libc/src/math/generic/log10f.cpp

   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 __llvm_libc {
  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   FPBits xbits(x);
 110   uint32_t x_u = xbits.uintval();
 111
 112   // Exact powers of 10 and other hard-to-round cases.
 113   if (LIBC_UNLIKELY((x_u & 0x3FF) == 0)) {
 114     switch (x_u) {
 115     case 0x3f80'0000U: // x = 1
 116       return 0.0f;
 117     case 0x4120'0000U: // x = 10
 118       return 1.0f;
 119     case 0x42c8'0000U: // x = 100
 120       return 2.0f;
 121     case 0x447a'0000U: // x = 1,000
 122       return 3.0f;
 123     case 0x461c'4000U: // x = 10,000
 124       return 4.0f;
 125     case 0x47c3'5000U: // x = 100,000
 126       return 5.0f;
 127     case 0x4974'2400U: // x = 1,000,000
 128       return 6.0f;
 129     }
 130   } else {
 131     switch (x_u) {
 132     case 0x4b18'9680U: // x = 10,000,000
 133       return 7.0f;
 134     case 0x4cbe'bc20U: // x = 100,000,000
 135       return 8.0f;
 136     case 0x4e6e'6b28U: // x = 1,000,000,000
 137       return 9.0f;
 138     case 0x5015'02f9U: // x = 10,000,000,000
 139       return 10.0f;
 140     case 0x0efe'ee7aU: // x = 0x1.fddcf4p-98f
 141       return fputil::round_result_slightly_up(-0x1.d33a46p+4f);
 142     case 0x3f5f'de1bU: // x = 0x1.bfbc36p-1f
 143       return fputil::round_result_slightly_up(-0x1.dd2c6ep-5f);
 144     case 0x3f80'70d8U: // x = 0x1.00e1bp0f
 145       return fputil::round_result_slightly_up(0x1.8762c4p-10f);
 146 #ifndef LIBC_TARGET_CPU_HAS_FMA
 147     case 0x08ae'a356U: // x = 0x1.5d46acp-110f
 148       return fputil::round_result_slightly_up(-0x1.07d3b4p+5f);
 149     case 0x120b'93dcU: // x = 0x1.1727b8p-91f
 150       return fputil::round_result_slightly_down(-0x1.b5b2aep+4f);
 151     case 0x13ae'78d3U: // x = 0x1.5cf1a6p-88f
 152       return fputil::round_result_slightly_down(-0x1.a5b2aep+4f);
 153     case 0x4f13'4f83U: // x = 2471461632.0
 154       return fputil::round_result_slightly_down(0x1.2c9314p+3f);
 155     case 0x7956'ba5eU: // x = 69683218960000541503257137270226944.0
 156       return fputil::round_result_slightly_up(0x1.16bebap+5f);
 157 #endif // LIBC_TARGET_CPU_HAS_FMA
 158     }
 159   }
 160
 161   int m = -FPBits::EXPONENT_BIAS;
 162
 163   if (LIBC_UNLIKELY(x_u < FPBits::MIN_NORMAL || x_u > FPBits::MAX_NORMAL)) {
 164     if (xbits.is_zero()) {
 165       // Return -inf and raise FE_DIVBYZERO
 166       fputil::set_errno_if_required(ERANGE);
 167       fputil::raise_except_if_required(FE_DIVBYZERO);
 168       return static_cast<float>(FPBits::neg_inf());
 169     }
 170     if (xbits.get_sign() && !xbits.is_nan()) {
 171       // Return NaN and raise FE_INVALID
 172       fputil::set_errno_if_required(EDOM);
 173       fputil::raise_except_if_required(FE_INVALID);
 174       return FPBits::build_quiet_nan(0);
 175     }
 176     if (xbits.is_inf_or_nan()) {
 177       return x;
 178     }
 179     // Normalize denormal inputs.
 180     xbits.set_val(xbits.get_val() * 0x1.0p23f);
 181     m -= 23;
 182     x_u = xbits.uintval();
 183   }
 184
 185   // Add unbiased exponent.
 186   m += static_cast<int>(x_u >> 23);
 187   // Extract 7 leading fractional bits of the mantissa
 188   int index = (x_u >> 16) & 0x7F;
 189   // Set bits to 1.m
 190   xbits.set_unbiased_exponent(0x7F);
 191
 192   float u = static_cast<float>(xbits);
 193   double v;
 194 #ifdef LIBC_TARGET_CPU_HAS_FMA
 195   v = static_cast<double>(fputil::multiply_add(u, R[index], -1.0f)); // Exact.
 196 #else
 197   v = fputil::multiply_add(static_cast<double>(u),
 198                            static_cast<double>(R[index]), -1.0); // Exact
 199 #endif // LIBC_TARGET_CPU_HAS_FMA
 200
 201   // Degree-5 polynomial approximation of log10 generated by:
 202   // > P = fpminimax(log10(1 + x)/x, 4, [|D...|], [-2^-8, 2^-7]);
 203   constexpr double COEFFS[5] = {0x1.bcb7b1526e2e5p-2, -0x1.bcb7b1528d43dp-3,
 204                                 0x1.287a77eb4ca0dp-3, -0x1.bcb8110a181b5p-4,
 205                                 0x1.60e7e3e747129p-4};
 206   double v2 = v * v; // Exact
 207   double p2 = fputil::multiply_add(v, COEFFS[4], COEFFS[3]);
 208   double p1 = fputil::multiply_add(v, COEFFS[2], COEFFS[1]);
 209   double p0 = fputil::multiply_add(v, COEFFS[0], LOG10_R[index]);
 210   double r = fputil::multiply_add(static_cast<double>(m), LOG10_2,
 211                                   fputil::polyeval(v2, p0, p1, p2));
 212
 213   return static_cast<float>(r);
 214 }
 215
 216 } // namespace __llvm_libc