1//===-- Single-precision cos 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/cosf.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/except_value_utils.h"
15#include "src/__support/FPUtil/multiply_add.h"
16#include "src/__support/common.h"
17#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY
18#include "src/__support/macros/properties/cpu_features.h" // LIBC_TARGET_CPU_HAS_FMA
19
20#include <errno.h>
21
22namespace LIBC_NAMESPACE {
23
24// Exceptional cases for cosf.
25static constexpr size_t N_EXCEPTS = 6;
26
27static constexpr fputil::ExceptValues<float, N_EXCEPTS> COSF_EXCEPTS{.values: {
28 // (inputs, RZ output, RU offset, RD offset, RN offset)
29 // x = 0x1.64a032p43, cos(x) = 0x1.9d4ba4p-1 (RZ)
30 {.input: 0x55325019, .rnd_towardzero_result: 0x3f4ea5d2, .rnd_upward_offset: 1, .rnd_downward_offset: 0, .rnd_tonearest_offset: 0},
31 // x = 0x1.4555p51, cos(x) = 0x1.115d7cp-1 (RZ)
32 {.input: 0x5922aa80, .rnd_towardzero_result: 0x3f08aebe, .rnd_upward_offset: 1, .rnd_downward_offset: 0, .rnd_tonearest_offset: 1},
33 // x = 0x1.48a858p54, cos(x) = 0x1.f48148p-2 (RZ)
34 {.input: 0x5aa4542c, .rnd_towardzero_result: 0x3efa40a4, .rnd_upward_offset: 1, .rnd_downward_offset: 0, .rnd_tonearest_offset: 0},
35 // x = 0x1.3170fp63, cos(x) = 0x1.fe2976p-1 (RZ)
36 {.input: 0x5f18b878, .rnd_towardzero_result: 0x3f7f14bb, .rnd_upward_offset: 1, .rnd_downward_offset: 0, .rnd_tonearest_offset: 0},
37 // x = 0x1.2b9622p67, cos(x) = 0x1.f0285cp-1 (RZ)
38 {.input: 0x6115cb11, .rnd_towardzero_result: 0x3f78142e, .rnd_upward_offset: 1, .rnd_downward_offset: 0, .rnd_tonearest_offset: 1},
39 // x = 0x1.ddebdep120, cos(x) = 0x1.114438p-1 (RZ)
40 {.input: 0x7beef5ef, .rnd_towardzero_result: 0x3f08a21c, .rnd_upward_offset: 1, .rnd_downward_offset: 0, .rnd_tonearest_offset: 0},
41}};
42
43LLVM_LIBC_FUNCTION(float, cosf, (float x)) {
44 using FPBits = typename fputil::FPBits<float>;
45
46 FPBits xbits(x);
47 xbits.set_sign(Sign::POS);
48
49 uint32_t x_abs = xbits.uintval();
50 double xd = static_cast<double>(xbits.get_val());
51
52 // Range reduction:
53 // For |x| > pi/16, we perform range reduction as follows:
54 // Find k and y such that:
55 // x = (k + y) * pi/32
56 // k is an integer
57 // |y| < 0.5
58 // For small range (|x| < 2^45 when FMA instructions are available, 2^22
59 // otherwise), this is done by performing:
60 // k = round(x * 32/pi)
61 // y = x * 32/pi - k
62 // For large range, we will omit all the higher parts of 16/pi such that the
63 // least significant bits of their full products with x are larger than 63,
64 // since cos((k + y + 64*i) * pi/32) = cos(x + i * 2pi) = cos(x).
65 //
66 // When FMA instructions are not available, we store the digits of 32/pi in
67 // chunks of 28-bit precision. This will make sure that the products:
68 // x * THIRTYTWO_OVER_PI_28[i] are all exact.
69 // When FMA instructions are available, we simply store the digits of 32/pi in
70 // chunks of doubles (53-bit of precision).
