1//===-- Single-precision sin 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/sinf.h"
10#include "sincosf_utils.h"
11#include "src/__support/FPUtil/BasicOperations.h"
12#include "src/__support/FPUtil/FEnvImpl.h"
13#include "src/__support/FPUtil/FPBits.h"
14#include "src/__support/FPUtil/PolyEval.h"
15#include "src/__support/FPUtil/multiply_add.h"
16#include "src/__support/FPUtil/rounding_mode.h"
17#include "src/__support/common.h"
18#include "src/__support/macros/config.h"
19#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY
20#include "src/__support/macros/properties/cpu_features.h" // LIBC_TARGET_CPU_HAS_FMA
21
22#if defined(LIBC_TARGET_CPU_HAS_FMA_DOUBLE)
23#include "range_reduction_fma.h"
24#else
25#include "range_reduction.h"
26#endif
27
28namespace LIBC_NAMESPACE_DECL {
29
30LLVM_LIBC_FUNCTION(float, sinf, (float x)) {
31 using FPBits = typename fputil::FPBits<float>;
32 FPBits xbits(x);
33
34 uint32_t x_u = xbits.uintval();
35 uint32_t x_abs = x_u & 0x7fff'ffffU;
36 double xd = static_cast<double>(x);
37
38 // Range reduction:
39 // For |x| > pi/32, we perform range reduction as follows:
40 // Find k and y such that:
41 // x = (k + y) * pi/32
42 // k is an integer
43 // |y| < 0.5
44 // For small range (|x| < 2^45 when FMA instructions are available, 2^22
45 // otherwise), this is done by performing:
46 // k = round(x * 32/pi)
47 // y = x * 32/pi - k
48 // For large range, we will omit all the higher parts of 32/pi such that the
49 // least significant bits of their full products with x are larger than 63,
50 // since sin((k + y + 64*i) * pi/32) = sin(x + i * 2pi) = sin(x).
51 //
52 // When FMA instructions are not available, we store the digits of 32/pi in
53 // chunks of 28-bit precision. This will make sure that the products:
54 // x * THIRTYTWO_OVER_PI_28[i] are all exact.
55 // When FMA instructions are available, we simply store the digits of 32/pi in
56 // chunks of doubles (53-bit of precision).
57 // So when multiplying by the largest values of single precision, the
58 // resulting output should be correct up to 2^(-208 + 128) ~ 2^-80. By the
59 // worst-case analysis of range reduction, |y| >= 2^-38, so this should give
60 // us more than 40 bits of accuracy. For the worst-case estimation of range
61 // reduction, see for instances:
62 // Elementary Functions by J-M. Muller, Chapter 11,
63 // Handbook of Floating-Point Arithmetic by J-M. Muller et. al.,
64 // Chapter 10.2.
65 //
66 // Once k and y are computed, we then deduce the answer by the sine of sum
67 // formula:
68 // sin(x) = sin((k + y)*pi/32)
69 // = sin(y*pi/32) * cos(k*pi/32) + cos(y*pi/32) * sin(k*pi/32)
70 // The values of sin(k*pi/32) and cos(k*pi/32) for k = 0..31 are precomputed
71 // and stored using a vector of 32 doubles. Sin(y*pi/32) and cos(y*pi/32) are
72 // computed using degree-7 and degree-6 minimax polynomials generated by
73 // Sollya respectively.
74
75 // |x| <= pi/16
76 if (LIBC_UNLIKELY(x_abs <= 0x3e49'0fdbU)) {
77
78 // |x| < 0x1.d12ed2p-12f
79 if (LIBC_UNLIKELY(x_abs < 0x39e8'9769U)) {
80 if (LIBC_UNLIKELY(x_abs == 0U)) {
81 // For signed zeros.
82 return x;
83 }
84 // When |x| < 2^-12, the relative error of the approximation sin(x) ~ x
85 // is:
86 // |sin(x) - x| / |sin(x)| < |x^3| / (6|x|)
87 // = x^2 / 6
88 // < 2^-25
89 // < epsilon(1)/2.
