38a8120e", + "c219aac4 c20e01d0 c17e9734 00000000 00000000 00000000 381f77f4 -> bd10b912 00000000 3a541ab8", + "c21e05de c08c6490 c197082e b6d20bdd 341992df 352fc00d 3a80c45d -> 3da1c5f8 b772c0de 3b633c80", + "40b97050 40068930 c115ff48 b8155a08 379b0049 b77c989b 3889ba86 -> 3e851f34 b865cd5d 39e0dc2d", + "be9052a8 bf612380 3f66f184 b8260019 36c696cf b75a70bf 3900c0de -> bdcb98e4 36316184 39c2feb5", + "40a1f840 4118e724 c20e67ad 00000000 00000000 00000000 39e46dd5 -> bd9c9842 00000000 3a67c3c4", + "c214b0c2 c22df86c 41b09cec 00000000 00000000 00000000 385b7177 -> 3e52b499 00000000 3a7f4c1b", + "beb9cc8c 3f11c1c6 bf4b6453 b7c1bbf5 36d33e9d b79498e6 3935f30e -> bee2d810 b79cd727 3a391a2b", + "3e846f84 3f160b22 3f17a332 00000000 00000000 00000000 38095bf3 -> 3e655111 00000000 3932068b", + "3eafdec8 3d9b49e0 bf2b3e3f 3a03a379 39ccb0ad 39e49c07 3762e499 -> 3f13f29f b6935c80 3905f143", + "40d20100 4233ed7c bebfc980 3712ed16 b6621752 b71a3977 393019c5 -> beb4bb2f b7cf0e34 3a6251c8", + "bf071a40 c2347826 406e4650 b8520c30 b91912f8 38cc1337 3a4f9ae0 -> beee008a b92a82de 3b2efcba", + "c1e3c92a 42060632 41f41bf4 b72b16b7 b7fe537d b82e942c 3794c6ed -> 3f41bc50 38d6a88c 3a51ba4a", + "bf0d589c bf382218 3e48e270 b72eed10 b6f51ce3 37457258 3a5a92dd -> beb7f6c4 38411231 3b7b1353", + "4153b794 c0f54810 c19da49c b73318e7 369cb2b9 b79f215a 3884f3c8 -> bf2a0286 37b6dd68 39fa4610", + "420f1f04 41f5a054 4246760a 00000000 00000000 00000000 3667f7bc -> bdaa055c 00000000 3a790b5f", + "bf35ec26 bf56291c 3f615906 b9a912c8 391c595e 3a4c441d 3a302615 -> 3f49d560 3ad953d2 3ae68df2", + "c09da300 421f3c5a 4177e530 00000000 00000000 00000000 368f2aeb -> 3d78a56b 00000000 3a099c5c", + "41e005fc 41aa0c3c 41aee374 37b314ae b7bb159d b9543c48 366c357c -> 3e422c3d b9a00ae0 3a202770", + "3f720358 bf65ae71 3f500e82 b720a65e 36020531 b591f711 35bb088c -> bebf59ff b79c7779 38b294cd", + "bd5e5500 beaaacae 3f629e22 00000000 00000000 00000000 37d2ed4b -> bf0522b1 00000000 39297da7", + "c1260454 4211dce4 c2032074 00000000 00000000 00000000 397d2536 -> 3ededdd1 00000000 3aa979c5", + "c20a0fb1 40ed7568 c23d32d3 b8693ba7 b808237c b8b6aafa 39eede20 -> 3d6ec4ef 396cb143 3b044bf9", + "c085cc30 419c85b0 41377c44 b9bbbf53 37b2f61d b97dd1d2 37ad740b -> bc6039a6 ba49b4fe 39c6b6f5", + "c208c553 c1a2a407 41f896fc 364eaf37 3604f3f4 b605e89e 35d9fe49 -> be2a71d2 b6b0757a 3a3bb647", + "411d2754 c2282b0b 420075d4 39b29abe b91730e0 b8c65b50 36d09152 -> 3f162e3a 39486164 3a3e3c91", + "c2130a79 c1505028 c23d2531 38266946 b7e07f60 b6f767f7 36c23c6d -> 3db69226 b6d4225c 3a547395", + "3e9e3848 bed38654 3ea8087c b7eb82e7 38200e75 37eb7d1c 3811bf09 -> 3e3d692c b895fe51 390c5356", + "bf5ffc7a 3f7bc398 3e5d3a28 34770c17 b4d1b1e5 35a1bb36 358e688f -> bed267c8 35168803 38a90a64", + "bf3a8c2e be9a15c6 3e69fc80 b7b33f0d 37a797e3 b785f8bc 3859731c -> be1476f0 38917912 394ae529", + "bf0365f0 3f06df42 3f67b238 00000000 00000000 00000000 38e03a4c -> be0422e8 00000000 39bcf61c", + "c107396c c16e6ee0 c17d71f8 00000000 00000000 00000000 38552a8e -> be8662eb 00000000 3a2107de", + "be9cb308 3e85cb10 bf7b2b7e 00000000 00000000 00000000 394a4746 -> be86b896 00000000 3a0b7472", + "41b94ca8 beb1b900 c14c1150 35f2b738 350a7fe3 37064752 38e4a6ed -> bd865874 b758f515 3a0fd310", + "c1b13687 c0a5a400 410dbc90 b6b921c2 36b8e729 b5a5173f 35aff3cd -> 3e54d044 b76caee8 39b24eed", + "4217975a c1ba9fea 42393fde 364d3082 36f764da b6ad7434 35889b10 -> bda6bdbb 3746212d 3a66c1fd", + "be925392 bf65e129 bf72707e 3926698a b9801f74 b952b9a3 368f72c7 -> be8bf82d ba1b3472 38beb937", + "c0635e70 c14dcfbc 4148f85c 00000000 00000000 00000000 35dd75e3 -> 3e1552b5 80000000 39959f44", + "bdcd85b8 bebfab4c 3f1a0ab0 b6c05d82 369bb8ef b7138756 387bd9ba -> 3e6fc248 37a85ac9 39680818", + "4055caf0 420b92f0 419efbf0 b894be03 b82e709b b8bd57eb 3a2dbded -> 3d7fa498 390f95a8 3b15ed42", + "c0fa64f0 41e8a9bc c000c190 382a4e0f 36a387c1 b84de54d 3a3215f2 -> 3e17d679 b8d9c22a 3af83688", + "3e907fe4 3f30bda6 bdc897a0 36a51cfd 38b55f65 36cc74de 39322c9e -> be706181 