0e26f1e059
CultMath.glsl is GlslLowering's output (CULTMATH_WRITE_GLSL=1, run on Yggdrasil); the fixture keeps every earlier case byte for byte (the platform-bound families stay as generated on Windows) and appends the three new families, which depend on no libm. Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
900 lines
38 KiB
GLSL
900 lines
38 KiB
GLSL
// 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. cultmath_cycles_divide, cultmath_cycles_power, cultmath_cycles_logarithm, cultmath_cycles_sqrt, cultmath_cycles_inverse_sqrt, cultmath_cycles_modulo, cultmath_cycles_floored_modulo, cultmath_cycles_wrap, cultmath_cycles_snap, cultmath_cycles_pingpong, cultmath_cycles_smooth_min, cultmath_cycles_smooth_max, cultmath_cycles_compare, cultmath_cycles_round, cultmath_cycles_trunc, cultmath_cycles_fract, cultmath_cycles_sign, cultmath_cycles_arcsine, cultmath_cycles_arccosine, cultmath_cycles_arctan2, cultmath_cycles_map_range_linear, cultmath_cycles_map_range_stepped, cultmath_cycles_map_range_smoothstep, cultmath_cycles_map_range_smootherstep, cultmath_cycles_clamp_minmax, cultmath_cycles_clamp_range, cultmath_cycles_mix_float, cultmath_cycles_mix_vector, cultmath_cycles_mix_blend, cultmath_cycles_mix_add, cultmath_cycles_mix_mul, cultmath_cycles_mix_screen, cultmath_cycles_mix_overlay, cultmath_cycles_mix_sub, cultmath_cycles_mix_div, cultmath_cycles_mix_diff, cultmath_cycles_mix_exclusion, cultmath_cycles_mix_dark, cultmath_cycles_mix_light, cultmath_cycles_mix_dodge, cultmath_cycles_mix_burn, cultmath_cycles_rgb_to_hsv, cultmath_cycles_hsv_to_rgb, cultmath_cycles_mix_hue, cultmath_cycles_mix_sat, cultmath_cycles_mix_val, cultmath_cycles_mix_color, cultmath_cycles_mix_soft, cultmath_cycles_mix_linear, cultmath_cycles_rgb_to_bw, cultmath_cycles_ramp_index is in CultMath.Cycles.glsl, to be concatenated after this file by consumers that call it.
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#ifndef CULTMATH_GLSL
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#define CULTMATH_GLSL
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// CultMath HLSL mirror.
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//
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// HLSL already owns the vec2/vec3/vec4 type names and most primitive
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// intrinsics. This include names the CultMath semantic surface explicitly so
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// shader code, C# wrappers, and Rust/native parity fixtures can share the same
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// small vocabulary without local helper drift.
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const float CULTMATH_PI = 3.14159265358979323846;
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const float CULTMATH_TAU = 6.28318530717958647692;
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const float CULTMATH_HALF_PI = 1.57079632679489661923;
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float cultmath_radians(float degrees_value) { return degrees_value * (CULTMATH_PI / 180.0); }
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float cultmath_degrees(float radians_value) { return radians_value * (180.0 / CULTMATH_PI); }
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float cultmath_frac(float value) { return value - floor(value); }
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vec2 cultmath_frac(vec2 value) { return value - floor(value); }
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vec3 cultmath_frac(vec3 value) { return value - floor(value); }
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vec4 cultmath_frac(vec4 value) { return value - floor(value); }
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float cultmath_clamp(float value, float minimum, float maximum) { return min(max(value, minimum), maximum); }
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vec2 cultmath_clamp(vec2 value, vec2 minimum, vec2 maximum) { return min(max(value, minimum), maximum); }
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vec3 cultmath_clamp(vec3 value, vec3 minimum, vec3 maximum) { return min(max(value, minimum), maximum); }
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vec4 cultmath_clamp(vec4 value, vec4 minimum, vec4 maximum) { return min(max(value, minimum), maximum); }
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// DXIL Saturate is FMin(1, FMax(0, x)); dxc lowers min/max to the same FMin/FMax, NaN rules included.
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float cultmath_saturate(float value) { return min(1.0, max(0.0, value)); }
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vec2 cultmath_saturate(vec2 value) { return min(vec2(1.0, 1.0), max(vec2(0.0, 0.0), value)); }
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vec3 cultmath_saturate(vec3 value) { return min(vec3(1.0, 1.0, 1.0), max(vec3(0.0, 0.0, 0.0), value)); }
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vec4 cultmath_saturate(vec4 value) { return min(vec4(1.0, 1.0, 1.0, 1.0), max(vec4(0.0, 0.0, 0.0, 0.0), value)); }
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float cultmath_lerp(float start, float end, float amount) { return start + (end - start) * amount; }
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vec2 cultmath_lerp(vec2 start, vec2 end, vec2 amount) { return start + (end - start) * amount; }
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vec3 cultmath_lerp(vec3 start, vec3 end, vec3 amount) { return start + (end - start) * amount; }
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vec4 cultmath_lerp(vec4 start, vec4 end, vec4 amount) { return start + (end - start) * amount; }
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float cultmath_step(float edge, float value) { return value < edge ? 0.0 : 1.0; }
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vec2 cultmath_step(vec2 edge, vec2 value) { return vec2(cultmath_step(edge.x, value.x), cultmath_step(edge.y, value.y)); }
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vec3 cultmath_step(vec3 edge, vec3 value) { return vec3(cultmath_step(edge.x, value.x), cultmath_step(edge.y, value.y), cultmath_step(edge.z, value.z)); }
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vec4 cultmath_step(vec4 edge, vec4 value) { return vec4(cultmath_step(edge.x, value.x), cultmath_step(edge.y, value.y), cultmath_step(edge.z, value.z), cultmath_step(edge.w, value.w)); }
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float cultmath_smoothstep(float minimum, float maximum, float value)
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{
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float t = cultmath_saturate((value - minimum) / (maximum - minimum));
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return t * t * (3.0 - 2.0 * t);
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}
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vec2 cultmath_smoothstep(vec2 minimum, vec2 maximum, vec2 value)
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{
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vec2 t = cultmath_saturate((value - minimum) / (maximum - minimum));
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return t * t * (vec2(3.0, 3.0) - vec2(2.0, 2.0) * t);
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}
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vec3 cultmath_smoothstep(vec3 minimum, vec3 maximum, vec3 value)
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{
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vec3 t = cultmath_saturate((value - minimum) / (maximum - minimum));
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return t * t * (vec3(3.0, 3.0, 3.0) - vec3(2.0, 2.0, 2.0) * t);
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}
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vec4 cultmath_smoothstep(vec4 minimum, vec4 maximum, vec4 value)
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{
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vec4 t = cultmath_saturate((value - minimum) / (maximum - minimum));
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return t * t * (vec4(3.0, 3.0, 3.0, 3.0) - vec4(2.0, 2.0, 2.0, 2.0) * t);
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}
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float cultmath_smoothstep01(float value) { return cultmath_smoothstep(0.0, 1.0, value); }
