1/* Quad-precision floating point sine and cosine on <-pi/4,pi/4>.
2 Copyright (C) 1999-2020 Free Software Foundation, Inc.
3 This file is part of the GNU C Library.
4 Contributed by Jakub Jelinek <jj@ultra.linux.cz>
5
6 The GNU C Library is free software; you can redistribute it and/or
7 modify it under the terms of the GNU Lesser General Public
8 License as published by the Free Software Foundation; either
9 version 2.1 of the License, or (at your option) any later version.
10
11 The GNU C Library is distributed in the hope that it will be useful,
12 but WITHOUT ANY WARRANTY; without even the implied warranty of
13 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
14 Lesser General Public License for more details.
15
16 You should have received a copy of the GNU Lesser General Public
17 License along with the GNU C Library; if not, see
18 <https://www.gnu.org/licenses/>. */
19
20#include <float.h>
21#include <math.h>
22#include <math_private.h>
23#include <math-underflow.h>
24
25static const _Float128 c[] = {
26#define ONE c[0]
27 L(1.00000000000000000000000000000000000E+00), /* 3fff0000000000000000000000000000 */
28
29/* cos x ~ ONE + x^2 ( SCOS1 + SCOS2 * x^2 + ... + SCOS4 * x^6 + SCOS5 * x^8 )
30 x in <0,1/256> */
31#define SCOS1 c[1]
32#define SCOS2 c[2]
33#define SCOS3 c[3]
34#define SCOS4 c[4]
35#define SCOS5 c[5]
36L(-5.00000000000000000000000000000000000E-01), /* bffe0000000000000000000000000000 */
37 L(4.16666666666666666666666666556146073E-02), /* 3ffa5555555555555555555555395023 */
38L(-1.38888888888888888888309442601939728E-03), /* bff56c16c16c16c16c16a566e42c0375 */
39 L(2.48015873015862382987049502531095061E-05), /* 3fefa01a01a019ee02dcf7da2d6d5444 */
40L(-2.75573112601362126593516899592158083E-07), /* bfe927e4f5dce637cb0b54908754bde0 */
41
42/* cos x ~ ONE + x^2 ( COS1 + COS2 * x^2 + ... + COS7 * x^12 + COS8 * x^14 )
43 x in <0,0.1484375> */
44#define COS1 c[6]
45#define COS2 c[7]
46#define COS3 c[8]
47#define COS4 c[9]
48#define COS5 c[10]
49#define COS6 c[11]
50#define COS7 c[12]
51#define COS8 c[13]
52L(-4.99999999999999999999999999999999759E-01), /* bffdfffffffffffffffffffffffffffb */
53 L(4.16666666666666666666666666651287795E-02), /* 3ffa5555555555555555555555516f30 */
54L(-1.38888888888888888888888742314300284E-03), /* bff56c16c16c16c16c16c16a463dfd0d */
55 L(2.48015873015873015867694002851118210E-05), /* 3fefa01a01a01a01a0195cebe6f3d3a5 */
56L(-2.75573192239858811636614709689300351E-07), /* bfe927e4fb7789f5aa8142a22044b51f */
57 L(2.08767569877762248667431926878073669E-09), /* 3fe21eed8eff881d1e9262d7adff4373 */
58L(-1.14707451049343817400420280514614892E-11), /* bfda9397496922a9601ed3d4ca48944b */
59 L(4.77810092804389587579843296923533297E-14), /* 3fd2ae5f8197cbcdcaf7c3fb4523414c */
60
61/* sin x ~ ONE * x + x^3 ( SSIN1 + SSIN2 * x^2 + ... + SSIN4 * x^6 + SSIN5 * x^8 )
62 x in <0,1/256> */
63#define SSIN1 c[14]
64#define SSIN2 c[15]
65#define SSIN3 c[16]
66#define SSIN4 c[17]
67#define SSIN5 c[18]
68L(-1.66666666666666666666666666666666659E-01), /* bffc5555555555555555555555555555 */
69 L(8.33333333333333333333333333146298442E-03), /* 3ff81111111111111111111110fe195d */
70L(-1.98412698412698412697726277416810661E-04), /* bff2a01a01a01a01a019e7121e080d88 */
71 L(2.75573192239848624174178393552189149E-06), /* 3fec71de3a556c640c6aaa51aa02ab41 */
72L(-2.50521016467996193495359189395805639E-08), /* bfe5ae644ee90c47dc71839de75b2787 */
73
74/* sin x ~ ONE * x + x^3 ( SIN1 + SIN2 * x^2 + ... + SIN7 * x^12 + SIN8 * x^14 )
