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// 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);
ASSERT_TRUE(worst_abs <= 1.0f);
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);
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);
ASSERT_TRUE(worst_abs <= 1.0f);
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);
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);
}
}