1//===-- Single-precision log(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/logf.h"
10#include "common_constants.h" // Lookup table for (1/f) and log(f)
11#include "src/__support/FPUtil/FEnvImpl.h"
12#include "src/__support/FPUtil/FPBits.h"
13#include "src/__support/FPUtil/PolyEval.h"
14#include "src/__support/FPUtil/except_value_utils.h"
15#include "src/__support/FPUtil/multiply_add.h"
16#include "src/__support/common.h"
17#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY
18#include "src/__support/macros/properties/cpu_features.h"
19
20// This is an algorithm for log(x) in single precision which is correctly
21// rounded for all rounding modes, based on the implementation of log(x) from
22// the RLIBM project at:
23// https://people.cs.rutgers.edu/~sn349/rlibm
24
25// Step 1 - Range reduction:
26// For x = 2^m * 1.mant, log(x) = m * log(2) + log(1.m)
27// If x is denormal, we normalize it by multiplying x by 2^23 and subtracting
28// m by 23.
29
30// Step 2 - Another range reduction:
31// To compute log(1.mant), let f be the highest 8 bits including the hidden
32// bit, and d be the difference (1.mant - f), i.e. the remaining 16 bits of the
33// mantissa. Then we have the following approximation formula:
34// log(1.mant) = log(f) + log(1.mant / f)
35// = log(f) + log(1 + d/f)
36// ~ log(f) + P(d/f)
37// since d/f is sufficiently small.
38// log(f) and 1/f are then stored in two 2^7 = 128 entries look-up tables.
39
40// Step 3 - Polynomial approximation:
41// To compute P(d/f), we use a single degree-5 polynomial in double precision
42// which provides correct rounding for all but few exception values.
43// For more detail about how this polynomial is obtained, please refer to the
44// paper:
45// Lim, J. and Nagarakatte, S., "One Polynomial Approximation to Produce
46// Correctly Rounded Results of an Elementary Function for Multiple
47// Representations and Rounding Modes", Proceedings of the 49th ACM SIGPLAN
48// Symposium on Principles of Programming Languages (POPL-2022), Philadelphia,
49// USA, January 16-22, 2022.
50// https://people.cs.rutgers.edu/~sn349/papers/rlibmall-popl-2022.pdf
51
52namespace LIBC_NAMESPACE {
53
54LLVM_LIBC_FUNCTION(float, logf, (float x)) {
55 constexpr double LOG_2 = 0x1.62e42fefa39efp-1;
56 using FPBits = typename fputil::FPBits<float>;
57
58 FPBits xbits(x);
59 uint32_t x_u = xbits.uintval();
60
61 int m = -FPBits::EXP_BIAS;
62
63 using fputil::round_result_slightly_down;
64 using fputil::round_result_slightly_up;
65
66 // Small inputs
67 if (x_u < 0x4c5d65a5U) {
68 // Hard-to-round cases.
69 switch (x_u) {
70 case 0x3f7f4d6fU: // x = 0x1.fe9adep-1f
71 return round_result_slightly_up(value_rn: -0x1.659ec8p-9f);
72 case 0x41178febU: // x = 0x1.2f1fd6p+3f
73 return round_result_slightly_up(value_rn: 0x1.1fcbcep+1f);
74#ifdef LIBC_TARGET_CPU_HAS_FMA
75 case 0x3f800000U: // x = 1.0f
76 return 0.0f;
77#else
78 case 0x1e88452dU: // x = 0x1.108a5ap-66f
79 return round_result_slightly_up(-0x1.6d7b18p+5f);
80#endif // LIBC_TARGET_CPU_HAS_FMA
81 }
82 // Subnormal inputs.
83 if (LIBC_UNLIKELY(x_u < FPBits::min_normal().uintval())) {
84 if (x_u == 0) {
85 // Return -inf and raise FE_DIVBYZERO
86 fputil::set_errno_if_required(ERANGE);
87 fputil::raise_except_if_required(FE_DIVBYZERO);
88 return FPBits::inf(sign: Sign::NEG).get_val();
89 }
90 // Normalize denormal inputs.
91 xbits = FPBits(xbits.get_val() * 0x1.0p23f);
92 m -= 23;
93 x_u = xbits.uintval();
94 }
95 } else {
96 // Hard-to-round cases.