75 x in <0,0.1484375> */
76#define SIN1 c[19]
77#define SIN2 c[20]
78#define SIN3 c[21]
79#define SIN4 c[22]
80#define SIN5 c[23]
81#define SIN6 c[24]
82#define SIN7 c[25]
83#define SIN8 c[26]
84L(-1.66666666666666666666666666666666538e-01), /* bffc5555555555555555555555555550 */
85 L(8.33333333333333333333333333307532934e-03), /* 3ff811111111111111111111110e7340 */
86L(-1.98412698412698412698412534478712057e-04), /* bff2a01a01a01a01a01a019e7a626296 */
87 L(2.75573192239858906520896496653095890e-06), /* 3fec71de3a556c7338fa38527474b8f5 */
88L(-2.50521083854417116999224301266655662e-08), /* bfe5ae64567f544e16c7de65c2ea551f */
89 L(1.60590438367608957516841576404938118e-10), /* 3fde6124613a811480538a9a41957115 */
90L(-7.64716343504264506714019494041582610e-13), /* bfd6ae7f3d5aef30c7bc660b060ef365 */
91 L(2.81068754939739570236322404393398135e-15), /* 3fce9510115aabf87aceb2022a9a9180 */
92};
93
94#define SINCOSL_COS_HI 0
95#define SINCOSL_COS_LO 1
96#define SINCOSL_SIN_HI 2
97#define SINCOSL_SIN_LO 3
98extern const _Float128 __sincosl_table[];
99
100void
101__kernel_sincosl(_Float128 x, _Float128 y, _Float128 *sinx, _Float128 *cosx, int iy)
102{
103 _Float128 h, l, z, sin_l, cos_l_m1;
104 int64_t ix;
105 uint32_t tix, hix, index;
106 GET_LDOUBLE_MSW64 (ix, x);
107 tix = ((uint64_t)ix) >> 32;
108 tix &= ~0x80000000; /* tix = |x|'s high 32 bits */
109 if (tix < 0x3ffc3000) /* |x| < 0.1484375 */
110 {
111 /* Argument is small enough to approximate it by a Chebyshev
112 polynomial of degree 16(17). */
113 if (tix < 0x3fc60000) /* |x| < 2^-57 */
114 {
115 math_check_force_underflow (x);
116 if (!((int)x)) /* generate inexact */
117 {
118 *sinx = x;
119 *cosx = ONE;
120 return;
121 }
122 }
123 z = x * x;
124 *sinx = x + (x * (z*(SIN1+z*(SIN2+z*(SIN3+z*(SIN4+
125 z*(SIN5+z*(SIN6+z*(SIN7+z*SIN8)))))))));
126 *cosx = ONE + (z*(COS1+z*(COS2+z*(COS3+z*(COS4+
127 z*(COS5+z*(COS6+z*(COS7+z*COS8))))))));
128 }
129 else
130 {
131 /* So that we don't have to use too large polynomial, we find
132 l and h such that x = l + h, where fabsl(l) <= 1.0/256 with 83
133 possible values for h. We look up cosl(h) and sinl(h) in
134 pre-computed tables, compute cosl(l) and sinl(l) using a
135 Chebyshev polynomial of degree 10(11) and compute
136 sinl(h+l) = sinl(h)cosl(l) + cosl(h)sinl(l) and
137 cosl(h+l) = cosl(h)cosl(l) - sinl(h)sinl(l). */
138 index = 0x3ffe - (tix >> 16);
139 hix = (tix + (0x200 << index)) & (0xfffffc00 << index);
140 if (signbit (x))
141 {
142 x = -x;
143 y = -y;
144 }
145 switch (index)
146 {
147 case 0: index = ((45 << 10) + hix - 0x3ffe0000) >> 8; break;
148 case 1: index = ((13 << 11) + hix - 0x3ffd0000) >> 9; break;
149 default:
150 case 2: index = (hix - 0x3ffc3000) >> 10; break;
151 }
152
153 SET_LDOUBLE_WORDS64(h, ((uint64_t)hix) << 32, 0);
154 if (iy)
155 l = y - (h - x);
156 else
157 l = x - h;
158 z = l * l;
159 sin_l = l*(ONE+z*(SSIN1+z*(SSIN2+z*(SSIN3+z*(SSIN4+z*SSIN5)))));
160 cos_l_m1 = z*(SCOS1+z*(SCOS2+z*(SCOS3+z*(SCOS4+z*SCOS5))));
161 z = __sincosl_table [index + SINCOSL_SIN_HI]
162 + (__sincosl_table [index + SINCOSL_SIN_LO]
163 + (__sincosl_table [index + SINCOSL_SIN_HI] * cos_l_m1)
164 + (__sincosl_table [index + SINCOSL_COS_HI] * sin_l));
165 *sinx = (ix < 0) ? -z : z;
166 *cosx = __sincosl_table [index + SINCOSL_COS_HI]
167 + (__sincosl_table [index + SINCOSL_COS_LO]
168 - (__sincosl_table [index + SINCOSL_SIN_HI] * sin_l
169 - __sincosl_table [index + SINCOSL_COS_HI] * cos_l_m1));
170 }
171}
172