slang-shaders/include/special-functions.h

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#ifndef SPECIAL_FUNCTIONS_H
#define SPECIAL_FUNCTIONS_H
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///////////////////////////////// MIT LICENSE ////////////////////////////////
// Copyright (C) 2014 TroggleMonkey
//
// Permission is hereby granted, free of charge, to any person obtaining a copy
// of this software and associated documentation files (the "Software"), to
// deal in the Software without restriction, including without limitation the
// rights to use, copy, modify, merge, publish, distribute, sublicense, and/or
// sell copies of the Software, and to permit persons to whom the Software is
// furnished to do so, subject to the following conditions:
//
// The above copyright notice and this permission notice shall be included in
// all copies or substantial portions of the Software.
//
// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
// IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
// FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
// AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
// LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
// FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS
// IN THE SOFTWARE.
///////////////////////////////// DESCRIPTION ////////////////////////////////
// This file implements the following mathematical special functions:
// 1.) erf() = 2/sqrt(pi) * indefinite_integral(e**(-x**2))
// 2.) gamma(s), a real-numbered extension of the integer factorial function
// It also implements normalized_ligamma(s, z), a normalized lower incomplete
// gamma function for s < 0.5 only. Both gamma() and normalized_ligamma() can
// be called with an _impl suffix to use an implementation version with a few
// extra precomputed parameters (which may be useful for the caller to reuse).
// See below for details.
//
// Design Rationale:
// Pretty much every line of code in this file is duplicated four times for
// different input types (float4/float3/float2/float). This is unfortunate,
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// but Cg doesn't allow function templates. Macros would be far less verbose,
// but they would make the code harder to document and read. I don't expect
// these functions will require a whole lot of maintenance changes unless
// someone ever has need for more robust incomplete gamma functions, so code
// duplication seems to be the lesser evil in this case.
/////////////////////////// GAUSSIAN ERROR FUNCTION //////////////////////////
float4 erf6(float4 x)
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{
// Requires: x is the standard parameter to erf().
// Returns: Return an Abramowitz/Stegun approximation of erf(), where:
// erf(x) = 2/sqrt(pi) * integral(e**(-x**2))
// This approximation has a max absolute error of 2.5*10**-5
// with solid numerical robustness and efficiency. See:
// https://en.wikipedia.org/wiki/Error_function#Approximation_with_elementary_functions
static const float4 one = float4(1.0);
const float4 sign_x = sign(x);
const float4 t = one/(one + 0.47047*abs(x));
const float4 result = one - t*(0.3480242 + t*(-0.0958798 + t*0.7478556))*
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exp(-(x*x));
return result * sign_x;
}
float3 erf6(const float3 x)
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{
// Float3 version:
static const float3 one = float3(1.0);
const float3 sign_x = sign(x);
const float3 t = one/(one + 0.47047*abs(x));
const float3 result = one - t*(0.3480242 + t*(-0.0958798 + t*0.7478556))*
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exp(-(x*x));
return result * sign_x;
}
float2 erf6(const float2 x)
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{
// Float2 version:
static const float2 one = float2(1.0);
const float2 sign_x = sign(x);
const float2 t = one/(one + 0.47047*abs(x));
const float2 result = one - t*(0.3480242 + t*(-0.0958798 + t*0.7478556))*
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exp(-(x*x));
return result * sign_x;
}
float erf6(const float x)
{
// Float version:
const float sign_x = sign(x);
const float t = 1.0/(1.0 + 0.47047*abs(x));
const float result = 1.0 - t*(0.3480242 + t*(-0.0958798 + t*0.7478556))*
exp(-(x*x));
return result * sign_x;
}
float4 erft(const float4 x)
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{
// Requires: x is the standard parameter to erf().
