1 //===-- Single-precision log10(x) 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-exception 6 // 7 //===----------------------------------------------------------------------===// 8 9 #include "src/math/log10f.h" 10 #include "common_constants.h" // Lookup table for (1/f) 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/except_value_utils.h" 16 #include "src/__support/FPUtil/multiply_add.h" 17 #include "src/__support/common.h" 18 #include "src/__support/macros/optimization.h" // LIBC_UNLIKELY 19 #include "src/__support/macros/properties/cpu_features.h" 20 21 // This is an algorithm for log10(x) in single precision which is 22 // correctly rounded for all rounding modes, based on the implementation of 23 // log10(x) from the RLIBM project at: 24 // https://people.cs.rutgers.edu/~sn349/rlibm 25 26 // Step 1 - Range reduction: 27 // For x = 2^m * 1.mant, log(x) = m * log10(2) + log10(1.m) 28 // If x is denormal, we normalize it by multiplying x by 2^23 and subtracting 29 // m by 23. 30 31 // Step 2 - Another range reduction: 32 // To compute log(1.mant), let f be the highest 8 bits including the hidden 33 // bit, and d be the difference (1.mant - f), i.e. the remaining 16 bits of the 34 // mantissa. Then we have the following approximation formula: 35 // log10(1.mant) = log10(f) + log10(1.mant / f) 36 // = log10(f) + log10(1 + d/f) 37 // ~ log10(f) + P(d/f) 38 // since d/f is sufficiently small. 39 // log10(f) and 1/f are then stored in two 2^7 = 128 entries look-up tables. 40 41 // Step 3 - Polynomial approximation: 42 // To compute P(d/f), we use a single degree-5 polynomial in double precision 43 // which provides correct rounding for all but few exception values. 44 // For more detail about how this polynomial is obtained, please refer to the 45 // papers: 46 // Lim, J. and Nagarakatte, S., "One Polynomial Approximation to Produce 47 // Correctly Rounded Results of an Elementary Function for Multiple 48 // Representations and Rounding Modes", Proceedings of the 49th ACM SIGPLAN 49 // Symposium on Principles of Programming Languages (POPL-2022), Philadelphia, 50 // USA, Jan. 16-22, 2022. 51 // https://people.cs.rutgers.edu/~sn349/papers/rlibmall-popl-2022.pdf 52 // Aanjaneya, M., Lim, J., and Nagarakatte, S., "RLibm-Prog: Progressive 53 // Polynomial Approximations for Fast Correctly Rounded Math Libraries", 54 // Dept. of Comp. Sci., Rutgets U., Technical Report DCS-TR-758, Nov. 2021. 55 // https://arxiv.org/pdf/2111.12852.pdf. 56 57 namespace LIBC_NAMESPACE { 58 59 // Lookup table for -log10(r) where r is defined in common_constants.cpp. 60 static constexpr double LOG10_R[128] = { 61 0x0.0000000000000p+0, 0x1.be76bd77b4fc3p-9, 0x1.c03a80ae5e054p-8, 62 0x1.51824c7587ebp-7, 0x1.c3d0837784c41p-7, 0x1.1b85d6044e9aep-6, 63 0x1.559bd2406c3bap-6, 0x1.902c31d62a843p-6, 0x1.cb38fccd8bfdbp-6, 64 0x1.e8eeb09f2f6cbp-6, 0x1.125d0432ea20ep-5, 0x1.30838cdc2fbfdp-5, 65 0x1.3faf7c663060ep-5, 0x1.5e3966b7e9295p-5, 0x1.7d070145f4fd7p-5, 66 0x1.8c878eeb05074p-5, 0x1.abbcebd84fcap-5, 0x1.bb7209d1e24e5p-5, 67 0x1.db11ed766abf4p-5, 0x1.eafd05035bd3bp-5, 0x1.0585283764178p-4, 68 0x1.0d966cc6500fap-4, 0x1.1dd5460c8b16fp-4, 0x1.2603072a25f82p-4, 