90 // So the correctly rounded values of sin(x) are:
91 // = x - sign(x)*eps(x) if rounding mode = FE_TOWARDZERO,
92 // or (rounding mode = FE_UPWARD and x is
93 // negative),
94 // = x otherwise.
95 // To simplify the rounding decision and make it more efficient, we use
96 // fma(x, -2^-25, x) instead.
97 // An exhaustive test shows that this formula work correctly for all
98 // rounding modes up to |x| < 0x1.c555dep-11f.
99 // Note: to use the formula x - 2^-25*x to decide the correct rounding, we
100 // do need fma(x, -2^-25, x) to prevent underflow caused by -2^-25*x when
101 // |x| < 2^-125. For targets without FMA instructions, we simply use
102 // double for intermediate results as it is more efficient than using an
103 // emulated version of FMA.
104#if defined(LIBC_TARGET_CPU_HAS_FMA_FLOAT)
105 return fputil::multiply_add(x, -0x1.0p-25f, x);
106#else
107 return static_cast<float>(fputil::multiply_add(xd, -0x1.0p-25, xd));
108#endif // LIBC_TARGET_CPU_HAS_FMA_FLOAT
109 }
110
111 // |x| < pi/16.
112 double xsq = xd * xd;
113
114 // Degree-9 polynomial approximation:
115 // sin(x) ~ x + a_3 x^3 + a_5 x^5 + a_7 x^7 + a_9 x^9
116 // = x (1 + a_3 x^2 + ... + a_9 x^8)
117 // = x * P(x^2)
118 // generated by Sollya with the following commands:
119 // > display = hexadecimal;
120 // > Q = fpminimax(sin(x)/x, [|0, 2, 4, 6, 8|], [|1, D...|], [0, pi/16]);
121 double result =
122 fputil::polyeval(xsq, 1.0, -0x1.55555555554c6p-3, 0x1.1111111085e65p-7,
123 -0x1.a019f70fb4d4fp-13, 0x1.718d179815e74p-19);
124 return static_cast<float>(xd * result);
125 }
126
127#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
128 if (LIBC_UNLIKELY(x_abs == 0x4619'9998U)) { // x = 0x1.33333p13
129 float r = -0x1.63f4bap-2f;
130 int rounding = fputil::quick_get_round();
131 if ((rounding == FE_DOWNWARD && xbits.is_pos()) ||
132 (rounding == FE_UPWARD && xbits.is_neg()))
133 r = -0x1.63f4bcp-2f;
134 return xbits.is_neg() ? -r : r;
135 }
136#endif // !LIBC_MATH_HAS_SKIP_ACCURATE_PASS
137
138 if (LIBC_UNLIKELY(x_abs >= 0x7f80'0000U)) {
139 if (xbits.is_signaling_nan()) {
140 fputil::raise_except_if_required(FE_INVALID);
141 return FPBits::quiet_nan().get_val();
142 }
143
144 if (x_abs == 0x7f80'0000U) {
145 fputil::set_errno_if_required(EDOM);
146 fputil::raise_except_if_required(FE_INVALID);
147 }
148 return x + FPBits::quiet_nan().get_val();
149 }
150
151 // Combine the results with the sine of sum formula:
152 // sin(x) = sin((k + y)*pi/32)
153 // = sin(y*pi/32) * cos(k*pi/32) + cos(y*pi/32) * sin(k*pi/32)
154 // = sin_y * cos_k + (1 + cosm1_y) * sin_k
155 // = sin_y * cos_k + (cosm1_y * sin_k + sin_k)
156 double sin_k, cos_k, sin_y, cosm1_y;
157
158 sincosf_eval(xd, x_abs, sin_k, cos_k, sin_y, cosm1_y);
159
160 return static_cast<float>(fputil::multiply_add(
161 sin_y, cos_k, fputil::multiply_add(cosm1_y, sin_k, sin_k)));
162}
163
164} // namespace LIBC_NAMESPACE_DECL
165

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