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#ifndef LLVM_LIBC_SRC___SUPPORT_MATH_SINF_H
10#define LLVM_LIBC_SRC___SUPPORT_MATH_SINF_H
11
12#include "src/__support/FPUtil/BasicOperations.h"
13#include "src/__support/FPUtil/FEnvImpl.h"
14#include "src/__support/FPUtil/FPBits.h"
15#include "src/__support/FPUtil/PolyEval.h"
16#include "src/__support/FPUtil/multiply_add.h"
17#include "src/__support/FPUtil/rounding_mode.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_MATH_HAS_SKIP_ACCURATE_PASS) && \
23 defined(LIBC_MATH_HAS_INTERMEDIATE_COMP_IN_FLOAT) && \
24 defined(LIBC_TARGET_CPU_HAS_FMA_FLOAT)
25
26#include "sincosf_float_eval.h"
27
28namespace LIBC_NAMESPACE_DECL {
29
30namespace math {
31
32LIBC_INLINE float sinf(float x) {
33 return math::sincosf_float_eval::sincosf_eval</*IS_SIN*/ true>(x);
34}
35
36} // namespace math
37
38} // namespace LIBC_NAMESPACE_DECL
39
40#else // !LIBC_MATH_HAS_INTERMEDIATE_COMP_IN_FLOAT
41
42#include "src/__support/math/sincosf_utils.h"
43
44#ifdef LIBC_TARGET_CPU_HAS_FMA_DOUBLE
45#include "src/__support/math/range_reduction_fma.h"
46#else // !LIBC_TARGET_CPU_HAS_FMA_DOUBLE
47#include "src/__support/math/range_reduction.h"
48#endif // LIBC_TARGET_CPU_HAS_FMA_DOUBLE
49
50namespace LIBC_NAMESPACE_DECL {
51
52namespace math {
53
54LIBC_INLINE float sinf(float x) {
55 using namespace sincosf_utils_internal;
56 using FPBits = typename fputil::FPBits<float>;
57 FPBits xbits(x);
58
59 uint32_t x_u = xbits.uintval();
60 uint32_t x_abs = x_u & 0x7fff'ffffU;
61 double xd = static_cast<double>(x);
62
63 // Range reduction:
64 // For |x| > pi/32, we perform range reduction as follows:
65 // Find k and y such that:
66 // x = (k + y) * pi/32
67 // k is an integer
68 // |y| < 0.5
69 // For small range (|x| < 2^45 when FMA instructions are available, 2^22
70 // otherwise), this is done by performing:
71 // k = round(x * 32/pi)
72 // y = x * 32/pi - k
73 // For large range, we will omit all the higher parts of 32/pi such that the
74 // least significant bits of their full products with x are larger than 63,
75 // since sin((k + y + 64*i) * pi/32) = sin(x + i * 2pi) = sin(x).
76 //
77 // When FMA instructions are not available, we store the digits of 32/pi in
78 // chunks of 28-bit precision. This will make sure that the products:
79 // x * THIRTYTWO_OVER_PI_28[i] are all exact.
80 // When FMA instructions are available, we simply store the digits of 32/pi in
81 // chunks of doubles (53-bit of precision).
82 // So when multiplying by the largest values of single precision, the
83 // resulting output should be correct up to 2^(-208 + 128) ~ 2^-80. By the
84 // worst-case analysis of range reduction, |y| >= 2^-38, so this should give
85 // us more than 40 bits of accuracy. For the worst-case estimation of range
86 // reduction, see for instances:
87 // Elementary Functions by J-M. Muller, Chapter 11,
88 // Handbook of Floating-Point Arithmetic by J-M. Muller et. al.,
89 // Chapter 10.2.
90 //
91 // Once k and y are computed, we then deduce the answer by the sine of sum
92 // formula:
93 // sin(x) = sin((k + y)*pi/32)
94 // = sin(y*pi/32) * cos(k*pi/32) + cos(y*pi/32) * sin(k*pi/32)
95 // The values of sin(k*pi/32) and cos(k*pi/32) for k = 0..31 are precomputed
96 // and stored using a vector of 32 doubles. Sin(y*pi/32) and cos(y*pi/32) are
97 // computed using degree-7 and degree-6 minimax polynomials generated by
98 // Sollya respectively.
99
100 // |x| <= pi/16
101 if (LIBC_UNLIKELY(x_abs <= 0x3e49'0fdbU)) {
102
103 // |x| < 0x1.d12ed2p-12f
104 if (LIBC_UNLIKELY(x_abs < 0x39e8'9769U)) {
105 if (LIBC_UNLIKELY(x_abs == 0U)) {
106 // For signed zeros.
