CultLib/packages/cultmath/docs/design.md

Design

CultMath exists to make renderer math readable in every place it needs to live: Rust hot kernels, C# host code, and HLSL shader code.

Parity Target: HLSL

HLSL is the semantic authority for every type and intrinsic HLSL defines. The goal is source compatibility: HLSL-shaped math compiles as C# under using static CultMath.math;. C# forces float literal suffixes (0.5f); the rest of the spelling should survive the move.

That makes these HLSL rules, not options:

• Every swizzle exists. float2/3/4, int2/3/4, and bool2/3/4 expose every read swizzle of length 2 to 4, repeats included (v.wwxy), in both xyzw and rgba naming (never mixed), plus single-component r/g/b/a. double2/3 get lengths 2 and 3 only, because there is no double4. Components and swizzles are writable: v.xz = f2;, c.rgb *= 0.5f;, and v[1] = s; work. Setters exist only for non-repeating masks, because HLSL rejects v.xx = ....
• Every HLSL mixed constructor exists, as a struct constructor and as a math function: float4(float2, float2), float4(float, float2, float), float3(float, float2), the copy form float3(float3), and the rest, for the float, int, and bool families (double gets constructors only; a math.double3 function would hide the double3 type name).
• Swizzles and mixed constructors are generated. tools/generate-swizzles.ps1 writes the checked-in src/CultMath/Swizzles.g.cs (partial structs plus a partial math). Edit the generator, rerun it, and commit both; never edit the output by hand.
• Comparison operators (<, >, <=, >=, ==, !=) are component-wise and return bool2/bool3/bool4. Reduce them with any and all. Equals/GetHashCode stay value-based so vectors still work as dictionary keys.
• select(condition, whenTrue, whenFalse) uses HLSL's argument order.
• Matrices are row-major: constructors take rows, m[i] returns a reference to row i (so m[1][2] = s; writes one element), _mRC is row R column C, and products go through mul (mul(m, v) treats v as a column, mul(v, m) as a row). * on matrices would be component-wise in HLSL, so CultMath does not define it.
• m[r][c] = v writes through locals, array elements, fields of classes, and ref locals. Through a readonly field, an in parameter, a property (auto-properties included), or a static readonly value such as float3x3.identity, C# takes a defensive copy: the write compiles and is silently lost. Copy the matrix to a local, write to it, and assign it back.
• Scalar/vector arithmetic and comparisons have explicit overloads on both sides rather than leaning on implicit scalar splat.
• Intrinsics return HLSL's types and edge cases: sign returns int/intN (NaN gives 0); step(y, x) is x < y ? 0 : 1 (NaN gives 1), which is how dxc lowers it, in C# and in the HLSL mirror; float and double min and max, and float clamp and saturate, follow the DXIL FMin/FMax/Saturate operations dxc emits for them: min(a, b) is a < b ? a : b, max(a, b) is a >= b ? a : b, a NaN operand returns the other (so saturate(NaN) is 0), clamp(x, a, b) is min(max(x, a), b), and saturate(x) is min(1, max(0, x)), which maps -0 to +0; any/all accept numeric vectors (a component is true when it is not 0); min, max, and abs on int vectors return int vectors; int clamp is IMin(IMax(x, a), b), so inverted bounds return b.
• Same-size vector conversions (intN(floatN), floatN(intN), boolN(floatN), boolN(intN), floatN(boolN), intN(boolN)) exist as constructors and math functions and follow dxc: float to int is fptosi, truncating toward zero; numeric to bool is != 0 (fcmp une, so NaN is true); bool to numeric is 0 or 1. DXIL leaves float-to-int undefined for NaN and out-of-range values, and so does C#; CultMath does not pin them.
• normalize(x) is x * Rsqrt(Dot(x, x)), which is how dxc lowers it, and DXIL.rst defines Rsqrt as 1 / sqrt(src). A zero vector therefore normalizes to NaN (0 * inf), and a NaN component makes every component NaN. (The special-value table under Rsqrt in DXIL.rst is a copy of the Round table and contradicts that definition, so CultMath follows the definition.) length and distance are Sqrt of the summed squares, so a zero vector gives 0. Callers that can meet zero-length vectors guard them before calling normalize.
• dxc demotes double frac, exp, lerp, and floor to float. CultMath's double overloads keep double precision instead: frac(double) is x - floor(x) in double. That is a deliberate divergence for CPU simulation time, which shaders do not carry.

dxc marks float arithmetic and compares fast (no NaNs) unless a value is precise, so drivers may optimize NaN handling away and NaN results on real GPUs are not guaranteed. CultMath defines the CPU result by dxc's lowering and the DXIL operation spec (DirectXShaderCompiler docs/DXIL.rst).

