tests/MathUnitTests/unittests.cpp - SwiftShader - Git at Google

 // Copyright 2019 The SwiftShader Authors. All Rights Reserved.
 //
 // Licensed under the Apache License, Version 2.0 (the "License");
 // you may not use this file except in compliance with the License.
 // You may obtain a copy of the License at
 //
 //    http://www.apache.org/licenses/LICENSE-2.0
 //
 // Unless required by applicable law or agreed to in writing, software
 // distributed under the License is distributed on an "AS IS" BASIS,
 // WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 // See the License for the specific language governing permissions and
 // limitations under the License.

 #include "System/CPUID.hpp"
 #include "System/Half.hpp"
 #include "System/Math.hpp"

 #include <gmock/gmock.h>
 #include <gtest/gtest.h>

 #include <cstdlib>
 #include <cmath>

 using std::isnan;
 using std::isinf;
 using std::signbit;

 using namespace sw;

 // Implementation of frexp() which satisfies C++ <cmath> requirements.
 float fast_frexp(float val, int *exp)
 {
 	int isNotZero = (val != 0.0f) ? 0xFFFFFFFF : 0x00000000;
 	int v = bit_cast<int>(val);
 	int isInfOrNaN = (v & 0x7F800000) == 0x7F800000 ? 0xFFFFFFFF : 0x00000000;

 	// When val is a subnormal value we can't directly use its mantissa to construct the significand in
 	// the range [0.5, 1.0). We need to multiply it by a factor that makes it normalized. For large
 	// values the factor must avoid overflow to inifity.
 	int factor = ((127 + 23) << 23) - (v & 0x3F800000);
 	int nval = bit_cast<int>(val * bit_cast<float>(factor));

 	// Extract the exponent of the normalized value and subtract the exponent of the normalizing factor.
 	int exponent = ((((nval & 0x7F800000) - factor) >> 23) + 1) & isNotZero;

 	// Substitute the exponent of 0.5f (if not zero) to obtain the significand.
 	float significand = bit_cast<float>((nval & 0x807FFFFF) | (0x3F000000 & isNotZero) | (0x7F800000 & isInfOrNaN));

 	*exp = exponent;
 	return significand;
 }

 TEST(MathTest, Frexp)
 {
 	for(bool flush : { false, true })
 	{
 		CPUID::setDenormalsAreZero(flush);
 		CPUID::setFlushToZero(flush);

 		std::vector<float> a = {
 			2.3f,
 			0.1f,
 			0.7f,
 			1.7f,
 			0.0f,
 			-2.3f,
 			-0.1f,
 			-0.7f,
 			-1.7f,
 			-0.0f,
 			100000000.0f,
 			-100000000.0f,
 			0.000000001f,
 			-0.000000001f,
 			FLT_MIN,
 			-FLT_MIN,
 			FLT_MAX,
 			-FLT_MAX,
 			FLT_TRUE_MIN,
 			-FLT_TRUE_MIN,
 			INFINITY,
 			-INFINITY,
 			NAN,
 			bit_cast<float>(0x007FFFFF),  // Largest subnormal
 			bit_cast<float>(0x807FFFFF),
 			bit_cast<float>(0x00000001),  // Smallest subnormal
 			bit_cast<float>(0x80000001),
 		};

 		for(float f : a)
 		{
 			int exp = -1000;
 			float sig = fast_frexp(f, &exp);

 			if(f == 0.0f)  // Could be subnormal if `flush` is true
 			{
 				// We don't rely on std::frexp here to produce a reference result because it may
 				// return non-zero significands and exponents for subnormal arguments., while our
 				// implementation is meant to respect denormals-are-zero / flush-to-zero.

 				ASSERT_EQ(sig, 0.0f) << "Argument: " << std::hexfloat << f;
 				ASSERT_TRUE(signbit(sig) == signbit(f)) << "Argument: " << std::hexfloat << f;
 				ASSERT_EQ(exp, 0) << "Argument: " << std::hexfloat << f;
 			}
 			else
 			{
 				int ref_exp = -1000;
 				float ref_sig = std::frexp(f, &ref_exp);

 				if(!isnan(f))
 				{
 					ASSERT_EQ(sig, ref_sig) << "Argument: " << std::hexfloat << f;
 				}
 				else
 				{
 					ASSERT_TRUE(isnan(sig)) << "Significand: " << std::hexfloat << sig;
 				}

 				if(!isinf(f) && !isnan(f))  // If the argument is NaN or Inf the exponent is unspecified.
 				{
 					ASSERT_EQ(exp, ref_exp) << "Argument: " << std::hexfloat << f;
 				}
 			}
 		}
 	}
 }

 // Returns the whole-number ULP error of `a` relative to `x`.
 // Use the doouble-precision version below. This just illustrates the principle.
 [[deprecated]] float ULP_32(float x, float a)
 {
 	// Flip the last mantissa bit to compute the 'unit in the last place' error.
 	float x1 = bit_cast<float>(bit_cast<uint32_t>(x) ^ 0x00000001);
 	float ulp = abs(x1 - x);

 	return abs(a - x) / ulp;
 }

 double ULP_32(double x, double a)
 {
 	// binary64 has 52 mantissa bits, while binary32 has 23, so the ULP for the latter is 29 bits shifted.
 	double x1 = bit_cast<double>(bit_cast<uint64_t>(x) ^ 0x0000000020000000ull);
 	double ulp = abs(x1 - x);

