212 lines · cpp
1//===-- Single-precision sincos function ----------------------------------===//2//3// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.4// See https://llvm.org/LICENSE.txt for license information.5// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception6//7//===----------------------------------------------------------------------===//8 9#include "src/math/sincosf.h"10#include "src/__support/FPUtil/FEnvImpl.h"11#include "src/__support/FPUtil/FPBits.h"12#include "src/__support/FPUtil/multiply_add.h"13#include "src/__support/FPUtil/rounding_mode.h"14#include "src/__support/common.h"15#include "src/__support/macros/config.h"16#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY17#include "src/__support/macros/properties/cpu_features.h" // LIBC_TARGET_CPU_HAS_FMA18#include "src/__support/math/sincosf_utils.h"19 20namespace LIBC_NAMESPACE_DECL {21 22#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS23// Exceptional values24static constexpr int N_EXCEPTS = 6;25 26static constexpr uint32_t EXCEPT_INPUTS[N_EXCEPTS] = {27 0x46199998, // x = 0x1.33333p13 x28 0x55325019, // x = 0x1.64a032p43 x29 0x5922aa80, // x = 0x1.4555p51 x30 0x5f18b878, // x = 0x1.3170fp63 x31 0x6115cb11, // x = 0x1.2b9622p67 x32 0x7beef5ef, // x = 0x1.ddebdep120 x33};34 35static constexpr uint32_t EXCEPT_OUTPUTS_SIN[N_EXCEPTS][4] = {36 {0xbeb1fa5d, 0, 1, 0}, // x = 0x1.33333p13, sin(x) = -0x1.63f4bap-2 (RZ)37 {0xbf171adf, 0, 1, 1}, // x = 0x1.64a032p43, sin(x) = -0x1.2e35bep-1 (RZ)38 {0xbf587521, 0, 1, 1}, // x = 0x1.4555p51, sin(x) = -0x1.b0ea42p-1 (RZ)39 {0x3dad60f6, 1, 0, 1}, // x = 0x1.3170fp63, sin(x) = 0x1.5ac1ecp-4 (RZ)40 {0xbe7cc1e0, 0, 1, 1}, // x = 0x1.2b9622p67, sin(x) = -0x1.f983cp-3 (RZ)41 {0xbf587d1b, 0, 1, 1}, // x = 0x1.ddebdep120, sin(x) = -0x1.b0fa36p-1 (RZ)42};43 44static constexpr uint32_t EXCEPT_OUTPUTS_COS[N_EXCEPTS][4] = {45 {0xbf70090b, 0, 1, 0}, // x = 0x1.33333p13, cos(x) = -0x1.e01216p-1 (RZ)46 {0x3f4ea5d2, 1, 0, 0}, // x = 0x1.64a032p43, cos(x) = 0x1.9d4ba4p-1 (RZ)47 {0x3f08aebe, 1, 0, 1}, // x = 0x1.4555p51, cos(x) = 0x1.115d7cp-1 (RZ)48 {0x3f7f14bb, 1, 0, 0}, // x = 0x1.3170fp63, cos(x) = 0x1.fe2976p-1 (RZ)49 {0x3f78142e, 1, 0, 1}, // x = 0x1.2b9622p67, cos(x) = 0x1.f0285cp-1 (RZ)50 {0x3f08a21c, 1, 0, 0}, // x = 0x1.ddebdep120, cos(x) = 0x1.114438p-1 (RZ)51};52#endif // !LIBC_MATH_HAS_SKIP_ACCURATE_PASS53 54LLVM_LIBC_FUNCTION(void, sincosf, (float x, float *sinp, float *cosp)) {55 using FPBits = typename fputil::FPBits<float>;56 FPBits xbits(x);57 58 uint32_t x_abs = xbits.uintval() & 0x7fff'ffffU;59 double xd = static_cast<double>(x);60 61 // Range reduction:62 // For |x| >= 2^-12, we perform range reduction as follows:63 // Find k and y such that:64 // x = (k + y) * pi/3265 // k is an integer66 // |y| < 0.567 // For small range (|x| < 2^45 when FMA instructions are