1//===-- Single-precision log2(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/log2f.h"
10#include "common_constants.h" // Lookup table for (1/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/config.h"
18#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY
19
20// This is a correctly-rounded algorithm for log2(x) in single precision with
21// round-to-nearest, tie-to-even mode from the RLIBM project at:
22// https://people.cs.rutgers.edu/~sn349/rlibm
23
24// Step 1 - Range reduction:
25// For x = 2^m * 1.mant, log2(x) = m + log2(1.m)
26// If x is denormal, we normalize it by multiplying x by 2^23 and subtracting
27// m by 23.
28
29// Step 2 - Another range reduction:
30// To compute log(1.mant), let f be the highest 8 bits including the hidden
31// bit, and d be the difference (1.mant - f), i.e. the remaining 16 bits of the
32// mantissa. Then we have the following approximation formula:
33// log2(1.mant) = log2(f) + log2(1.mant / f)
34// = log2(f) + log2(1 + d/f)
35// ~ log2(f) + P(d/f)
36// since d/f is sufficiently small.
37// log2(f) and 1/f are then stored in two 2^7 = 128 entries look-up tables.
38
39// Step 3 - Polynomial approximation:
40// To compute P(d/f), we use a single degree-5 polynomial in double precision
41// which provides correct rounding for all but few exception values.
42// For more detail about how this polynomial is obtained, please refer to the
43// papers:
44// Lim, J. and Nagarakatte, S., "One Polynomial Approximation to Produce
45// Correctly Rounded Results of an Elementary Function for Multiple
46// Representations and Rounding Modes", Proceedings of the 49th ACM SIGPLAN
47// Symposium on Principles of Programming Languages (POPL-2022), Philadelphia,
48// USA, Jan. 16-22, 2022.
49// https://people.cs.rutgers.edu/~sn349/papers/rlibmall-popl-2022.pdf
50// Aanjaneya, M., Lim, J., and Nagarakatte, S., "RLibm-Prog: Progressive
51// Polynomial Approximations for Fast Correctly Rounded Math Libraries",
52// Dept. of Comp. Sci., Rutgets U., Technical Report DCS-TR-758, Nov. 2021.
53// https://arxiv.org/pdf/2111.12852.pdf.
54
55namespace LIBC_NAMESPACE_DECL {
56
57LLVM_LIBC_FUNCTION(float, log2f, (float x)) {
58 using FPBits = typename fputil::FPBits<float>;
59
60 FPBits xbits(x);
61 uint32_t x_u = xbits.uintval();
62
63 // Hard to round value(s).
64 using fputil::round_result_slightly_up;
65
66 int m = -FPBits::EXP_BIAS;
67
68 // log2(1.0f) = 0.0f.
69 if (LIBC_UNLIKELY(x_u == 0x3f80'0000U))
70 return 0.0f;
71
72 // Exceptional inputs.
73 if (LIBC_UNLIKELY(x_u < FPBits::min_normal().uintval() ||
74 x_u > FPBits::max_normal().uintval())) {
75 if (x == 0.0f) {
76 fputil::set_errno_if_required(ERANGE);
77 fputil::raise_except_if_required(FE_DIVBYZERO);
78 return FPBits::inf(Sign::NEG).get_val();
79 }
80 if (xbits.is_neg() && !xbits.is_nan()) {
81 fputil::set_errno_if_required(EDOM);
82 fputil::raise_except(FE_INVALID);
83 return FPBits::quiet_nan().get_val();
84 }
85 if (xbits.is_inf_or_nan()) {
86 return x;
87 }
88 // Normalize denormal inputs.
89 xbits = FPBits(xbits.get_val() * 0x1.0p23f);
90 m -= 23;
91 }
92
93 m += xbits.get_biased_exponent();
94 int index = xbits.get_mantissa() >> 16;
95 // Set bits to 1.m
96 xbits.set_biased_exponent(0x7F);
97
98 float u = xbits.get_val();
99 double v;
100#ifdef LIBC_TARGET_CPU_HAS_FMA_FLOAT
101 v = static_cast<double>(fputil::multiply_add(u, R[index], -1.0f)); // Exact.
102#else
103 v = fputil::multiply_add(static_cast<double>(u), RD[index], -1.0); // Exact
104#endif // LIBC_TARGET_CPU_HAS_FMA_FLOAT
105
106 double extra_factor = static_cast<double>(m) + LOG2_R[index];
107
108 // Degree-5 polynomial approximation of log2 generated by Sollya with:
109 // > P = fpminimax(log2(1 + x)/x, 4, [|1, D...|], [-2^-8, 2^-7]);
110 constexpr double COEFFS[5] = {0x1.71547652b8133p0, -0x1.71547652d1e33p-1,
111 0x1.ec70a098473dep-2, -0x1.7154c5ccdf121p-2,
112 0x1.2514fd90a130ap-2};
113
114 double vsq = v * v; // Exact
115 double c0 = fputil::multiply_add(v, COEFFS[0], extra_factor);
116 double c1 = fputil::multiply_add(v, COEFFS[2], COEFFS[1]);
117 double c2 = fputil::multiply_add(v, COEFFS[4], COEFFS[3]);
118
119 double r = fputil::polyeval(vsq, c0, c1, c2);
120
121 return static_cast<float>(r);
122}
123
124} // namespace LIBC_NAMESPACE_DECL
125

source code of libc/src/math/generic/log2f.cpp