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 _Float128 c[] = {
25 #define ONE c[0]
26 L(1.00000000000000000000000000000000000E+00), /* 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 L(-5.00000000000000000000000000000000000E-01), /* bffe0000000000000000000000000000 */
36 L(4.16666666666666666666666666556146073E-02), /* 3ffa5555555555555555555555395023 */
37 L(-1.38888888888888888888309442601939728E-03), /* bff56c16c16c16c16c16a566e42c0375 */
38 L(2.48015873015862382987049502531095061E-05), /* 3fefa01a01a019ee02dcf7da2d6d5444 */
39 L(-2.75573112601362126593516899592158083E-07), /* 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 L(-4.99999999999999999999999999999999759E-01), /* bffdfffffffffffffffffffffffffffb */
52 L(4.16666666666666666666666666651287795E-02), /* 3ffa5555555555555555555555516f30 */
53 L(-1.38888888888888888888888742314300284E-03), /* bff56c16c16c16c16c16c16a463dfd0d */
54 L(2.48015873015873015867694002851118210E-05), /* 3fefa01a01a01a01a0195cebe6f3d3a5 */
55 L(-2.75573192239858811636614709689300351E-07), /* bfe927e4fb7789f5aa8142a22044b51f */
56 L(2.08767569877762248667431926878073669E-09), /* 3fe21eed8eff881d1e9262d7adff4373 */
57 L(-1.14707451049343817400420280514614892E-11), /* bfda9397496922a9601ed3d4ca48944b */
58 L(4.77810092804389587579843296923533297E-14), /* 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 L(-1.66666666666666666666666666666666659E-01), /* bffc5555555555555555555555555555 */
68 L(8.33333333333333333333333333146298442E-03), /* 3ff81111111111111111111110fe195d */
69 L(-1.98412698412698412697726277416810661E-04), /* bff2a01a01a01a01a019e7121e080d88 */
70 L(2.75573192239848624174178393552189149E-06), /* 3fec71de3a556c640c6aaa51aa02ab41 */
71 L(-2.50521016467996193495359189395805639E-08), /* 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 L(-1.66666666666666666666666666666666538e-01), /* bffc5555555555555555555555555550 */
84 L(8.33333333333333333333333333307532934e-03), /* 3ff811111111111111111111110e7340 */
85 L(-1.98412698412698412698412534478712057e-04), /* bff2a01a01a01a01a01a019e7a626296 */
86 L(2.75573192239858906520896496653095890e-06), /* 3fec71de3a556c7338fa38527474b8f5 */
87 L(-2.50521083854417116999224301266655662e-08), /* bfe5ae64567f544e16c7de65c2ea551f */
88 L(1.60590438367608957516841576404938118e-10), /* 3fde6124613a811480538a9a41957115 */
89 L(-7.64716343504264506714019494041582610e-13), /* bfd6ae7f3d5aef30c7bc660b060ef365 */
90 L(2.81068754939739570236322404393398135e-15), /* 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 _Float128 __sincosl_table[];
98
99 void
100 __kernel_sincosl(_Float128 x, _Float128 y, _Float128 *sinx, _Float128 *cosx, int iy)
101 {
102 _Float128 h, l, z, sin_l, cos_l_m1;
103 int64_t ix;
104 uint32_t tix, hix, index;
105 GET_LDOUBLE_MSW64 (ix, x);
106 tix = ((uint64_t)ix) >> 32;
107 tix &= ~0x80000000; /* tix = |x|'s high 32 bits */
108 if (tix < 0x3ffc3000) /* |x| < 0.1484375 */
109 {
110 /* Argument is small enough to approximate it by a Chebyshev
111 polynomial of degree 16(17). */
112 if (tix < 0x3fc60000) /* |x| < 2^-57 */
113 {
114 math_check_force_underflow (x);
115 if (!((int)x)) /* generate inexact */
116 {
117 *sinx = x;
118 *cosx = ONE;
119 return;
120 }
121 }
122 z = x * x;
123 *sinx = x + (x * (z*(SIN1+z*(SIN2+z*(SIN3+z*(SIN4+
124 z*(SIN5+z*(SIN6+z*(SIN7+z*SIN8)))))))));
125 *cosx = ONE + (z*(COS1+z*(COS2+z*(COS3+z*(COS4+
126 z*(COS5+z*(COS6+z*(COS7+z*COS8))))))));
127 }
128 else
129 {
130 /* So that we don't have to use too large polynomial, we find
131 l and h such that x = l + h, where fabsl(l) <= 1.0/256 with 83
132 possible values for h. We look up cosl(h) and sinl(h) in
133 pre-computed tables, compute cosl(l) and sinl(l) using a
134 Chebyshev polynomial of degree 10(11) and compute
135 sinl(h+l) = sinl(h)cosl(l) + cosl(h)sinl(l) and
136 cosl(h+l) = cosl(h)cosl(l) - sinl(h)sinl(l). */
137 index = 0x3ffe - (tix >> 16);
138 hix = (tix + (0x200 << index)) & (0xfffffc00 << index);
139 if (signbit (x))
140 {
141 x = -x;
142 y = -y;
143 }
144 switch (index)
145 {
146 case 0: index = ((45 << 10) + hix - 0x3ffe0000) >> 8; break;
147 case 1: index = ((13 << 11) + hix - 0x3ffd0000) >> 9; break;
148 default:
149 case 2: index = (hix - 0x3ffc3000) >> 10; break;
150 }
151
152 SET_LDOUBLE_WORDS64(h, ((uint64_t)hix) << 32, 0);
153 if (iy)
154 l = y - (h - x);
155 else
156 l = x - h;
157 z = l * l;
158 sin_l = l*(ONE+z*(SSIN1+z*(SSIN2+z*(SSIN3+z*(SSIN4+z*SSIN5)))));
159 cos_l_m1 = z*(SCOS1+z*(SCOS2+z*(SCOS3+z*(SCOS4+z*SCOS5))));
160 z = __sincosl_table [index + SINCOSL_SIN_HI]
161 + (__sincosl_table [index + SINCOSL_SIN_LO]
162 + (__sincosl_table [index + SINCOSL_SIN_HI] * cos_l_m1)
163 + (__sincosl_table [index + SINCOSL_COS_HI] * sin_l));
164 *sinx = (ix < 0) ? -z : z;
165 *cosx = __sincosl_table [index + SINCOSL_COS_HI]
166 + (__sincosl_table [index + SINCOSL_COS_LO]
167 - (__sincosl_table [index + SINCOSL_SIN_HI] * sin_l
168 - __sincosl_table [index + SINCOSL_COS_HI] * cos_l_m1));
169 }
170 }