1//===-- Implementation header for cos ---------------------------*- 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_COS_H
10#define LLVM_LIBC_SRC___SUPPORT_MATH_COS_H
11
12#include "range_reduction_double_common.h"
13#include "sincos_eval.h"
14#include "src/__support/FPUtil/FEnvImpl.h"
15#include "src/__support/FPUtil/FPBits.h"
16#include "src/__support/FPUtil/double_double.h"
17#include "src/__support/FPUtil/dyadic_float.h"
18#include "src/__support/FPUtil/except_value_utils.h"
19#include "src/__support/macros/config.h"
20#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY
21#include "src/__support/macros/properties/cpu_features.h" // LIBC_TARGET_CPU_HAS_FMA
22
23#ifdef LIBC_TARGET_CPU_HAS_FMA_DOUBLE
24#include "range_reduction_double_fma.h"
25#else
26#include "range_reduction_double_nofma.h"
27#endif // LIBC_TARGET_CPU_HAS_FMA_DOUBLE
28
29namespace LIBC_NAMESPACE_DECL {
30
31namespace math {
32
33#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
34LIBC_INLINE double
35cos_accurate(double x, uint16_t x_e, unsigned k,
36 const range_reduction_double_internal::LargeRangeReduction
37 &range_reduction_large) {
38 using namespace math::range_reduction_double_internal;
39 using FPBits = typename fputil::FPBits<double>;
40
41 DFloat128 u_f128, sin_u, cos_u;
42 if (LIBC_LIKELY(x_e < FPBits::EXP_BIAS + FAST_PASS_EXPONENT))
43 u_f128 = range_reduction_small_f128(x);
44 else
45 u_f128 = range_reduction_large.accurate();
46
47 math::sincos_eval_internal::sincos_eval(u: u_f128, sin_u, cos_u);
48
49 auto get_sin_k = [](unsigned kk) -> DFloat128 {
50 unsigned idx = (kk & 64) ? 64 - (kk & 63) : (kk & 63);
51 DFloat128 ans = SIN_K_PI_OVER_128_F128[idx];
52 if (kk & 128)
53 ans.sign = Sign::NEG;
54 return ans;
55 };
56
57 // -sin(k * pi/128) = sin((k + 128) * pi/128)
58 // cos(k * pi/128) = sin(k * pi/128 + pi/2) = sin((k + 64) * pi/128).
59 DFloat128 msin_k_f128 = get_sin_k(k + 128);
60 DFloat128 cos_k_f128 = get_sin_k(k + 64);
61
62 // cos(x) = cos((k * pi/128 + u)
63 // = cos(u) * cos(k*pi/128) - sin(u) * sin(k*pi/128)
64 DFloat128 r = fputil::quick_add(a: fputil::quick_mul(a: cos_k_f128, b: cos_u),
65 b: fputil::quick_mul(a: msin_k_f128, b: sin_u));
66
67 // TODO: Add assertion if Ziv's accuracy tests fail in debug mode.
68 // https://github.com/llvm/llvm-project/issues/96452.
69
70 return static_cast<double>(r);
71}
72#endif // !LIBC_MATH_HAS_SKIP_ACCURATE_PASS
73
74LIBC_INLINE double cos(double x) {
75 using namespace range_reduction_double_internal;
76 using FPBits = typename fputil::FPBits<double>;
77 FPBits xbits(x);
78
79 uint16_t x_e = xbits.get_biased_exponent();
80
81 DoubleDouble y;
82 unsigned k = 0;
83 LargeRangeReduction range_reduction_large;
84
85 // |x| < 2^16.
86 if (LIBC_LIKELY(x_e < FPBits::EXP_BIAS + FAST_PASS_EXPONENT)) {
87 // |x| < 2^-4
88 if (LIBC_UNLIKELY(x_e < FPBits::EXP_BIAS - 4)) {
89 // |x| < 2^-27
90 if (LIBC_UNLIKELY(x_e < FPBits::EXP_BIAS - 27)) {
91 // Signed zeros.
92 if (LIBC_UNLIKELY(x == 0.0))
93 return 1.0;
94
95 // For |x| < 2^-27, |cos(x) - 1| < |x|^2/2 < 2^-54 = ulp(1 - 2^-53)/2.
96 return fputil::round_result_slightly_down(value_rn: 1.0);
97 }
98 // No range reduction needed.
99
100 // Use degree-8 polynomial approximation:
101 // cos(x) ~ 1 + a1 * x^2 + a2 * x^4 + a3 * x^6 + a4 * x^8
102 // ~ 1 + x^2 * Q(x^2).
103 // > P = fpminimax(cos(x), [|0, 2, 4, 6, 8|], [|1, D...|], [0, 2^-4]);
104 // > dirtyinfnorm(cos(x) - P, [-2^-4, 2^-4]);
105 // 0x1.3cfe...p-70
106 // > P;
107 constexpr double COEFFS[] = {-0x1p-1, 0x1.5555555555262p-5,
108 -0x1.6c16c1508bff1p-10,
109 0x1.a00ffd769159ap-16};
110 double x_sq = x * x;
111 double c0 = fputil::multiply_add(x: x_sq, y: COEFFS[1], z: COEFFS[0]);
112 double c1 = fputil::multiply_add(x: x_sq, y: COEFFS[3], z: COEFFS[2]);
113 double x4 = x_sq * x_sq;
114 double r_lo = fputil::multiply_add(x: x4, y: c1, z: c0) * x_sq;
115
116#ifdef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
117 return 1.0 + r_lo;
118#else
119 // Overall errors <= ulp(x^2/2) + 2^-69.