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float cultmath_smootherstep(float value)
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{
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value = cultmath_saturate(value);
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return value * value * value * (value * (value * 6.0 - 15.0) + 10.0);
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}
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vec2 cultmath_snoise_mod289(vec2 value) { return value - floor(value * (1.0 / 289.0)) * 289.0; }
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vec3 cultmath_snoise_mod289(vec3 value) { return value - floor(value * (1.0 / 289.0)) * 289.0; }
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vec4 cultmath_snoise_mod289(vec4 value) { return value - floor(value * (1.0 / 289.0)) * 289.0; }
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vec3 cultmath_snoise_permute(vec3 value) { return cultmath_snoise_mod289(((value * 34.0) + 10.0) * value); }
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vec4 cultmath_snoise_permute(vec4 value) { return cultmath_snoise_mod289(((value * 34.0) + 10.0) * value); }
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// Ashima Arts / Ian McEwan 3D simplex noise (MIT), same float32 evaluation
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// order as the C# math.snoise(vec3) mirror. Ties resolve in simplexnoise1234's
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// total order; see math.snoise.
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float cultmath_snoise(vec3 value)
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{
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vec2 c = vec2(1.0 / 6.0, 1.0 / 3.0);
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vec3 i = floor(value + dot(value, c.yyy));
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vec3 x0 = value - i + dot(i, c.xxx);
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vec3 g = vec3(step(x0.y, x0.x), step(x0.z, x0.y), 1.0 - step(x0.z, x0.x));
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vec3 l = 1.0 - g;
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vec3 i1 = min(g.xyz, l.zxy);
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vec3 i2 = max(g.xyz, l.zxy);
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vec3 x1 = x0 - i1 + c.xxx;
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vec3 x2 = x0 - i2 + c.yyy;
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vec3 x3 = x0 - 0.5;
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i = cultmath_snoise_mod289(i);
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vec4 p = cultmath_snoise_permute(cultmath_snoise_permute(cultmath_snoise_permute(
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i.z + vec4(0.0, i1.z, i2.z, 1.0)) + i.y + vec4(0.0, i1.y, i2.y, 1.0)) + i.x + vec4(0.0, i1.x, i2.x, 1.0));
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const float n_ = 0.142857142857;
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vec3 ns = vec3(2.0 * n_, 0.5 * n_ - 1.0, n_);
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vec4 j = p - 49.0 * floor(p * ns.z * ns.z);
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vec4 x_ = floor(j * ns.z);
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vec4 y_ = floor(j - 7.0 * x_);
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vec4 x = x_ * ns.x + ns.yyyy;
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vec4 y = y_ * ns.x + ns.yyyy;
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vec4 h = 1.0 - abs(x) - abs(y);
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vec4 b0 = vec4(x.xy, y.xy);
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vec4 b1 = vec4(x.zw, y.zw);
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vec4 s0 = floor(b0) * 2.0 + 1.0;
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vec4 s1 = floor(b1) * 2.0 + 1.0;
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vec4 sh = -step(h, vec4(0.0, 0.0, 0.0, 0.0));
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vec4 a0 = b0.xzyw + s0.xzyw * sh.xxyy;
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vec4 a1 = b1.xzyw + s1.xzyw * sh.zzww;
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vec3 p0 = vec3(a0.xy, h.x);
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vec3 p1 = vec3(a0.zw, h.y);
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vec3 p2 = vec3(a1.xy, h.z);
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vec3 p3 = vec3(a1.zw, h.w);
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vec4 norm = 1.79284291400159 - 0.85373472095314 * vec4(dot(p0, p0), dot(p1, p1), dot(p2, p2), dot(p3, p3));
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p0 *= norm.x;
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p1 *= norm.y;
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p2 *= norm.z;
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p3 *= norm.w;
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vec4 m = max(0.5 - vec4(dot(x0, x0), dot(x1, x1), dot(x2, x2), dot(x3, x3)), 0.0);
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m = m * m;
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return 105.0 * dot(m * m, vec4(dot(p0, x0), dot(p1, x1), dot(p2, x2), dot(p3, x3)));
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}
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float cultmath_snoise(vec2 value)
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{
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vec4 c = vec4(
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0.211324865405187,
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0.366025403784439,
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-0.577350269189626,
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0.024390243902439);
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vec2 i = floor(value + dot(value, c.yy));
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vec2 x0 = value - i + dot(i, c.xx);
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vec2 i1 = x0.x > x0.y ? vec2(1.0, 0.0) : vec2(0.0, 1.0);
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vec4 x12 = vec4(x0.x + c.x - i1.x, x0.y + c.x - i1.y, x0.x + c.z, x0.y + c.z);
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i = cultmath_snoise_mod289(i);
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vec3 p = cultmath_snoise_permute(
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cultmath_snoise_permute(i.y + vec3(0.0, i1.y, 1.0)) + i.x + vec3(0.0, i1.x, 1.0));
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vec3 m = max(0.5 - vec3(dot(x0, x0), dot(x12.xy, x12.xy), dot(x12.zw, x12.zw)), 0.0);
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m *= m;
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m *= m;
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vec3 x = 2.0 * fract(p * c.w) - 1.0;
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vec3 h = abs(x) - 0.5;
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vec3 ox = floor(x + 0.5);
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vec3 a0 = x - ox;
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m *= 1.79284291400159 - 0.85373472095314 * (a0 * a0 + h * h);
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vec3 g = vec3(
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a0.x * x0.x + h.x * x0.y,
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a0.y * x12.x + h.y * x12.y,
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a0.z * x12.z + h.z * x12.w);
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return 130.0 * dot(m, g);
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}
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// Analytic gradient of cultmath_snoise(vec3) (invariant 8), following webgl-noise
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// src/noise3Dgrad.glsl (Ashima Arts / McEwan, MIT); same kernel, permutation and scale as
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// cultmath_snoise(vec3) (stegu/webgl-noise 22434e04d7). See the C# math.snoise_grad comment for the
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// derivation. Every local up through p0..p3 and x0..x3 is cultmath_snoise(vec3)'s own derivation
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// verbatim.