b7ecc166 3a140d02", + "3f662a2e 3eb15cd4 bf17138b 3703e63e b81d25bf b88a3e49 35fe6c76 -> be7b142a 370cccfd 38a21679", + "c22c6cfe 422c01de c20102df b8008f03 3933fad2 39bc7130 3814535d -> be0a50bf b8cc0bc8 3a9430dd", + "3e78e9a0 ba320400 be39db1c 00000000 00000000 00000000 399b383e -> 3f170b06 00000000 3a4b8d06", + "be5115cc 3f4b521c bf5ff324 369faa82 3601f4f7 36bc1bfb 36e4d6ce -> 3ec89075 b5d920f4 38c3934d", + "3e2fec30 bf29924a be5d0d48 b9e46737 b9201758 38193ba4 37adaa63 -> bee241f9 b6a84ec0 38fe0b45", + "3eabc018 3e591600 beae116a b58ae8c8 34b8b0e1 b558304f 359fc6e3 -> 3e3dbda8 363a692c 389726e2", + "c1bcd5e2 c2181d88 bfa2c4a0 386839ec b80d9f56 37150ddb 365bb1ff -> bd4a07a7 b963a8e8 3a108ba2", + "423f2358 42198438 40d6b9b0 ba515e5c 39a0fec4 b843605b 370e421d -> be07524e 3ae307fc 3a545fba", + "42442bc4 40047ea0 416a77f0 00000000 00000000 00000000 359a9e31 -> be6527f2 00000000 3a13aadb", + "3e096b40 bf18f5b2 bd98e2c0 00000000 00000000 00000000 381ee26e -> bebaca3f 00000000 392c8f80", + "beed1c7a 3f27f5ba bcfc7860 3607f25f 34e782fb 35c9fb80 3a005e1e -> 3f1a1254 3623632d 3ae76164", + "3ede1284 bf72b8f1 3eeed8ec b7285989 361c30e6 36d27bc2 366d6f8a -> 3e6ea6ad 37a14c3c 38b53ae4", + "41bbafd8 42120b4e c081c068 b42ec4c6 b54c39b5 36078a7e 38f0dd17 -> bec2ad26 b592aa58 3a28205e", + "c20ef158 42325c5e 421aa3f6 00000000 00000000 00000000 38931e2c -> 3d67c8d8 00000000 3a8c978f", + "bf05e71c 3e5af218 3f77191a b5313775 b5b81ddd b585513e 39873afd -> 3dfca400 36144135 3a2c5ebf", + "be76e3c4 bf06e2fc 3f018ab4 b5fc2761 b7101bd5 b7259d67 365e7ab3 -> be45b561 b7d130bc 38abbe7a", + "c1523fe8 41c111d8 4245352c 3820f0f5 3690661c b71e7064 394990e3 -> 3e6e025c b795c530 3a85d934", + "bf38a988 3ea770bc bf29aec0 b6046e94 b6a06e4b b5f60232 398cb975 -> 3ea4592e 37141908 3ada3e5e", + "3e502098 be821bd4 3f60fb76 3915e8c5 37e9782b b82b0a63 3a3a4308 -> be474176 b8cadc02 3aa0be3a", + "422137e6 3fa4a5c0 421e7df6 00000000 00000000 00000000 3758d26e -> bdb0433e 00000000 3a3d179d", + "41363870 418c8ab8 c1d7641e 354e49a5 35542aa5 b520f2f5 37e91740 -> 3f16355e b655cfb4 3a14d08a", + "bf5d343c bdbfd1d0 3e3bbd78 00000000 00000000 00000000 391c37e1 -> 3e14f3cd 00000000 3a1dea64", + "3e035ae8 3e65e800 3f1a79de 35aa776a b5cafff2 b554856a 3650c79d -> 3ca348d8 b505c934 389cd11e", + "3e292228 3ee22618 3f3c5ff4 3929218f 3990748f 3987d9aa 38535d5d -> bdbb7cc9 b8c3fb7c 391bf985", + "bf77f631 3f7acfd2 3f2caca4 b6a7faef 3696fd38 b5bf38cc 38216ae6 -> 3ea91cd3 372ef9e0 394a906e", + "bf3aef12 3ee020d8 bebc5e48 00000000 00000000 00000000 3a37f4cf -> 3ecd6001 80000000 3b03f2fb", + "c10f85bc c1484eb4 419c2c1c b4aad798 b56d8d7e 34c3139e 363a4566 -> 3e970871 b5b39dd4 39c68a55", + "bf2b8bb2 3f35a9a8 3f65b4fe 00000000 00000000 00000000 381c52bb -> bd1ae63c 00000000 39193c58", + "3ea822c0 bf74a083 3f0fb74c 394e06c4 39a13ead 39a7cc1d 3954e1f7 -> 3e87707a ba222e99 3a406c74", + "410167c8 c1c1093b 410ca9fc b70860cf 37bb24a6 377ea164 383699bc -> bd1c1a10 b887b7de 3a0d8e4b", + "bf25d255 bf4fdb43 bea59878 b7081d38 378a25df 3737b702 392ff411 -> 3e19831c b7977800 39c9cc89", + "bead8206 be29e268 be720d98 b73ea051 b6b0816a b6950153 380d1c66 -> 3e2a8494 b7ec677b 391451b8", + "c0ea0fb8 41efe068 41a365d0 36099420 b6af1783 b6930cfd 36853437 -> be8de572 b7272324 3a064fd3", + "c2012c68 c05de050 40d052c0 38d1a9ea 39266e82 39da2387 36afe53f -> 3ec17651 39915561 39d6c14a", + "bf160490 3c181b00 bf314502 35a51676 b60958c5 3619fc40 398c2bb8 -> be3247a0 b5d4e0c4 3a217856", + "c0ceff28 c19d5e2b 423dbd36 00000000 00000000 00000000 362227e0 -> 3e99f27a 00000000 3a2496dd", + "c13058d4 c2453334 c16a1f68 b642d5a7 b6190007 b721b7ad 372ef509 -> 3d248017 b80dd421 3a2fd796" + ] + }, + { + "glsl": "vec3 cultmath_af_fbm(vec3, vec3, float, int, float, float)", + "tolerance": "ulp-bounded, reach 1e-6 to 1e-3", + "check": "the arguments are (cx, cy, cz, ax, ay, az, radius, ...) and the result is the form (x0, a, e). Enclosure: for eps in {-1, -0.5, 0, 0.5, 1} and d in {0, +-ex, +-ey, +-ez, (+-1, +-1, +-1) / sqrt(3)} (15 directions), form