// Returns: Approximate erf() with the hyperbolic tangent. The error is
// visually noticeable, but it's blazing fast and perceptually
// close...at least on ATI hardware. See:
// http://www.maplesoft.com/applications/view.aspx?SID=5525&view=html
// Warning: Only use this if your hardware drivers correctly implement
// tanh(): My nVidia 8800GTS returns garbage output.
return tanh(1.202760580 * x);
}
float3 erft(const float3 x)
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{
// Float3 version:
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return tanh(1.202760580 * x);
}
float2 erft(const float2 x)
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{
// Float2 version:
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return tanh(1.202760580 * x);
}
float erft(const float x)
{
// Float version:
return tanh(1.202760580 * x);
}
inline float4 erf(const float4 x)
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{
// Requires: x is the standard parameter to erf().
// Returns: Some approximation of erf(x), depending on user settings.
#ifdef ERF_FAST_APPROXIMATION
return erft(x);
#else
return erf6(x);
#endif
}
inline float3 erf(const float3 x)
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{
// Float3 version:
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#ifdef ERF_FAST_APPROXIMATION
return erft(x);
#else
return erf6(x);
#endif
}
inline float2 erf(const float2 x)
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{
// Float2 version:
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#ifdef ERF_FAST_APPROXIMATION
return erft(x);
#else
return erf6(x);
#endif
}
inline float erf(const float x)
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{
// Float version:
#ifdef ERF_FAST_APPROXIMATION
return erft(x);
#else
return erf6(x);
#endif
}
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/////////////////////////// COMPLETE GAMMA FUNCTION //////////////////////////
float4 gamma_impl(const float4 s, const float4 s_inv)
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{
// Requires: 1.) s is the standard parameter to the gamma function, and
// it should lie in the [0, 36] range.
// 2.) s_inv = 1.0/s. This implementation function requires
// the caller to precompute this value, giving users the
// opportunity to reuse it.
// Returns: Return approximate gamma function (real-numbered factorial)
// output using the Lanczos approximation with two coefficients
// calculated using Paul Godfrey's method here:
// http://my.fit.edu/~gabdo/gamma.txt
// An optimal g value for s in [0, 36] is ~1.12906830989, with
// a maximum relative error of 0.000463 for 2**16 equally
// evals. We could use three coeffs (0.0000346 error) without
// hurting latency, but this allows more parallelism with
// outside instructions.
static const float4 g = float4(1.12906830989);
static const float4 c0 = float4(0.8109119309638332633713423362694399653724431);
static const float4 c1 = float4(0.4808354605142681877121661197951496120000040);
static const float4 e = float4(2.71828182845904523536028747135266249775724709);
const float4 sph = s + float4(0.5);
const float4 lanczos_sum = c0 + c1/(s + float4(1.0));
const float4 base = (sph + g)/e; // or (s + g + float4(0.5))/e
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// gamma(s + 1) = base**sph * lanczos_sum; divide by s for gamma(s).
// This has less error for small s's than (s -= 1.0) at the beginning.
return (pow(base, sph) * lanczos_sum) * s_inv;
}
float3 gamma_impl(const float3 s, const float3 s_inv)
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{
// Float3 version:
static const float3 g = float3(1.12906830989);
static const float3 c0 = float3(0.8109119309638332633713423362694399653724431);
static const float3 c1 = float3(0.4808354605142681877121661197951496120000040);
static const float3 e = float3(2.71828182845904523536028747135266249775724709);
const float3 sph = s + float3(0.5);
const float3 lanczos_sum = c0 + c1/(s + float3(1.0));
const float3 base = (sph + g)/e;
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return (pow(base, sph) * lanczos_sum) * s_inv;
}
float2 gamma_impl(const float2 s, const float2 s_inv)
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{
// Float2 version:
static const float2 g = float2(1.12906830989);
static const float2 c0 = float2(0.8109119309638332633713423362694399653724431);
static const float2 c1 = float2(0.4808354605142681877121661197951496120000040);
static const float2 e = float2(2.71828182845904523536028747135266249775724709);
const float2 sph = s + float2(0.5);
const float2 lanczos_sum = c0 + c1/(s + float2(1.0));
const float2 base = (sph + g)/e;
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return (pow(base, sph) * lanczos_sum) * s_inv;
}
float gamma_impl(const float s, const float s_inv)
{
// Float version:
static const float g = 1.12906830989;
static const float c0 = 0.8109119309638332633713423362694399653724431;
static const float c1 = 0.4808354605142681877121661197951496120000040;
static const float e = 2.71828182845904523536028747135266249775724709;
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const float sph = s + 0.5;
const float lanczos_sum = c0 + c1/(s + 1.0);
const float base = (sph + g)/e;
return (pow(base, sph) * lanczos_sum) * s_inv;
}
float4 gamma(const float4 s)
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{
// Requires: s is the standard parameter to the gamma function, and it
// should lie in the [0, 36] range.