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vec4 cultmath_snoise_grad(vec3 value)
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{
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vec2 c = vec2(1.0 / 6.0, 1.0 / 3.0);
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vec3 i = floor(value + dot(value, c.yyy));
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vec3 x0 = value - i + dot(i, c.xxx);
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vec3 g = vec3(step(x0.y, x0.x), step(x0.z, x0.y), 1.0 - step(x0.z, x0.x));
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vec3 l = 1.0 - g;
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vec3 i1 = min(g.xyz, l.zxy);
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vec3 i2 = max(g.xyz, l.zxy);
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vec3 x1 = x0 - i1 + c.xxx;
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vec3 x2 = x0 - i2 + c.yyy;
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vec3 x3 = x0 - 0.5;
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i = cultmath_snoise_mod289(i);
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vec4 p = cultmath_snoise_permute(cultmath_snoise_permute(cultmath_snoise_permute(
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i.z + vec4(0.0, i1.z, i2.z, 1.0)) + i.y + vec4(0.0, i1.y, i2.y, 1.0)) + i.x + vec4(0.0, i1.x, i2.x, 1.0));
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const float n_ = 0.142857142857;
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vec3 ns = vec3(2.0 * n_, 0.5 * n_ - 1.0, n_);
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vec4 j = p - 49.0 * floor(p * ns.z * ns.z);
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vec4 x_ = floor(j * ns.z);
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vec4 y_ = floor(j - 7.0 * x_);
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vec4 x = x_ * ns.x + ns.yyyy;
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vec4 y = y_ * ns.x + ns.yyyy;
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vec4 h = 1.0 - abs(x) - abs(y);
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vec4 b0 = vec4(x.xy, y.xy);
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vec4 b1 = vec4(x.zw, y.zw);
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vec4 s0 = floor(b0) * 2.0 + 1.0;
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vec4 s1 = floor(b1) * 2.0 + 1.0;
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vec4 sh = -step(h, vec4(0.0, 0.0, 0.0, 0.0));
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vec4 a0 = b0.xzyw + s0.xzyw * sh.xxyy;
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vec4 a1 = b1.xzyw + s1.xzyw * sh.zzww;
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vec3 p0 = vec3(a0.xy, h.x);
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vec3 p1 = vec3(a0.zw, h.y);
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vec3 p2 = vec3(a1.xy, h.z);
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vec3 p3 = vec3(a1.zw, h.w);
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vec4 norm = 1.79284291400159 - 0.85373472095314 * vec4(dot(p0, p0), dot(p1, p1), dot(p2, p2), dot(p3, p3));
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p0 *= norm.x;
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p1 *= norm.y;
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p2 *= norm.z;
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p3 *= norm.w;
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vec4 m0 = max(0.5 - vec4(dot(x0, x0), dot(x1, x1), dot(x2, x2), dot(x3, x3)), 0.0);
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vec4 m2 = m0 * m0;
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vec4 m3 = m2 * m0;
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vec4 m4 = m2 * m2;
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vec4 px = vec4(dot(p0, x0), dot(p1, x1), dot(p2, x2), dot(p3, x3));
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float value2 = 105.0 * dot(m4, px);
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vec3 grad = -8.0 * m3.x * x0 * px.x + m4.x * p0
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+ -8.0 * m3.y * x1 * px.y + m4.y * p1
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+ -8.0 * m3.z * x2 * px.z + m4.z * p2
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+ -8.0 * m3.w * x3 * px.w + m4.w * p3;
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grad *= 105.0;
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return vec4(grad, value2);
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}
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// Analytic Hessian of cultmath_snoise(vec3); see the C# math.snoise_hessian comment for the
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// derivation. Every local up through p0..p3 and x0..x3 is cultmath_snoise_grad's own. The matrix is
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// built from nine scalars, which read as rows in HLSL and as columns in GLSL; it is symmetric, so the
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// two agree.