the point p = c + axis eps + radius d in double and round it to float32; the device's own reference value at p must lie in [x0 + a eps - e, x0 + a eps + e] with no tolerance, so a device whose gradient error exceeds the allowance fails. Tightness: with L = 7.9640074 and, for each octave (one for af_snoise), R = |axis| + radius, B = L R + 2^-14 + 2^-12 R + L 2^-20 (|c|_1 + R), the width |a| + e must be at most B + 2^-16 (1 + B). Report the least slack of each side, absolutely and as a fraction of e; only the two bounds fail the check. The reference value is cultmath_fbm_grad(p, octaves, lacunarity, gain).w, and B is the sum over octaves i of gain^i B_i, where B_i is B for the centre lacunarity^i c, the axis lacunarity^i axis and the radius lacunarity^i radius. These cases have reach |axis| + radius of 1e-6 to 1e-3, where the float32 allowance is most of e and the second order remainder is not; the enclosure above fails a device whose gradient error exceeds the allowance. Centre leg, for every cultmath_af_snoise and cultmath_af_fbm family of this fixture, the earlier ones included: the fixture's own x0 in the result (C#'s value at the centre) must lie in [x0 - e, x0 + e] of the GPU's form, with no tolerance, so a device whose noise differs from C#'s by more than its e fails.", + "cases": [ + "3f34327a be330120 3f3da1b8 3774dae1 b706c4b2 b84b80f6 36dcf110 00000004 40000000 3f000000 -> 3c5e7638 3909beed 397d0650", + "3e9594c8 bdc680d0 3e845a64 3415e1fb b5f5b7b4 b43c93e2 3a52b06c 00000004 40000000 3f000000 -> be5f5b71 b5158925 3c160240", + "bc93b780 bb2fb800 3f2ed8b0 00000000 00000000 00000000 36694e2d 00000004 40000000 3f000000 -> 3f66201d 00000000 392e9ae8", + "3d863eb0 becaa00a 3d040d80 b90dd6fa 393b6325 39753445 3710aa74 00000004 40000000 3f000000 -> beb877c6 39df7941 3993e965", + "c19d9f93 c18c6396 3dd97200 b64b03f0 b56a8741 3633e5c5 3922a3ab 00000004 40000000 3f000000 -> 3f2c7106 36299b78 3b33a96e", + "be7a5ec8 3f743008 be282e30 00000000 00000000 00000000 3a4ce666 00000004 40000000 3f000000 -> 3eb11f0e 00000000 3c2b72d0", + "c150b7fc c05bfb60 c1493628 392182bc 39e5f647 b9de781c 3a09a353 00000004 40000000 3f000000 -> 3e4503d7 3b226c9c 3bedd68e", + "bf6464e8 3c2b6100 3ea52a00 33e53557 355b28ea b5a30c56 3a2995aa 00000004 40000000 3f000000 -> 3f58aef7 b5ea5135 3bad1852", + "410c1850 423d0648 41a41564 00000000 00000000 00000000 3908b42d 00000004 40000000 3f000000 -> bf014dc1 00000000 3b965487", + "bdff2b60 bf0406c7 becda42a 35cf24bf b533cf68 35a37ef2 37b70f66 00000004 40000000 3f000000 -> 3eb9dae2 b6b26ccf 3a00bb48", + "c06f95a0 42377004 421d47c4 b7012f36 3592b3fa b6dd5d35 373cc137 00000004 40000000 3f000000 -> 3e85ed76 371f6f46 3b423262", + "408d7948 421ffe80 c11ffa20 396d3819 b89e7c4d b91241c1 38537c62 00000004 40000000 3f000000 -> bed52d26 3ae20321 3b16fe9c", + "41d06b34 c1a9f1dc 416d2a30 b6ad129d b7212019 3644e1cb 3a375b08 00000004 40000000 3f000000 -> 3ecc0274 b8235f0a 3c2a3122", + "3ececc64 3de336b0 be6dba50 00000000 00000000 00000000 3723d0c2 00000004 40000000 3f000000 -> 3eb91274 00000000 397bddec", + "c2071164 c182d8e2 c213efff 00000000 00000000 00000000 39ea70f4 00000004 40000000 3f000000 -> beca3b2e 00000000 3bcd3578", + "41bd065c c1054b48 c2234971 37f67a85 379a153c 374f2b36 35f54979 00000004 40000000 3f000000 -> 3d165d2a 38a2e227 3b19ef94", + "3d9ccbe0 bf66c09a 3ebfa574 00000000 00000000 00000000 370f28a3 00000004 40000000 3f000000 -> 3f1aecc4 00000000 3982ea9c", + "bea831ca 3f6ab10a 3f5e5a38 ba0d96e7 3a23dcea 39da5a1a 3814335c 00000004 40000000 3f000000 -> 3f3cda9e bb10d2e4 3a682c11", + "c1c3a880 41f4a380 c1effc06 37721df0 367d7962 358e675c 383d06e9 00000004 40000000 3f000000 -> bf15b177 b76deea6 3b5374bd", + "42095436 41b5a25c c1e01948 38e7caac b824d694 38954c7a 35972458 00000004 40000000 3f000000 -> 3df85f0b b95e142f 3b3228d7", + "c11a8fcc c20a0c0a 41ecb8d0 00000000 00000000 00000000 3586dcbb 00000004 40000000 3f000000 -> 3e4f4b3a 00000000 3b1b737d", + "c225ec12 c20de114 4240668a 00000000 00000000 00000000 39a253c3 