 	return abs(a - x) / ulp;
 }

 float ULP_16(float x, float a)
 {
 	// binary32 has 23 mantissa bits, while binary16 has 10, so the ULP for the latter is 13 bits shifted.
 	double x1 = bit_cast<float>(bit_cast<uint32_t>(x) ^ 0x00002000);
 	float ulp = abs(x1 - x);

 	return abs(a - x) / ulp;
 }

 // lolremez --float -d 2 -r "0:2^23" "(log2(x/2^23+1)-x/2^23)/x" "1/x"
 // ULP-16: 0.797363281, abs: 0.0991751999
 float f(float x)
 {
 	float u = 2.8017103e-22f;
 	u = u * x + -8.373131e-15f;
 	return u * x + 5.0615534e-8f;
 }

 float Log2Relaxed(float x)
 {
 	// Reinterpretation as an integer provides a piecewise linear
 	// approximation of log2(). Scale to the radix and subtract exponent bias.
 	int im = bit_cast<int>(x);
 	float y = (float)im * (1.0f / (1 << 23)) - 127.0f;

 	// Handle log2(inf) = inf.
 	if(im == 0x7F800000) y = INFINITY;

 	float m = (float)(im & 0x007FFFFF);  // Unnormalized mantissa of x.

 	// Add a polynomial approximation of log2(m+1)-m to the result's mantissa.
 	return f(m) * m + y;
 }

 TEST(MathTest, Log2RelaxedExhaustive)
 {
 	CPUID::setDenormalsAreZero(true);
 	CPUID::setFlushToZero(true);

 	float worst_margin = 0;
 	float worst_ulp = 0;
 	float worst_x = 0;
 	float worst_val = 0;
 	float worst_ref = 0;

 	float worst_abs = 0;

 	for(float x = 0.0f; x <= INFINITY; x = inc(x))
 	{
 		float val = Log2Relaxed(x);

 		double ref = log2((double)x);

 		if(ref == (int)ref)
 		{
 			ASSERT_EQ(val, ref);
 		}
 		else if(x >= 0.5f && x <= 2.0f)
 		{
 			const float tolerance = pow(2.0f, -7.0f);  // Absolute

 			float margin = abs(val - ref) / tolerance;

 			if(margin > worst_abs)
 			{
 				worst_abs = margin;
 			}
 		}
 		else
 		{
 			const float tolerance = 3;  // ULP

 			float ulp = (float)ULP_16(ref, (double)val);
 			float margin = ulp / tolerance;

 			if(margin > worst_margin)
 			{
 				worst_margin = margin;
 				worst_ulp = ulp;
 				worst_x = x;
 				worst_val = val;
 				worst_ref = ref;
 			}
 		}
 	}

 	ASSERT_TRUE(worst_margin < 1.0f) << " worst_x " << worst_x << " worst_val " << worst_val << " worst_ref " << worst_ref << " worst_ulp " << worst_ulp;
 	ASSERT_TRUE(worst_abs <= 1.0f) << " worst_x " << worst_x << " worst_val " << worst_val << " worst_ref " << worst_ref << " worst_ulp " << worst_ulp;

 	CPUID::setDenormalsAreZero(false);
 	CPUID::setFlushToZero(false);
 }

 // lolremez --float -d 2 -r "0:1" "(2^x-x-1)/x" "1/x"
 // ULP-16: 0.130859017
 float Pr(float x)
 {
 	float u = 7.8145574e-2f;
 	u = u * x + 2.2617357e-1f;
 	return u * x + -3.0444314e-1f;
 }

 float Exp2Relaxed(float x)
 {
 	x = min(x, 128.0f);
 	x = max(x, bit_cast<float>(int(0xC2FDFFFF)));  // -126.999992

 	// 2^f - f - 1 as P(f) * f
 	// This is a correction term to be added to 1+x to obtain 2^x.
 	float f = x - floor(x);
 	float y = Pr(f) * f + x;

 	// bit_cast<float>(int(x * 2^23)) is a piecewise linear approximation of 2^(x-127).
 	// See "Fast Exponential Computation on SIMD Architectures" by Malossi et al.
 	return bit_cast<float>(int((1 << 23) * y + (127 << 23)));
 }

 TEST(MathTest, Exp2RelaxedExhaustive)
 {
 	CPUID::setDenormalsAreZero(true);
 	CPUID::setFlushToZero(true);

 	float worst_margin = 0;
 	float worst_ulp = 0;
 	float worst_x = 0;
 	float worst_val = 0;
 	float worst_ref = 0;

 	for(float x = -10; x <= 10; x = inc(x))
 	{
 		float val = Exp2Relaxed(x);

 		double ref = exp2((double)x);

 		if(x == (int)x)
 		{
 			ASSERT_EQ(val, ref);
 		}

 		const float tolerance = (1 + 2 * abs(x));
 		float ulp = ULP_16((float)ref, val);
 		float margin = ulp / tolerance;

 		if(margin > worst_margin)
 		{
 			worst_margin = margin;
 			worst_ulp = ulp;
 			worst_x = x;
 			worst_val = val;
 			worst_ref = ref;
 		}
 	}

 	ASSERT_TRUE(worst_margin <= 1.0f) << " worst_x " << worst_x << " worst_val " << worst_val << " worst_ref " << worst_ref << " worst_ulp " << worst_ulp;

 	CPUID::setDenormalsAreZero(false);
 	CPUID::setFlushToZero(false);
 }

 // lolremez --float -d 7 -r "0:1" "(log2(x+1)-x)/x" "1/x"
 // ULP-32: 1.69571960, abs: 0.360798746
 float Pl(float x)
 {
 	float u = -9.3091638e-3f;
 	u = u * x + 5.2059003e-2f;
 	u = u * x + -1.3752135e-1f;
 	u = u * x + 2.4186478e-1f;
 	u = u * x + -3.4730109e-1f;
 	u = u * x + 4.786837e-1f;
 	u = u * x + -7.2116581e-1f;
 	return u * x + 4.4268988e-1f;
 }

 float Log2(float x)
 {
 	// Reinterpretation as an integer provides a piecewise linear
 	// approximation of log2(). Scale to the radix and subtract exponent bias.
 	int im = bit_cast<int>(x);
 	float y = (float)(im - (127 << 23)) * (1.0f / (1 << 23));

 	// Handle log2(inf) = inf.
 	if(im == 0x7F800000) y = INFINITY;

 	float m = (float)(im & 0x007FFFFF) * (1.0f / (1 << 23));  // Normalized mantissa of x.