69 0x1.367ba3aaa1883p-4, 0x1.3ec6ad5407868p-4, 0x1.4f7aad9bbcbafp-4, 70 0x1.57e3d47c3af7bp-4, 0x1.605735ee985f1p-4, 0x1.715d0ce367afcp-4, 71 0x1.79efb57b0f803p-4, 0x1.828cfed29a215p-4, 0x1.93e7de0fc3e8p-4, 72 0x1.9ca5aa1729f45p-4, 0x1.a56e8325f5c87p-4, 0x1.ae4285509950bp-4, 73 0x1.b721cd17157e3p-4, 0x1.c902a19e65111p-4, 0x1.d204698cb42bdp-4, 74 0x1.db11ed766abf4p-4, 0x1.e42b4c16caaf3p-4, 0x1.ed50a4a26eafcp-4, 75 0x1.ffbfc2bbc7803p-4, 0x1.0484e4942aa43p-3, 0x1.093025a19976cp-3, 76 0x1.0de1b56356b04p-3, 0x1.1299a4fb3e306p-3, 0x1.175805d1587c1p-3, 77 0x1.1c1ce9955c0c6p-3, 0x1.20e8624038fedp-3, 0x1.25ba8215af7fcp-3, 78 0x1.2a935ba5f1479p-3, 0x1.2f7301cf4e87bp-3, 0x1.345987bfeea91p-3, 79 0x1.394700f7953fdp-3, 0x1.3e3b8149739d4p-3, 0x1.43371cde076c2p-3, 80 0x1.4839e83506c87p-3, 0x1.4d43f8275a483p-3, 0x1.525561e9256eep-3, 81 0x1.576e3b0bde0a7p-3, 0x1.5c8e998072fe2p-3, 0x1.61b6939983048p-3, 82 0x1.66e6400da3f77p-3, 0x1.6c1db5f9bb336p-3, 0x1.6c1db5f9bb336p-3, 83 0x1.715d0ce367afcp-3, 0x1.76a45cbb7e6ffp-3, 0x1.7bf3bde099f3p-3, 84 0x1.814b4921bd52bp-3, 0x1.86ab17c10bc7fp-3, 0x1.86ab17c10bc7fp-3, 85 0x1.8c13437695532p-3, 0x1.9183e673394fap-3, 0x1.96fd1b639fc09p-3, 86 0x1.9c7efd734a2f9p-3, 0x1.a209a84fbcff8p-3, 0x1.a209a84fbcff8p-3, 87 0x1.a79d382bc21d9p-3, 0x1.ad39c9c2c608p-3, 0x1.b2df7a5c50299p-3, 88 0x1.b2df7a5c50299p-3, 0x1.b88e67cf9798p-3, 0x1.be46b087354bcp-3, 89 0x1.c4087384f4f8p-3, 0x1.c4087384f4f8p-3, 0x1.c9d3d065c5b42p-3, 90 0x1.cfa8e765cbb72p-3, 0x1.cfa8e765cbb72p-3, 0x1.d587d96494759p-3, 91 0x1.db70c7e96e7f3p-3, 0x1.db70c7e96e7f3p-3, 0x1.e163d527e68cfp-3, 92 0x1.e76124046b3f3p-3, 0x1.e76124046b3f3p-3, 0x1.ed68d819191fcp-3, 93 0x1.f37b15bab08d1p-3, 0x1.f37b15bab08d1p-3, 0x1.f99801fdb749dp-3, 94 0x1.ffbfc2bbc7803p-3, 0x1.ffbfc2bbc7803p-3, 0x1.02f93f4c87101p-2, 95 0x1.06182e84fd4acp-2, 0x1.06182e84fd4acp-2, 0x1.093cc32c90f84p-2, 96 0x1.093cc32c90f84p-2, 0x1.0c6711d6abd7ap-2, 0x1.0f972f87ff3d6p-2, 97 0x1.0f972f87ff3d6p-2, 0x1.12cd31b9c99ffp-2, 0x1.12cd31b9c99ffp-2, 98 0x1.16092e5d3a9a6p-2, 0x1.194b3bdef6b9ep-2, 0x1.194b3bdef6b9ep-2, 99 0x1.1c93712abc7ffp-2, 0x1.1c93712abc7ffp-2, 0x1.1fe1e5af2c141p-2, 100 0x1.1fe1e5af2c141p-2, 0x1.2336b161b3337p-2, 0x1.2336b161b3337p-2, 101 0x1.2691ecc29f042p-2, 0x1.2691ecc29f042p-2, 0x1.29f3b0e15584bp-2, 102 0x1.29f3b0e15584bp-2, 0x1.2d5c1760b86bbp-2, 0x1.2d5c1760b86bbp-2, 103 0x1.30cb3a7bb3625p-2, 0x1.34413509f79ffp-2}; 104 105 LLVM_LIBC_FUNCTION(float, log10f, (float x)) { 106 constexpr double LOG10_2 = 0x1.34413509f79ffp-2; 107 108 using FPBits = typename fputil::FPBits<float>; 109 110 FPBits xbits(x); 111 uint32_t x_u = xbits.uintval(); 112 113 // Exact powers of 10 and other hard-to-round cases. 114 if (LIBC_UNLIKELY((x_u & 0x3FF) == 0)) { 115 switch (x_u) { 116 case 0x3f80'0000U: // x = 1 117 return 0.0f; 118 case 0x4120'0000U: // x = 10 119 return 1.0f; 120 case 0x42c8'0000U: // x = 100 121 return 2.0f; 122 case 0x447a'0000U: // x = 1,000 123 return 3.0f; 124 case 0x461c'4000U: // x = 10,000 125 return 4.0f; 126 case 0x47c3'5000U: // x = 100,000 127 return 5.0f; 128 case 0x4974'2400U: // x = 1,000,000 129 return 6.0f; 130 } 131 } else { 132 switch (x_u) { 133 case 0x4b18'9680U: // x = 10,000,000 