NaN does not reliably propagate through CultMath, just as it does not on the GPU. Because min, max, and saturate return the non-NaN operand, some functions turn NaN or infinity into numbers: smoothstep(a, a, a) is 0; infinite inputs to smoothstep and the intercept helpers give numbers; the bezier, smoothstep01, and smootherstep functions return finite values for NaN input. Callers that must detect bad data should check isnan before calling these functions.

Deferred until a consumer needs them: float4x4, transpose, determinant, and HLSL's one-based _11.._33 element names.

HLSL Target: Source Transformations

HlslSourceCompatibilityTests compiles the real shaders/CultMath.hlsl with Roslyn against CultMath. These transformations are the complete list; if a body needs anything else, the test fails:

1. Include guards (#ifndef/#define/#endif) are dropped.
2. Functions taking Texture2D/SamplerState are dropped; GPU resources have no CultMath analog.
3. File-scope static const becomes const.
4. Floating literals gain f (0.5 becomes 0.5f).
5. Struct fields gain public. HLSL struct members have no access-modifier concept; a C# struct's fields default to private, which would hide them from the mirror test's reflection-based field comparison.
6. The file body is wrapped in a class, since C# has no free functions; HLSL functions become private instance methods.

HLSL constructs that stay outside the target and must not appear in the mirror: const locals of vector type (C# const is primitives only; write a plain local) and writes through a swizzle component (v.xy.x = s, which C# rejects as modifying a temporary).

The HLSL side is checked too: tools/get-dxc.ps1 fetches Microsoft's DirectXShaderCompiler, and dxc -T lib_6_3 -HV 2021 shaders/CultMath.hlsl must compile cleanly.

The mirror test compares bit patterns (any NaN equals any NaN; -0 differs from 0). It proves the text of CultMath.hlsl computes the same float32 results as C# math on the CPU, not that a GPU agrees: driver sin precision alone makes cultmath_hash and value noise differ bit for bit on hardware. The mirror is compiled under using static CultMath.math;, so its intrinsics are C# math itself: the test proves the composition in the text, not the intrinsic rules, which only HlslSemanticsTests pins.

The comparison also allows exactly one struct return shape: a mirror function whose HLSL return type is a plain struct of float/floatN/int/intN fields is compared to its C# counterpart field by field, recursing into each field's own components, down to bit-for-bit scalars, matching each leaf's declaration path (field names, not just position) and declared type as well as its value. CultCellular (math.cs, cellular) and CultPhasor (math.Phacelle.cs, phacelle) are its consumers. A struct return with a mismatched, reordered-by-name, or retyped field does not get a second comparison path or a shape of its own; it fails the same walk that already handles a bare vector return.

Invariant 8: Value-and-Gradient Primitives

A primitive tagged invariant 8 returns its value and analytic gradient together, laid out as float4(gradient.xyz, value.w), with no value-only twin (a value-only form would be a second path with no consumer). smin_grad and cellular are the first primitives in this family:

• smin_grad(float4 a, float4 b, float k) is Inigo Quilez's quadratic-polynomial smooth minimum ("smooth minimum", https://iquilezles.org/articles/smin/), carried to value-and-gradient form. Its blend factor h is affine in b.w - a.w, so differentiating the value with respect to position, the terms carrying dh/dp cancel exactly; the surviving gradient is lerp(∇b, ∇a, h), the same h the value uses. This is the analytic gradient, not an approximation, and it is continuous across the |a.w - b.w| = k seam where h saturates to 0 or 1.

• cellular(float3 p) is Worley's cellular texture basis function ("A Cellular Texture Basis Function", SIGGRAPH 1996), searched over the jittered 5×5×5 neighbourhood of feature points selected by pcg3d (never the sin-based hash), with cells whose position-only lower bound already exceeds the running F2 skipped before their hash is even computed. It returns CultCellular { nearest, edge, id }: nearest is (∇F1, F1), edge is (∇F2 - ∇F1, F2 - F1), and id is the nearest cell's pcg4d hash (over the cell coordinate alone, never the pcg3d hash that produced its jitter) mapped to [0, 1). ∇F1 and ∇F2 are each undefined exactly at their own feature point (F1 = 0 or F2 = 0); cellular follows the repo's existing degenerate-normal convention (GameCult.Geometry.CultGeometryIsoSurface.EmitOrientedTriangle, which guards a zero-length normal and returns the zero vector instead of the NaN a bare normalize gives there) and returns the zero vector rather than NaN for either. F1 = 0 is ordinary (p sits on a feature point); F2 = 0 needs two distinct cells' pcg3d-jittered feature points to land on the same float32 value in every component, which is unreachable at ordinary magnitudes but confirmed reachable (see "Precision domain" below). The identity of the nearest and second-nearest feature point changes discontinuously across the F1 = F2 set (a genuine kink, not a numerical artifact), so ∇F1 and ∇F2 are only continuous away from that set.

