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1//===-- Implementation header for atanhf16 ----------------------*- 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_ATANHF16_H10#define LLVM_LIBC_SRC___SUPPORT_MATH_ATANHF16_H11 12#include "include/llvm-libc-macros/float16-macros.h"13 14#ifdef LIBC_TYPES_HAS_FLOAT1615 16#include "src/__support/FPUtil/FEnvImpl.h"17#include "src/__support/FPUtil/FPBits.h"18#include "src/__support/FPUtil/PolyEval.h"19#include "src/__support/FPUtil/cast.h"20#include "src/__support/FPUtil/except_value_utils.h"21#include "src/__support/FPUtil/multiply_add.h"22#include "src/__support/macros/config.h"23#include "src/__support/macros/optimization.h"24 25namespace LIBC_NAMESPACE_DECL {26 27namespace math {28 29namespace atanhf16_internal {30 31// Lookup table for logf(f) = logf(1 + n*2^(-7)) where n = 0..127,32// computed and stored as float precision constants.33// Generated by Sollya with the following commands:34// display = hexadecimal;35// for n from 0 to 127 do { print(single(1 / (1 + n / 128.0))); };36static constexpr float ONE_OVER_F_FLOAT[128] = {37 0x1p0f, 0x1.fc07fp-1f, 0x1.f81f82p-1f, 0x1.f4465ap-1f,38 0x1.f07c2p-1f, 0x1.ecc07cp-1f, 0x1.e9131ap-1f, 0x1.e573acp-1f,39 0x1.e1e1e2p-1f, 0x1.de5d6ep-1f, 0x1.dae608p-1f, 0x1.d77b66p-1f,40 0x1.d41d42p-1f, 0x1.d0cb58p-1f, 0x1.cd8568p-1f, 0x1.ca4b3p-1f,41 0x1.c71c72p-1f, 0x1.c3f8fp-1f, 0x1.c0e07p-1f, 0x1.bdd2b8p-1f,42 0x1.bacf92p-1f, 0x1.b7d6c4p-1f, 0x1.b4e81cp-1f, 0x1.b20364p-1f,43 0x1.af286cp-1f, 0x1.ac5702p-1f, 0x1.a98ef6p-1f, 0x1.a6d01ap-1f,44 0x1.a41a42p-1f, 0x1.a16d4p-1f, 0x1.9ec8eap-1f, 0x1.9c2d14p-1f,45 0x1.99999ap-1f, 0x1.970e5p-1f, 0x1.948b1p-1f, 0x1.920fb4p-1f,46 0x1.8f9c18p-1f, 0x1.8d3018p-1f, 0x1.8acb9p-1f, 0x1.886e6p-1f,47 0x1.861862p-1f, 0x1.83c978p-1f, 0x1.818182p-1f, 0x1.7f406p-1f,48 0x1.7d05f4p-1f, 0x1.7ad22p-1f, 0x1.78a4c8p-1f, 0x1.767dcep-1f,49 0x1.745d18p-1f, 0x1.724288p-1f, 0x1.702e06p-1f, 0x1.6e1f76p-1f,50 0x1.6c16c2p-1f, 0x1.6a13cep-1f, 0x1.681682p-1f, 0x1.661ec6p-1f,51 0x1.642c86p-1f, 0x1.623fa8p-1f, 0x1.605816p-1f, 0x1.5e75bcp-1f,52 0x1.5c9882p-1f, 0x1.5ac056p-1f, 0x1.58ed24p-1f, 0x1.571ed4p-1f,53 0x1.555556p-1f, 0x1.539094p-1f, 0x1.51d07ep-1f, 0x1.501502p-1f,54 0x1.4e5e0ap-1f, 0x1.4cab88p-1f, 0x1.4afd6ap-1f, 0x1.49539ep-1f,55 0x1.47ae14p-1f, 0x1.460cbcp-1f, 0x1.446f86p-1f, 0x1.42d662p-1f,56 0x1.414142p-1f, 0x1.3fb014p-1f, 0x1.3e22ccp-1f, 0x1.3c995ap-1f,57 0x1.3b13b2p-1f, 0x1.3991c2p-1f, 0x1.381382p-1f, 0x1.3698ep-1f,58 0x1.3521dp-1f, 