// Returns: Return approximate gamma function output with a maximum
// relative error of 0.000463. See gamma_impl for details.
return gamma_impl(s, float4(1.0)/s);
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}
float3 gamma(const float3 s)
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{
// Float3 version:
return gamma_impl(s, float3(1.0)/s);
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}
float2 gamma(const float2 s)
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{
// Float2 version:
return gamma_impl(s, float2(1.0)/s);
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}
float gamma(const float s)
{
// Float version:
return gamma_impl(s, 1.0/s);
}
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//////////////// INCOMPLETE GAMMA FUNCTIONS (RESTRICTED INPUT) ///////////////
// Lower incomplete gamma function for small s and z (implementation):
float4 ligamma_small_z_impl(const float4 s, const float4 z, const float4 s_inv)
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{
// Requires: 1.) s < ~0.5
// 2.) z <= ~0.775075
// 3.) s_inv = 1.0/s (precomputed for outside reuse)
// Returns: A series representation for the lower incomplete gamma
// function for small s and small z (4 terms).
// The actual "rolled up" summation looks like:
// last_sign = 1.0; last_pow = 1.0; last_factorial = 1.0;
// sum = last_sign * last_pow / ((s + k) * last_factorial)
// for(int i = 0; i < 4; ++i)
// {
// last_sign *= -1.0; last_pow *= z; last_factorial *= i;
// sum += last_sign * last_pow / ((s + k) * last_factorial);
// }
// Unrolled, constant-unfolded and arranged for madds and parallelism:
const float4 scale = pow(z, s);
float4 sum = s_inv; // Summation iteration 0 result
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// Summation iterations 1, 2, and 3:
const float4 z_sq = z*z;
const float4 denom1 = s + float4(1.0);
const float4 denom2 = 2.0*s + float4(4.0);
const float4 denom3 = 6.0*s + float4(18.0);
//float4 denom4 = 24.0*s + float4(96.0);
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sum -= z/denom1;
sum += z_sq/denom2;
sum -= z * z_sq/denom3;
//sum += z_sq * z_sq / denom4;
// Scale and return:
return scale * sum;
}
float3 ligamma_small_z_impl(const float3 s, const float3 z, const float3 s_inv)
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{
// Float3 version:
const float3 scale = pow(z, s);
float3 sum = s_inv;
const float3 z_sq = z*z;
const float3 denom1 = s + float3(1.0);
const float3 denom2 = 2.0*s + float3(4.0);
const float3 denom3 = 6.0*s + float3(18.0);
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sum -= z/denom1;
sum += z_sq/denom2;
sum -= z * z_sq/denom3;
return scale * sum;
}
float2 ligamma_small_z_impl(const float2 s, const float2 z, const float2 s_inv)
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{
// Float2 version:
const float2 scale = pow(z, s);
float2 sum = s_inv;
const float2 z_sq = z*z;
const float2 denom1 = s + float2(1.0);
const float2 denom2 = 2.0*s + float2(4.0);
const float2 denom3 = 6.0*s + float2(18.0);
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sum -= z/denom1;
sum += z_sq/denom2;
sum -= z * z_sq/denom3;
return scale * sum;
}
float ligamma_small_z_impl(const float s, const float z, const float s_inv)
{
// Float version:
const float scale = pow(z, s);
float sum = s_inv;
const float z_sq = z*z;
const float denom1 = s + 1.0;
const float denom2 = 2.0*s + 4.0;
const float denom3 = 6.0*s + 18.0;
sum -= z/denom1;
sum += z_sq/denom2;
sum -= z * z_sq/denom3;
return scale * sum;
}
// Upper incomplete gamma function for small s and large z (implementation):
float4 uigamma_large_z_impl(const float4 s, const float4 z)
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{
// Requires: 1.) s < ~0.5
// 2.) z > ~0.775075
// Returns: Gauss's continued fraction representation for the upper
// incomplete gamma function (4 terms).