 	// Add a polynomial approximation of log2(m+1)-m to the result's mantissa.
 	return Pl(m) * m + y;
 }

 TEST(MathTest, Log2Exhaustive)
 {
 	CPUID::setDenormalsAreZero(true);
 	CPUID::setFlushToZero(true);

 	float worst_margin = 0;
 	float worst_ulp = 0;
 	float worst_x = 0;
 	float worst_val = 0;
 	float worst_ref = 0;

 	float worst_abs = 0;

 	for(float x = 0.0f; x <= INFINITY; x = inc(x))
 	{
 		float val = Log2(x);

 		double ref = log2((double)x);

 		if(ref == (int)ref)
 		{
 			ASSERT_EQ(val, ref);
 		}
 		else if(x >= 0.5f && x <= 2.0f)
 		{
 			const float tolerance = pow(2.0f, -21.0f);  // Absolute

 			float margin = abs(val - ref) / tolerance;

 			if(margin > worst_abs)
 			{
 				worst_abs = margin;
 			}
 		}
 		else
 		{
 			const float tolerance = 3;  // ULP

 			float ulp = (float)ULP_32(ref, (double)val);
 			float margin = ulp / tolerance;

 			if(margin > worst_margin)
 			{
 				worst_margin = margin;
 				worst_ulp = ulp;
 				worst_x = x;
 				worst_val = val;
 				worst_ref = ref;
 			}
 		}
 	}

 	ASSERT_TRUE(worst_margin < 1.0f) << " worst_x " << worst_x << " worst_val " << worst_val << " worst_ref " << worst_ref << " worst_ulp " << worst_ulp;
 	ASSERT_TRUE(worst_abs <= 1.0f) << " worst_x " << worst_x << " worst_val " << worst_val << " worst_ref " << worst_ref << " worst_ulp " << worst_ulp;

 	CPUID::setDenormalsAreZero(false);
 	CPUID::setFlushToZero(false);
 }

 // lolremez --float -d 4 -r "0:1" "(2^x-x-1)/x" "1/x"
 // ULP_32: 2.14694786, Vulkan margin: 0.686957061
 float P(float x)
 {
 	float u = 1.8852974e-3f;
 	u = u * x + 8.9733787e-3f;
 	u = u * x + 5.5835927e-2f;
 	u = u * x + 2.4015281e-1f;
 	return u * x + -3.0684753e-1f;
 }

 float Exp2(float x)
 {
 	x = min(x, 128.0f);
 	x = max(x, bit_cast<float>(0xC2FDFFFF));  // -126.999992

 	// 2^f - f - 1 as P(f) * f
 	// This is a correction term to be added to 1+x to obtain 2^x.
 	float f = x - floor(x);
 	float y = P(f) * f + x;

 	// bit_cast<float>(int(x * 2^23)) is a piecewise linear approximation of 2^(x-127).
 	// See "Fast Exponential Computation on SIMD Architectures" by Malossi et al.
 	return bit_cast<float>(int(y * (1 << 23)) + (127 << 23));
 }

 TEST(MathTest, Exp2Exhaustive)
 {
 	CPUID::setDenormalsAreZero(true);
 	CPUID::setFlushToZero(true);

 	float worst_margin = 0;
 	float worst_ulp = 0;
 	float worst_x = 0;
 	float worst_val = 0;
 	float worst_ref = 0;

 	for(float x = -10; x <= 10; x = inc(x))
 	{
 		float val = Exp2(x);

 		double ref = exp2((double)x);

 		if(x == (int)x)
 		{
 			ASSERT_EQ(val, ref);
 		}

 		const float tolerance = (3 + 2 * abs(x));
 		float ulp = (float)ULP_32(ref, (double)val);
 		float margin = ulp / tolerance;

 		if(margin > worst_margin)
 		{
 			worst_margin = margin;
 			worst_ulp = ulp;
 			worst_x = x;
 			worst_val = val;
 			worst_ref = ref;
 		}
 	}

 	ASSERT_TRUE(worst_margin <= 1.0f) << " worst_x " << worst_x << " worst_val " << worst_val << " worst_ref " << worst_ref << " worst_ulp " << worst_ulp;