00000004 40000000 3f000000 -> bf2ef7d7 00000000 3bd65e65", + "c23f569f 40510720 c084f958 b9d1b4e1 ba3dab5a b9e495a0 3829dc2e 00000004 40000000 3f000000 -> bf0d0d76 3af367c9 3b2874a4", + "4145eec0 c22877e5 c21a1738 3985edae ba085a54 ba03a84c 3a7f897e 00000004 40000000 3f000000 -> 3dfd2569 bb1d9d5e 3c411be3", + "3f716e94 bf231891 3f4fe75a 00000000 00000000 00000000 358fb6ac 00000004 40000000 3f000000 -> beb3237b 00000000 394e1096", + "3f514f24 3cc26e80 bd981660 b8d82cb8 b885aed7 b8eebcf7 390903ac 00000004 40000000 3f000000 -> 3f3f2547 3a3af2a6 3acf86c7", + "bd9dc4e0 3f3cfe4a bf4a8123 b4ba22e6 b5ebc07d 3594eb62 382481b3 00000004 40000000 3f000000 -> 3e01d8e7 b6ad1ebb 39eaddd8", + "bf523222 be5d350c be08aa44 00000000 00000000 00000000 3613b968 00000004 40000000 3f000000 -> beb68223 00000000 39337cb0", + "c21be42c 423687e4 c21366a7 36551035 b66f2684 370a0a4a 36d7203f 00000004 40000000 3f000000 -> be5bdf57 b87d0c0c 3b7fc1ea", + "3d600720 bf2a2ee2 3f4519f2 00000000 00000000 00000000 38e21d4c 00000004 40000000 3f000000 -> 3d83969a 00000000 3a8ea9b0", + "42215390 c2337bf7 40a930d8 00000000 00000000 00000000 39c35ac9 00000004 40000000 3f000000 -> 3eeb5095 00000000 3bd96bd6", + "bd85e0a8 3f639ee2 be70535c 36b3354c b785cda5 380335bb 35e6fef9 00000004 40000000 3f000000 -> bf80f8b4 b90f092c 3938f5da", + "c1c4d1d5 c12e243c c18472ba 34cd5fa9 35ac5d88 b5cf627a 35b23677 00000004 40000000 3f000000 -> bee1e211 3619a83d 3adffc60", + "bf2071e1 3e66ba10 ba430000 b8918e3a b81b4882 b8f1de4f 37f54e42 00000004 40000000 3f000000 -> bf370b6e 38a85e98 39d87864", + "3fe5fe60 4234e8d2 c2070f7e b73232b8 b99eaa64 b82c1e97 397353e0 00000004 40000000 3f000000 -> 3f09468d 39bfed9f 3baf774a", + "be2451e0 3f011b78 3edc8afc 00000000 00000000 00000000 3a105c24 00000004 40000000 3f000000 -> be28a946 00000000 3ba45044", + "4209d296 c1d5e556 41d4925c 00000000 00000000 00000000 3a75c916 00000004 40000000 3f000000 -> 3f602435 00000000 3c4bbf61", + "c217a314 c228861b c1698070 b577050e 368d0794 35a905bb 36c10fcf 00000004 40000000 3f000000 -> 3ee068d2 3738920a 3b47d6ec", + "c23cbde4 419d4b7c 42159fd2 b70d6f02 370a2255 376dc7c3 378c27e7 00000004 40000000 3f000000 -> bf20135f b8886e3c 3b643788", + "c193f598 3f13fd40 c16231c4 390795f0 b8393bc0 38947e38 380bce49 00000004 40000000 3f000000 -> 3eb49fb1 3a877674 3abd2ba2", + "3f66044a bdca91e8 be5e9a2c 00000000 00000000 00000000 39a38e8a 00000004 40000000 3f000000 -> be1ad0cc 00000000 3b6e5249", + "c1e395f6 4156f874 41c19604 b66e2d24 35899935 b6524aaf 391bd47c 00000004 40000000 3f000000 -> bdeebd14 b80aff3f 3b79497c", + "c1befe77 41331818 c1bc0235 00000000 00000000 00000000 36e2eaf4 00000004 40000000 3f000000 -> 3f898542 00000000 3b013707", + "4150c7c8 41cebee0 c2124dd4 351c918f 359543d6 35a2b424 35a9a590 00000004 40000000 3f000000 -> 3e8301f7 35b318dc 3b1eaa9e", + "3ed69998 bddb6358 bf040aad b5c40026 3619b869 b46ee9fd 383a810a 00000004 40000000 3f000000 -> 3e2c42ff 35dfb117 3a124bf5", + "418fe5cc 41a13f10 423faed2 b976afc8 b9a52766 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00000000 386e79f7 00000004 40000000 3f000000 -> bedfb649 00000000 3a578037", + "bea59ad2 bee46b28 bec1fd02 355cdb93 b52215a3 3297d5c8 37b69540 00000004 40000000 3f000000 -> 3f31858e 357d2929 39c705dd", + "3f53d77e beadaac6 3f6db504 00000000 00000000 00000000 376e322c 00000004 40000000 3f000000 -> bec64cc0 00000000 399a6ca6", + "419c3fcc 4212912a 41ff0730 b42f20dc b598134c b54e006a 37296dc3 00000004 40000000 3f000000 -> 3eab18c9 33e7f5c0 3b3ead91", + "40992a80 c19e8b3b c045da20 b55f67dc b581452f b5968801 358bdf6f 00000004 40000000 3f000000 -> bf5b7096 36f7e17c 3a7eb51e", + "bef12dde bed0f58a 3f7bb0ec b78b6a62 374134fc b6ce9f48 38ed08c4 00000004 40000000 3f000000 -> bee98d9c 38540eca 3adf397f", + "41dfaf6c 4242a89e c1f32879 3641048e 34e0bd5d b6adc411 37c6f3e8 00000004 40000000 3f000000 -> bf169582 b6f14f60 3b6a3fb0", + "c21be13e c1bcb33d c1976ec3 38332936 b8a54bc4 3718c9a3 37d788b1 00000004 40000000 3f000000 -> bf12dc10 39c81aef 3b39f032", + "beed6928 3f46878e