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mat3 cultmath_snoise_hessian(vec3 value)
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{
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vec2 c = vec2(1.0 / 6.0, 1.0 / 3.0);
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vec3 i = floor(value + dot(value, c.yyy));
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vec3 x0 = value - i + dot(i, c.xxx);
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vec3 g = vec3(step(x0.y, x0.x), step(x0.z, x0.y), 1.0 - step(x0.z, x0.x));
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vec3 l = 1.0 - g;
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vec3 i1 = min(g.xyz, l.zxy);
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vec3 i2 = max(g.xyz, l.zxy);
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vec3 x1 = x0 - i1 + c.xxx;
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vec3 x2 = x0 - i2 + c.yyy;
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vec3 x3 = x0 - 0.5;
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i = cultmath_snoise_mod289(i);
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vec4 p = cultmath_snoise_permute(cultmath_snoise_permute(cultmath_snoise_permute(
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i.z + vec4(0.0, i1.z, i2.z, 1.0)) + i.y + vec4(0.0, i1.y, i2.y, 1.0)) + i.x + vec4(0.0, i1.x, i2.x, 1.0));
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const float n_ = 0.142857142857;
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vec3 ns = vec3(2.0 * n_, 0.5 * n_ - 1.0, n_);
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vec4 j = p - 49.0 * floor(p * ns.z * ns.z);
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vec4 x_ = floor(j * ns.z);
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vec4 y_ = floor(j - 7.0 * x_);
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vec4 x = x_ * ns.x + ns.yyyy;
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vec4 y = y_ * ns.x + ns.yyyy;
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vec4 h = 1.0 - abs(x) - abs(y);
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vec4 b0 = vec4(x.xy, y.xy);
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vec4 b1 = vec4(x.zw, y.zw);
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vec4 s0 = floor(b0) * 2.0 + 1.0;
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vec4 s1 = floor(b1) * 2.0 + 1.0;
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vec4 sh = -step(h, vec4(0.0, 0.0, 0.0, 0.0));
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vec4 a0 = b0.xzyw + s0.xzyw * sh.xxyy;
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vec4 a1 = b1.xzyw + s1.xzyw * sh.zzww;
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vec3 p0 = vec3(a0.xy, h.x);
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vec3 p1 = vec3(a0.zw, h.y);
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vec3 p2 = vec3(a1.xy, h.z);
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vec3 p3 = vec3(a1.zw, h.w);
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vec4 norm = 1.79284291400159 - 0.85373472095314 * vec4(dot(p0, p0), dot(p1, p1), dot(p2, p2), dot(p3, p3));
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p0 *= norm.x;
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p1 *= norm.y;
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p2 *= norm.z;
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p3 *= norm.w;
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vec4 m0 = max(0.5 - vec4(dot(x0, x0), dot(x1, x1), dot(x2, x2), dot(x3, x3)), 0.0);
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vec4 m2 = m0 * m0;
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vec4 m3 = m2 * m0;
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vec4 px = vec4(dot(p0, x0), dot(p1, x1), dot(p2, x2), dot(p3, x3));
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vec4 wide = 48.0 * m2 * px;
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vec4 narrow = 8.0 * m3;
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vec4 cxx = vec4(x0.x, x1.x, x2.x, x3.x);
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vec4 cxy = vec4(x0.y, x1.y, x2.y, x3.y);
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vec4 cxz = vec4(x0.z, x1.z, x2.z, x3.z);
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vec4 cpx = vec4(p0.x, p1.x, p2.x, p3.x);
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vec4 cpy = vec4(p0.y, p1.y, p2.y, p3.y);
|
|
vec4 cpz = vec4(p0.z, p1.z, p2.z, p3.z);
|
|
|
|
float hxx = 105.0 * (dot(wide * cxx, cxx) - dot(narrow, 2.0 * cpx * cxx + px));
|
|
float hyy = 105.0 * (dot(wide * cxy, cxy) - dot(narrow, 2.0 * cpy * cxy + px));
|
|
float hzz = 105.0 * (dot(wide * cxz, cxz) - dot(narrow, 2.0 * cpz * cxz + px));
|
|
float hxy = 105.0 * (dot(wide * cxx, cxy) - dot(narrow, cpx * cxy + cpy * cxx));
|
|
float hxz = 105.0 * (dot(wide * cxx, cxz) - dot(narrow, cpx * cxz + cpz * cxx));
|
|
float hyz = 105.0 * (dot(wide * cxy, cxz) - dot(narrow, cpy * cxz + cpz * cxy));
|
|
return mat3(hxx, hxy, hxz, hxy, hyy, hyz, hxz, hyz, hzz);
|
|
}
|
|
|
|
// The integral over [t0, t1] of clamp(a + b t, lo, hi); see the C# math.integral_clamp_linear comment.
|
|
float cultmath_integral_clamp_linear(float a, float b, float lo, float hi, float t0, float t1)
|
|
{
|
|
if (b == 0.0)
|
|
return clamp(a, lo, hi) * (t1 - t0);
|
|
|
|
lo = min(lo, hi);
|
|
float direction = t1 < t0 ? -1.0 : 1.0;
|
|
float start = min(t0, t1);
|
|
float stop = max(t0, t1);
|
|
float atLo = (lo - a) / b;
|
|
float atHi = (hi - a) / b;
|
|
float enter = clamp(min(atLo, atHi), start, stop);
|
|
float leave = clamp(max(atLo, atHi), start, stop);
|
|
float before = b > 0.0 ? lo : hi;
|
|
float after = b > 0.0 ? hi : lo;
|
|
float total = before * (enter - start) + (a + b * (0.5 * (enter + leave))) * (leave - enter) + after * (stop - leave);
|
|
return direction * total;
|
|
}
|
|
|
|
// Octave sum of cultmath_snoise_grad (invariant 8); see the C# math.fbm_grad comment for the
|
|
// per-octave frequency/amplitude derivation and why octaves is clamped.
|
|
vec4 cultmath_fbm_grad(vec3 p, int octaves, float lacunarity, float gain)
|
|
{
|
|
octaves = clamp(octaves, 0, 16);
|
|
float amplitude = 1.0;
|
|
float frequency = 1.0;
|
|
float value = 0.0;
|
|
vec3 gradient = vec3(0.0, 0.0, 0.0);
|
|
for (int i = 0; i < octaves; i++)
|
|
{
|
|
vec4 n = cultmath_snoise_grad(p * frequency);
|
|
value += amplitude * n.w;
|
|
gradient += amplitude * frequency * n.xyz;
|
|
frequency *= lacunarity;
|
|
amplitude *= gain;
|
|
}
|
|
|
|
return vec4(gradient, value);
|
|
}
|
|
|
|
// Ridged multifractal noise (Musgrave, Texturing and Modeling, ch. 16); see the C# math.ridged_grad
|
|
// comment for the fold/sign derivation, the deliberate crease at n = 0, and why octaves is clamped.
