1 /* Quad-precision floating point sine and cosine on <-pi/4,pi/4>.
2 Copyright (C) 1999-2023 Free Software Foundation, Inc.
3 This file is part of the GNU C Library.
4
5 The GNU C Library is free software; you can redistribute it and/or
6 modify it under the terms of the GNU Lesser General Public
7 License as published by the Free Software Foundation; either
8 version 2.1 of the License, or (at your option) any later version.
9
10 The GNU C Library is distributed in the hope that it will be useful,
11 but WITHOUT ANY WARRANTY; without even the implied warranty of
12 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
13 Lesser General Public License for more details.
14
15 You should have received a copy of the GNU Lesser General Public
16 License along with the GNU C Library; if not, see
17 <https://www.gnu.org/licenses/>. */
18
19 #include <float.h>
20 #include <math.h>
21 #include <math_private.h>
22 #include <math-underflow.h>
23
24 static const long double c[] = {
25 #define ONE c[0]
26 1.00000000000000000000000000000000000E+00L, /* 3fff0000000000000000000000000000 */
27
28 /* cos x ~ ONE + x^2 ( SCOS1 + SCOS2 * x^2 + ... + SCOS4 * x^6 + SCOS5 * x^8 )
29 x in <0,1/256> */
30 #define SCOS1 c[1]
31 #define SCOS2 c[2]
32 #define SCOS3 c[3]
33 #define SCOS4 c[4]
34 #define SCOS5 c[5]
35 -5.00000000000000000000000000000000000E-01L, /* bffe0000000000000000000000000000 */
36 4.16666666666666666666666666556146073E-02L, /* 3ffa5555555555555555555555395023 */
37 -1.38888888888888888888309442601939728E-03L, /* bff56c16c16c16c16c16a566e42c0375 */
38 2.48015873015862382987049502531095061E-05L, /* 3fefa01a01a019ee02dcf7da2d6d5444 */
39 -2.75573112601362126593516899592158083E-07L, /* bfe927e4f5dce637cb0b54908754bde0 */
40
41 /* cos x ~ ONE + x^2 ( COS1 + COS2 * x^2 + ... + COS7 * x^12 + COS8 * x^14 )
42 x in <0,0.1484375> */
43 #define COS1 c[6]
44 #define COS2 c[7]
45 #define COS3 c[8]
46 #define COS4 c[9]
47 #define COS5 c[10]
48 #define COS6 c[11]
49 #define COS7 c[12]
50 #define COS8 c[13]
51 -4.99999999999999999999999999999999759E-01L, /* bffdfffffffffffffffffffffffffffb */
52 4.16666666666666666666666666651287795E-02L, /* 3ffa5555555555555555555555516f30 */
53 -1.38888888888888888888888742314300284E-03L, /* bff56c16c16c16c16c16c16a463dfd0d */
54 2.48015873015873015867694002851118210E-05L, /* 3fefa01a01a01a01a0195cebe6f3d3a5 */
55 -2.75573192239858811636614709689300351E-07L, /* bfe927e4fb7789f5aa8142a22044b51f */
56 2.08767569877762248667431926878073669E-09L, /* 3fe21eed8eff881d1e9262d7adff4373 */
57 -1.14707451049343817400420280514614892E-11L, /* bfda9397496922a9601ed3d4ca48944b */
58 4.77810092804389587579843296923533297E-14L, /* 3fd2ae5f8197cbcdcaf7c3fb4523414c */
59
60 /* sin x ~ ONE * x + x^3 ( SSIN1 + SSIN2 * x^2 + ... + SSIN4 * x^6 + SSIN5 * x^8 )
61 x in <0,1/256> */
62 #define SSIN1 c[14]
63 #define SSIN2 c[15]
64 #define SSIN3 c[16]
65 #define SSIN4 c[17]
66 #define SSIN5 c[18]
67 -1.66666666666666666666666666666666659E-01L, /* bffc5555555555555555555555555555 */
68 8.33333333333333333333333333146298442E-03L, /* 3ff81111111111111111111110fe195d */
69 -1.98412698412698412697726277416810661E-04L, /* bff2a01a01a01a01a019e7121e080d88 */
70 2.75573192239848624174178393552189149E-06L, /* 3fec71de3a556c640c6aaa51aa02ab41 */
71 -2.50521016467996193495359189395805639E-08L, /* bfe5ae644ee90c47dc71839de75b2787 */
72
73 /* sin x ~ ONE * x + x^3 ( SIN1 + SIN2 * x^2 + ... + SIN7 * x^12 + SIN8 * x^14 )
74 x in <0,0.1484375> */
75 #define SIN1 c[19]
76 #define SIN2 c[20]
77 #define SIN3 c[21]
78 #define SIN4 c[22]
79 #define SIN5 c[23]
80 #define SIN6 c[24]
81 #define SIN7 c[25]
82 #define SIN8 c[26]
83 -1.66666666666666666666666666666666538e-01L, /* bffc5555555555555555555555555550 */