available, 2^2268 // otherwise), this is done by performing:69 // k = round(x * 32/pi)70 // y = x * 32/pi - k71 // For large range, we will omit all the higher parts of 32/pi such that the72 // least significant bits of their full products with x are larger than 63,73 // since:74 // sin((k + y + 64*i) * pi/32) = sin(x + i * 2pi) = sin(x), and75 // cos((k + y + 64*i) * pi/32) = cos(x + i * 2pi) = cos(x).76 //77 // When FMA instructions are not available, we store the digits of 32/pi in78 // chunks of 28-bit precision. This will make sure that the products:79 // x * THIRTYTWO_OVER_PI_28[i] are all exact.80 // When FMA instructions are available, we simply store the digits of326/pi in81 // chunks of doubles (53-bit of precision).82 // So when multiplying by the largest values of single precision, the83 // resulting output should be correct up to 2^(-208 + 128) ~ 2^-80. By the84 // worst-case analysis of range reduction, |y| >= 2^-38, so this should give85 // us more than 40 bits of accuracy. For the worst-case estimation of range86 // reduction, see for instances:87 // Elementary Functions by J-M. Muller, Chapter 11,88 // Handbook of Floating-Point Arithmetic by J-M. Muller et. al.,89 // Chapter 10.2.90 //91 // Once k and y are computed, we then deduce the answer by the sine and cosine92 // of sum formulas:93 // sin(x) = sin((k + y)*pi/32)94 // = sin(y*pi/32) * cos(k*pi/32) + cos(y*pi/32) * sin(k*pi/32)95 // cos(x) = cos((k + y)*pi/32)96 // = cos(y*pi/32) * cos(k*pi/32) - sin(y*pi/32) * sin(k*pi/32)97 // The values of sin(k*pi/32) and cos(k*pi/32) for k = 0..63 are precomputed98 // and stored using a vector of 32 doubles. Sin(y*pi/32) and cos(y*pi/32) are99 // computed using degree-7 and degree-6 minimax polynomials generated by100 // Sollya respectively.101 102 // |x| < 0x1.0p-12f103 if (LIBC_UNLIKELY(x_abs < 0x3980'0000U)) {104 if (LIBC_UNLIKELY(x_abs == 0U)) {105 // For signed zeros.106 *sinp = x;107 *cosp = 1.0f;108 return;109 }110 // When |x| < 2^-12, the relative errors of the approximations111 // sin(x) ~ x, cos(x) ~ 1112 // are:113 // |sin(x) - x| / |sin(x)| < |x^3| / (6|x|)114 // = x^2 / 6115 // < 2^-25116 // < epsilon(1)/2.117 // |cos(x) - 1| < |x^2 / 2| = 2^-25 < epsilon(1)/2.118 // So the correctly rounded values of sin(x) and cos(x) are:119 // sin(x) = x - sign(x)*eps(x) if rounding mode = FE_TOWARDZERO,120 // or (rounding mode = FE_UPWARD and x is121 // negative),122 // = x otherwise.123 // cos(x) = 1 - eps(x) if rounding mode = FE_TOWARDZERO or FE_DOWWARD,124 // = 1 otherwise.125 // To simplify the rounding decision and make it more efficient and to126 // prevent compiler to perform constant folding, we use127 // sin(x) = fma(x, -2^-25, x),128 // cos(x) = fma(x*0.5f, -x, 1)129 // instead.130 // Note: to use the formula x - 2^-25*x to decide the correct rounding, we131 // do need