175 lines · cpp
1//===-- Single-precision e^x - 1 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/expm1f.h"10#include "src/__support/FPUtil/BasicOperations.h"11#include "src/__support/FPUtil/FEnvImpl.h"12#include "src/__support/FPUtil/FMA.h"13#include "src/__support/FPUtil/FPBits.h"14#include "src/__support/FPUtil/PolyEval.h"15#include "src/__support/FPUtil/multiply_add.h"16#include "src/__support/FPUtil/nearest_integer.h"17#include "src/__support/FPUtil/rounding_mode.h"18#include "src/__support/common.h"19#include "src/__support/macros/config.h"20#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY21#include "src/__support/macros/properties/cpu_features.h" // LIBC_TARGET_CPU_HAS_FMA22#include "src/__support/math/common_constants.h" // Lookup tables EXP_M1 and EXP_M2.23 24namespace LIBC_NAMESPACE_DECL {25 26LLVM_LIBC_FUNCTION(float, expm1f, (float x)) {27 using namespace common_constants_internal;28 using FPBits = typename fputil::FPBits<float>;29 FPBits xbits(x);30 31 uint32_t x_u = xbits.uintval();32 uint32_t x_abs = x_u & 0x7fff'ffffU;33 34#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS35 // Exceptional value36 if (LIBC_UNLIKELY(x_u == 0x3e35'bec5U)) { // x = 0x1.6b7d8ap-3f37 int round_mode = fputil::quick_get_round();38 if (round_mode == FE_TONEAREST || round_mode == FE_UPWARD)39 return 0x1.8dbe64p-3f;40 return 0x1.8dbe62p-3f;41 }42#if !defined(LIBC_TARGET_CPU_HAS_FMA_DOUBLE)43 if (LIBC_UNLIKELY(x_u == 0xbdc1'c6cbU)) { // x = -0x1.838d96p-4f44 int round_mode = fputil::quick_get_round();45 if (round_mode == FE_TONEAREST || round_mode == FE_DOWNWARD)46 return -0x1.71c884p-4f;47 return -0x1.71c882p-4f;48 }49#endif // LIBC_TARGET_CPU_HAS_FMA_DOUBLE50#endif // !LIBC_MATH_HAS_SKIP_ACCURATE_PASS51 52 // When |x| > 25*log(2), or nan53 if (LIBC_UNLIKELY(x_abs >= 0x418a'a123U)) {54 // x < log(2^-25)55 if (xbits.is_neg()) {56 // exp(-Inf) = 057 if (xbits.is_inf())58 return -1.0f;59 // exp(nan) = nan60 if (xbits.is_nan())61 return x;62 int round_mode = fputil::quick_get_round();63 if (round_mode == FE_UPWARD || round_mode == FE_TOWARDZERO)64 return -0x1.ffff'fep-1f; // -1.0f + 0x1.0p-24f65 return -1.0f;66 } else {67 // x >= 89 or nan68 if (xbits.uintval() >= 0x42b2'0000) {69 if (xbits.uintval() < 0x7f80'0000U) {70 int rounding = fputil::quick_get_round();71 if (rounding == FE_DOWNWARD || rounding == FE_TOWARDZERO)72 return FPBits::max_normal().get_val();73 74 fputil::set_errno_if_required(ERANGE);75 fputil::raise_except_if_required(FE_OVERFLOW);76 }77 return x + FPBits::inf().get_val();78 }79 }80 }81 82 // |x| < 2^-483 if (x_abs < 0x3d80'0000U) {84 // |x| < 2^-2585 if (x_abs < 0x3300'0000U) {86 // x = -0.0f87 if (LIBC_UNLIKELY(xbits.uintval() == 0x8000'0000U))88 return x;89 // When |x| < 2^-25, the relative error of the approximation e^x - 1 ~ x90 // is:91 // |(e^x - 1) - x| / |e^x - 1| < |x^2| / |x|92 // = |x|93 // < 2^-2594 // < epsilon(1)/2.95 // So the correctly rounded values of expm1(x) are:96 // = x + eps(x) if rounding mode = FE_UPWARD,97 // or (rounding mode = FE_TOWARDZERO and x is98 // negative),99 // = x otherwise.100 // To simplify the rounding decision and make it more efficient, we use101 // fma(x, x, x) ~ x + x^2 instead.102 // Note: to use the formula x + x^2 to decide the correct rounding, we103 // do need fma(x, x, x) to prevent underflow caused by x*x when |x| <104 // 2^-76. For targets without FMA instructions, we simply use double for105 // intermediate results as it is more efficient than using an emulated106 // version of FMA.107#if defined(LIBC_TARGET_CPU_HAS_FMA_FLOAT)108 return fputil::multiply_add(x, x, x);109#else110 double xd = x;111 return static_cast<float>(fputil::multiply_add(xd, xd, xd));112#endif // LIBC_TARGET_CPU_HAS_FMA_FLOAT113 }114 115 constexpr double COEFFS[] = {0x1p-1,116 0x1.55555555557ddp-3,117 0x1.55555555552fap-5,118 0x1.111110fcd58b7p-7,119 0x1.6c16c1717660bp-10,120 0x1.a0241f0006d62p-13,121 0x1.a01e3f8d3c06p-16};122 123 // 2^-25 <= |x| < 2^-4124 double xd = static_cast<double>(x);125 double xsq = xd * xd;126 // Degree-8 minimax polynomial generated by Sollya with:127 // > display = hexadecimal;128 // > P = fpminimax((expm1(x) - x)/x^2, 6, [|D...|], [-2^-4, 2^-4]);129 130 double c0 = fputil::multiply_add(xd, COEFFS[1], COEFFS[0]);131 double c1 = fputil::multiply_add(xd, COEFFS[3], COEFFS[2]);132 double c2 = fputil::multiply_add(xd, COEFFS[5], COEFFS[4]);133 134 double r = fputil::polyeval(xsq, c0, c1, c2, COEFFS[6]);135 return static_cast<float>(fputil::multiply_add(r, xsq, xd));136 }137 138 // For -18 < x < 89, to compute expm1(x), we perform the following range139 // reduction: find hi, mid, lo such that:140 // x = hi + mid + lo, in which141 // hi is an integer,142 // mid * 2^7 is an integer143 // -2^(-8) <= lo < 2^-8.144 // In particular,145 // hi + mid = round(x * 2^7) * 2^(-7).146 // Then,147 // expm1(x) = exp(hi + mid + lo) - 1 = exp(hi) * exp(mid) * exp(lo) - 1.148 // We store exp(hi) and exp(mid) in the lookup tables EXP_M1 and EXP_M2149 // respectively. exp(lo) is computed using a degree-4 minimax polynomial150 // generated by Sollya.151 152 // x_hi = hi + mid.153 float kf = fputil::nearest_integer(x * 0x1.0p7f);154 int x_hi = static_cast<int>(kf);155 // Subtract (hi + mid) from x to get lo.156 double xd = static_cast<double>(fputil::multiply_add(kf, -0x1.0p-7f, x));157 x_hi += 104 << 7;158 // hi = x_hi >> 7159 double exp_hi = EXP_M1[x_hi >> 7];160 // lo = x_hi & 0x0000'007fU;161 double exp_mid = EXP_M2[x_hi & 0x7f];162 double exp_hi_mid = exp_hi * exp_mid;163 // Degree-4 minimax polynomial generated by Sollya with the following164 // commands:165 // > display = hexadecimal;166 // > Q = fpminimax(expm1(x)/x, 3, [|D...|], [-2^-8, 2^-8]);167 // > Q;168 double exp_lo =169 fputil::polyeval(xd, 0x1.0p0, 0x1.ffffffffff777p-1, 0x1.000000000071cp-1,170 0x1.555566668e5e7p-3, 0x1.55555555ef243p-5);171 return static_cast<float>(fputil::multiply_add(exp_hi_mid, exp_lo, -1.0));172}173 174} // namespace LIBC_NAMESPACE_DECL175