|
|
vec4 cultmath_ridged_grad(vec3 p, int octaves, float lacunarity, float gain)
|
|
{
|
|
octaves = clamp(octaves, 0, 16);
|
|
float amplitude = 1.0;
|
|
float frequency = 1.0;
|
|
float value = 0.0;
|
|
vec3 gradient = vec3(0.0, 0.0, 0.0);
|
|
for (int i = 0; i < octaves; i++)
|
|
{
|
|
vec4 n = cultmath_snoise_grad(p * frequency);
|
|
float s = sign(n.w);
|
|
value += amplitude * (1.0 - abs(n.w));
|
|
gradient += -amplitude * frequency * s * n.xyz;
|
|
frequency *= lacunarity;
|
|
amplitude *= gain;
|
|
}
|
|
|
|
return vec4(gradient, value);
|
|
}
|
|
|
|
float cultmath_lengthsq(vec2 value) { return dot(value, value); }
|
|
float cultmath_lengthsq(vec3 value) { return dot(value, value); }
|
|
float cultmath_lengthsq(vec4 value) { return dot(value, value); }
|
|
|
|
float cultmath_distance(vec2 left, vec2 right) { return length(left - right); }
|
|
float cultmath_distance(vec3 left, vec3 right) { return length(left - right); }
|
|
float cultmath_distance(vec4 left, vec4 right) { return length(left - right); }
|
|
|
|
vec2 cultmath_reflect(vec2 incident, vec2 normal_value)
|
|
{
|
|
return incident - 2.0 * dot(normal_value, incident) * normal_value;
|
|
}
|
|
|
|
vec3 cultmath_reflect(vec3 incident, vec3 normal_value)
|
|
{
|
|
return incident - 2.0 * dot(normal_value, incident) * normal_value;
|
|
}
|
|
|
|
vec2 cultmath_rotate(vec2 value, float radians_value)
|
|
{
|
|
float s = sin(radians_value);
|
|
float c = cos(radians_value);
|
|
return vec2(value.x * c - value.y * s, value.x * s + value.y * c);
|
|
}
|
|
|
|
vec2 cultmath_rotate_degrees(vec2 value, float degrees_value)
|
|
{
|
|
return cultmath_rotate(value, cultmath_radians(degrees_value));
|
|
}
|
|
|
|
float cultmath_csum(vec2 value) { return value.x + value.y; }
|
|
float cultmath_csum(vec3 value) { return value.x + value.y + value.z; }
|
|
float cultmath_csum(vec4 value) { return value.x + value.y + value.z + value.w; }
|
|
|
|
float cultmath_decay(float source, float lambda_value, float dt) { return source * exp(-lambda_value * dt); }
|
|
vec2 cultmath_decay(vec2 source, float lambda_value, float dt) { return source * exp(-lambda_value * dt); }
|
|
vec3 cultmath_decay(vec3 source, float lambda_value, float dt) { return source * exp(-lambda_value * dt); }
|
|
vec4 cultmath_decay(vec4 source, float lambda_value, float dt) { return source * exp(-lambda_value * dt); }
|
|
|
|
float cultmath_damp(float start, float end, float lambda_value, float dt)
|
|
{
|
|
return cultmath_lerp(start, end, 1.0 - exp(-lambda_value * dt));
|
|
}
|
|
|
|
vec2 cultmath_damp(vec2 start, vec2 end, float lambda_value, float dt)
|
|
{
|
|
float amount = 1.0 - exp(-lambda_value * dt);
|
|
return cultmath_lerp(start, end, vec2(amount, amount));
|
|
}
|
|
|
|
vec3 cultmath_damp(vec3 start, vec3 end, float lambda_value, float dt)
|
|
{
|
|
float amount = 1.0 - exp(-lambda_value * dt);
|
|
return cultmath_lerp(start, end, vec3(amount, amount, amount));
|
|
}
|
|
|
|
vec4 cultmath_damp(vec4 start, vec4 end, float lambda_value, float dt)
|
|
{
|
|
float amount = 1.0 - exp(-lambda_value * dt);
|
|
return cultmath_lerp(start, end, vec4(amount, amount, amount, amount));
|
|
}
|
|
|
|
float cultmath_catmullrom(float p0, float p1, float p2, float p3, float t)
|
|
{
|
|
float t2 = t * t;
|
|
float t3 = t2 * t;
|
|
return 0.5 * ((2.0 * p1) + (p2 - p0) * t + (2.0 * p0 - 5.0 * p1 + 4.0 * p2 - p3) * t2 + (-p0 + 3.0 * p1 - 3.0 * p2 + p3) * t3);
|
|
}
|
|
|
|
vec2 cultmath_catmullrom(vec2 p0, vec2 p1, vec2 p2, vec2 p3, float t)
|
|
{
|
|
float t2 = t * t;
|
|
float t3 = t2 * t;
|
|
return 0.5 * ((2.0 * p1) + (p2 - p0) * t + (2.0 * p0 - 5.0 * p1 + 4.0 * p2 - p3) * t2 + (-p0 + 3.0 * p1 - 3.0 * p2 + p3) * t3);
|
|
}
|
|
|
|
vec3 cultmath_catmullrom(vec3 p0, vec3 p1, vec3 p2, vec3 p3, float t)
|
|
{
|
|
float t2 = t * t;
|
|
float t3 = t2 * t;
|
|
return 0.5 * ((2.0 * p1) + (p2 - p0) * t + (2.0 * p0 - 5.0 * p1 + 4.0 * p2 - p3) * t2 + (-p0 + 3.0 * p1 - 3.0 * p2 + p3) * t3);
|
|
}
|
|
|
|
vec4 cultmath_catmullrom(vec4 p0, vec4 p1, vec4 p2, vec4 p3, float t)
|
|
{
|
|
float t2 = t * t;
|
|
float t3 = t2 * t;
|
|
return 0.5 * ((2.0 * p1) + (p2 - p0) * t + (2.0 * p0 - 5.0 * p1 + 4.0 * p2 - p3) * t2 + (-p0 + 3.0 * p1 - 3.0 * p2 + p3) * t3);
|
|
}
|
|
|
|
float cultmath_quadratic_bezier(float p0, float p1, float p2, float t)
|
|
{
|
|
t = cultmath_saturate(t);
|
|
float inv = 1.0 - t;
|
|
return inv * inv * p0 + 2.0 * inv * t * p1 + t * t * p2;
|
|
}
|
|
|
|
vec3 cultmath_quadratic_bezier(vec3 p0, vec3 p1, vec3 p2, float t)
|
|
{
|
|
t = cultmath_saturate(t);
|
|
float inv = 1.0 - t;
|
|
return inv * inv * p0 + 2.0 * inv * t * p1 + t * t * p2;
|
|
}
|
|
|
|
float cultmath_cubic_bezier(float p0, float p1, float p2, float p3, float t)
|
|
{
|
|
t = cultmath_saturate(t);
|
|
float inv = 1.0 - t;
|
|
return inv * inv * inv * p0 + 3.0 * inv * inv * t * p1 + 3.0 * inv * t * t * p2 + t * t * t * p3;
|
|
}
|
|
|
|
vec3 cultmath_cubic_bezier(vec3 p0, vec3 p1, vec3 p2, vec3 p3, float t)
|
|
{
|
|
t = cultmath_saturate(t);
|
|
float inv = 1.0 - t;
|
|
return inv * inv * inv * p0 + 3.0 * inv * inv * t * p1 + 3.0 * inv * t * t * p2 + t * t * t * p3;
|
|
}
|
|
|
|
float cultmath_hash(float value) { return cultmath_frac(sin(value) * 43758.5453); }
|
|
float cultmath_hash(vec2 value) { return cultmath_hash(dot(value, vec2(127.1, 311.7))); }
|
|
float cultmath_hash(vec3 value) { return cultmath_hash(dot(value, vec3(127.1, 311.7, 74.7))); }
|
|
|
|
// PCG hashes (Jarzynski and Olano, JCGT 2020), written over scalar uints so CultMath's ivec3/ivec4 carry the bits.