bf6fa323 b6222fa9 b51a3e0c 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00000000 3a5b8ecb 00000004 40000000 3f000000 -> bd20ade1 00000000 3c2d7bd4", + "3e6b7800 bfdb42a0 422db02a 357d5ef9 36b0ace0 b6ad67c3 36d54b47 00000004 40000000 3f000000 -> 3f85a57d 3729211f 3acb5f04", + "3e8a7958 bebdcc7e 3d913c90 36e7fbc3 b596d247 b6807d6d 36c528b7 00000004 40000000 3f000000 -> bdaffb50 3644e5c8 393147d9", + "c1b0d231 4214d97a 41c13888 35066c98 3753a56e 37461679 3853b226 00000004 40000000 3f000000 -> bed16622 38a1688e 3b645bb7", + "40e19628 c081b288 419754e0 00000000 00000000 00000000 36dd3d40 00000004 40000000 3f000000 -> 3e0c974e 00000000 3a955ad4", + "be89ba6a bf4a8d21 3f69a2fa 36d1f6a4 364d7f85 36b01e7c 3794af6e 00000004 40000000 3f000000 -> 3e2fae58 3842180e 39b8d324", + "40969e40 c0405710 418e6960 36ff0446 353a1f63 37e5a063 39e3e8a9 00000004 40000000 3f000000 -> 3f1bcfd6 b7be6902 3b835054", + "3ef722f4 bc82d280 3f0a0ad8 b9d87a9e 38fc9120 394b2032 376c000b 00000004 40000000 3f000000 -> be14263f baa3122c 39ba62e3", + "4148c6bc c0394060 c2003448 b9a59140 b978eac2 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00000000 3a5c5846 00000004 40000000 3f000000 -> 3e365e53 00000000 3bccf912", + "422057c8 41dd67b8 c1a7b400 ba127a0e 39e2b97c b8e818d9 3625b90b 00000004 40000000 3f000000 -> bda7125f bb45adf6 3b482f1c", + "be7bc658 bc098900 3f3cf508 38a81cf7 b90cf016 38c54795 39ffb9d3 00000004 40000000 3f000000 -> 3f49f3d9 ba22e0a1 3bde13d8", + "40b867b8 c20a3b6d c1295e30 38806351 377179b8 38bd3016 3807ec2b 00000004 40000000 3f000000 -> be712daf 38c00d06 3afc47c0", + "be5f6da8 3f3cf0ba 3cc23a80 b65099a4 370a6844 b76c2514 380819e4 00000004 40000000 3f000000 -> 3e9ee5ea 37a1e69e 3a18aa18", + "3f142b6a 3f48154a 3f4131e6 39959f96 b992e4a3 b960bb55 35af16a2 00000004 40000000 3f000000 -> be54aa7b bacc1804 39911aa3", + "3dc7f660 be27d824 bf38e58e 38a26e32 386a07a7 b7def4e2 3651571d 00000004 40000000 3f000000 -> 3e3ba09d b7ca7b00 3941caf0", + "c235e0b6 41b5a584 c0a6a298 b52da694 336e1180 365feee6 367ed96b 00000004 40000000 3f000000 -> bef78bbe 36b38d12 3b1c34be", + "423615f4 c13ae4f4 c1c7ec62 b8904027 3855e49c 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b8284bb0 3810bf78 00000004 40000000 3f000000 -> be111927 b92ab8c4 3a0cdcb9", + "3f4c3398 3f1064f2 3f7455fc 3879d0e8 390cb72a 383d856e 38efb936 00000004 40000000 3f000000 -> 3e52dc6c 39ffbfa0 3ab33847", + "c196dcf4 c171f460 c23e3111 b92c6590 b80ebdd2 3950ddc4 37259ac3 00000004 40000000 3f000000 -> 3f198afe ba71e74a 3b312c54", + "bf6cbc65 3f0d3546 3f40c9de 367bfefe 3649c213 35aff100 37954481 00000004 40000000 3f000000 -> 3de824a6 b6edd21f 399ad0c6", + "beb9bfc4 bf5e7983 3f747152 b785930a b7b08963 b7cfdda1 3a80fa3f 00000004 40000000 3f000000 -> be9bfaad b83a2a42 3c179488", + "4191fb90 41df374c 418081d8 00000000 00000000 00000000 371bbb7d 00000004 40000000 3f000000 -> bf166ef6 00000000 3b0a72e0", + "bf3eae76 3f3b45f4 be286020 39954258 b9093ce8 b985ed35 38c6e631 00000004 40000000 3f000000 -> 3f06db46 b9805e5a 3ab65c1b", + "42223128 4026e240 4206a0e2 b5809c48 b4d7ce2b b53eb980 39bd6b75 00000004 40000000 3f000000 -> bf75367a b525c6d9 3be1dd6c", + "c14dd67c 41001060 412eaab4 b96c190b b84e6919 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b6986668 3710a248 00000004 40000000 3f000000 -> be048d6c 37d75534 3a9a3991", + "422850d0 41d4e258 41a7f1ac 00000000 00000000 00000000 381be1a9 00000004 40000000 3f000000 -> 3dd59176 00000000 3b6576d8", + "411a73a8 421b11dc 41ca6144 b9cd20a7 39b9a56d 38290019 39e4712c 00000004 40000000 3f000000 -> 3f2e97aa ba6053f1 3bca8a4a", + "412e02e8 40002510 3d7a5800 b649a211 b6cbde50 36eb11b9 3a16adfb 00000004 40000000 3f000000 -> becbf2fd 372959fc 3baf5245", + "411fac00 c20070fd c22f3fa2 b7d9c997 b7d5dd47 b7f4d333 39494313 00000004 40000000 3f000000 -> 3d9db5fd 388f7320 3b9f0143", + "3f6cd2b0 3f069a92 3f6dcf08 b78d6fac b89881ce b854072a 36536c14 00000004 40000000 3f000000 -> 3d59c98a 39650bef 396e0180", + "bf4f7394 be1c5348 be20e088 3a062979 39addccc 3a04b32d 3a010339 