0x1.33ae46p-1f, 0x1.323e34p-1f, 0x1.30d19p-1f,59 0x1.2f684cp-1f, 0x1.2e025cp-1f, 0x1.2c9fb4p-1f, 0x1.2b404ap-1f,60 0x1.29e412p-1f, 0x1.288b02p-1f, 0x1.27350cp-1f, 0x1.25e228p-1f,61 0x1.24924ap-1f, 0x1.234568p-1f, 0x1.21fb78p-1f, 0x1.20b47p-1f,62 0x1.1f7048p-1f, 0x1.1e2ef4p-1f, 0x1.1cf06ap-1f, 0x1.1bb4a4p-1f,63 0x1.1a7b96p-1f, 0x1.194538p-1f, 0x1.181182p-1f, 0x1.16e068p-1f,64 0x1.15b1e6p-1f, 0x1.1485fp-1f, 0x1.135c82p-1f, 0x1.12358ep-1f,65 0x1.111112p-1f, 0x1.0fef02p-1f, 0x1.0ecf56p-1f, 0x1.0db20ap-1f,66 0x1.0c9714p-1f, 0x1.0b7e6ep-1f, 0x1.0a681p-1f, 0x1.0953f4p-1f,67 0x1.08421p-1f, 0x1.07326p-1f, 0x1.0624dep-1f, 0x1.05198p-1f,68 0x1.041042p-1f, 0x1.03091cp-1f, 0x1.020408p-1f, 0x1.010102p-1f};69 70// Lookup table for log(f) = log(1 + n*2^(-7)) where n = 0..127,71// computed and stored as float precision constants.72// Generated by Sollya with the following commands:73// display = hexadecimal;74// for n from 0 to 127 do { print(single(log(1 + n / 128.0))); };75static constexpr float LOG_F_FLOAT[128] = {76 0.0f, 0x1.fe02a6p-8f, 0x1.fc0a8cp-7f, 0x1.7b91bp-6f,77 0x1.f829bp-6f, 0x1.39e87cp-5f, 0x1.77459p-5f, 0x1.b42dd8p-5f,78 0x1.f0a30cp-5f, 0x1.16536ep-4f, 0x1.341d7ap-4f, 0x1.51b074p-4f,79 0x1.6f0d28p-4f, 0x1.8c345ep-4f, 0x1.a926d4p-4f, 0x1.c5e548p-4f,80 0x1.e27076p-4f, 0x1.fec914p-4f, 0x1.0d77e8p-3f, 0x1.1b72aep-3f,81 0x1.29553p-3f, 0x1.371fc2p-3f, 0x1.44d2b6p-3f, 0x1.526e5ep-3f,82 0x1.5ff308p-3f, 0x1.6d60fep-3f, 0x1.7ab89p-3f, 0x1.87fa06p-3f,83 0x1.9525aap-3f, 0x1.a23bc2p-3f, 0x1.af3c94p-3f, 0x1.bc2868p-3f,84 0x1.c8ff7cp-3f, 0x1.d5c216p-3f, 0x1.e27076p-3f, 0x1.ef0adcp-3f,85 0x1.fb9186p-3f, 0x1.04025ap-2f, 0x1.0a324ep-2f, 0x1.1058cp-2f,86 0x1.1675cap-2f, 0x1.1c898cp-2f, 0x1.22942p-2f, 0x1.2895a2p-2f,87 0x1.2e8e2cp-2f, 0x1.347ddap-2f, 0x1.3a64c6p-2f, 0x1.404308p-2f,88 0x1.4618bcp-2f, 0x1.4be5fap-2f, 0x1.51aad8p-2f, 0x1.576772p-2f,89 0x1.5d1bdcp-2f, 0x1.62c83p-2f, 0x1.686c82p-2f, 0x1.6e08eap-2f,90 0x1.739d8p-2f, 0x1.792a56p-2f, 0x1.7eaf84p-2f, 0x1.842d1ep-2f,91 0x1.89a338p-2f, 0x1.8f11e8p-2f, 0x1.947942p-2f, 0x1.99d958p-2f,92 0x1.9f323ep-2f, 0x1.a4840ap-2f, 0x1.a9cecap-2f, 0x1.af1294p-2f,93 0x1.b44f78p-2f, 0x1.b9858ap-2f, 0x1.beb4dap-2f, 0x1.c3dd7ap-2f,94 0x1.c8ff7cp-2f, 0x1.ce1afp-2f, 0x1.d32fe8p-2f, 0x1.d83e72p-2f,95 0x1.dd46ap-2f, 0x1.e24882p-2f, 0x1.e74426p-2f, 0x1.ec399ep-2f,96 0x1.f128f6p-2f, 0x1.f6124p-2f, 0x1.faf588p-2f, 0x1.ffd2ep-2f,97 0x1.02552ap-1f, 0x1.04bdfap-1f, 0x1.0723e6p-1f, 0x1.0986f4p-1f,98 0x1.0be72ep-1f, 0x1.0e4498p-1f, 