|
|
uint cultmath_pcg(uint value)
|
|
{
|
|
uint state = value * 747796405u + 2891336453u;
|
|
uint word = ((state >> int((state >> 28) + 4u)) ^ state) * 277803737u;
|
|
return (word >> 22) ^ word;
|
|
}
|
|
|
|
ivec3 cultmath_pcg3d(ivec3 value)
|
|
{
|
|
uint x = uint(value.x) * 1664525u + 1013904223u, y = uint(value.y) * 1664525u + 1013904223u, z = uint(value.z) * 1664525u + 1013904223u;
|
|
x += y * z; y += z * x; z += x * y;
|
|
x ^= x >> 16; y ^= y >> 16; z ^= z >> 16;
|
|
x += y * z; y += z * x; z += x * y;
|
|
return ivec3(int(x), int(y), int(z));
|
|
}
|
|
|
|
ivec3 cultmath_pcg3d(vec3 value) { return cultmath_pcg3d(ivec3(int(floatBitsToUint(value.x)), int(floatBitsToUint(value.y)), int(floatBitsToUint(value.z)))); }
|
|
ivec3 cultmath_pcg3d(vec2 value) { return cultmath_pcg3d(vec3(value, 0.0)); }
|
|
|
|
ivec4 cultmath_pcg4d(ivec4 value)
|
|
{
|
|
uint x = uint(value.x) * 1664525u + 1013904223u, y = uint(value.y) * 1664525u + 1013904223u;
|
|
uint z = uint(value.z) * 1664525u + 1013904223u, w = uint(value.w) * 1664525u + 1013904223u;
|
|
x += y * w; y += z * x; z += x * y; w += y * z;
|
|
x ^= x >> 16; y ^= y >> 16; z ^= z >> 16; w ^= w >> 16;
|
|
x += y * w; y += z * x; z += x * y; w += y * z;
|
|
return ivec4(int(x), int(y), int(z), int(w));
|
|
}
|
|
|
|
ivec4 cultmath_pcg4d(vec4 value) { return cultmath_pcg4d(ivec4(int(floatBitsToUint(value.x)), int(floatBitsToUint(value.y)), int(floatBitsToUint(value.z)), int(floatBitsToUint(value.w)))); }
|
|
|
|
// Inigo Quilez, "smooth minimum": quadratic-polynomial smin carried to value-and-gradient form. See
|
|
// math.smin_grad's comment (math.cs) for why the gradient blend is exact, not an approximation.
|
|
vec4 cultmath_smin_grad(vec4 a, vec4 b, float k)
|
|
{
|
|
float h = clamp(0.5 + 0.5 * (b.w - a.w) / k, 0.0, 1.0);
|
|
float value = mix(b.w, a.w, h) - k * h * (1.0 - h);
|
|
vec3 gradient = mix(b.xyz, a.xyz, h);
|
|
return vec4(gradient, value);
|
|
}
|
|
|
|
// Invariant 8's one struct return shape (design.md, "HLSL Target: Source Transformations"):
|
|
// compared field by field and bit for bit against CultMath.CultCellular, same field order.
|
|
struct CultCellular
|
|
{
|
|
vec4 nearest;
|
|
vec4 edge;
|
|
float id;
|
|
};
|
|
|
|
// Worley, "A Cellular Texture Basis Function" (SIGGRAPH 1996). See math.cellular's comment
|
|
// (math.cs) for the hashing, gradient, search-radius exactness proof, id source and
|
|
// degenerate-point conventions this mirrors exactly, including the 5x5x5 lower-bound prune.