 	CPUID::setDenormalsAreZero(false);
 	CPUID::setFlushToZero(false);
 }

 // Polynomial approximation of order 5 for sin(x * 2 * pi) in the range [-1/4, 1/4]
 static float sin5(float x)
 {
 	// A * x^5 + B * x^3 + C * x
 	// Exact at x = 0, 1/12, 1/6, 1/4, and their negatives, which correspond to x * 2 * pi = 0, pi/6, pi/3, pi/2
 	const float A = (36288 - 20736 * sqrt(3)) / 5;
 	const float B = 288 * sqrt(3) - 540;
 	const float C = (47 - 9 * sqrt(3)) / 5;

 	float x2 = x * x;

 	return ((A * x2 + B) * x2 + C) * x;
 }

 TEST(MathTest, SinExhaustive)
 {
 	const float tolerance = powf(2.0f, -12.0f);  // Vulkan requires absolute error <= 2^−11 inside the range [−pi, pi]
 	const float pi = 3.1415926535f;

 	for(float x = -pi; x <= pi; x = inc(x))
 	{
 		// Range reduction and mirroring
 		float x_2 = 0.25f - x * (0.5f / pi);
 		float z = 0.25f - fabs(x_2 - round(x_2));

 		float val = sin5(z);

 		ASSERT_NEAR(val, sinf(x), tolerance);
 	}
 }

 TEST(MathTest, CosExhaustive)
 {
 	const float tolerance = powf(2.0f, -12.0f);  // Vulkan requires absolute error <= 2^−11 inside the range [−pi, pi]
 	const float pi = 3.1415926535f;

 	for(float x = -pi; x <= pi; x = inc(x))
 	{
 		// Phase shift, range reduction, and mirroring
 		float x_2 = x * (0.5f / pi);
 		float z = 0.25f - fabs(x_2 - round(x_2));

 		float val = sin5(z);

 		ASSERT_NEAR(val, cosf(x), tolerance);
 	}
 }

 TEST(MathTest, UnsignedFloat11_10)
 {
 	// Test the largest value which causes underflow to 0, and the smallest value
 	// which produces a denormalized result.

 	EXPECT_EQ(R11G11B10F::float32ToFloat11(bit_cast<float>(0x3500007F)), 0x0000);
 	EXPECT_EQ(R11G11B10F::float32ToFloat11(bit_cast<float>(0x35000080)), 0x0001);

 	EXPECT_EQ(R11G11B10F::float32ToFloat10(bit_cast<float>(0x3580003F)), 0x0000);
 	EXPECT_EQ(R11G11B10F::float32ToFloat10(bit_cast<float>(0x35800040)), 0x0001);
 }

 // Clamps to the [0, hi] range. NaN input produces 0, hi must be non-NaN.
 float clamp0hi(float x, float hi)
 {
 	// If x=NaN, x > 0 will compare false and we return 0.
 	if(!(x > 0))
 	{
 		return 0;
 	}

 	// x is non-NaN at this point, so std::min() is safe for non-NaN hi.
 	return std::min(x, hi);
 }

 unsigned int RGB9E5_reference(float r, float g, float b)
 {
 	// Vulkan 1.1.117 section 15.2.1 RGB to Shared Exponent Conversion

 	// B is the exponent bias (15)
 	constexpr int g_sharedexp_bias = 15;

 	// N is the number of mantissa bits per component (9)
 	constexpr int g_sharedexp_mantissabits = 9;

 	// Emax is the maximum allowed biased exponent value (31)
 	constexpr int g_sharedexp_maxexponent = 31;

 	constexpr float g_sharedexp_max =
 	    ((static_cast<float>(1 << g_sharedexp_mantissabits) - 1) /
 	     static_cast<float>(1 << g_sharedexp_mantissabits)) *
 	    static_cast<float>(1 << (g_sharedexp_maxexponent - g_sharedexp_bias));

 	const float red_c = clamp0hi(r, g_sharedexp_max);
 	const float green_c = clamp0hi(g, g_sharedexp_max);
 	const float blue_c = clamp0hi(b, g_sharedexp_max);

 	const float max_c = fmax(fmax(red_c, green_c), blue_c);
 	const float exp_p = fmax(-g_sharedexp_bias - 1, floor(log2(max_c))) + 1 + g_sharedexp_bias;
 	const int max_s = static_cast<int>(floor((max_c / exp2(exp_p - g_sharedexp_bias - g_sharedexp_mantissabits)) + 0.5f));
 	const int exp_s = static_cast<int>((max_s < exp2(g_sharedexp_mantissabits)) ? exp_p : exp_p + 1);

 	unsigned int R = static_cast<unsigned int>(floor((red_c / exp2(exp_s - g_sharedexp_bias - g_sharedexp_mantissabits)) + 0.5f));
 	unsigned int G = static_cast<unsigned int>(floor((green_c / exp2(exp_s - g_sharedexp_bias - g_sharedexp_mantissabits)) + 0.5f));
 	unsigned int B = static_cast<unsigned int>(floor((blue_c / exp2(exp_s - g_sharedexp_bias - g_sharedexp_mantissabits)) + 0.5f));
 	unsigned int E = exp_s;

 	return (E << 27) | (B << 18) | (G << 9) | R;
 }

 TEST(MathTest, SharedExponentSparse)
 {
 	for(uint64_t i = 0; i < 0x0000000100000000; i += 0x400)
 	{
 		float f = bit_cast<float>(i);

 		unsigned int ref = RGB9E5_reference(f, 0.0f, 0.0f);
 		unsigned int val = RGB9E5(f, 0.0f, 0.0f);

 		EXPECT_EQ(ref, val);
 	}
 }

 TEST(MathTest, SharedExponentRandom)
 {
 	srand(0);

 	unsigned int x = 0;
 	unsigned int y = 0;
 	unsigned int z = 0;

 	for(int i = 0; i < 10000000; i++)
 	{
 		float r = bit_cast<float>(x);
 		float g = bit_cast<float>(y);
 		float b = bit_cast<float>(z);

 		unsigned int ref = RGB9E5_reference(r, g, b);
 		unsigned int val = RGB9E5(r, g, b);