84 8.33333333333333333333333333307532934e-03L, /* 3ff811111111111111111111110e7340 */
85 -1.98412698412698412698412534478712057e-04L, /* bff2a01a01a01a01a01a019e7a626296 */
86 2.75573192239858906520896496653095890e-06L, /* 3fec71de3a556c7338fa38527474b8f5 */
87 -2.50521083854417116999224301266655662e-08L, /* bfe5ae64567f544e16c7de65c2ea551f */
88 1.60590438367608957516841576404938118e-10L, /* 3fde6124613a811480538a9a41957115 */
89 -7.64716343504264506714019494041582610e-13L, /* bfd6ae7f3d5aef30c7bc660b060ef365 */
90 2.81068754939739570236322404393398135e-15L, /* 3fce9510115aabf87aceb2022a9a9180 */
91 };
92
93 #define SINCOSL_COS_HI 0
94 #define SINCOSL_COS_LO 1
95 #define SINCOSL_SIN_HI 2
96 #define SINCOSL_SIN_LO 3
97 extern const long double __sincosl_table[];
98
99 void
100 __kernel_sincosl(long double x, long double y, long double *sinx, long double *cosx, int iy)
101 {
102 long double h, l, z, sin_l, cos_l_m1;
103 int64_t ix;
104 uint32_t tix, hix, index;
105 double xhi, hhi;
106
107 xhi = ldbl_high (x);
108 EXTRACT_WORDS64 (ix, xhi);
109 tix = ((uint64_t)ix) >> 32;
110 tix &= ~0x80000000; /* tix = |x|'s high 32 bits */
111 if (tix < 0x3fc30000) /* |x| < 0.1484375 */
112 {
113 /* Argument is small enough to approximate it by a Chebyshev
114 polynomial of degree 16(17). */
115 if (tix < 0x3c600000) /* |x| < 2^-57 */
116 {
117 math_check_force_underflow (x);
118 if (!((int)x)) /* generate inexact */
119 {
120 *sinx = x;
121 *cosx = ONE;
122 return;
123 }
124 }
125 z = x * x;
126 *sinx = x + (x * (z*(SIN1+z*(SIN2+z*(SIN3+z*(SIN4+
127 z*(SIN5+z*(SIN6+z*(SIN7+z*SIN8)))))))));
128 *cosx = ONE + (z*(COS1+z*(COS2+z*(COS3+z*(COS4+
129 z*(COS5+z*(COS6+z*(COS7+z*COS8))))))));
130 }
131 else
132 {
133 /* So that we don't have to use too large polynomial, we find
134 l and h such that x = l + h, where fabsl(l) <= 1.0/256 with 83
135 possible values for h. We look up cosl(h) and sinl(h) in
136 pre-computed tables, compute cosl(l) and sinl(l) using a
137 Chebyshev polynomial of degree 10(11) and compute
138 sinl(h+l) = sinl(h)cosl(l) + cosl(h)sinl(l) and
139 cosl(h+l) = cosl(h)cosl(l) - sinl(h)sinl(l). */
140 int six = tix;
141 tix = ((six - 0x3ff00000) >> 4) + 0x3fff0000;
142 index = 0x3ffe - (tix >> 16);
143 hix = (tix + (0x200 << index)) & (0xfffffc00 << index);
144 x = fabsl (x);
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 hix = (hix << 4) & 0x3fffffff;
153 /*
154 The following should work for double but generates the wrong index.
155 For now the code above converts double to ieee extended to compute
156 the index back to double for the h value.
157
158
159 index = 0x3fe - (tix >> 20);
160 hix = (tix + (0x2000 << index)) & (0xffffc000 << index);
161 if (signbit (x))
162 {
163 x = -x;
164 y = -y;
165 }
166 switch (index)
167 {
168 case 0: index = ((45 << 14) + hix - 0x3fe00000) >> 12; break;
169 case 1: index = ((13 << 15) + hix - 0x3fd00000) >> 13; break;
170 default:
171 case 2: index = (hix - 0x3fc30000) >> 14; break;
172 }
173 */
174 INSERT_WORDS64 (hhi, ((uint64_t)hix) << 32);
175 h = hhi;
176 if (iy)
177 l = y - (h - x);
178 else
179 l = x - h;
180 z = l * l;
181 sin_l = l*(ONE+z*(SSIN1+z*(SSIN2+z*(SSIN3+z*(SSIN4+z*SSIN5)))));
182 cos_l_m1 = z*(SCOS1+z*(SCOS2+z*(SCOS3+z*(SCOS4+z*SCOS5))));
183 z = __sincosl_table [index + SINCOSL_SIN_HI]
184 + (__sincosl_table [index + SINCOSL_SIN_LO]
185 + (__sincosl_table [index + SINCOSL_SIN_HI] * cos_l_m1)
186 + (__sincosl_table [index + SINCOSL_COS_HI] * sin_l));
187 *sinx = (ix < 0) ? -z : z;
188 *cosx = __sincosl_table [index + SINCOSL_COS_HI]
189 + (__sincosl_table [index + SINCOSL_COS_LO]
190 - (__sincosl_table [index + SINCOSL_SIN_HI] * sin_l
191 - __sincosl_table [index + SINCOSL_COS_HI] * cos_l_m1));
192 }
193 }