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1//===-- Implementation header for exp2m1f ------------------------*- C++-*-===//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#ifndef LLVM_LIBC_SRC___SUPPORT_MATH_EXP2M1F_H10#define LLVM_LIBC_SRC___SUPPORT_MATH_EXP2M1F_H11 12#include "exp10f_utils.h"13#include "src/__support/FPUtil/FEnvImpl.h"14#include "src/__support/FPUtil/FPBits.h"15#include "src/__support/FPUtil/PolyEval.h"16#include "src/__support/FPUtil/except_value_utils.h"17#include "src/__support/FPUtil/multiply_add.h"18#include "src/__support/FPUtil/rounding_mode.h"19#include "src/__support/common.h"20#include "src/__support/libc_errno.h"21#include "src/__support/macros/config.h"22#include "src/__support/macros/optimization.h"23#include "src/__support/macros/properties/cpu_features.h"24 25namespace LIBC_NAMESPACE_DECL {26 27namespace math {28 29LIBC_INLINE static constexpr float exp2m1f(float x) {30#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS31 constexpr size_t N_EXCEPTS_LO = 8;32 33 constexpr fputil::ExceptValues<float, N_EXCEPTS_LO> EXP2M1F_EXCEPTS_LO = {{34 // (input, RZ output, RU offset, RD offset, RN offset)35 // x = 0x1.36dc8ep-36, exp2m1f(x) = 0x1.aef212p-37 (RZ)36 {0x2d9b'6e47U, 0x2d57'7909U, 1U, 0U, 0U},37 // x = 0x1.224936p-19, exp2m1f(x) = 0x1.926c0ep-20 (RZ)38 {0x3611'249bU, 0x35c9'3607U, 1U, 0U, 1U},39 // x = 0x1.d16d2p-20, exp2m1f(x) = 0x1.429becp-20 (RZ)40 {0x35e8'b690U, 0x35a1'4df6U, 1U, 0U, 1U},41 // x = 0x1.17949ep-14, exp2m1f(x) = 0x1.8397p-15 (RZ)42 {0x388b'ca4fU, 0x3841'cb80U, 1U, 0U, 1U},43 // x = -0x1.9c3e1ep-38, exp2m1f(x) = -0x1.1dbeacp-38 (RZ)44 {0xacce'1f0fU, 0xac8e'df56U, 0U, 1U, 0U},45 // x = -0x1.4d89b4p-32, exp2m1f(x) = -0x1.ce61b6p-33 (RZ)46 {0xafa6'c4daU, 0xaf67'30dbU, 0U, 1U, 1U},47 // x = -0x1.a6eac4p-10, exp2m1f(x) = -0x1.24fadap-10 (RZ)48 {0xbad3'7562U, 0xba92'7d6dU, 0U, 1U, 1U},49 // x = -0x1.e7526ep-6, exp2m1f(x) = -0x1.4e53dep-6 (RZ)50 {0xbcf3'a937U, 0xbca7'29efU, 0U, 1U, 1U},51 }};52 53 constexpr size_t N_EXCEPTS_HI = 3;54 55 constexpr fputil::ExceptValues<float, N_EXCEPTS_HI> EXP2M1F_EXCEPTS_HI = {{56 // (input, RZ output, RU offset, RD offset, RN offset)57 // x = 0x1.16a972p-1, exp2m1f(x) = 0x1.d545b2p-2 (RZ)58 {0x3f0b'54b9U, 0x3eea'a2d9U, 1U, 0U, 0U},59 // x = -0x1.9f12acp-5, exp2m1f(x) = -0x1.1ab68cp-5 (RZ)60 {0xbd4f'8956U, 0xbd0d'5b46U, 0U, 1U, 0U},61 // x = -0x1.de7b9cp-5, exp2m1f(x) = -0x1.4508f4p-5 (RZ)62 {0xbd6f'3dceU, 0xbd22'847aU, 0U, 1U, 1U},63 }};64#endif // !LIBC_MATH_HAS_SKIP_ACCURATE_PASS65 66 using FPBits = fputil::FPBits<float>;67 FPBits xbits(x);68 69 uint32_t x_u = xbits.uintval();70 uint32_t x_abs = x_u & 0x7fff'ffffU;71 72 // When |x| >= 128, or x is nan, or |x| <= 2^-573 if (LIBC_UNLIKELY(x_abs >= 0x4300'0000U || x_abs <= 0x3d00'0000U)) {74 // |x| <= 2^-575 if (x_abs <= 0x3d00'0000U) {76#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS77 if (auto r = EXP2M1F_EXCEPTS_LO.lookup(x_u); LIBC_UNLIKELY(r.has_value()))78 return r.value();79#endif // !LIBC_MATH_HAS_SKIP_ACCURATE_PASS80 81 // Minimax polynomial generated by Sollya with:82 // > display = hexadecimal;83 // > fpminimax((2^x - 1)/x, 5, [|D...