1//===-- Single-precision sincos 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/sincosf.h"
10#include "sincosf_utils.h"
11#include "src/__support/FPUtil/FEnvImpl.h"
12#include "src/__support/FPUtil/FPBits.h"
13#include "src/__support/FPUtil/multiply_add.h"
14#include "src/__support/FPUtil/rounding_mode.h"
15#include "src/__support/common.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
20namespace LIBC_NAMESPACE_DECL {
21
22#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
23// Exceptional values
24static constexpr int N_EXCEPTS = 6;
25
26static constexpr uint32_t EXCEPT_INPUTS[N_EXCEPTS] = {
27 0x46199998, // x = 0x1.33333p13 x
28 0x55325019, // x = 0x1.64a032p43 x
29 0x5922aa80, // x = 0x1.4555p51 x
30 0x5f18b878, // x = 0x1.3170fp63 x
31 0x6115cb11, // x = 0x1.2b9622p67 x
32 0x7beef5ef, // x = 0x1.ddebdep120 x
33};
34
35static constexpr uint32_t EXCEPT_OUTPUTS_SIN[N_EXCEPTS][4] = {
36 {0xbeb1fa5d, 0, 1, 0}, // x = 0x1.33333p13, sin(x) = -0x1.63f4bap-2 (RZ)
37 {0xbf171adf, 0, 1, 1}, // x = 0x1.64a032p43, sin(x) = -0x1.2e35bep-1 (RZ)
38 {0xbf587521, 0, 1, 1}, // x = 0x1.4555p51, sin(x) = -0x1.b0ea42p-1 (RZ)
39 {0x3dad60f6, 1, 0, 1}, // x = 0x1.3170fp63, sin(x) = 0x1.5ac1ecp-4 (RZ)
40 {0xbe7cc1e0, 0, 1, 1}, // x = 0x1.2b9622p67, sin(x) = -0x1.f983cp-3 (RZ)
41 {0xbf587d1b, 0, 1, 1}, // x = 0x1.ddebdep120, sin(x) = -0x1.b0fa36p-1 (RZ)
42};
43
44static constexpr uint32_t EXCEPT_OUTPUTS_COS[N_EXCEPTS][4] = {
45 {0xbf70090b, 0, 1, 0}, // x = 0x1.33333p13, cos(x) = -0x1.e01216p-1 (RZ)
46 {0x3f4ea5d2, 1, 0, 0}, // x = 0x1.64a032p43, cos(x) = 0x1.9d4ba4p-1 (RZ)
47 {0x3f08aebe, 1, 0, 1}, // x = 0x1.4555p51, cos(x) = 0x1.115d7cp-1 (RZ)
48 {0x3f7f14bb, 1, 0, 0}, // x = 0x1.3170fp63, cos(x) = 0x1.fe2976p-1 (RZ)
49 {0x3f78142e, 1, 0, 1}, // x = 0x1.2b9622p67, cos(x) = 0x1.f0285cp-1 (RZ)
50 {0x3f08a21c, 1, 0, 0}, // x = 0x1.ddebdep120, cos(x) = 0x1.114438p-1 (RZ)
51};
52#endif // !LIBC_MATH_HAS_SKIP_ACCURATE_PASS
53
54LLVM_LIBC_FUNCTION(void, sincosf, (float x, float *sinp, float *cosp)) {
55 using FPBits = typename fputil::FPBits<float>;
56 FPBits xbits(x);
57
58 uint32_t x_abs = xbits.uintval() & 0x7fff'ffffU;
59 double xd = static_cast<double>(x);
60
61 // Range reduction:
62 // For |x| >= 2^-12, we perform range reduction as follows:
63 // Find k and y such that:
64 // x = (k + y) * pi/32
65 // k is an integer
66 // |y| < 0.5
67 // For small range (|x| < 2^45 when FMA instructions are available, 2^22
68 // otherwise), this is done by performing:
69 // k = round(x * 32/pi)
70 // y = x * 32/pi - k
71 // For large range, we will omit all the higher parts of 32/pi such that the
72 // least significant bits of their full products with x are larger than 63,
73 // since:
74 // sin((k + y + 64*i) * pi/32) = sin(x + i * 2pi) = sin(x), and
75 // cos((k + y + 64*i) * pi/32) = cos(x + i * 2pi) = cos(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 of326/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 and cosine
92 // of sum formulas:
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 // cos(x) = cos((k + y)*pi/32)
96 // = cos(y*pi/32) * cos(k*pi/32) - sin(y*pi/32) * sin(k*pi/32)
97 // The values of sin(k*pi/32) and cos(k*pi/32) for k = 0..63 are precomputed
98 // and stored using a vector of 32 doubles. Sin(y*pi/32) and cos(y*pi/32) are
99 // computed using degree-7 and degree-6 minimax polynomials generated by
100 // Sollya respectively.
101
102 // |x| < 0x1.0p-12f
103 if (LIBC_UNLIKELY(x_abs < 0x3980'0000U)) {
104 if (LIBC_UNLIKELY(x_abs == 0U)) {
105 // For signed zeros.
106 *sinp = x;
107 *cosp = 1.0f;
108 return;
109 }
110 // When |x| < 2^-12, the relative errors of the approximations
111 // sin(x) ~ x, cos(x) ~ 1
112 // are:
113 // |sin(x) - x| / |sin(x)| < |x^3| / (6|x|)
114 // = x^2 / 6
115 // < 2^-25
116 // < epsilon(1)/2.