 		EXPECT_EQ(ref, val);

 		x += rand();
 		y += rand();
 		z += rand();
 	}
 }

 TEST(MathTest, SharedExponentExhaustive)
 {
 	for(uint64_t i = 0; i < 0x0000000100000000; i += 1)
 	{
 		float f = bit_cast<float>(i);

 		unsigned int ref = RGB9E5_reference(f, 0.0f, 0.0f);
 		unsigned int val = RGB9E5(f, 0.0f, 0.0f);

 		EXPECT_EQ(ref, val);
 	}
 }
	// Copyright 2019 The SwiftShader Authors. All Rights Reserved.
	//
	// Licensed under the Apache License, Version 2.0 (the "License");
	// you may not use this file except in compliance with the License.
	// You may obtain a copy of the License at
	//
	// http://www.apache.org/licenses/LICENSE-2.0
	//
	// Unless required by applicable law or agreed to in writing, software
	// distributed under the License is distributed on an "AS IS" BASIS,
	// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
	// See the License for the specific language governing permissions and
	// limitations under the License.

	#include "System/CPUID.hpp"
	#include "System/Half.hpp"
	#include "System/Math.hpp"

	#include <gmock/gmock.h>
	#include <gtest/gtest.h>

	#include <cstdlib>
	#include <cmath>

	using std::isnan;
	using std::isinf;
	using std::signbit;

	using namespace sw;

	// Implementation of frexp() which satisfies C++ <cmath> requirements.
	float fast_frexp(float val, int *exp)
	{
	int isNotZero = (val != 0.0f) ? 0xFFFFFFFF : 0x00000000;
	int v = bit_cast<int>(val);
	int isInfOrNaN = (v & 0x7F800000) == 0x7F800000 ? 0xFFFFFFFF : 0x00000000;

	// When val is a subnormal value we can't directly use its mantissa to construct the significand in
	// the range [0.5, 1.0). We need to multiply it by a factor that makes it normalized. For large
	// values the factor must avoid overflow to inifity.
	int factor = ((127 + 23) << 23) - (v & 0x3F800000);
	int nval = bit_cast<int>(val * bit_cast<float>(factor));

	// Extract the exponent of the normalized value and subtract the exponent of the normalizing factor.
	int exponent = ((((nval & 0x7F800000) - factor) >> 23) + 1) & isNotZero;

	// Substitute the exponent of 0.5f (if not zero) to obtain the significand.
	float significand = bit_cast<float>((nval & 0x807FFFFF) \| (0x3F000000 & isNotZero) \| (0x7F800000 & isInfOrNaN));

	*exp = exponent;
	return significand;
	}

	TEST(MathTest, Frexp)
	{
	for(bool flush : { false, true })
	{
	CPUID::setDenormalsAreZero(flush);
	CPUID::setFlushToZero(flush);

	std::vector<float> a = {
	2.3f,
	0.1f,
	0.7f,
	1.7f,
	0.0f,
	-2.3f,
	-0.1f,
	-0.7f,
	-1.7f,
	-0.0f,
	100000000.0f,
	-100000000.0f,
	0.000000001f,
	-0.000000001f,
	FLT_MIN,
	-FLT_MIN,
	FLT_MAX,
	-FLT_MAX,
	FLT_TRUE_MIN,
	-FLT_TRUE_MIN,
	INFINITY,
	-INFINITY,
	NAN,
	bit_cast<float>(0x007FFFFF), // Largest subnormal
	bit_cast<float>(0x807FFFFF),
	bit_cast<float>(0x00000001), // Smallest subnormal
	bit_cast<float>(0x80000001),
	};

	for(float f : a)
	{
	int exp = -1000;
	float sig = fast_frexp(f, &exp);

	if(f == 0.0f) // Could be subnormal if `flush` is true
	{
	// We don't rely on std::frexp here to produce a reference result because it may
	// return non-zero significands and exponents for subnormal arguments., while our
	// implementation is meant to respect denormals-are-zero / flush-to-zero.

	ASSERT_EQ(sig, 0.0f) << "Argument: " << std::hexfloat << f;
	ASSERT_TRUE(signbit(sig) == signbit(f)) << "Argument: " << std::hexfloat << f;
	ASSERT_EQ(exp, 0) << "Argument: " << std::hexfloat << f;
	}
	else
	{
	int ref_exp = -1000;
	float ref_sig = std::frexp(f, &ref_exp);

	if(!isnan(f))
	{
	ASSERT_EQ(sig, ref_sig) << "Argument: " << std::hexfloat << f;
	}
	else
	{
	ASSERT_TRUE(isnan(sig)) << "Significand: " << std::hexfloat << sig;
	}

	if(!isinf(f) && !isnan(f)) // If the argument is NaN or Inf the exponent is unspecified.
	{
	ASSERT_EQ(exp, ref_exp) << "Argument: " << std::hexfloat << f;
	}
	}
	}
	}
	}