|
|
CultCellular cultmath_cellular(vec3 p)
|
|
{
|
|
vec3 cell = floor(p);
|
|
float f1 = 1.0e30;
|
|
float f2 = 1.0e30;
|
|
vec3 c1 = vec3(0.0, 0.0, 0.0);
|
|
vec3 c2 = vec3(0.0, 0.0, 0.0);
|
|
vec3 cellId = vec3(0.0, 0.0, 0.0);
|
|
|
|
for (int dz = -2; dz <= 2; dz++)
|
|
{
|
|
float lz = max(0.0, abs(float(dz)) - 1.0);
|
|
for (int dy = -2; dy <= 2; dy++)
|
|
{
|
|
float ly = max(0.0, abs(float(dy)) - 1.0);
|
|
for (int dx = -2; dx <= 2; dx++)
|
|
{
|
|
float lx = max(0.0, abs(float(dx)) - 1.0);
|
|
float lowerBound = sqrt(lx * lx + ly * ly + lz * lz);
|
|
if (lowerBound >= f2)
|
|
continue;
|
|
|
|
vec3 neighbor = cell + vec3(dx, dy, dz);
|
|
ivec3 hash = cultmath_pcg3d(ivec3(neighbor));
|
|
vec3 jitter = vec3(
|
|
float(uint(hash.x) >> 8) * (1.0 / 16777216.0),
|
|
float(uint(hash.y) >> 8) * (1.0 / 16777216.0),
|
|
float(uint(hash.z) >> 8) * (1.0 / 16777216.0));
|
|
vec3 feature = neighbor + jitter;
|
|
float d = cultmath_distance(p, feature);
|
|
|
|
if (d < f1)
|
|
{
|
|
f2 = f1; c2 = c1;
|
|
f1 = d; c1 = feature; cellId = neighbor;
|
|
}
|
|
else if (d < f2)
|
|
{
|
|
f2 = d; c2 = feature;
|
|
}
|
|
}
|
|
}
|
|
}
|
|
|
|
// F1 = 0 and F2 = 0 share the same degenerate-gradient guard; see math.cellular's comment
|
|
// (math.cs) for why F2 = 0 is reachable past roughly |p| = 2^23.
|
|
vec3 grad1 = f1 > 0.0 ? (p - c1) / f1 : vec3(0.0, 0.0, 0.0);
|
|
vec3 grad2 = f2 > 0.0 ? (p - c2) / f2 : vec3(0.0, 0.0, 0.0);
|
|
|
|
// id comes from pcg4d over the winning cell's integer coordinate alone (same int-hashing rule
|
|
// as the jitter above), never from the pcg3d hash that produced its jitter (F4).
|
|
ivec4 idHash = cultmath_pcg4d(ivec4(ivec3(cellId), 0));
|
|
|
|
CultCellular result;
|
|
result.nearest = vec4(grad1, f1);
|
|
result.edge = vec4(grad2 - grad1, f2 - f1);
|
|
result.id = float(uint(idHash.w) >> 8) * (1.0 / 16777216.0);
|
|
return result;
|
|
}
|
|
|
|
|
|
|
|
|
|
// Interval arithmetic over vec2(lo, hi), lo <= hi, both finite. Every cultmath_iv_* function returns
|
|
// an interval containing f(x) for every x in its input intervals, mirroring math.Interval.cs bit for
|
|
// bit; see that file and docs/design.md ("Intervals") for each op's enclosure and its ulp widening.
|
|
// Included by CultMath.hlsl after its own definitions of cultmath_smoothstep and cultmath_snoise.
|
|
|
|
// 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); }
|
|
vec2 cultmath_iv_neg(vec2 a) { return vec2(-a.y, -a.x); }
|
|
|
|
vec2 cultmath_iv_mul(vec2 a, vec2 b)
|
|
{
|
|
float p0 = a.x * b.x;
|
|
float p1 = a.x * b.y;
|
|
float p2 = a.y * b.x;
|
|
float p3 = a.y * b.y;
|
|
return vec2(min(min(p0, p1), min(p2, p3)), max(max(p0, p1), max(p2, p3)));
|
|
}
|
|
|
|
vec2 cultmath_iv_scale(vec2 a, float s) { return s >= 0.0 ? vec2(a.x * s, a.y * s) : vec2(a.y * s, a.x * s); }
|
|
|
|
vec2 cultmath_iv_abs(vec2 a)
|
|
{
|
|
if (a.x >= 0.0)
|
|
return a;
|
|
if (a.y <= 0.0)
|
|
return vec2(-a.y, -a.x);
|
|
return vec2(0.0, max(-a.x, a.y));
|
|
}
|
|
|
|
vec2 cultmath_iv_min(vec2 a, vec2 b) { return vec2(min(a.x, b.x), min(a.y, b.y)); }
|
|
vec2 cultmath_iv_max(vec2 a, vec2 b) { return vec2(max(a.x, b.x), max(a.y, b.y)); }
|
|
|
|
vec2 cultmath_iv_sqr(vec2 a)
|
|
{
|
|
float l = a.x * a.x;
|
|
float h = a.y * a.y;
|
|
if (a.x >= 0.0)
|
|
return vec2(l, h);
|
|
if (a.y <= 0.0)
|
|
return vec2(h, l);
|
|
return vec2(0.0, max(l, h));
|
|
}
|
|
|
|
vec2 cultmath_iv_sqrt(vec2 a) { return vec2(sqrt(max(a.x, 0.0)), sqrt(max(a.y, 0.0))); }
|
|
|
|
// Widened by one ulp on each side: exp is not required to be correctly rounded.
|
|
vec2 cultmath_iv_exp(vec2 a)
|
|
{
|
|
float lo = exp(a.x);
|
|
float hi = exp(a.y);
|
|
return vec2(lo > 0.0 ? uintBitsToFloat(floatBitsToUint(lo) - 1u) : 0.0, uintBitsToFloat(floatBitsToUint(hi) + 1u));
|
|
}
|
|
|
|
vec2 cultmath_iv_clamp(vec2 a, float minimum, float maximum)
|
|
{
|
|
return vec2(cultmath_clamp(a.x, minimum, maximum), cultmath_clamp(a.y, minimum, maximum));
|
|
}
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|
|
|
vec2 cultmath_iv_saturate(vec2 a) { return vec2(cultmath_saturate(a.x), cultmath_saturate(a.y)); }
|
|
|
|
// The endpoints, ordered, widened by one ulp inside [0, 1]: the float32 evaluation of smoothstep is
|
|
// monotone only to within one ulp.