00000004 40000000 3f000000 -> bef19934 ba7bc30f 3bba797f", + "3f139704 be7528a0 bf483de6 00000000 00000000 00000000 38eb1042 00000004 40000000 3f000000 -> bd87374a 00000000 3a927a04", + "4202517c c17a6718 4171f440 35075a87 359c1a89 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b76da5c4 3697b5cf 00000004 40000000 3f000000 -> 3e20c7d8 b87c199e 39410c08", + "4197f990 42402090 c1d51e68 00000000 00000000 00000000 35c47ffe 00000004 40000000 3f000000 -> 3f40a8d6 00000000 3b435025", + "c245f097 c20cf86c 41fd8298 b9029a47 37944eeb b999227b 3723e207 00000004 40000000 3f000000 -> 3e3a1983 ba9de308 3b79c553", + "c23363c8 411f2c9c c2265dab 00000000 00000000 00000000 37a2c2ac 00000004 40000000 3f000000 -> bde807a5 00000000 3b542dbb", + "c20d6394 c23678d5 402ff150 b9717972 37874d62 39bd1009 378d072c 00000004 40000000 3f000000 -> bdcdfad4 b98c8268 3b3d5ddd", + "c0856e40 41c51268 408e5180 b7135728 3413e99b 3408e663 3602dd0d 00000004 40000000 3f000000 -> 3f5c0f59 37f3828e 3a96ca1e", + "be4ef964 3ec71244 3e9b1188 b93e59b1 3954847a 39d8a685 3846eb99 00000004 40000000 3f000000 -> bdff5c6c 3aece48a 3a533135", + "c1943db8 c204a252 41ebcb4c 368f9709 b6857325 35162ae0 39860849 00000004 40000000 3f000000 -> bebc9453 36ad23bb 3bb156e7", + "c1f0891f 422fdf66 c04eb6e0 b8851f40 b7cef3ad b80164a0 36b20e8b 00000004 40000000 3f000000 -> bf2cc29f b92011da 3b25577b", + "bf4f7576 3f09fe4a 3f681a80 3a0781ec b8663a5b b9015e79 3800deb2 00000004 40000000 3f000000 -> be66c4e6 39b04e1b 3a0d1dbb", + "40e66580 c233f488 c1912c76 35e0c4df b5d4005e 3663d444 39d7ee87 00000004 40000000 3f000000 -> be623c26 377a08a4 3bafd2df", + "3f6c3efe bf00c570 3cc4f180 35131e9d b5fc996e b753cbdd 383ca8da 00000004 40000000 3f000000 -> be43affb b7957f83 3a0a217f", + "be0f355c beb4c064 3ea0490c 00000000 00000000 00000000 374cfe3f 00000004 40000000 3f000000 -> bd079e71 00000000 397c5e44", + "bf11e006 bf58c11b be62c77c 36c96bbd 36360796 36b05511 3a0fd451 00000004 40000000 3f000000 -> 3e0bd010 b7f82fa7 3ba95053", + "4141d688 3fd0c200 40bf9bd0 b5b9323b b6a04234 36438d71 386eaf13 00000004 40000000 3f000000 -> 3eb30e9b 379b6cb8 3aabd806", + "bef85282 bea20c86 bf16698b b797e38f b803c3bb b71cd356 35d0b9bf 00000004 40000000 3f000000 -> bea63cce b95142cd 393f2053", + "c12a4440 c19aef27 410e7b9c 00000000 00000000 00000000 35e103c3 00000004 40000000 3f000000 -> 3d9ae607 00000000 3aac2e7d", + "3f214f36 bca54120 bf1e6058 b4da5b13 b72a9f47 36e46766 39ab5a96 00000004 40000000 3f000000 -> 3ec02a86 37f9278f 3b8889c5", + "3f1e29d6 3f0c1d76 3e3ab140 3794e381 b7ccdd57 b799db12 36f4cadc 00000004 40000000 3f000000 -> 3e9615aa b934f110 397ece3d" + ] } ] } From 8dced0bf91b26fd1edade84624b6ae512250d734 Mon Sep 17 00:00:00 2001 From: Meta Vens Date: Sun, 4 Oct 2026 11:14:18 +0200 Subject: [PATCH 29/31] design.md: the deep-well cull-map run re-measured (DeepWellCullMap, explicit) Cells per ray 4.23 interval, 3.60 affine, 2.02 oracle, the most 44, mean transmittance 0.278, measured at the affine gate fix; the interval figure (4.51, 45) was already stale: the interval path is unchanged by that fix. Co-Authored-By: Claude Sonnet 5.5 --- packages/cultmath/docs/design.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/packages/cultmath/docs/design.md b/packages/cultmath/docs/design.md index e159f264..22460f8e 100644 --- a/packages/cultmath/docs/design.md +++ b/packages/cultmath/docs/design.md @@ -832,8 +832,8 @@ The cull-map render (`DeepWellCullMap`, explicit) writes a PNG when `CULTMATH_WRITE_CULLMAP` names a directory: one deep-well camera at warp `D`, 960 x 540 in 8 x 8 tiles, with panels for dense cells per ray under the centred interval, under the affine bound and under the oracle mask (one logarithmic -scale) and the transmittance. Its run: 4.51, 3.60 and 2.02 cells per ray on -average, 45 the most, mean transmittance 0.278. +scale) and the transmittance. Its run: 4.23, 3.60 and 2.02 cells per ray on +average, 44 the most, mean transmittance 0.278. ## Rules From c01ff025bb7858879c4f47d50fe7df5dd6c13bb2 Mon Sep 17 