	// Returns the whole-number ULP error of `a` relative to `x`.
	// Use the doouble-precision version below. This just illustrates the principle.
	[[deprecated]] float ULP_32(float x, float a)
	{
	// Flip the last mantissa bit to compute the 'unit in the last place' error.
	float x1 = bit_cast<float>(bit_cast<uint32_t>(x) ^ 0x00000001);
	float ulp = abs(x1 - x);

	return abs(a - x) / ulp;
	}

	double ULP_32(double x, double a)
	{
	// binary64 has 52 mantissa bits, while binary32 has 23, so the ULP for the latter is 29 bits shifted.
	double x1 = bit_cast<double>(bit_cast<uint64_t>(x) ^ 0x0000000020000000ull);
	double ulp = abs(x1 - x);

	return abs(a - x) / ulp;
	}

	float ULP_16(float x, float a)
	{
	// binary32 has 23 mantissa bits, while binary16 has 10, so the ULP for the latter is 13 bits shifted.
	double x1 = bit_cast<float>(bit_cast<uint32_t>(x) ^ 0x00002000);
	float ulp = abs(x1 - x);

	return abs(a - x) / ulp;
	}

	// lolremez --float -d 2 -r "0:2^23" "(log2(x/2^23+1)-x/2^23)/x" "1/x"
	// ULP-16: 0.797363281, abs: 0.0991751999
	float f(float x)
	{
	float u = 2.8017103e-22f;
	u = u * x + -8.373131e-15f;
	return u * x + 5.0615534e-8f;
	}

	float Log2Relaxed(float x)
	{
	// Reinterpretation as an integer provides a piecewise linear
	// approximation of log2(). Scale to the radix and subtract exponent bias.
	int im = bit_cast<int>(x);
	float y = (float)im * (1.0f / (1 << 23)) - 127.0f;

	// Handle log2(inf) = inf.
	if(im == 0x7F800000) y = INFINITY;

	float m = (float)(im & 0x007FFFFF); // Unnormalized mantissa of x.

	// Add a polynomial approximation of log2(m+1)-m to the result's mantissa.
	return f(m) * m + y;
	}

	TEST(MathTest, Log2RelaxedExhaustive)
	{
	CPUID::setDenormalsAreZero(true);
	CPUID::setFlushToZero(true);

	float worst_margin = 0;
	float worst_ulp = 0;
	float worst_x = 0;
	float worst_val = 0;
	float worst_ref = 0;

	float worst_abs = 0;

	for(float x = 0.0f; x <= INFINITY; x = inc(x))
	{
	float val = Log2Relaxed(x);

	double ref = log2((double)x);

	if(ref == (int)ref)
	{
	ASSERT_EQ(val, ref);
	}
	else if(x >= 0.5f && x <= 2.0f)
	{
	const float tolerance = pow(2.0f, -7.0f); // Absolute

	float margin = abs(val - ref) / tolerance;

	if(margin > worst_abs)
	{
	worst_abs = margin;
	}
	}
	else
	{
	const float tolerance = 3; // ULP

	float ulp = (float)ULP_16(ref, (double)val);
	float margin = ulp / tolerance;

	if(margin > worst_margin)
	{
	worst_margin = margin;
	worst_ulp = ulp;
	worst_x = x;
	worst_val = val;
	worst_ref = ref;
	}
	}
	}

	ASSERT_TRUE(worst_margin < 1.0f) << " worst_x " << worst_x << " worst_val " << worst_val << " worst_ref " << worst_ref << " worst_ulp " << worst_ulp;
	ASSERT_TRUE(worst_abs <= 1.0f) << " worst_x " << worst_x << " worst_val " << worst_val << " worst_ref " << worst_ref << " worst_ulp " << worst_ulp;

	CPUID::setDenormalsAreZero(false);
	CPUID::setFlushToZero(false);
	}

	// lolremez --float -d 2 -r "0:1" "(2^x-x-1)/x" "1/x"
	// ULP-16: 0.130859017
	float Pr(float x)
	{
	float u = 7.8145574e-2f;
	u = u * x + 2.2617357e-1f;
	return u * x + -3.0444314e-1f;
	}

	float Exp2Relaxed(float x)
	{
	x = min(x, 128.0f);
	x = max(x, bit_cast<float>(int(0xC2FDFFFF))); // -126.999992

	// 2^f - f - 1 as P(f) * f
	// This is a correction term to be added to 1+x to obtain 2^x.
	float f = x - floor(x);
	float y = Pr(f) * f + x;

	// bit_cast<float>(int(x * 2^23)) is a piecewise linear approximation of 2^(x-127).
	// See "Fast Exponential Computation on SIMD Architectures" by Malossi et al.
	return bit_cast<float>(int((1 << 23) * y + (127 << 23)));
	}

	TEST(MathTest, Exp2RelaxedExhaustive)
	{
	CPUID::setDenormalsAreZero(true);
	CPUID::setFlushToZero(true);

	float worst_margin = 0;
	float worst_ulp = 0;
	float worst_x = 0;
	float worst_val = 0;
	float worst_ref = 0;

	for(float x = -10; x <= 10; x = inc(x))
	{
	float val = Exp2Relaxed(x);

	double ref = exp2((double)x);

	if(x == (int)x)
	{
	ASSERT_EQ(val, ref);
	}

	const float tolerance = (1 + 2 * abs(x));
	float ulp = ULP_16((float)ref, val);
	float margin = ulp / tolerance;

	if(margin > worst_margin)
	{
	worst_margin = margin;
	worst_ulp = ulp;
	worst_x = x;
	worst_val = val;
	worst_ref = ref;
	}
	}

	ASSERT_TRUE(worst_margin <= 1.0f) << " worst_x " << worst_x << " worst_val " << worst_val << " worst_ref " << worst_ref << " worst_ulp " << worst_ulp;

	CPUID::setDenormalsAreZero(false);
	CPUID::setFlushToZero(false);
	}

	// lolremez --float -d 7 -r "0:1" "(log2(x+1)-x)/x" "1/x"
	// ULP-32: 1.69571960, abs: 0.360798746
	float Pl(float x)
	{
	float u = -9.3091638e-3f;
	u = u * x + 5.2059003e-2f;
	u = u * x + -1.3752135e-1f;
	u = u * x + 2.4186478e-1f;
	u = u * x + -3.4730109e-1f;
	u = u * x + 4.786837e-1f;
	u = u * x + -7.2116581e-1f;
	return u * x + 4.4268988e-1f;
	}

	float Log2(float x)
	{
	// Reinterpretation as an integer provides a piecewise linear
	// approximation of log2(). Scale to the radix and subtract exponent bias.
	int im = bit_cast<int>(x);
	float y = (float)(im - (127 << 23)) * (1.0f / (1 << 23));

	// Handle log2(inf) = inf.
	if(im == 0x7F800000) y = INFINITY;

	float m = (float)(im & 0x007FFFFF) * (1.0f / (1 << 23)); // Normalized mantissa of x.

