1//===-- Implementation header for cosf --------------------------*- C++ -*-===//
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_COSF_H
10#define LLVM_LIBC_SRC___SUPPORT_MATH_COSF_H
11
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/macros/config.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#if defined(LIBC_MATH_HAS_SKIP_ACCURATE_PASS) && \
21 defined(LIBC_MATH_HAS_INTERMEDIATE_COMP_IN_FLOAT) && \
22 defined(LIBC_TARGET_CPU_HAS_FMA_FLOAT)
23
24#include "sincosf_float_eval.h"
25
26namespace LIBC_NAMESPACE_DECL {
27namespace math {
28
29LIBC_INLINE float cosf(float x) {
30 return sincosf_float_eval::sincosf_eval</*IS_SIN*/ false>(x);
31}
32
33} // namespace math
34} // namespace LIBC_NAMESPACE_DECL
35
36#else // !LIBC_MATH_HAS_INTERMEDIATE_COMP_IN_FLOAT
37
38#include "sincosf_utils.h"
39
40namespace LIBC_NAMESPACE_DECL {
41
42namespace math {
43
44LIBC_INLINE float cosf(float x) {
45 using namespace sincosf_utils_internal;
46
47#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
48 // Exceptional cases for cosf.
49 constexpr size_t N_EXCEPTS = 6;
50
51 constexpr fputil::ExceptValues<float, N_EXCEPTS> COSF_EXCEPTS{.values: {
52 // (inputs, RZ output, RU offset, RD offset, RN offset)
53 // x = 0x1.64a032p43, cos(x) = 0x1.9d4ba4p-1 (RZ)
54 {.input: 0x55325019, .rnd_towardzero_result: 0x3f4ea5d2, .rnd_upward_offset: 1, .rnd_downward_offset: 0, .rnd_tonearest_offset: 0},
55 // x = 0x1.4555p51, cos(x) = 0x1.115d7cp-1 (RZ)
56 {.input: 0x5922aa80, .rnd_towardzero_result: 0x3f08aebe, .rnd_upward_offset: 1, .rnd_downward_offset: 0, .rnd_tonearest_offset: 1},
57 // x = 0x1.48a858p54, cos(x) = 0x1.f48148p-2 (RZ)
58 {.input: 0x5aa4542c, .rnd_towardzero_result: 0x3efa40a4, .rnd_upward_offset: 1, .rnd_downward_offset: 0, .rnd_tonearest_offset: 0},
59 // x = 0x1.3170fp63, cos(x) = 0x1.fe2976p-1 (RZ)
60 {.input: 0x5f18b878, .rnd_towardzero_result: 0x3f7f14bb, .rnd_upward_offset: 1, .rnd_downward_offset: 0, .rnd_tonearest_offset: 0},
61 // x = 0x1.2b9622p67, cos(x) = 0x1.f0285cp-1 (RZ)
62 {.input: 0x6115cb11, .rnd_towardzero_result: 0x3f78142e, .rnd_upward_offset: 1, .rnd_downward_offset: 0, .rnd_tonearest_offset: 1},
63 // x = 0x1.ddebdep120, cos(x) = 0x1.114438p-1 (RZ)
64 {.input: 0x7beef5ef, .rnd_towardzero_result: 0x3f08a21c, .rnd_upward_offset: 1, .rnd_downward_offset: 0, .rnd_tonearest_offset: 0},
65 }};
66#endif // !LIBC_MATH_HAS_SKIP_ACCURATE_PASS
67
68 using FPBits = typename fputil::FPBits<float>;
69
70 FPBits xbits(x);
71 xbits.set_sign(Sign::POS);
72
73 uint32_t x_abs = xbits.uintval();
74
75 // Range reduction:
76 // For |x| > pi/16, we perform range reduction as follows:
77 // Find k and y such that:
78 // x = (k + y) * pi/32
79 // k is an integer
80 // |y| < 0.5
81 // For small range (|x| < 2^45 when FMA instructions are available, 2^22
82 // otherwise), this is done by performing:
83 // k = round(x * 32/pi)
84 // y = x * 32/pi - k
85 // For large range, we will omit all the higher parts of 16/pi such that the
86 // least significant bits of their full products with x are larger than 63,
87 // since cos((k + y + 64*i) * pi/32) = cos(x + i * 2pi) = cos(x).
88 //
89 // When FMA instructions are not available, we store the digits of 32/pi in
90 // chunks of 28-bit precision. This will make sure that the products:
91 // x * THIRTYTWO_OVER_PI_28[i] are all exact.