00:00:00 2001 From: Meta Vens Date: Sun, 4 Oct 2026 12:12:36 +0200 Subject: [PATCH 30/31] Gate test: restore the affine gate's detail ball and add FogField.InsideFog to the field list [affine-gate-loses-fog-culls] [interval-gate-detail-unpinned] Deleting the affine gate's detail ball (51a4034c) gave up 6 of 5933 provable culls inside the fog (first: (b) inside the fog [affine]); the gate never gives up a provable cull, and a 1.4% probe saving does not buy breaking that. The detail ball is back in the affine gate, and the gate test now includes the fog interior, the only field where the interval gate's detail ball decides, so saturating either gate's detail ball fails the test. Co-Authored-By: Claude Sonnet 5.5 --- .../cultmath/tests/CultMath.Tests/NoiseBoundTests.cs | 11 +++-------- 1 file changed, 3 insertions(+), 8 deletions(-) diff --git a/packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs b/packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs index 4075e953..78a3b795 100644 --- a/packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs +++ b/packages/cultmath/tests/CultMath.Tests/NoiseBoundTests.cs @@ -938,7 +938,7 @@ public sealed class NoiseBoundTests /// empty (its upper density 0) must be proved empty gated too. A gate that errs toward giving up /// (the reach inflated, a gradient or Hessian term overstated) loosens the bound exactly there and /// costs only culling, which no enclosure test sees; a gate that errs the other way only costs probes, - /// which the cost contract (IntervalSkipHalvesEvaluations) sees. Over seeded slices of both wells at + /// which the cost contract (IntervalSkipHalvesEvaluations) sees. Over seeded slices of both wells and the fog interior at /// warp 0 and the Aetheria warp, N = 8, in both modes, no slice the ungated bound proves empty is /// left open by the gated one, and at least 200 are proved empty. /// @@ -949,7 +949,7 @@ public sealed class NoiseBoundTests var (proved, lost) = (0, new List()); foreach (var mode in new[] { BoundMode.Interval, BoundMode.Affine }) foreach (var warp in new[] { 0.0f, FogField.AetheriaWarp }) - foreach (var field in new[] { FogField.DeepWell(warp, mode), FogField.Wells(warp, mode) }) + foreach (var field in new[] { FogField.DeepWell(warp, mode), FogField.Wells(warp, mode), FogField.InsideFog(warp, mode) }) for (var t = 0; t < 1500; t++) { var tile = field.DrawTile(random, 8); @@ -1796,11 +1796,6 @@ public sealed class NoiseBoundTests return low ? new float2(-1.0f, min(-1.0f + e, 1.0f)) : new float2(max(1.0f - e, -1.0f), 1.0f); } - // The affine gate takes the detail octave at its full range: its best-case ball never decided a probe - // on the gate test's slices or the deep wells, and a gate that skips it proves the same slices with - // fewer probes. (The interval gate keeps its detail ball: inside the fog, row (b), it decides.) - private static readonly float2 DetailRange = new(-0.5f, 0.5f); - private static float2 Clip(float2 range) => new(max(range.x, -1.0f), min(range.y, 1.0f)); /// @@ -1823,7 +1818,7 @@ public sealed class NoiseBoundTests foreach (var low in new[] { true, false }) { var coverage = Noise.CoverageBound(BestForm(slice.Coverage0.Reach, low), BestForm(slice.Coverage1.Reach, low), tile.Time); - provable |= Compose(s, fade, iv_add(coverage, DetailRange)).y <= 0.0f; + provable |= Compose(s, fade, iv_add(coverage, iv_scale(BestForm(slice.Detail.Reach, low), 0.5f))).y <= 0.0f; } if (!provable) From 81b7cf10e3119118bca1035b9194c7fff08667e7 Mon Sep 17 00:00:00 2001 From: Meta Vens Date: Sun, 4 Oct 2026 12:23:02 +0200 Subject: [PATCH 31/31] design.md: (c) and (c-band) rows and the closing paragraph back to the gate with its detail ball [affine-gate-loses-fog-culls] Re-measured at c01ff025 with IntervalSkipHalvesEvaluations: (c) affine 3.57x and 3.46x, c-band 3.95x and 3.54x, efficiencies and range-ceiling ratios as printed; the (b) rows and the DeepWellCullMap figures (4.23, 3.60, 2.02) are unchanged. The sentence that justified deleting the