134 return 7.0f; 135 case 0x4cbe'bc20U: // x = 100,000,000 136 return 8.0f; 137 case 0x4e6e'6b28U: // x = 1,000,000,000 138 return 9.0f; 139 case 0x5015'02f9U: // x = 10,000,000,000 140 return 10.0f; 141 case 0x0efe'ee7aU: // x = 0x1.fddcf4p-98f 142 return fputil::round_result_slightly_up(-0x1.d33a46p+4f); 143 case 0x3f5f'de1bU: // x = 0x1.bfbc36p-1f 144 return fputil::round_result_slightly_up(-0x1.dd2c6ep-5f); 145 case 0x3f80'70d8U: // x = 0x1.00e1bp0f 146 return fputil::round_result_slightly_up(0x1.8762c4p-10f); 147 #ifndef LIBC_TARGET_CPU_HAS_FMA 148 case 0x08ae'a356U: // x = 0x1.5d46acp-110f 149 return fputil::round_result_slightly_up(-0x1.07d3b4p+5f); 150 case 0x120b'93dcU: // x = 0x1.1727b8p-91f 151 return fputil::round_result_slightly_down(-0x1.b5b2aep+4f); 152 case 0x13ae'78d3U: // x = 0x1.5cf1a6p-88f 153 return fputil::round_result_slightly_down(-0x1.a5b2aep+4f); 154 case 0x4f13'4f83U: // x = 2471461632.0 155 return fputil::round_result_slightly_down(0x1.2c9314p+3f); 156 case 0x7956'ba5eU: // x = 69683218960000541503257137270226944.0 157 return fputil::round_result_slightly_up(0x1.16bebap+5f); 158 #endif // LIBC_TARGET_CPU_HAS_FMA 159 } 160 } 161 162 int m = -FPBits::EXP_BIAS; 163 164 if (LIBC_UNLIKELY(x_u < FPBits::min_normal().uintval() || 165 x_u > FPBits::max_normal().uintval())) { 166 if (xbits.is_zero()) { 167 // Return -inf and raise FE_DIVBYZERO 168 fputil::set_errno_if_required(ERANGE); 169 fputil::raise_except_if_required(FE_DIVBYZERO); 170 return FPBits::inf(Sign::NEG).get_val(); 171 } 172 if (xbits.is_neg() && !xbits.is_nan()) { 173 // Return NaN and raise FE_INVALID 174 fputil::set_errno_if_required(EDOM); 175 fputil::raise_except_if_required(FE_INVALID); 176 return FPBits::quiet_nan().get_val(); 177 } 178 if (xbits.is_inf_or_nan()) { 179 return x; 180 } 181 // Normalize denormal inputs. 182 xbits = FPBits(xbits.get_val() * 0x1.0p23f); 183 m -= 23; 184 x_u = xbits.uintval(); 185 } 186 187 // Add unbiased exponent. 188 m += static_cast<int>(x_u >> 23); 189 // Extract 7 leading fractional bits of the mantissa 190 int index = (x_u >> 16) & 0x7F; 191 // Set bits to 1.m 192 xbits.set_biased_exponent(0x7F); 193 194 float u = xbits.get_val(); 195 double v; 196 #ifdef LIBC_TARGET_CPU_HAS_FMA 197 v = static_cast<double>(fputil::multiply_add(u, R[index], -1.0f)); // Exact. 198 #else 199 v = fputil::multiply_add(static_cast<double>(u), 200 static_cast<double>(R[index]), -1.0); // Exact 201 #endif // LIBC_TARGET_CPU_HAS_FMA 202 203 // Degree-5 polynomial approximation of log10 generated by: 204 // > P = fpminimax(log10(1 + x)/x, 4, [|D...|], [-2^-8, 2^-7]); 205 constexpr double COEFFS[5] = {0x1.bcb7b1526e2e5p-2, -0x1.bcb7b1528d43dp-3, 206 0x1.287a77eb4ca0dp-3, -0x1.bcb8110a181b5p-4, 207 0x1.60e7e3e747129p-4}; 208 double v2 = v * v; // Exact 209 double p2 = fputil::multiply_add(v, COEFFS[4], COEFFS[3]); 210 double p1 = fputil::multiply_add(v, COEFFS[2], COEFFS[1]); 211 double p0 = fputil::multiply_add(v, COEFFS[0], LOG10_R[index]); 212 double r = fputil::multiply_add(static_cast<double>(m), LOG10_2, 213 fputil::polyeval(v2, p0, p1, p2)); 214 215 return static_cast<float>(r); 216 } 217 218 } // namespace LIBC_NAMESPACE 219