1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
|
//===-- Single-precision sin function -------------------------------------===//
//
// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
// See https://llvm.org/LICENSE.txt for license information.
// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
//
//===----------------------------------------------------------------------===//
#include "src/math/sinf.h"
#include "sincosf_utils.h"
#include "src/__support/FPUtil/BasicOperations.h"
#include "src/__support/FPUtil/FEnvImpl.h"
#include "src/__support/FPUtil/FPBits.h"
#include "src/__support/FPUtil/PolyEval.h"
#include "src/__support/FPUtil/multiply_add.h"
#include "src/__support/FPUtil/rounding_mode.h"
#include "src/__support/common.h"
#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY
#include "src/__support/macros/properties/cpu_features.h" // LIBC_TARGET_CPU_HAS_FMA
#include <errno.h>
#if defined(LIBC_TARGET_CPU_HAS_FMA)
#include "range_reduction_fma.h"
#else
#include "range_reduction.h"
#endif
namespace __llvm_libc {
LLVM_LIBC_FUNCTION(float, sinf, (float x)) {
using FPBits = typename fputil::FPBits<float>;
FPBits xbits(x);
uint32_t x_u = xbits.uintval();
uint32_t x_abs = x_u & 0x7fff'ffffU;
double xd = static_cast<double>(x);
// Range reduction:
// For |x| > pi/32, we perform range reduction as follows:
// Find k and y such that:
// x = (k + y) * pi/32
// k is an integer
// |y| < 0.5
// For small range (|x| < 2^45 when FMA instructions are available, 2^22
// otherwise), this is done by performing:
// k = round(x * 32/pi)
// y = x * 32/pi - k
// For large range, we will omit all the higher parts of 32/pi such that the
// least significant bits of their full products with x are larger than 63,
// since sin((k + y + 64*i) * pi/32) = sin(x + i * 2pi) = sin(x).
//
// When FMA instructions are not available, we store the digits of 32/pi in
// chunks of 28-bit precision. This will make sure that the products:
// x * THIRTYTWO_OVER_PI_28[i] are all exact.
// When FMA instructions are available, we simply store the digits of 32/pi in
// chunks of doubles (53-bit of precision).
// So when multiplying by the largest values of single precision, the
// resulting output should be correct up to 2^(-208 + 128) ~ 2^-80. By the
// worst-case analysis of range reduction, |y| >= 2^-38, so this should give
// us more than 40 bits of accuracy. For the worst-case estimation of range
// reduction, see for instances:
// Elementary Functions by J-M. Muller, Chapter 11,
// Handbook of Floating-Point Arithmetic by J-M. Muller et. al.,
// Chapter 10.2.
//
// Once k and y are computed, we then deduce the answer by the sine of sum
// formula:
// sin(x) = sin((k + y)*pi/32)
// = sin(y*pi/32) * cos(k*pi/32) + cos(y*pi/32) * sin(k*pi/32)
// The values of sin(k*pi/32) and cos(k*pi/32) for k = 0..31 are precomputed
// and stored using a vector of 32 doubles. Sin(y*pi/32) and cos(y*pi/32) are
// computed using degree-7 and degree-6 minimax polynomials generated by
// Sollya respectively.
// |x| <= pi/16
if (LIBC_UNLIKELY(x_abs <= 0x3e49'0fdbU)) {
// |x| < 0x1.d12ed2p-12f
if (LIBC_UNLIKELY(x_abs < 0x39e8'9769U)) {
if (LIBC_UNLIKELY(x_abs == 0U)) {
// For signed zeros.
return x;
}
// When |x| < 2^-12, the relative error of the approximation sin(x) ~ x
// is:
// |sin(x) - x| / |sin(x)| < |x^3| / (6|x|)
// = x^2 / 6
// < 2^-25
// < epsilon(1)/2.
// So the correctly rounded values of sin(x) are:
// = x - sign(x)*eps(x) if rounding mode = FE_TOWARDZERO,
// or (rounding mode = FE_UPWARD and x is
// negative),
// = x otherwise.
// To simplify the rounding decision and make it more efficient, we use
// fma(x, -2^-25, x) instead.
// An exhaustive test shows that this formula work correctly for all
// rounding modes up to |x| < 0x1.c555dep-11f.
// Note: to use the formula x - 2^-25*x to decide the correct rounding, we
// do need fma(x, -2^-25, x) to prevent underflow caused by -2^-25*x when
// |x| < 2^-125. For targets without FMA instructions, we simply use
// double for intermediate results as it is more efficient than using an
// emulated version of FMA.
#if defined(LIBC_TARGET_CPU_HAS_FMA)
return fputil::multiply_add(x, -0x1.0p-25f, x);
#else
return static_cast<float>(fputil::multiply_add(xd, -0x1.0p-25, xd));
#endif // LIBC_TARGET_CPU_HAS_FMA
}
// |x| < pi/16.
double xsq = xd * xd;
// Degree-9 polynomial approximation:
// sin(x) ~ x + a_3 x^3 + a_5 x^5 + a_7 x^7 + a_9 x^9
// = x (1 + a_3 x^2 + ... + a_9 x^8)
// = x * P(x^2)
// generated by Sollya with the following commands:
// > display = hexadecimal;
// > Q = fpminimax(sin(x)/x, [|0, 2, 4, 6, 8|], [|1, D...|], [0, pi/16]);
double result =
fputil::polyeval(xsq, 1.0, -0x1.55555555554c6p-3, 0x1.1111111085e65p-7,
-0x1.a019f70fb4d4fp-13, 0x1.718d179815e74p-19);
return static_cast<float>(xd * result);
}
if (LIBC_UNLIKELY(x_abs == 0x4619'9998U)) { // x = 0x1.33333p13
float r = -0x1.63f4bap-2f;
int rounding = fputil::quick_get_round();
bool sign = xbits.get_sign();
if ((rounding == FE_DOWNWARD && !sign) || (rounding == FE_UPWARD && sign))
r = -0x1.63f4bcp-2f;
return xbits.get_sign() ? -r : r;
}
if (LIBC_UNLIKELY(x_abs >= 0x7f80'0000U)) {
if (x_abs == 0x7f80'0000U) {
fputil::set_errno_if_required(EDOM);
fputil::raise_except_if_required(FE_INVALID);
}
return x + FPBits::build_quiet_nan(0);
}
// Combine the results with the sine of sum formula:
// sin(x) = sin((k + y)*pi/32)
// = sin(y*pi/32) * cos(k*pi/32) + cos(y*pi/32) * sin(k*pi/32)
// = sin_y * cos_k + (1 + cosm1_y) * sin_k
// = sin_y * cos_k + (cosm1_y * sin_k + sin_k)
double sin_k, cos_k, sin_y, cosm1_y;
sincosf_eval(xd, x_abs, sin_k, cos_k, sin_y, cosm1_y);
return static_cast<float>(fputil::multiply_add(
sin_y, cos_k, fputil::multiply_add(cosm1_y, sin_k, sin_k)));
}
} // namespace __llvm_libc
|