fma(x, -2^-25, x) to prevent underflow caused by -2^-25*x when132 // |x| < 2^-125. For targets without FMA instructions, we simply use133 // double for intermediate results as it is more efficient than using an134 // emulated version of FMA.135#if defined(LIBC_TARGET_CPU_HAS_FMA_FLOAT)136 *sinp = fputil::multiply_add(x, -0x1.0p-25f, x);137 *cosp = fputil::multiply_add(FPBits(x_abs).get_val(), -0x1.0p-25f, 1.0f);138#else139 *sinp = static_cast<float>(fputil::multiply_add(xd, -0x1.0p-25, xd));140 *cosp = static_cast<float>(fputil::multiply_add(141 static_cast<double>(FPBits(x_abs).get_val()), -0x1.0p-25, 1.0));142#endif // LIBC_TARGET_CPU_HAS_FMA_FLOAT143 return;144 }145 146 // x is inf or nan.147 if (LIBC_UNLIKELY(x_abs >= 0x7f80'0000U)) {148 if (xbits.is_signaling_nan()) {149 fputil::raise_except_if_required(FE_INVALID);150 *sinp = *cosp = FPBits::quiet_nan().get_val();151 return;152 }153 154 if (x_abs == 0x7f80'0000U) {155 fputil::set_errno_if_required(EDOM);156 fputil::raise_except_if_required(FE_INVALID);157 }158 *sinp = FPBits::quiet_nan().get_val();159 *cosp = *sinp;160 return;161 }162 163#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS164 // Check exceptional values.165 for (int i = 0; i < N_EXCEPTS; ++i) {166 if (LIBC_UNLIKELY(x_abs == EXCEPT_INPUTS[i])) {167 uint32_t s = EXCEPT_OUTPUTS_SIN[i][0]; // FE_TOWARDZERO168 uint32_t c = EXCEPT_OUTPUTS_COS[i][0]; // FE_TOWARDZERO169 bool x_sign = x < 0;170 switch (fputil::quick_get_round()) {171 case FE_UPWARD:172 s += x_sign ? EXCEPT_OUTPUTS_SIN[i][2] : EXCEPT_OUTPUTS_SIN[i][1];173 c += EXCEPT_OUTPUTS_COS[i][1];174 break;175 case FE_DOWNWARD:176 s += x_sign ? EXCEPT_OUTPUTS_SIN[i][1] : EXCEPT_OUTPUTS_SIN[i][2];177 c += EXCEPT_OUTPUTS_COS[i][2];178 break;179 case FE_TONEAREST:180 s += EXCEPT_OUTPUTS_SIN[i][3];181 c += EXCEPT_OUTPUTS_COS[i][3];182 break;183 }184 *sinp = x_sign ? -FPBits(s).get_val() : FPBits(s).get_val();185 *cosp = FPBits(c).get_val();186 187 return;188 }189 }190#endif // !LIBC_MATH_HAS_SKIP_ACCURATE_PASS191 192 // Combine the results with the sine and cosine of sum formulas:193 // sin(x) = sin((k + y)*pi/32)194 // = sin(y*pi/32) * cos(k*pi/32) + cos(y*pi/32) * sin(k*pi/32)195 // = sin_y * cos_k + (1 + cosm1_y) * sin_k196 // = sin_y * cos_k + (cosm1_y * sin_k + sin_k)197 // cos(x) = cos((k + y)*pi/32)198 // = cos(y*pi/32) * cos(k*pi/32) - sin(y*pi/32) * sin(k*pi/32)199 // = cosm1_y * cos_k + sin_y * sin_k200 // = (cosm1_y * cos_k + cos_k) + sin_y * sin_k201 double sin_k, cos_k, sin_y, cosm1_y;202 203 sincosf_eval(xd, x_abs, sin_k, cos_k, sin_y, cosm1_y);204 205 *sinp = static_cast<float>(fputil::multiply_add(206 sin_y, cos_k, fputil::multiply_add(cosm1_y, sin_k, sin_k)));207 *cosp = static_cast<float>(fputil::multiply_add(208 sin_y, -sin_k, fputil::multiply_add(cosm1_y, cos_k, cos_k)));209}210 211} // namespace LIBC_NAMESPACE_DECL212