	// Add a polynomial approximation of log2(m+1)-m to the result's mantissa.
	return Pl(m) * m + y;
	}

	TEST(MathTest, Log2Exhaustive)
	{
	CPUID::setDenormalsAreZero(true);
	CPUID::setFlushToZero(true);

	float worst_margin = 0;
	float worst_ulp = 0;
	float worst_x = 0;
	float worst_val = 0;
	float worst_ref = 0;

	float worst_abs = 0;

	for(float x = 0.0f; x <= INFINITY; x = inc(x))
	{
	float val = Log2(x);

	double ref = log2((double)x);

	if(ref == (int)ref)
	{
	ASSERT_EQ(val, ref);
	}
	else if(x >= 0.5f && x <= 2.0f)
	{
	const float tolerance = pow(2.0f, -21.0f); // Absolute

	float margin = abs(val - ref) / tolerance;

	if(margin > worst_abs)
	{
	worst_abs = margin;
	}
	}
	else
	{
	const float tolerance = 3; // ULP

	float ulp = (float)ULP_32(ref, (double)val);
	float margin = ulp / tolerance;

	if(margin > worst_margin)
	{
	worst_margin = margin;
	worst_ulp = ulp;
	worst_x = x;
	worst_val = val;
	worst_ref = ref;
	}
	}
	}

	ASSERT_TRUE(worst_margin < 1.0f) << " worst_x " << worst_x << " worst_val " << worst_val << " worst_ref " << worst_ref << " worst_ulp " << worst_ulp;
	ASSERT_TRUE(worst_abs <= 1.0f) << " worst_x " << worst_x << " worst_val " << worst_val << " worst_ref " << worst_ref << " worst_ulp " << worst_ulp;

	CPUID::setDenormalsAreZero(false);
	CPUID::setFlushToZero(false);
	}

	// lolremez --float -d 4 -r "0:1" "(2^x-x-1)/x" "1/x"
	// ULP_32: 2.14694786, Vulkan margin: 0.686957061
	float P(float x)
	{
	float u = 1.8852974e-3f;
	u = u * x + 8.9733787e-3f;
	u = u * x + 5.5835927e-2f;
	u = u * x + 2.4015281e-1f;
	return u * x + -3.0684753e-1f;
	}

	float Exp2(float x)
	{
	x = min(x, 128.0f);
	x = max(x, bit_cast<float>(0xC2FDFFFF)); // -126.999992

	// 2^f - f - 1 as P(f) * f
	// This is a correction term to be added to 1+x to obtain 2^x.
	float f = x - floor(x);
	float y = P(f) * f + x;

	// bit_cast<float>(int(x * 2^23)) is a piecewise linear approximation of 2^(x-127).
	// See "Fast Exponential Computation on SIMD Architectures" by Malossi et al.
	return bit_cast<float>(int(y * (1 << 23)) + (127 << 23));
	}

	TEST(MathTest, Exp2Exhaustive)
	{
	CPUID::setDenormalsAreZero(true);
	CPUID::setFlushToZero(true);

	float worst_margin = 0;
	float worst_ulp = 0;
	float worst_x = 0;
	float worst_val = 0;
	float worst_ref = 0;

	for(float x = -10; x <= 10; x = inc(x))
	{
	float val = Exp2(x);

	double ref = exp2((double)x);

	if(x == (int)x)
	{
	ASSERT_EQ(val, ref);
	}

	const float tolerance = (3 + 2 * abs(x));
	float ulp = (float)ULP_32(ref, (double)val);
	float margin = ulp / tolerance;

	if(margin > worst_margin)
	{
	worst_margin = margin;
	worst_ulp = ulp;
	worst_x = x;
	worst_val = val;
	worst_ref = ref;
	}
	}

	ASSERT_TRUE(worst_margin <= 1.0f) << " worst_x " << worst_x << " worst_val " << worst_val << " worst_ref " << worst_ref << " worst_ulp " << worst_ulp;

	CPUID::setDenormalsAreZero(false);
	CPUID::setFlushToZero(false);
	}

	// Polynomial approximation of order 5 for sin(x * 2 * pi) in the range [-1/4, 1/4]
	static float sin5(float x)
	{
	// A * x^5 + B * x^3 + C * x
	// Exact at x = 0, 1/12, 1/6, 1/4, and their negatives, which correspond to x * 2 * pi = 0, pi/6, pi/3, pi/2
	const float A = (36288 - 20736 * sqrt(3)) / 5;
	const float B = 288 * sqrt(3) - 540;
	const float C = (47 - 9 * sqrt(3)) / 5;

	float x2 = x * x;

	return ((A * x2 + B) * x2 + C) * x;
	}

	TEST(MathTest, SinExhaustive)
	{
	const float tolerance = powf(2.0f, -12.0f); // Vulkan requires absolute error <= 2^−11 inside the range [−pi, pi]
	const float pi = 3.1415926535f;

	for(float x = -pi; x <= pi; x = inc(x))
	{
	// Range reduction and mirroring
	float x_2 = 0.25f - x * (0.5f / pi);
	float z = 0.25f - fabs(x_2 - round(x_2));

	float val = sin5(z);