// The "rolled up" continued fraction looks like this. The denominator
// is truncated, and it's calculated "from the bottom up:"
// denom = float4('inf');
// float4 one = float4(1.0);
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// for(int i = 4; i > 0; --i)
// {
// denom = ((i * 2.0) - one) + z - s + (i * (s - i))/denom;
// }
// Unrolled and constant-unfolded for madds and parallelism:
const float4 numerator = pow(z, s) * exp(-z);
float4 denom = float4(7.0) + z - s;
denom = float4(5.0) + z - s + (3.0*s - float4(9.0))/denom;
denom = float4(3.0) + z - s + (2.0*s - float4(4.0))/denom;
denom = float4(1.0) + z - s + (s - float4(1.0))/denom;
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return numerator / denom;
}
float3 uigamma_large_z_impl(const float3 s, const float3 z)
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{
// Float3 version:
const float3 numerator = pow(z, s) * exp(-z);
float3 denom = float3(7.0) + z - s;
denom = float3(5.0) + z - s + (3.0*s - float3(9.0))/denom;
denom = float3(3.0) + z - s + (2.0*s - float3(4.0))/denom;
denom = float3(1.0) + z - s + (s - float3(1.0))/denom;
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return numerator / denom;
}
float2 uigamma_large_z_impl(const float2 s, const float2 z)
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{
// Float2 version:
const float2 numerator = pow(z, s) * exp(-z);
float2 denom = float2(7.0) + z - s;
denom = float2(5.0) + z - s + (3.0*s - float2(9.0))/denom;
denom = float2(3.0) + z - s + (2.0*s - float2(4.0))/denom;
denom = float2(1.0) + z - s + (s - float2(1.0))/denom;
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return numerator / denom;
}
float uigamma_large_z_impl(const float s, const float z)
{
// Float version:
const float numerator = pow(z, s) * exp(-z);
float denom = 7.0 + z - s;
denom = 5.0 + z - s + (3.0*s - 9.0)/denom;
denom = 3.0 + z - s + (2.0*s - 4.0)/denom;
denom = 1.0 + z - s + (s - 1.0)/denom;
return numerator / denom;
}
// Normalized lower incomplete gamma function for small s (implementation):
float4 normalized_ligamma_impl(const float4 s, const float4 z,
const float4 s_inv, const float4 gamma_s_inv)
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{
// Requires: 1.) s < ~0.5
// 2.) s_inv = 1/s (precomputed for outside reuse)
// 3.) gamma_s_inv = 1/gamma(s) (precomputed for outside reuse)
// Returns: Approximate the normalized lower incomplete gamma function
// for s < 0.5. Since we only care about s < 0.5, we only need
// to evaluate two branches (not four) based on z. Each branch
// uses four terms, with a max relative error of ~0.00182. The
// branch threshold and specifics were adapted for fewer terms
// from Gil/Segura/Temme's paper here:
// http://oai.cwi.nl/oai/asset/20433/20433B.pdf
// Evaluate both branches: Real branches test slower even when available.