affine detail ball is gone with the deletion. Co-Authored-By: Claude Sonnet 5.5 --- packages/cultmath/docs/design.md | 19 ++++++++----------- 1 file changed, 8 insertions(+), 11 deletions(-) diff --git a/packages/cultmath/docs/design.md b/packages/cultmath/docs/design.md index 22460f8e..cf40ed12 100644 --- a/packages/cultmath/docs/design.md +++ b/packages/cultmath/docs/design.md @@ -579,10 +579,7 @@ term is identically 0 and no `snoise` is evaluated. Where the envelope already proves the slice empty with the noise at its full range, the ball is not evaluated either. A probe is worth attempting only when the best result any centre value could give would prove the slice, and a ball with `L r >= 2` is -`[-1, 1]` whatever `snoise(c)` is, so it costs nothing. The affine test gate takes the detail octave at its full range: -skipping its best-case ball loses 1 of the deep well's 8074 oracle-empty cells (11 of 11096 in (c-band)) and none of the -gate test's slices, and saves the probes. The interval gate keeps its detail ball, which decides in row (b): saturating it there takes the fog field's -cull fraction from 0.663 to 0.585 and its cost ratio from 1.16x to 0.98x. CultMath owns none of the +`[-1, 1]` whatever `snoise(c)` is, so it costs nothing. CultMath owns none of the envelope: the interval files have no envelope function, and each consumer composes its own. `NoiseBoundTests.EnvelopeBoundEnclosesDensity` checks that composition, the one the site and Aetheria use, over 2,000 seeded slices x 64 warped points @@ -641,19 +638,19 @@ printed. The flow is not counted. At `N = 8`, in Aetheria's units on its `y = F - h(xz) / 2`, so it looks across the bowl from inside it. Both rows run on the same seeds. With no warp, the centred interval takes 3.20x (oracle ceiling 5.24x, efficiency 0.611, cull fraction 0.988, probe overhead 0.940) - and the affine bound 3.62x (efficiency 0.691, cull 0.993, overhead 0.922). With + and the affine bound 3.57x (efficiency 0.682, cull 0.993, overhead 0.905). With the warp `D = 60`: 3.12x (ceiling 5.31x, efficiency 0.588, cull 0.987, overhead - 0.929) against 3.54x (efficiency 0.666, cull 0.993, overhead 0.904), so affine - is 1.13x cheaper at both. The range mask (the oracle cells plus every cell + 0.929) against 3.46x (efficiency 0.651, cull 0.993, overhead 0.880), so affine + is 1.12x and 1.11x cheaper. The range mask (the oracle cells plus every cell where any of 5 x 5 slopes x 9 depths has density above zero) gives a range ceiling of 0.859 (0.848 with the warp), the cost a bound that proves exactly the range-empty cells would reach against the oracle's; the efficiencies - against it are 0.805 and 0.712 (warp 0), 0.785 and 0.694 (warp `D`). + against it are 0.794 and 0.712 (warp 0), 0.768 and 0.694 (warp `D`). - (c-band) The shallow wells, four mass-derived wells in the zone bowl with the camera gazing across, whose geometry is band-dominated, printed only. Centred interval 3.01x (efficiency 0.414, cull 0.960, overhead 0.958), affine - 3.96x (0.545, 0.979, 0.923); with the warp 2.75x (0.383, 0.954, 0.944) against - 3.60x (0.503, 0.975, 0.893). + 3.95x (0.544, 0.980, 0.901); with the warp 2.75x (0.383, 0.954, 0.944) against + 3.54x (0.494, 0.976, 0.865). - (d) r1's uniform slab, which has no envelope: 1.38x with the warp 0.05, and 1.67x with no warp. - (e) The void at the shipped point (ruling `operator-shipped-void`): `Rh = 198`, @@ -709,7 +706,7 @@ The contracts, at `N = 8`, the lower of warp 0 and `D`: converged march's by at most twice the fixed-step march's error; and it takes at most half the reference's steps per pixel. -This run meets every contract: (a) 16384x, (c) affine 3.62x and 3.54x against +This run meets every contract: (a) 16384x, (c) affine 3.57x and 3.46x against the centred interval's 3.20x and 3.12x, with a cull fraction of 0.987 at the lowest, (e) 2.69x fewer steps and a depth error 1.42x the fixed-step march's. The fields, the grids, the 0.2 envelope weight and the 2.0 gradient weight are