71 // So when multiplying by the largest values of single precision, the
72 // resulting output should be correct up to 2^(-208 + 128) ~ 2^-80. By the
73 // worst-case analysis of range reduction, |y| >= 2^-38, so this should give
74 // us more than 40 bits of accuracy. For the worst-case estimation of range
75 // reduction, see for instances:
76 // Elementary Functions by J-M. Muller, Chapter 11,
77 // Handbook of Floating-Point Arithmetic by J-M. Muller et. al.,
78 // Chapter 10.2.
79 //
80 // Once k and y are computed, we then deduce the answer by the cosine of sum
81 // formula:
82 // cos(x) = cos((k + y)*pi/32)
83 // = cos(y*pi/32) * cos(k*pi/32) - sin(y*pi/32) * sin(k*pi/32)
84 // The values of sin(k*pi/32) and cos(k*pi/32) for k = 0..63 are precomputed
85 // and stored using a vector of 32 doubles. Sin(y*pi/32) and cos(y*pi/32) are
86 // computed using degree-7 and degree-6 minimax polynomials generated by
87 // Sollya respectively.
88
89 // |x| < 0x1.0p-12f
90 if (LIBC_UNLIKELY(x_abs < 0x3980'0000U)) {
91 // When |x| < 2^-12, the relative error of the approximation cos(x) ~ 1
92 // is:
93 // |cos(x) - 1| < |x^2 / 2| = 2^-25 < epsilon(1)/2.
94 // So the correctly rounded values of cos(x) are:
95 // = 1 - eps(x) if rounding mode = FE_TOWARDZERO or FE_DOWWARD,
96 // = 1 otherwise.
97 // To simplify the rounding decision and make it more efficient and to
98 // prevent compiler to perform constant folding, we use
99 // fma(x, -2^-25, 1) instead.
100 // Note: to use the formula 1 - 2^-25*x to decide the correct rounding, we
101 // do need fma(x, -2^-25, 1) to prevent underflow caused by -2^-25*x when
102 // |x| < 2^-125. For targets without FMA instructions, we simply use
103 // double for intermediate results as it is more efficient than using an
104 // emulated version of FMA.
105#if defined(LIBC_TARGET_CPU_HAS_FMA)
106 return fputil::multiply_add(x: xbits.get_val(), y: -0x1.0p-25f, z: 1.0f);
107#else
108 return static_cast<float>(fputil::multiply_add(xd, -0x1.0p-25, 1.0));
109#endif // LIBC_TARGET_CPU_HAS_FMA
110 }
111
112 if (auto r = COSF_EXCEPTS.lookup(x_bits: x_abs); LIBC_UNLIKELY(r.has_value()))
113 return r.value();
114
115 // x is inf or nan.
116 if (LIBC_UNLIKELY(x_abs >= 0x7f80'0000U)) {
117 if (x_abs == 0x7f80'0000U) {
118 fputil::set_errno_if_required(EDOM);
119 fputil::raise_except_if_required(FE_INVALID);
120 }
121 return x + FPBits::quiet_nan().get_val();
122 }
123
124 // Combine the results with the sine of sum formula:
125 // cos(x) = cos((k + y)*pi/32)
126 // = cos(y*pi/32) * cos(k*pi/32) - sin(y*pi/32) * sin(k*pi/32)
127 // = cosm1_y * cos_k + sin_y * sin_k
128 // = (cosm1_y * cos_k + cos_k) + sin_y * sin_k
129 double sin_k, cos_k, sin_y, cosm1_y;
130
131 sincosf_eval(xd, x_abs, sin_k, cos_k, sin_y, cosm1_y);
132
133 return static_cast<float>(fputil::multiply_add(
134 x: sin_y, y: -sin_k, z: fputil::multiply_add(x: cosm1_y, y: cos_k, z: cos_k)));
135}
136
137} // namespace LIBC_NAMESPACE
138

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