183 lines · cpp
1//===-- Single-precision sin 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/sinf.h"10#include "src/__support/FPUtil/BasicOperations.h"11#include "src/__support/FPUtil/FEnvImpl.h"12#include "src/__support/FPUtil/FPBits.h"13#include "src/__support/FPUtil/PolyEval.h"14#include "src/__support/FPUtil/multiply_add.h"15#include "src/__support/FPUtil/rounding_mode.h"16#include "src/__support/common.h"17#include "src/__support/macros/config.h"18#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY19#include "src/__support/macros/properties/cpu_features.h" // LIBC_TARGET_CPU_HAS_FMA20 21#if defined(LIBC_MATH_HAS_SKIP_ACCURATE_PASS) && \22 defined(LIBC_MATH_HAS_INTERMEDIATE_COMP_IN_FLOAT) && \23 defined(LIBC_TARGET_CPU_HAS_FMA_FLOAT)24 25#include "src/__support/math/sincosf_float_eval.h"26 27namespace LIBC_NAMESPACE_DECL {28 29LLVM_LIBC_FUNCTION(float, sinf, (float x)) {30 return math::sincosf_float_eval::sincosf_eval</*IS_SIN*/ true>(x);31}32 33} // namespace LIBC_NAMESPACE_DECL34 35#else // !LIBC_MATH_HAS_INTERMEDIATE_COMP_IN_FLOAT36 37#include "src/__support/math/sincosf_utils.h"38 39#ifdef LIBC_TARGET_CPU_HAS_FMA_DOUBLE40#include "src/__support/math/range_reduction_fma.h"41#else // !LIBC_TARGET_CPU_HAS_FMA_DOUBLE42#include "src/__support/math/range_reduction.h"43#endif // LIBC_TARGET_CPU_HAS_FMA_DOUBLE44 45namespace LIBC_NAMESPACE_DECL {46 47LLVM_LIBC_FUNCTION(float, sinf, (float x)) {48 using FPBits = typename fputil::FPBits<float>;49 FPBits xbits(x);50 51 uint32_t x_u = xbits.uintval();52 uint32_t x_abs = x_u & 0x7fff'ffffU;53 double xd = static_cast<double>(x);54 55 // Range reduction:56 // For |x| > pi/32, we perform range reduction as follows:57 // Find k and y such that:58 // x = (k + y) * pi/3259 // k is an integer60 // |y| < 0.561 // For small range (|x| < 2^45 when FMA instructions are available, 2^2262 // otherwise), this is done by performing:63 // k = round(x * 32/pi)64 // y = x * 32/pi - k65 // For large range, we will omit all the higher parts of 32/pi such that the66 // least significant bits of their full products with x are larger than 63,67 // since sin((k + y + 64*i) * pi/32) = sin(x + i * 2pi) = sin(x).68 //69 // When FMA instructions are not available, we store the digits of 32/pi in70 // chunks of 28-bit precision. This will make sure that the products:71 // x * THIRTYTWO_OVER_PI_28[i] are all exact.72 // When FMA instructions are available, we simply store the digits of 32/pi in73 // chunks of doubles (53-bit of precision).74 // So when multiplying by the largest values of single precision, the75 // resulting output should be correct up to 2^(-208 + 128) ~ 2^-80. By the76 // worst-case analysis of range reduction, |y| >= 2^-38, so this should give77 // us more than 40 bits of accuracy. For the worst-case estimation of range78 // reduction, see for instances:79 // Elementary Functions by J-M. Muller, Chapter 11,80 // Handbook of Floating-Point Arithmetic by J-M. Muller et. al.,81 // Chapter 10.2.82 //83 // Once k and y are computed, we then deduce the answer by the sine of sum84 // formula:85 // sin(x) = sin((k + y)*pi/32)86 // = sin(y*pi/32) * cos(k*pi/32) + cos(y*pi/32) * sin(k*pi/32)87 // The values of sin(k*pi/32) and cos(k*pi/32) for k = 0..31 are precomputed88 // and stored using a vector of 32 doubles. Sin(y*pi/32) and cos(y*pi/32) are89 // computed using degree-7 