107 return x;
108 }
109 // When |x| < 2^-12, the relative error of the approximation sin(x) ~ x
110 // is:
111 // |sin(x) - x| / |sin(x)| < |x^3| / (6|x|)
112 // = x^2 / 6
113 // < 2^-25
114 // < epsilon(1)/2.
115 // So the correctly rounded values of sin(x) are:
116 // = x - sign(x)*eps(x) if rounding mode = FE_TOWARDZERO,
117 // or (rounding mode = FE_UPWARD and x is
118 // negative),
119 // = x otherwise.
120 // To simplify the rounding decision and make it more efficient, we use
121 // fma(x, -2^-25, x) instead.
122 // An exhaustive test shows that this formula work correctly for all
123 // rounding modes up to |x| < 0x1.c555dep-11f.
124 // Note: to use the formula x - 2^-25*x to decide the correct rounding, we
125 // do need fma(x, -2^-25, x) to prevent underflow caused by -2^-25*x when
126 // |x| < 2^-125. For targets without FMA instructions, we simply use
127 // double for intermediate results as it is more efficient than using an
128 // emulated version of FMA.
129#if defined(LIBC_TARGET_CPU_HAS_FMA_FLOAT)
130 return fputil::multiply_add(x, -0x1.0p-25f, x);
131#else
132 return static_cast<float>(fputil::multiply_add(x: xd, y: -0x1.0p-25, z: xd));
133#endif // LIBC_TARGET_CPU_HAS_FMA_FLOAT
134 }
135
136 // |x| < pi/16.
137 double xsq = xd * xd;
138
139 // Degree-9 polynomial approximation:
140 // sin(x) ~ x + a_3 x^3 + a_5 x^5 + a_7 x^7 + a_9 x^9
141 // = x (1 + a_3 x^2 + ... + a_9 x^8)
142 // = x * P(x^2)
143 // generated by Sollya with the following commands:
144 // > display = hexadecimal;
145 // > Q = fpminimax(sin(x)/x, [|0, 2, 4, 6, 8|], [|1, D...|], [0, pi/16]);
146 double result =
147 fputil::polyeval(x: xsq, a0: 1.0, a: -0x1.55555555554c6p-3, a: 0x1.1111111085e65p-7,
148 a: -0x1.a019f70fb4d4fp-13, a: 0x1.718d179815e74p-19);
149 return static_cast<float>(xd * result);
150 }
151
152#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
153 if (LIBC_UNLIKELY(x_abs == 0x4619'9998U)) { // x = 0x1.33333p13
154 float r = -0x1.63f4bap-2f;
155#ifndef LIBC_MATH_HAS_ASSUME_ROUND_NEAREST_ONLY
156 int rounding = fputil::quick_get_round();
157 if ((rounding == FE_DOWNWARD && xbits.is_pos()) ||
158 (rounding == FE_UPWARD && xbits.is_neg()))
159 r = -0x1.63f4bcp-2f;
160#endif // !LIBC_MATH_HAS_ASSUME_ROUND_NEAREST_ONLY
161 return xbits.is_neg() ? -r : r;
162 }
163#endif // !LIBC_MATH_HAS_SKIP_ACCURATE_PASS
164
165 if (LIBC_UNLIKELY(x_abs >= 0x7f80'0000U)) {
166 if (xbits.is_signaling_nan()) {
167 fputil::raise_except_if_required(FE_INVALID);
168 return FPBits::quiet_nan().get_val();
169 }
170
171 if (x_abs == 0x7f80'0000U) {
172 fputil::set_errno_if_required(EDOM);
173 fputil::raise_except_if_required(FE_INVALID);
174 }
175 return x + FPBits::quiet_nan().get_val();
176 }
177
178 // Combine the results with the sine of sum formula:
179 // sin(x) = sin((k + y)*pi/32)
180 // = sin(y*pi/32) * cos(k*pi/32) + cos(y*pi/32) * sin(k*pi/32)
181 // = sin_y * cos_k + (1 + cosm1_y) * sin_k
182 // = sin_y * cos_k + (cosm1_y * sin_k + sin_k)
183 double sin_k, cos_k, sin_y, cosm1_y;
184
185 sincosf_eval(xd, x_abs, sin_k, cos_k, sin_y, cosm1_y);
186
187 return static_cast<float>(fputil::multiply_add(
188 x: sin_y, y: cos_k, z: fputil::multiply_add(x: cosm1_y, y: sin_k, z: sin_k)));
189}
190
191} // namespace math
192
193} // namespace LIBC_NAMESPACE_DECL
194
195#endif // LIBC_MATH_HAS_INTERMEDIATE_COMP_IN_FLOAT
196
197#endif // LLVM_LIBC_SRC___SUPPORT_MATH_SINF_H
198