117 // |cos(x) - 1| < |x^2 / 2| = 2^-25 < epsilon(1)/2.
118 // So the correctly rounded values of sin(x) and cos(x) are:
119 // sin(x) = x - sign(x)*eps(x) if rounding mode = FE_TOWARDZERO,
120 // or (rounding mode = FE_UPWARD and x is
121 // negative),
122 // = x otherwise.
123 // cos(x) = 1 - eps(x) if rounding mode = FE_TOWARDZERO or FE_DOWWARD,
124 // = 1 otherwise.
125 // To simplify the rounding decision and make it more efficient and to
126 // prevent compiler to perform constant folding, we use
127 // sin(x) = fma(x, -2^-25, x),
128 // cos(x) = fma(x*0.5f, -x, 1)
129 // instead.
130 // Note: to use the formula x - 2^-25*x to decide the correct rounding, we
131 // do need fma(x, -2^-25, x) to prevent underflow caused by -2^-25*x when
132 // |x| < 2^-125. For targets without FMA instructions, we simply use
133 // double for intermediate results as it is more efficient than using an
134 // emulated version of FMA.
135#if defined(LIBC_TARGET_CPU_HAS_FMA_FLOAT)
136 *sinp = fputil::multiply_add(x, -0x1.0p-25f, x);
137 *cosp = fputil::multiply_add(FPBits(x_abs).get_val(), -0x1.0p-25f, 1.0f);
138#else
139 *sinp = static_cast<float>(fputil::multiply_add(xd, -0x1.0p-25, xd));
140 *cosp = static_cast<float>(fputil::multiply_add(
141 static_cast<double>(FPBits(x_abs).get_val()), -0x1.0p-25, 1.0));
142#endif // LIBC_TARGET_CPU_HAS_FMA_FLOAT
143 return;
144 }
145
146 // x is inf or nan.
147 if (LIBC_UNLIKELY(x_abs >= 0x7f80'0000U)) {
148 if (xbits.is_signaling_nan()) {
149 fputil::raise_except_if_required(FE_INVALID);
150 *sinp = *cosp = FPBits::quiet_nan().get_val();
151 return;
152 }
153
154 if (x_abs == 0x7f80'0000U) {
155 fputil::set_errno_if_required(EDOM);
156 fputil::raise_except_if_required(FE_INVALID);
157 }
158 *sinp = FPBits::quiet_nan().get_val();
159 *cosp = *sinp;
160 return;
161 }
162
163#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
164 // Check exceptional values.
165 for (int i = 0; i < N_EXCEPTS; ++i) {
166 if (LIBC_UNLIKELY(x_abs == EXCEPT_INPUTS[i])) {
167 uint32_t s = EXCEPT_OUTPUTS_SIN[i][0]; // FE_TOWARDZERO
168 uint32_t c = EXCEPT_OUTPUTS_COS[i][0]; // FE_TOWARDZERO
169 bool x_sign = x < 0;
170 switch (fputil::quick_get_round()) {
171 case FE_UPWARD:
172 s += x_sign ? EXCEPT_OUTPUTS_SIN[i][2] : EXCEPT_OUTPUTS_SIN[i][1];
173 c += EXCEPT_OUTPUTS_COS[i][1];
174 break;
175 case FE_DOWNWARD:
176 s += x_sign ? EXCEPT_OUTPUTS_SIN[i][1] : EXCEPT_OUTPUTS_SIN[i][2];
177 c += EXCEPT_OUTPUTS_COS[i][2];
178 break;
179 case FE_TONEAREST:
180 s += EXCEPT_OUTPUTS_SIN[i][3];
181 c += EXCEPT_OUTPUTS_COS[i][3];
182 break;
183 }
184 *sinp = x_sign ? -FPBits(s).get_val() : FPBits(s).get_val();
185 *cosp = FPBits(c).get_val();
186
187 return;
188 }
189 }
190#endif // !LIBC_MATH_HAS_SKIP_ACCURATE_PASS
191
192 // Combine the results with the sine and cosine of sum formulas:
193 // sin(x) = sin((k + y)*pi/32)
194 // = sin(y*pi/32) * cos(k*pi/32) + cos(y*pi/32) * sin(k*pi/32)
195 // = sin_y * cos_k + (1 + cosm1_y) * sin_k
196 // = sin_y * cos_k + (cosm1_y * sin_k + sin_k)
197 // cos(x) = cos((k + y)*pi/32)
198 // = cos(y*pi/32) * cos(k*pi/32) - sin(y*pi/32) * sin(k*pi/32)
199 // = cosm1_y * cos_k + sin_y * sin_k
200 // = (cosm1_y * cos_k + cos_k) + sin_y * sin_k
201 double sin_k, cos_k, sin_y, cosm1_y;
202
203 sincosf_eval(xd, x_abs, sin_k, cos_k, sin_y, cosm1_y);
204
205 *sinp = static_cast<float>(fputil::multiply_add(
206 sin_y, cos_k, fputil::multiply_add(cosm1_y, sin_k, sin_k)));
207 *cosp = static_cast<float>(fputil::multiply_add(
208 sin_y, -sin_k, fputil::multiply_add(cosm1_y, cos_k, cos_k)));
209}
210
211} // namespace LIBC_NAMESPACE_DECL
212

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