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
|
//===-- Single-precision cos 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/cosf.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/except_value_utils.h"
#include "src/__support/FPUtil/multiply_add.h"
#include "src/__support/common.h"
#include "src/__support/macros/config.h"
#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY
#include "src/__support/macros/properties/cpu_features.h" // LIBC_TARGET_CPU_HAS_FMA
namespace LIBC_NAMESPACE_DECL {
#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
// Exceptional cases for cosf.
static constexpr size_t N_EXCEPTS = 6;
static constexpr fputil::ExceptValues<float, N_EXCEPTS> COSF_EXCEPTS{{
// (inputs, RZ output, RU offset, RD offset, RN offset)
// x = 0x1.64a032p43, cos(x) = 0x1.9d4ba4p-1 (RZ)
{0x55325019, 0x3f4ea5d2, 1, 0, 0},
// x = 0x1.4555p51, cos(x) = 0x1.115d7cp-1 (RZ)
{0x5922aa80, 0x3f08aebe, 1, 0, 1},
// x = 0x1.48a858p54, cos(x) = 0x1.f48148p-2 (RZ)
{0x5aa4542c, 0x3efa40a4, 1, 0, 0},
// x = 0x1.3170fp63, cos(x) = 0x1.fe2976p-1 (RZ)
{0x5f18b878, 0x3f7f14bb, 1, 0, 0},
// x = 0x1.2b9622p67, cos(x) = 0x1.f0285cp-1 (RZ)
{0x6115cb11, 0x3f78142e, 1, 0, 1},
// x = 0x1.ddebdep120, cos(x) = 0x1.114438p-1 (RZ)
{0x7beef5ef, 0x3f08a21c, 1, 0, 0},
}};
#endif // !LIBC_MATH_HAS_SKIP_ACCURATE_PASS
LLVM_LIBC_FUNCTION(float, cosf, (float x)) {
using FPBits = typename fputil::FPBits<float>;
FPBits xbits(x);
xbits.set_sign(Sign::POS);
uint32_t x_abs = xbits.uintval();
double xd = static_cast<double>(xbits.get_val());
// Range reduction:
// For |x| > pi/16, 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 16/pi such that the
// least significant bits of their full products with x are larger than 63,
// since cos((k + y + 64*i) * pi/32) = cos(x + i * 2pi) = cos(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 cosine of sum
// formula:
// cos(x) = cos((k + y)*pi/32)
// = cos(y*pi/32) * cos(k*pi/32) - sin(y*pi/32) * sin(k*pi/32)
// The values of sin(k*pi/32) and cos(k*pi/32) for k = 0..63 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| < 0x1.0p-12f
if (LIBC_UNLIKELY(x_abs < 0x3980'0000U)) {
// When |x| < 2^-12, the relative error of the approximation cos(x) ~ 1
// is:
// |cos(x) - 1| < |x^2 / 2| = 2^-25 < epsilon(1)/2.
// So the correctly rounded values of cos(x) are:
// = 1 - eps(x) if rounding mode = FE_TOWARDZERO or FE_DOWWARD,
// = 1 otherwise.
// To simplify the rounding decision and make it more efficient and to
// prevent compiler to perform constant folding, we use
// fma(x, -2^-25, 1) instead.
// Note: to use the formula 1 - 2^-25*x to decide the correct rounding, we
// do need fma(x, -2^-25, 1) 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_FLOAT)
return fputil::multiply_add(xbits.get_val(), -0x1.0p-25f, 1.0f);
#else
return static_cast<float>(fputil::multiply_add(xd, -0x1.0p-25, 1.0));
#endif // LIBC_TARGET_CPU_HAS_FMA_FLOAT
}
#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
if (auto r = COSF_EXCEPTS.lookup(x_abs); LIBC_UNLIKELY(r.has_value()))
return r.value();
#endif // !LIBC_MATH_HAS_SKIP_ACCURATE_PASS
// x is inf or nan.
if (LIBC_UNLIKELY(x_abs >= 0x7f80'0000U)) {
if (xbits.is_signaling_nan()) {
fputil::raise_except_if_required(FE_INVALID);
return FPBits::quiet_nan().get_val();
}
if (x_abs == 0x7f80'0000U) {
fputil::set_errno_if_required(EDOM);
fputil::raise_except_if_required(FE_INVALID);
}
return x + FPBits::quiet_nan().get_val();
}
// Combine the results with the sine of sum formula:
// cos(x) = cos((k + y)*pi/32)
// = cos(y*pi/32) * cos(k*pi/32) - sin(y*pi/32) * sin(k*pi/32)
// = cosm1_y * cos_k + sin_y * sin_k
// = (cosm1_y * cos_k + cos_k) + sin_y * 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, -sin_k, fputil::multiply_add(cosm1_y, cos_k, cos_k)));
}
} // namespace LIBC_NAMESPACE_DECL
|