  The 5×5×5 radius is exact, not merely safer than the classic 3×3×3 radius: for a query at position u inside its own cell (u in [0, 1)^3 per component) and a candidate feature at integer cell offset d with jitter j in [0, 1)^3, the per-axis displacement (u_i - j_i) - d_i is always strictly greater in magnitude than max(0, |d_i| - 1), because u_i - j_i is always strictly inside (-1, 1). So any cell with |d_i| >= 3 on some axis is strictly farther than 2 from the query. Two real, always-present candidates bound F1 and F2: the query's own cell, whose distance is always strictly less than sqrt(3) (every axis strictly under 1); and the near-side neighbour on whichever single axis has u_i closest to a cell boundary. Let m = min(u_i, 1 - u_i) on that chosen axis, so m is in [0, 1/2] by construction. Stepping one cell towards that nearer boundary puts the query within 1 + m of the far face on the offset axis and within 1 - m of the near face on each of the other two axes, so that neighbour's squared distance is at most (1 + m)^2 + 2*(1 - m)^2 = 3 - 2m + 3m^2 <= 3 for every m in [0, 1/2] (3 at m = 0, dipping to a minimum of 8/3 ~= 2.667 at m = 1/3, then rising back to 2.75 at m = 1/2), and strictly below 3 because the jitter never reaches exactly 0 or 1. F2 is at most the larger of these two real candidates (any two real candidates upper-bound the 2nd-smallest value over the full infinite candidate set), so F1 < sqrt(3) and F2 < sqrt(3) always, comfortably under the radius-2 exclusion distance of 2, with margin 2 - sqrt(3) ~= 0.268 (Soul's adversarial check, cut 2a-i: the worst realizable in-box F2 is exactly sqrt(3), reached only in the unattainable limit at a cell corner). 3×3×3's matching argument only reaches radius 1, whose excluded cells start at distance 1, short of the sqrt(3) ceiling — the gap its measured failures (9 of 400,000 points in [-50, 50]^3, worst error 0.088) live in.

  An earlier version of this proof picked the near-side neighbour on a fixed axis instead of whichever axis sits closest to its own boundary, and that version is wrong (Soul, cut 2a-i): u = (0.5, 0, 0) with the x-neighbour's jittered feature at (-1, 1 - 2^-24, 1 - 2^-24) gives a distance of about 2.06, outside the radius-2 exclusion the proof needs — u.x = 0.5 is the point on the cell farthest from any boundary, so x was the wrong axis to fix there. Choosing the axis by proximity to its own boundary is what keeps m <= 1/2 and the bound at sqrt(3).

  CellularAndSminGradTests pins both the prune's bit-identical equivalence to an unpruned 5×5×5 reference and the search's exact agreement with an independent 7×7×7 brute-force oracle.

  Precision domain: cellular's exactness above is a statement about the search radius, not about float32. Past |p| of roughly 2^23, adjacent integer cells stop being distinguishable from their neighbours at float32 precision in some directions, and distinct cells' jittered features can round onto the same value, making F1 = F2 ties (and, per-component, F2 = 0) increasingly common from about 1e6 and confirmed at 2^24 (measured: 4096 of 4096 sampled integer points at 2^24 hit this before the guard above). cellular stays finite there; it does not stay accurate in the sense of matching the idealized real-valued Worley field.

  The lower-bound prune's bit-identity to the unpruned 5×5×5 search also has a domain: it holds for |p| < 2^25. Above that, cell + d (an integer cell coordinate plus a small integer offset) starts to round in float32, so a pruned cell's own recomputed offset can land on a different neighbour than the unpruned loop visits at that same nominal (dx, dy, dz), and the two search paths diverge (Soul, cut 2a-i: measured 180 of 1,100,000 points differ at |p| ~= 3.4e7). This is a float32-precision limit on the prune's equivalence proof, not a bug in the prune itself: below 2^25 the two paths are proven bit-identical (PrunedSearchIsBitIdenticalToTheUnprunedFiveCubedSearch, which samples [-50, 50]^3, is comfortably inside that domain).