92 // When FMA instructions are available, we simply store the digits of 32/pi in
93 // chunks of doubles (53-bit of precision).
94 // So when multiplying by the largest values of single precision, the
95 // resulting output should be correct up to 2^(-208 + 128) ~ 2^-80. By the
96 // worst-case analysis of range reduction, |y| >= 2^-38, so this should give
97 // us more than 40 bits of accuracy. For the worst-case estimation of range
98 // reduction, see for instances:
99 // Elementary Functions by J-M. Muller, Chapter 11,
100 // Handbook of Floating-Point Arithmetic by J-M. Muller et. al.,
101 // Chapter 10.2.
102 //
103 // Once k and y are computed, we then deduce the answer by the cosine of sum
104 // formula:
105 // cos(x) = cos((k + y)*pi/32)
106 // = cos(y*pi/32) * cos(k*pi/32) - sin(y*pi/32) * sin(k*pi/32)
107 // The values of sin(k*pi/32) and cos(k*pi/32) for k = 0..63 are precomputed
108 // and stored using a vector of 32 doubles. Sin(y*pi/32) and cos(y*pi/32) are
109 // computed using degree-7 and degree-6 minimax polynomials generated by
110 // Sollya respectively.
111
112#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
113 // |x| < 0x1.0p-12f
114 if (LIBC_UNLIKELY(x_abs < 0x3980'0000U)) {
115 // When |x| < 2^-12, the relative error of the approximation cos(x) ~ 1
116 // is:
117 // |cos(x) - 1| < |x^2 / 2| = 2^-25 < epsilon(1)/2.
118 // So the correctly rounded values of cos(x) are:
119 // = 1 - eps(x) if rounding mode = FE_TOWARDZERO or FE_DOWWARD,
120 // = 1 otherwise.
121 // To simplify the rounding decision and make it more efficient and to
122 // prevent compiler to perform constant folding, we use
123 // fma(x, -2^-25, 1) instead.
124 // Note: to use the formula 1 - 2^-25*x to decide the correct rounding, we
125 // do need fma(x, -2^-25, 1) 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(xbits.get_val(), -0x1.0p-25f, 1.0f);
131#else // !LIBC_TARGET_CPU_HAS_FMA_FLOAT
132 double xd = static_cast<double>(xbits.get_val());
133 return static_cast<float>(fputil::multiply_add(x: xd, y: -0x1.0p-25, z: 1.0));
134#endif // LIBC_TARGET_CPU_HAS_FMA_FLOAT
135 }
136
137 if (auto r = COSF_EXCEPTS.lookup(x_bits: x_abs); LIBC_UNLIKELY(r.has_value()))
138 return r.value();
139#endif // !LIBC_MATH_HAS_SKIP_ACCURATE_PASS
140
141 // x is inf or nan.
142 if (LIBC_UNLIKELY(x_abs >= 0x7f80'0000U)) {
143 if (xbits.is_signaling_nan()) {
144 fputil::raise_except_if_required(FE_INVALID);
145 return FPBits::quiet_nan().get_val();
146 }
147
148 if (x_abs == 0x7f80'0000U) {
149 fputil::set_errno_if_required(EDOM);
150 fputil::raise_except_if_required(FE_INVALID);
151 }
152 return x + FPBits::quiet_nan().get_val();
153 }
154
155 double xd = static_cast<double>(xbits.get_val());
156 // Combine the results with the sine of sum formula:
157 // cos(x) = cos((k + y)*pi/32)
158 // = cos(y*pi/32) * cos(k*pi/32) - sin(y*pi/32) * sin(k*pi/32)
159 // = cosm1_y * cos_k + sin_y * sin_k
160 // = (cosm1_y * cos_k + cos_k) + sin_y * sin_k
161 double sin_k = 0, cos_k = 0, sin_y = 0, cosm1_y = 0;
162
163 sincosf_eval(xd, x_abs, sin_k, cos_k, sin_y, cosm1_y);
164
165 return static_cast<float>(fputil::multiply_add(
166 x: sin_y, y: -sin_k, z: fputil::multiply_add(x: cosm1_y, y: cos_k, z: cos_k)));
167}
168
169} // namespace math
170
171} // namespace LIBC_NAMESPACE_DECL
172
173#endif // LIBC_MATH_HAS_INTERMEDIATE_COMP_IN_FLOAT
174
175#endif // LLVM_LIBC_SRC___SUPPORT_MATH_COSF_H
176