static const float4 thresh = float4(0.775075);
bool4 z_is_large;
z_is_large.x = z.x > thresh.x;
z_is_large.y = z.y > thresh.y;
z_is_large.z = z.z > thresh.z;
z_is_large.w = z.w > thresh.w;
const float4 large_z = float4(1.0) - uigamma_large_z_impl(s, z) * gamma_s_inv;
const float4 small_z = ligamma_small_z_impl(s, z, s_inv) * gamma_s_inv;
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// Combine the results from both branches:
bool4 inverse_z_is_large = not(z_is_large);
return large_z * float4(z_is_large) + small_z * float4(inverse_z_is_large);
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}
float3 normalized_ligamma_impl(const float3 s, const float3 z,
const float3 s_inv, const float3 gamma_s_inv)
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{
// Float3 version:
static const float3 thresh = float3(0.775075);
bool3 z_is_large;
z_is_large.x = z.x > thresh.x;
z_is_large.y = z.y > thresh.y;
z_is_large.z = z.z > thresh.z;
const float3 large_z = float3(1.0) - uigamma_large_z_impl(s, z) * gamma_s_inv;
const float3 small_z = ligamma_small_z_impl(s, z, s_inv) * gamma_s_inv;
bool3 inverse_z_is_large = not(z_is_large);
return large_z * float3(z_is_large) + small_z * float3(inverse_z_is_large);
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}
float2 normalized_ligamma_impl(const float2 s, const float2 z,
const float2 s_inv, const float2 gamma_s_inv)
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{
// Float2 version:
static const float2 thresh = float2(0.775075);
bool2 z_is_large;
z_is_large.x = z.x > thresh.x;
z_is_large.y = z.y > thresh.y;
const float2 large_z = float2(1.0) - uigamma_large_z_impl(s, z) * gamma_s_inv;
const float2 small_z = ligamma_small_z_impl(s, z, s_inv) * gamma_s_inv;
bool2 inverse_z_is_large = not(z_is_large);
return large_z * float2(z_is_large) + small_z * float2(inverse_z_is_large);
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}
float normalized_ligamma_impl(const float s, const float z,
const float s_inv, const float gamma_s_inv)
{
// Float version:
static const float thresh = 0.775075;
const bool z_is_large = z > thresh;
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const float large_z = 1.0 - uigamma_large_z_impl(s, z) * gamma_s_inv;
const float small_z = ligamma_small_z_impl(s, z, s_inv) * gamma_s_inv;
return large_z * float(z_is_large) + small_z * float(!z_is_large);
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}
// Normalized lower incomplete gamma function for small s:
float4 normalized_ligamma(const float4 s, const float4 z)
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{
// Requires: s < ~0.5
// Returns: Approximate the normalized lower incomplete gamma function
// for s < 0.5. See normalized_ligamma_impl() for details.
const float4 s_inv = float4(1.0)/s;
const float4 gamma_s_inv = float4(1.0)/gamma_impl(s, s_inv);
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return normalized_ligamma_impl(s, z, s_inv, gamma_s_inv);
}
float3 normalized_ligamma(const float3 s, const float3 z)
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{
// Float3 version:
const float3 s_inv = float3(1.0)/s;
const float3 gamma_s_inv = float3(1.0)/gamma_impl(s, s_inv);
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return normalized_ligamma_impl(s, z, s_inv, gamma_s_inv);
}
float2 normalized_ligamma(const float2 s, const float2 z)
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{
// Float2 version:
const float2 s_inv = float2(1.0)/s;
const float2 gamma_s_inv = float2(1.0)/gamma_impl(s, s_inv);
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return normalized_ligamma_impl(s, z, s_inv, gamma_s_inv);
}
float normalized_ligamma(const float s, const float z)
{
// Float version:
const float s_inv = 1.0/s;
const float gamma_s_inv = 1.0/gamma_impl(s, s_inv);
return normalized_ligamma_impl(s, z, s_inv, gamma_s_inv);
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}
#endif // SPECIAL_FUNCTIONS_H