97 switch (x_u) {
98 case 0x4c5d65a5U: // x = 0x1.bacb4ap+25f
99 return round_result_slightly_down(value_rn: 0x1.1e0696p+4f);
100 case 0x65d890d3U: // x = 0x1.b121a6p+76f
101 return round_result_slightly_down(value_rn: 0x1.a9a3f2p+5f);
102 case 0x6f31a8ecU: // x = 0x1.6351d8p+95f
103 return round_result_slightly_down(value_rn: 0x1.08b512p+6f);
104 case 0x7a17f30aU: // x = 0x1.2fe614p+117f
105 return round_result_slightly_up(value_rn: 0x1.451436p+6f);
106#ifndef LIBC_TARGET_CPU_HAS_FMA
107 case 0x500ffb03U: // x = 0x1.1ff606p+33f
108 return round_result_slightly_up(0x1.6fdd34p+4f);
109 case 0x5cd69e88U: // x = 0x1.ad3d1p+58f
110 return round_result_slightly_up(0x1.45c146p+5f);
111 case 0x5ee8984eU: // x = 0x1.d1309cp+62f;
112 return round_result_slightly_up(0x1.5c9442p+5f);
113#endif // LIBC_TARGET_CPU_HAS_FMA
114 }
115 // Exceptional inputs.
116 if (LIBC_UNLIKELY(x_u > FPBits::max_normal().uintval())) {
117 if (x_u == 0x8000'0000U) {
118 // Return -inf and raise FE_DIVBYZERO
119 fputil::set_errno_if_required(ERANGE);
120 fputil::raise_except_if_required(FE_DIVBYZERO);
121 return FPBits::inf(sign: Sign::NEG).get_val();
122 }
123 if (xbits.is_neg() && !xbits.is_nan()) {
124 // Return NaN and raise FE_INVALID
125 fputil::set_errno_if_required(EDOM);
126 fputil::raise_except_if_required(FE_INVALID);
127 return FPBits::quiet_nan().get_val();
128 }
129 // x is +inf or nan
130 return x;
131 }
132 }
133
134#ifndef LIBC_TARGET_CPU_HAS_FMA
135 // Returning the correct +0 when x = 1.0 for non-FMA targets with FE_DOWNWARD
136 // rounding mode.
137 if (LIBC_UNLIKELY((x_u & 0x007f'ffffU) == 0))
138 return static_cast<float>(
139 static_cast<double>(m + xbits.get_biased_exponent()) * LOG_2);
140#endif // LIBC_TARGET_CPU_HAS_FMA
141
142 uint32_t mant = xbits.get_mantissa();
143 // Extract 7 leading fractional bits of the mantissa
144 int index = mant >> 16;
145 // Add unbiased exponent. Add an extra 1 if the 7 leading fractional bits are
146 // all 1's.
147 m += static_cast<int>((x_u + (1 << 16)) >> 23);
148
149 // Set bits to 1.m
150 xbits.set_biased_exponent(0x7F);
151
152 float u = xbits.get_val();
153 double v;
154#ifdef LIBC_TARGET_CPU_HAS_FMA
155 v = static_cast<double>(fputil::multiply_add(x: u, y: R[index], z: -1.0f)); // Exact.
156#else
157 v = fputil::multiply_add(static_cast<double>(u), RD[index], -1.0); // Exact
158#endif // LIBC_TARGET_CPU_HAS_FMA
159
160 // Degree-5 polynomial approximation of log generated by Sollya with:
161 // > P = fpminimax(log(1 + x)/x, 4, [|1, D...|], [-2^-8, 2^-7]);
162 constexpr double COEFFS[4] = {-0x1.000000000fe63p-1, 0x1.555556e963c16p-2,
163 -0x1.000028dedf986p-2, 0x1.966681bfda7f7p-3};
164 double v2 = v * v; // Exact
165 double p2 = fputil::multiply_add(x: v, y: COEFFS[3], z: COEFFS[2]);
166 double p1 = fputil::multiply_add(x: v, y: COEFFS[1], z: COEFFS[0]);
167 double p0 = LOG_R[index] + v;
168 double r = fputil::multiply_add(x: static_cast<double>(m), y: LOG_2,
169 z: fputil::polyeval(x: v2, a0: p0, a: p1, a: p2));
170 return static_cast<float>(r);
171}
172
173} // namespace LIBC_NAMESPACE
174

source code of libc/src/math/generic/logf.cpp