0x1.109f3ap-1f, 0x1.12f71ap-1f,99 0x1.154c3ep-1f, 0x1.179eacp-1f, 0x1.19ee6cp-1f, 0x1.1c3b82p-1f,100 0x1.1e85f6p-1f, 0x1.20cdcep-1f, 0x1.23130ep-1f, 0x1.2555bcp-1f,101 0x1.2795e2p-1f, 0x1.29d38p-1f, 0x1.2c0e9ep-1f, 0x1.2e4744p-1f,102 0x1.307d74p-1f, 0x1.32b134p-1f, 0x1.34e28ap-1f, 0x1.37117cp-1f,103 0x1.393e0ep-1f, 0x1.3b6844p-1f, 0x1.3d9026p-1f, 0x1.3fb5b8p-1f,104 0x1.41d8fep-1f, 0x1.43f9fep-1f, 0x1.4618bcp-1f, 0x1.48353ep-1f,105 0x1.4a4f86p-1f, 0x1.4c679ap-1f, 0x1.4e7d82p-1f, 0x1.50913cp-1f,106 0x1.52a2d2p-1f, 0x1.54b246p-1f, 0x1.56bf9ep-1f, 0x1.58cadcp-1f,107 0x1.5ad404p-1f, 0x1.5cdb1ep-1f, 0x1.5ee02ap-1f, 0x1.60e33p-1f};108 109// x should be positive, normal finite value110// TODO: Simplify range reduction and polynomial degree for float16.111// See issue #137190.112LIBC_INLINE static float log_eval_f(float x) {113 // For x = 2^ex * (1 + mx), logf(x) = ex * logf(2) + logf(1 + mx).114 using FPBits = fputil::FPBits<float>;115 FPBits xbits(x);116 117 float ex = static_cast<float>(xbits.get_exponent());118 // p1 is the leading 7 bits of mx, i.e.119 // p1 * 2^(-7) <= m_x < (p1 + 1) * 2^(-7).120 int p1 = static_cast<int>(xbits.get_mantissa() >> (FPBits::FRACTION_LEN - 7));121 122 // Set bits to (1 + (mx - p1*2^(-7)))123 xbits.set_uintval(xbits.uintval() & (FPBits::FRACTION_MASK >> 7));124 xbits.set_biased_exponent(FPBits::EXP_BIAS);125 // dx = (mx - p1*2^(-7)) / (1 + p1*2^(-7)).126 float dx = (xbits.get_val() - 1.0f) * ONE_OVER_F_FLOAT[p1];127 128 // Minimax polynomial for log(1 + dx), generated using Sollya:129 // > P = fpminimax(log(1 + x)/x, 6, [|SG...|], [0, 2^-7]);130 // > Q = (P - 1) / x;131 // > for i from 0 to degree(Q) do print(coeff(Q, i));132 constexpr float COEFFS[6] = {-0x1p-1f, 0x1.555556p-2f, -0x1.00022ep-2f,133 0x1.9ea056p-3f, -0x1.e50324p-2f, 0x1.c018fp3f};134 135 float dx2 = dx * dx;136 137 float c1 = fputil::multiply_add(dx, COEFFS[1], COEFFS[0]);138 float c2 = fputil::multiply_add(dx, COEFFS[3], COEFFS[2]);139 float c3 = fputil::multiply_add(dx, COEFFS[5], COEFFS[4]);140 141 float p = fputil::polyeval(dx2, dx, c1, c2, c3);142 143 // Generated by Sollya with the following commands:144 // > display = hexadecimal;145 // > round(log(2), SG, RN);146 constexpr float LOGF_2 = 0x1.62e43p-1f;147 148 float result = fputil::multiply_add(ex, LOGF_2, LOG_F_FLOAT[p1] + p);149 return result;150}151 152} // namespace atanhf16_internal153 154LIBC_INLINE static constexpr float16 atanhf16(float16 x) {155 constexpr size_t N_EXCEPTS = 1;156 constexpr