120 double err = fputil::multiply_add(x: x_sq, y: 0x1.0p-53, z: 0x1.0p-69);
121 double r_lo_u = r_lo + err;
122 double r_lo_l = r_lo - err;
123 double r_upper = 1.0 + r_lo_u;
124 double r_lower = 1.0 + r_lo_l;
125
126 if (LIBC_LIKELY(r_upper == r_lower))
127 return r_upper;
128
129 k = range_reduction_small(x, u&: y);
130 return cos_accurate(x, x_e, k, range_reduction_large);
131#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS
132 } else {
133 // Small range reduction.
134 k = range_reduction_small(x, u&: y);
135 }
136 } else {
137 // Inf or NaN
138 if (LIBC_UNLIKELY(x_e > 2 * FPBits::EXP_BIAS)) {
139 if (xbits.is_signaling_nan()) {
140 fputil::raise_except_if_required(FE_INVALID);
141 return FPBits::quiet_nan().get_val();
142 }
143 // cos(+-Inf) = NaN
144 if (xbits.get_mantissa() == 0) {
145 fputil::set_errno_if_required(EDOM);
146 fputil::raise_except_if_required(FE_INVALID);
147 }
148 return x + FPBits::quiet_nan().get_val();
149 }
150
151 // Large range reduction.
152 k = range_reduction_large.fast(x, u&: y);
153 }
154
155 DoubleDouble sin_y, cos_y;
156
157 [[maybe_unused]] double err =
158 math::sincos_eval_internal::sincos_eval(u: y, sin_u&: sin_y, cos_u&: cos_y);
159
160 // Look up sin(k * pi/128) and cos(k * pi/128)
161#ifdef LIBC_MATH_HAS_SMALL_TABLES
162 // Memory saving versions. Use 65-entry table.
163 auto get_idx_dd = [](unsigned kk) -> DoubleDouble {
164 unsigned idx = (kk & 64) ? 64 - (kk & 63) : (kk & 63);
165 DoubleDouble ans = SIN_K_PI_OVER_128[idx];
166 if (kk & 128) {
167 ans.hi = -ans.hi;
168 ans.lo = -ans.lo;
169 }
170 return ans;
171 };
172 DoubleDouble msin_k = get_idx_dd(k + 128);
173 DoubleDouble cos_k = get_idx_dd(k + 64);
174#else
175 // Fast look up version, but needs 256-entry table.
176 // -sin(k * pi/128) = sin((k + 128) * pi/128)
177 // cos(k * pi/128) = sin(k * pi/128 + pi/2) = sin((k + 64) * pi/128).
178 DoubleDouble msin_k = SIN_K_PI_OVER_128[(k + 128) & 255];
179 DoubleDouble cos_k = SIN_K_PI_OVER_128[(k + 64) & 255];
180#endif // LIBC_MATH_HAS_SMALL_TABLES
181
182 // After range reduction, k = round(x * 128 / pi) and y = x - k * (pi / 128).
183 // So k is an integer and -pi / 256 <= y <= pi / 256.
184 // Then cos(x) = cos((k * pi/128 + y)
185 // = cos(y) * cos(k*pi/128) - sin(y) * sin(k*pi/128)
186 DoubleDouble cos_k_cos_y = fputil::quick_mult(a: cos_y, b: cos_k);
187 DoubleDouble msin_k_sin_y = fputil::quick_mult(a: sin_y, b: msin_k);
188 // When k != 64 mod 128,
189 // |cos( k * pi/128 )| > pi/128 - epsilon > |y| >= |sin(y)|,
190 // and cos(y) > 1 - pi/128. So we can use Fast2Sum for the subtraction:
191 // cos(y) * cos(k*pi/128) - sin(y) * sin(k*pi/128).
192 DoubleDouble rr = fputil::exact_add(a: cos_k_cos_y.hi, b: msin_k_sin_y.hi);
193 rr.lo += msin_k_sin_y.lo + cos_k_cos_y.lo;
194
195#ifdef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
196 return rr.hi + rr.lo;
197#else
198 double rlp = rr.lo + err;
199 double rlm = rr.lo - err;
200
201 double r_upper = rr.hi + rlp; // (rr.lo + ERR);
202 double r_lower = rr.hi + rlm; // (rr.lo - ERR);
203
204 // Ziv's rounding test.
205 if (LIBC_LIKELY(r_upper == r_lower))
206 return r_upper;
207
208 return cos_accurate(x, x_e, k, range_reduction_large);
209#endif // !LIBC_MATH_HAS_SKIP_ACCURATE_PASS
210}
211
212} // namespace math
213
214} // namespace LIBC_NAMESPACE_DECL
215
216#endif // LLVM_LIBC_SRC___SUPPORT_MATH_COS_H
217