• snoise(float3) follows webgl-noise src/noise3D.glsl as of stegu/webgl-noise 22434e04d7: kernel radius squared 0.5, output scale 105, permutation mod289((34x+10)x). Upstream's 21d9fe23d7 (2020-10-14, "an age-old bug that caused slight discontinuities along simplex boundaries") moved the radius from 0.6 to 0.5; at 0.6 a lattice vertex outside the four summed corners can lie inside the kernel, leaving a value jump at the cell boundary (3.1e-3 over the committed probe's 400 lines). ff3b5d34ea set the scale to 105 and a3e6d57095 (2021-06-30) the permutation; the 2D snoise shares that permutation helper, so its values changed too, while its own kernel (0.5) and scale (130) already matched upstream. NoiseGradTests.SnoiseAndItsGradientAreContinuousAcrossSimplexCellBoundaries pins the continuity.

• snoise_grad(float3 p) carries snoise(float3) to value-and-gradient form, following the differentiation in webgl-noise src/noise3Dgrad.glsl (Ashima Arts / Ian McEwan, MIT), which uses the same radius, permutation and scale as snoise(float3): per corner, m0 = max(0.5 - dot(x,x), 0), so d(m0^4 * dot(p,x))/dx = -8*m0^3*dot(p,x)*x + m0^4*p, summed over the four corners and scaled by the same constant as the value, so .w is exactly snoise(float3). fbm_grad and ridged_grad are octave sums of snoise_grad: each octave samples at p * frequency, so by the chain rule its gradient scales by that same frequency, and both amplitude (gain) and frequency (lacunarity) compound per octave rather than applying once to the whole sum. ridged_grad folds each octave about zero (Σ aᵢ(1 − |nᵢ|), gradient −Σ aᵢfᵢ sign(nᵢ)∇nᵢ, Musgrave's ridged multifractal, Texturing and Modeling ch. 16); the crease at nᵢ = 0 is deliberate (the operator's dune term, 2026-09-25), not a bug. Both clamp octaves to [0, 16]: HlslSourceCompatibilityTests's bit-parity oracle drives every mirrored int parameter across the full int32 range, which is the right domain for pcg3d/pcg4d's O(1) hash inputs but would turn an unclamped octave count into a multi-billion-iteration loop for these two mirrors.

• phacelle(float3 p, float3 side, float offset, float normalization) is Rune Skovbo Johansen's Phacelle noise ("phase + cell": a stripe pattern built by blending one cosine/sine wave per jittered cell), generalised to 3D cells. It returns CultPhasor { cos, sin }, each (∇, value). side is the stripe wave vector, whose length is 2π times the stripe frequency; in 3D the perpendicular to the flow is not unique, so the caller chooses it (on a unit sphere, cross(p̂, flow)). offset is the phase in cycles. It visits 4×4×4 cells with pcg3d-hashed jitter in [-0.5, 0.5), weights each wave by max(0, exp(-2d²) - 0.01111), and normalizes the blended phasor as upstream does (length 1 where the raw length is at least 1 - normalization, scaled by 1 / (1 - normalization) below). The gradient is exact, weight derivatives and normalization included, rather than upstream's ∇cos ≈ -sin·side, which ignores the weight gradient. It is discontinuous where the raw length crosses the normalization threshold and where a cell's weight leaves its support (d² = 2.24995, distance 1.49998); the value is continuous at both. At the support edge ∇w jumps from -4·0.01111·v to 0, a per-cell jump of length 0.0444·|v| (0.0667 at |v| = 1.5); measured over random points, the output gradient jumps by a median 1.5% of its length and at most about 27%. normalization is meant to lie in [0, 1]: 1 or more saturates (every output is stretched to length 1), a negative value scales every output down by 1 / (1 - normalization), and the normalized gradient scales as 1 / |I| for the blended phasor I, so it grows where the blend nearly cancels. The total weight W is positive for every pcg3d input (the smallest found by hill-climbing is 0.056); W = 0, and a NaN output, needs adversarial jitter no hash yields. Cells whose per-axis lower bound Σ max(0, |local - g| - 0.5)² is at least 2.25 lie outside the support and have weight exactly 0, so they are skipped, which changes no output bit. Cells whose bound sits just under 2.25 can still carry a nonzero weight, so the cut cannot be lowered; PhacelleTests pins the cut with points next to such cells, against an internal overload with the prune disabled, and the HLSL mirror test pins it on the shader side. The prune keeps about 45 of the 64 cells on average (a saving of about 30%); about 14 have a nonzero weight. The port sits in its own MPL-2.0 files (math.Phacelle.cs, CultPhasor.cs, CultMath.Phacelle.hlsl); see THIRD-PARTY-NOTICES.md. No 2D variant exists; nothing consumes one.