fputil::ExceptValues<float16, N_EXCEPTS> ATANHF16_EXCEPTS{{157 // (input, RZ output, RU offset, RD offset, RN offset)158 // x = 0x1.a5cp-4, atanhf16(x) = 0x1.a74p-4 (RZ)159 {0x2E97, 0x2E9D, 1, 0, 0},160 }};161 162 using namespace atanhf16_internal;163 using FPBits = fputil::FPBits<float16>;164 165 FPBits xbits(x);166 Sign sign = xbits.sign();167 uint16_t x_abs = xbits.abs().uintval();168 169 // |x| >= 1170 if (LIBC_UNLIKELY(x_abs >= 0x3c00U)) {171 if (xbits.is_nan()) {172 if (xbits.is_signaling_nan()) {173 fputil::raise_except_if_required(FE_INVALID);174 return FPBits::quiet_nan().get_val();175 }176 return x;177 }178 179 // |x| == 1.0180 if (x_abs == 0x3c00U) {181 fputil::set_errno_if_required(ERANGE);182 fputil::raise_except_if_required(FE_DIVBYZERO);183 return FPBits::inf(sign).get_val();184 }185 // |x| > 1.0186 fputil::set_errno_if_required(EDOM);187 fputil::raise_except_if_required(FE_INVALID);188 return FPBits::quiet_nan().get_val();189 }190 191 if (auto r = ATANHF16_EXCEPTS.lookup(xbits.uintval());192 LIBC_UNLIKELY(r.has_value()))193 return r.value();194 195 // For |x| less than approximately 0.24196 if (LIBC_UNLIKELY(x_abs <= 0x33f3U)) {197 // atanh(+/-0) = +/-0198 if (LIBC_UNLIKELY(x_abs == 0U))199 return x;200 // The Taylor expansion of atanh(x) is:201 // atanh(x) = x + x^3/3 + x^5/5 + x^7/7 + x^9/9 + x^11/11202 // = x * [1 + x^2/3 + x^4/5 + x^6/7 + x^8/9 + x^10/11]203 // When |x| < 2^-5 (0x0800U), this can be approximated by:204 // atanh(x) ≈ x + (1/3)*x^3205 if (LIBC_UNLIKELY(x_abs < 0x0800U)) {206 float xf = x;207 return fputil::cast<float16>(xf + 0x1.555556p-2f * xf * xf * xf);208 }209 210 // For 2^-5 <= |x| <= 0x1.fccp-3 (~0.24):211 // Let t = x^2.212 // Define P(t) ≈ (1/3)*t + (1/5)*t^2 + (1/7)*t^3 + (1/9)*t^4 + (1/11)*t^5.213 // Coefficients (from Sollya, RN, hexadecimal):214 // 1/3 = 0x1.555556p-2, 1/5 = 0x1.99999ap-3, 1/7 = 0x1.24924ap-3,215 // 1/9 = 0x1.c71c72p-4, 1/11 = 0x1.745d18p-4216 // Thus, atanh(x) ≈ x * (1 + P(x^2)).217 float xf = x;218 float x2 = xf * xf;219 float pe = fputil::polyeval(x2, 0.0f, 0x1.555556p-2f, 0x1.99999ap-3f,220 0x1.24924ap-3f, 0x1.c71c72p-4f, 0x1.745d18p-4f);221 return fputil::cast<float16>(fputil::multiply_add(xf, pe, xf));222 }223 224 float xf = x;225 return fputil::cast<float16>(0.5 * log_eval_f((xf + 1.0f) / (xf - 1.0f)));226}227 228} // namespace math229 230} // namespace LIBC_NAMESPACE_DECL231 232#endif // LIBC_TYPES_HAS_FLOAT16233 234#endif // LLVM_LIBC_SRC___SUPPORT_MATH_ATANHF16_H235