and degree-6 minimax polynomials generated by90 // Sollya respectively.91 92 // |x| <= pi/1693 if (LIBC_UNLIKELY(x_abs <= 0x3e49'0fdbU)) {94 95 // |x| < 0x1.d12ed2p-12f96 if (LIBC_UNLIKELY(x_abs < 0x39e8'9769U)) {97 if (LIBC_UNLIKELY(x_abs == 0U)) {98 // For signed zeros.99 return x;100 }101 // When |x| < 2^-12, the relative error of the approximation sin(x) ~ x102 // is:103 // |sin(x) - x| / |sin(x)| < |x^3| / (6|x|)104 // = x^2 / 6105 // < 2^-25106 // < epsilon(1)/2.107 // So the correctly rounded values of sin(x) are:108 // = x - sign(x)*eps(x) if rounding mode = FE_TOWARDZERO,109 // or (rounding mode = FE_UPWARD and x is110 // negative),111 // = x otherwise.112 // To simplify the rounding decision and make it more efficient, we use113 // fma(x, -2^-25, x) instead.114 // An exhaustive test shows that this formula work correctly for all115 // rounding modes up to |x| < 0x1.c555dep-11f.116 // Note: to use the formula x - 2^-25*x to decide the correct rounding, we117 // do need fma(x, -2^-25, x) to prevent underflow caused by -2^-25*x when118 // |x| < 2^-125. For targets without FMA instructions, we simply use119 // double for intermediate results as it is more efficient than using an120 // emulated version of FMA.121#if defined(LIBC_TARGET_CPU_HAS_FMA_FLOAT)122 return fputil::multiply_add(x, -0x1.0p-25f, x);123#else124 return static_cast<float>(fputil::multiply_add(xd, -0x1.0p-25, xd));125#endif // LIBC_TARGET_CPU_HAS_FMA_FLOAT126 }127 128 // |x| < pi/16.129 double xsq = xd * xd;130 131 // Degree-9 polynomial approximation:132 // sin(x) ~ x + a_3 x^3 + a_5 x^5 + a_7 x^7 + a_9 x^9133 // = x (1 + a_3 x^2 + ... + a_9 x^8)134 // = x * P(x^2)135 // generated by Sollya with the following commands:136 // > display = hexadecimal;137 // > Q = fpminimax(sin(x)/x, [|0, 2, 4, 6, 8|], [|1, D...|], [0, pi/16]);138 double result =139 fputil::polyeval(xsq, 1.0, -0x1.55555555554c6p-3, 0x1.1111111085e65p-7,140 -0x1.a019f70fb4d4fp-13, 0x1.718d179815e74p-19);141 return static_cast<float>(xd * result);142 }143 144#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS145 if (LIBC_UNLIKELY(x_abs == 0x4619'9998U)) { // x = 0x1.33333p13146 float r = -0x1.63f4bap-2f;147 int rounding = fputil::quick_get_round();148 if ((rounding == FE_DOWNWARD && xbits.is_pos()) ||149 (rounding == FE_UPWARD && xbits.is_neg()))150 r = -0x1.63f4bcp-2f;151 return xbits.is_neg() ? -r : r;152 }153#endif // !LIBC_MATH_HAS_SKIP_ACCURATE_PASS154 155 if (LIBC_UNLIKELY(x_abs >= 0x7f80'0000U)) {156 if (xbits.is_signaling_nan()) {157 fputil::raise_except_if_required(FE_INVALID);158 return FPBits::quiet_nan().get_val();159 }160 161 if (x_abs == 0x7f80'0000U) {162 fputil::set_errno_if_required(EDOM);163 fputil::raise_except_if_required(FE_INVALID);164 }165 return x + FPBits::quiet_nan().get_val();166 }167 168 // Combine the results with the sine of sum formula:169 // sin(x) = sin((k + y)*pi/32)170 // = sin(y*pi/32) * cos(k*pi/32) + cos(y*pi/32) * sin(k*pi/32)171 // = sin_y * cos_k + (1 + cosm1_y) * sin_k172 // = sin_y * cos_k + (cosm1_y * sin_k + sin_k)173 double sin_k, cos_k, sin_y, cosm1_y;174 175 sincosf_eval(xd, x_abs, sin_k, cos_k, sin_y, cosm1_y);176 177 return static_cast<float>(fputil::multiply_add(178 sin_y, cos_k, fputil::multiply_add(cosm1_y, sin_k, sin_k)));179}180 181} // namespace LIBC_NAMESPACE_DECL182#endif // LIBC_MATH_HAS_INTERMEDIATE_COMP_IN_FLOAT183