Integer hashing uses the PCG hashes from Jarzynski and Olano, "Hash Functions for GPU Rendering" (JCGT 9(3), 2020): pcg(uint) is O'Neill's RXS-M-XS 32/32 permutation over one LCG step, and pcg3d/pcg4d are the paper's (3 → 3) and (4 → 4) hashes. They use only uint multiply, add, xor, and shifts. dxc wraps uint multiply and add modulo 2^32 exactly as C# unchecked arithmetic does, so they are integer-exact in HLSL and on the CPU, GPU included, unlike the sin-based hash. CultMath has no uint vectors, so pcg3d(int3) and pcg4d(int4) carry uint bit patterns in int components. The float2/float3/float4 overloads hash the IEEE-754 bits (asuint), so -0 and 0 hash differently; they exist for callers that already hold a hash key as float bits, and pcg3d(float2) holds z at 0, which is the paper's route for (2 → N) inputs. Seeds and other scalar uses take one output component.

cellular's jitter and id always hash the integer cell coordinate directly (pcg3d(int3(neighbor)), pcg4d(int4(int3(cellId), 0))), never a cell coordinate's float32 bit pattern. Hashing the bit pattern of an integer-valued float is not the same permutation as hashing the integer: two adjacent cells' bit patterns share more structure (both have the same exponent and a mantissa that differs by one increment) than two adjacent integers do going into pcg3d's own mixing, and that residual structure survived into the jitter. Hashing the integer cell coordinate directly is exact and runtime-independent, and it is the HLSL mirror's own hash, so C# and shader agree bit-for-bit. What pins this is the oracle and mirror tests, not a uniformity measurement: reverting to hashing the float bits (or the sin-based hash) is caught by ProductionSearchMatchesTheBruteForceOracleExactly, EveryMirrorFunctionMatchesCSharpMath, F1NeverExceedsF2, and the other tests that key off the integer-hash fixtures (cut 2a-i). CellularAndSminGradTests (JitterAndIdPassAChiSquareUniformityTest) checks the uniformity of the hash as cellular uses it: the jitter marginals and the id pass a chi-square test at a stated significance against pcg3d(int3(...)).

Where HLSL is silent, CultMath keeps its own decisions and does not defer to Unity.Mathematics: hash returns float, and Random is CultMath's own xorshift32. Engine-shaped helpers that HLSL lacks (float2x2.Rotate, float3x3.Euler, quaternion, snoise) enter only when a consumer needs them, with their semantics stated in code.

Known C# friction: under using static CultMath.math; the constructor functions (float3(...), float3x3(...)) hide the type names in member access, so static members need a qualified type (CultMath.float3x3.Euler(...)).

Rules

• Keep type names HLSL-shaped: float2, float3, float4, float3x3, and math.
• Keep Rust as the owner of native hot-path kernels and parity fixtures.
• Keep C#, shader, and scripting surfaces as runtime wrappers, not competing math authorities.
• Keep shaders/CultMath.hlsl as the canonical shader mirror for shared CultMath functions HLSL does not already provide (HLSL intrinsics such as mul, any, and log need no mirror). Project-local shader helpers should move here once more than one organ needs them or once CPU/GPU parity matters.
• Implement public shader semantics directly; do not copy Unity.Mathematics.
• Prefer plain mutable value types with public component fields, as HLSL does.
• Prefer component-wise overloads over clever generic machinery until the API earns that complexity.
• Keep the surface boring and exact. Fill what a consumer uses, guided by HLSL semantics; noise primitives, matrices, quaternions, and packing helpers need an owner before they enter.
• Keep the core assembly engine-free. Unity conversions live in the Unity package's CultMath.UnityBridge assembly.

Non-Goals

• Compiling C# to shaders.
• Generating HLSL from Rust or Rust from HLSL.
• Replacing System.Numerics for general .NET work.
• Recreating every Unity.Mathematics type because a spreadsheet somewhere says “coverage.” That way lies decorative obesity.