	ASSERT_NEAR(val, sinf(x), tolerance);
	}
	}

	TEST(MathTest, CosExhaustive)
	{
	const float tolerance = powf(2.0f, -12.0f); // Vulkan requires absolute error <= 2^−11 inside the range [−pi, pi]
	const float pi = 3.1415926535f;

	for(float x = -pi; x <= pi; x = inc(x))
	{
	// Phase shift, range reduction, and mirroring
	float x_2 = x * (0.5f / pi);
	float z = 0.25f - fabs(x_2 - round(x_2));

	float val = sin5(z);

	ASSERT_NEAR(val, cosf(x), tolerance);
	}
	}

	TEST(MathTest, UnsignedFloat11_10)
	{
	// Test the largest value which causes underflow to 0, and the smallest value
	// which produces a denormalized result.

	EXPECT_EQ(R11G11B10F::float32ToFloat11(bit_cast<float>(0x3500007F)), 0x0000);
	EXPECT_EQ(R11G11B10F::float32ToFloat11(bit_cast<float>(0x35000080)), 0x0001);

	EXPECT_EQ(R11G11B10F::float32ToFloat10(bit_cast<float>(0x3580003F)), 0x0000);
	EXPECT_EQ(R11G11B10F::float32ToFloat10(bit_cast<float>(0x35800040)), 0x0001);
	}

	// Clamps to the [0, hi] range. NaN input produces 0, hi must be non-NaN.
	float clamp0hi(float x, float hi)
	{
	// If x=NaN, x > 0 will compare false and we return 0.
	if(!(x > 0))
	{
	return 0;
	}

	// x is non-NaN at this point, so std::min() is safe for non-NaN hi.
	return std::min(x, hi);
	}

	unsigned int RGB9E5_reference(float r, float g, float b)
	{
	// Vulkan 1.1.117 section 15.2.1 RGB to Shared Exponent Conversion

	// B is the exponent bias (15)
	constexpr int g_sharedexp_bias = 15;

	// N is the number of mantissa bits per component (9)
	constexpr int g_sharedexp_mantissabits = 9;

	// Emax is the maximum allowed biased exponent value (31)
	constexpr int g_sharedexp_maxexponent = 31;

	constexpr float g_sharedexp_max =
	((static_cast<float>(1 << g_sharedexp_mantissabits) - 1) /
	static_cast<float>(1 << g_sharedexp_mantissabits)) *
	static_cast<float>(1 << (g_sharedexp_maxexponent - g_sharedexp_bias));

	const float red_c = clamp0hi(r, g_sharedexp_max);
	const float green_c = clamp0hi(g, g_sharedexp_max);
	const float blue_c = clamp0hi(b, g_sharedexp_max);

	const float max_c = fmax(fmax(red_c, green_c), blue_c);
	const float exp_p = fmax(-g_sharedexp_bias - 1, floor(log2(max_c))) + 1 + g_sharedexp_bias;
	const int max_s = static_cast<int>(floor((max_c / exp2(exp_p - g_sharedexp_bias - g_sharedexp_mantissabits)) + 0.5f));
	const int exp_s = static_cast<int>((max_s < exp2(g_sharedexp_mantissabits)) ? exp_p : exp_p + 1);

	unsigned int R = static_cast<unsigned int>(floor((red_c / exp2(exp_s - g_sharedexp_bias - g_sharedexp_mantissabits)) + 0.5f));
	unsigned int G = static_cast<unsigned int>(floor((green_c / exp2(exp_s - g_sharedexp_bias - g_sharedexp_mantissabits)) + 0.5f));
	unsigned int B = static_cast<unsigned int>(floor((blue_c / exp2(exp_s - g_sharedexp_bias - g_sharedexp_mantissabits)) + 0.5f));
	unsigned int E = exp_s;

	return (E << 27) \| (B << 18) \| (G << 9) \| R;
	}

	TEST(MathTest, SharedExponentSparse)
	{
	for(uint64_t i = 0; i < 0x0000000100000000; i += 0x400)
	{
	float f = bit_cast<float>(i);

	unsigned int ref = RGB9E5_reference(f, 0.0f, 0.0f);
	unsigned int val = RGB9E5(f, 0.0f, 0.0f);

	EXPECT_EQ(ref, val);
	}
	}

	TEST(MathTest, SharedExponentRandom)
	{
	srand(0);

	unsigned int x = 0;
	unsigned int y = 0;
	unsigned int z = 0;

	for(int i = 0; i < 10000000; i++)
	{
	float r = bit_cast<float>(x);
	float g = bit_cast<float>(y);
	float b = bit_cast<float>(z);

	unsigned int ref = RGB9E5_reference(r, g, b);
	unsigned int val = RGB9E5(r, g, b);

	EXPECT_EQ(ref, val);

	x += rand();
	y += rand();
	z += rand();
	}
	}

	TEST(MathTest, SharedExponentExhaustive)
	{
	for(uint64_t i = 0; i < 0x0000000100000000; i += 1)
	{
	float f = bit_cast<float>(i);

	unsigned int ref = RGB9E5_reference(f, 0.0f, 0.0f);
	unsigned int val = RGB9E5(f, 0.0f, 0.0f);

	EXPECT_EQ(ref, val);
	}
	}