|], [-2^-5, 2^-5]);84 constexpr double COEFFS[] = {85 0x1.62e42fefa39f3p-1, 0x1.ebfbdff82c57bp-3, 0x1.c6b08d6f2d7aap-5,86 0x1.3b2ab6fc92f5dp-7, 0x1.5d897cfe27125p-10, 0x1.43090e61e6af1p-13};87 double xd = x;88 double xsq = xd * xd;89 double c0 = fputil::multiply_add(xd, COEFFS[1], COEFFS[0]);90 double c1 = fputil::multiply_add(xd, COEFFS[3], COEFFS[2]);91 double c2 = fputil::multiply_add(xd, COEFFS[5], COEFFS[4]);92 double p = fputil::polyeval(xsq, c0, c1, c2);93 return static_cast<float>(p * xd);94 }95 96 // x >= 128, or x is nan97 if (xbits.is_pos()) {98 if (xbits.is_finite()) {99 int rounding = fputil::quick_get_round();100 if (rounding == FE_DOWNWARD || rounding == FE_TOWARDZERO)101 return FPBits::max_normal().get_val();102 103 fputil::set_errno_if_required(ERANGE);104 fputil::raise_except_if_required(FE_OVERFLOW);105 }106 107 // x >= 128 and 2^x - 1 rounds to +inf, or x is +inf or nan108 return x + FPBits::inf().get_val();109 }110 }111 112 if (LIBC_UNLIKELY(x <= -25.0f)) {113 // 2^(-inf) - 1 = -1114 if (xbits.is_inf())115 return -1.0f;116 // 2^nan - 1 = nan117 if (xbits.is_nan())118 return x;119 120 int rounding = fputil::quick_get_round();121 if (rounding == FE_UPWARD || rounding == FE_TOWARDZERO)122 return -0x1.ffff'fep-1f; // -1.0f + 0x1.0p-24f123 124 fputil::set_errno_if_required(ERANGE);125 fputil::raise_except_if_required(FE_UNDERFLOW);126 return -1.0f;127 }128 129#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS130 if (auto r = EXP2M1F_EXCEPTS_HI.lookup(x_u); LIBC_UNLIKELY(r.has_value()))131 return r.value();132#endif // !LIBC_MATH_HAS_SKIP_ACCURATE_PASS133 134 // For -25 < x < 128, to compute 2^x, we perform the following range135 // reduction: find hi, mid, lo such that:136 // x = hi + mid + lo, in which:137 // hi is an integer,138 // 0 <= mid * 2^5 < 32 is an integer,139 // -2^(-6) <= lo <= 2^(-6).140 // In particular,141 // hi + mid = round(x * 2^5) * 2^(-5).142 // Then,143 // 2^x = 2^(hi + mid + lo) = 2^hi * 2^mid * 2^lo.144 // 2^mid is stored in the lookup table of 32 elements.145 // 2^lo is computed using a degree-4 minimax polynomial generated by Sollya.146 // We perform 2^hi * 2^mid by simply add hi to the exponent field of 2^mid.147 148 // kf = (hi + mid) * 2^5 = round(x * 2^5)149 float kf = 0;150 int k = 0;151#ifdef LIBC_TARGET_CPU_HAS_NEAREST_INT152 kf = fputil::nearest_integer(x * 32.0f);153 k = static_cast<int>(kf);154#else155 constexpr float HALF[2] = {0.5f, -0.5f};156 k = static_cast<int>(fputil::multiply_add(x, 32.0f, HALF[x < 0.0f]));157 kf = static_cast<float>(k);158#endif // LIBC_TARGET_CPU_HAS_NEAREST_INT159 160 // lo = x - (hi + mid) = x - kf * 2^(-5)161 double lo = fputil::multiply_add(-0x1.0p-5f, kf, x);162 163 // hi = floor(kf * 2^(-4))164 // exp2_hi = shift hi to the exponent field of double precision.165 int64_t exp2_hi =166 static_cast<int64_t>(static_cast<uint64_t>(k >> ExpBase::MID_BITS)167 << fputil::FPBits<double>::FRACTION_LEN);168 // mh = 2^hi * 2^mid169 // mh_bits = bit field of mh170 int64_t mh_bits = ExpBase::EXP_2_MID[k & ExpBase::MID_MASK] + exp2_hi;171 double mh = fputil::FPBits<double>(static_cast<uint64_t>(mh_bits)).get_val();172 173 // Degree-4 polynomial approximating (2^x - 1)/x generated by Sollya with:174 // > display = hexadecimal;175 // > fpminimax((2^x - 1)/x, 4, [|D...|], [-2^-6, 2^-6]);176 constexpr double COEFFS[5] = {0x1.62e42fefa39efp-1, 0x1.ebfbdff8131c4p-3,177 0x1.c6b08d7061695p-5, 0x1.3b2b1bee74b2ap-7,178 0x1.5d88091198529p-10};179 double lo_sq = lo * lo;180 double c1 = fputil::multiply_add(lo, COEFFS[0], 1.0);181 double c2 = fputil::multiply_add(lo, COEFFS[2], COEFFS[1]);182 double c3 = fputil::multiply_add(lo, COEFFS[4], COEFFS[3]);183 double exp2_lo = fputil::polyeval(lo_sq, c1, c2, c3);184 // 2^x - 1 = 2^(hi + mid + lo) - 1185 // = 2^(hi + mid) * 2^lo - 1186 // ~ mh * (1 + lo * P(lo)) - 1187 // = mh * exp2_lo - 1188 return static_cast<float>(fputil::multiply_add(exp2_lo, mh, -1.0));189}190 191} // namespace math192 193} // namespace LIBC_NAMESPACE_DECL194 195#endif // LLVM_LIBC_SRC___SUPPORT_MATH_EXP2M1F_H196