|
|
vec2 cultmath_iv_smoothstep(float minimum, float maximum, vec2 a)
|
|
{
|
|
float s0 = cultmath_smoothstep(minimum, maximum, a.x);
|
|
float s1 = cultmath_smoothstep(minimum, maximum, a.y);
|
|
float lo = min(s0, s1);
|
|
float hi = max(s0, s1);
|
|
return vec2(lo > 0.0 ? uintBitsToFloat(floatBitsToUint(lo) - 1u) : 0.0, hi < 1.0 ? uintBitsToFloat(floatBitsToUint(hi) + 1u) : 1.0);
|
|
}
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|
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|
// The natural interval extension of x + (y - x) * amount, which is how cultmath_lerp evaluates.
|
|
vec2 cultmath_iv_lerp(vec2 a, vec2 b, float amount) { return cultmath_iv_add(a, cultmath_iv_scale(cultmath_iv_sub(b, a), amount)); }
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// [n - L r, n + L r] intersected with [-1, 1], n = cultmath_snoise(centre). One snoise, value only.
|
|
vec2 cultmath_iv_snoise_ball(vec3 centre, float radius)
|
|
{
|
|
float n = cultmath_snoise(centre);
|
|
float e = CULTMATH_SNOISE_LIPSCHITZ * radius;
|
|
return vec2(max(n - e, -1.0), min(n + e, 1.0));
|
|
}
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// 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));
|
|
}
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|
|
|
// 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)
|
|
{
|
|
octaves = clamp(octaves, 0, 16);
|
|
float amplitude = 1.0;
|
|
float frequency = 1.0;
|
|
vec2 sum = vec2(0.0, 0.0);
|
|
for (int i = 0; i < octaves; i++)
|
|
{
|
|
sum = cultmath_iv_add(sum, cultmath_iv_scale(cultmath_iv_snoise_ball(centre * frequency, radius * abs(frequency)), amplitude));
|
|
frequency *= lacunarity;
|
|
amplitude *= gain;
|
|
}
|
|
|
|
return sum;
|
|
}
|
|
|
|
// A ball (xyz centre, w radius) enclosing every point of every ray of a screen tile over the depth segment
|
|
// [z0, z1], warped by at most warp. Camera frame: pinhole at the origin looking down +z, ray (m z, z).
|
|
// footprintPerDepth = N / (sqrt(2) f) for an N x N tile at focal length f. The radius is widened by
|
|
// (|c|_1 + radius) 2^-20, which covers the ball's own float32 rounding. Derivation in math.Interval.cs.
|
|
vec4 cultmath_iv_frustum_ball(vec2 centreSlope, float z0, float z1, float footprintPerDepth, float warp)
|
|
{
|
|
float zm = (z0 + z1) * 0.5;
|
|
float axis = sqrt(centreSlope.x * centreSlope.x + centreSlope.y * centreSlope.y + 1.0);
|
|
float cx = centreSlope.x * zm;
|
|
float cy = centreSlope.y * zm;
|
|
float radius = (z1 - z0) * 0.5 * axis + z1 * footprintPerDepth + warp;
|
|
radius += (abs(cx) + abs(cy) + abs(zm) + radius) * 9.5367431640625e-7;
|
|
return vec4(cx, cy, zm, radius);
|
|
}
|
|
|
|
|
|
|
|
|
|
|
|
|
|
// 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 = 55.642143;
|
|
|
|
// An upper bound on the operator norm of the third-derivative tensor of cultmath_snoise(vec3), the
|
|
// next order after CULTMATH_SNOISE_HESSIAN; provenance in math.Span.cs (SNOISE_THIRD).
|
|
const float CULTMATH_SNOISE_THIRD = 536.9152;
|
|
|
|
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. 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((floatBitsToUint(lo) & 0x7fffffffu) > 0x7f800000u ? uintBitsToFloat(0xff800000u) : lo,
|
|
(floatBitsToUint(hi) & 0x7fffffffu) > 0x7f800000u ? uintBitsToFloat(0x7f800000u) : hi);
|
|
}
|
|
|
|
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);
|
|
vec2 local = cultmath_frac(position);
|
|
vec2 u = local * local * (vec2(3.0, 3.0) - vec2(2.0, 2.0) * local);
|
|
|
|
float a = cultmath_hash(cell);
|
|
float b = cultmath_hash(cell + vec2(1.0, 0.0));
|
|
float c = cultmath_hash(cell + vec2(0.0, 1.0));
|
|
float d = cultmath_hash(cell + vec2(1.0, 1.0));
|
|
|
|
return cultmath_lerp(cultmath_lerp(a, b, u.x), cultmath_lerp(c, d, u.x), u.y);
|
|
}
|
|
|
|
float cultmath_value_noise_bicubic(vec2 position)
|
|
{
|
|
vec2 cell = floor(position);
|
|
vec2 local = cultmath_frac(position);
|
|
|
|
float y0 = cultmath_catmullrom(cultmath_hash(cell + vec2(-1.0, -1.0)), cultmath_hash(cell + vec2(0.0, -1.0)), cultmath_hash(cell + vec2(1.0, -1.0)), cultmath_hash(cell + vec2(2.0, -1.0)), local.x);
|
|
float y1 = cultmath_catmullrom(cultmath_hash(cell + vec2(-1.0, 0.0)), cultmath_hash(cell + vec2(0.0, 0.0)), cultmath_hash(cell + vec2(1.0, 0.0)), cultmath_hash(cell + vec2(2.0, 0.0)), local.x);
|
|
float y2 = cultmath_catmullrom(cultmath_hash(cell + vec2(-1.0, 1.0)), cultmath_hash(cell + vec2(0.0, 1.0)), cultmath_hash(cell + vec2(1.0, 1.0)), cultmath_hash(cell + vec2(2.0, 1.0)), local.x);
|
|
float y3 = cultmath_catmullrom(cultmath_hash(cell + vec2(-1.0, 2.0)), cultmath_hash(cell + vec2(0.0, 2.0)), cultmath_hash(cell + vec2(1.0, 2.0)), cultmath_hash(cell + vec2(2.0, 2.0)), local.x);
|
|
|
|
return cultmath_catmullrom(y0, y1, y2, y3, local.y);
|
|
}
|
|
|
|
#endif
|