1/* Quad-precision floating point cosine on <-pi/4,pi/4>.
2 Copyright (C) 1999-2024 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 <math.h>
20#include <math_private.h>
21
22static const long double c[] = {
23#define ONE c[0]
24 1.00000000000000000000000000000000000E+00L, /* 3fff0000000000000000000000000000 */
25
26/* cos x ~ ONE + x^2 ( SCOS1 + SCOS2 * x^2 + ... + SCOS4 * x^6 + SCOS5 * x^8 )
27 x in <0,1/256> */
28#define SCOS1 c[1]
29#define SCOS2 c[2]
30#define SCOS3 c[3]
31#define SCOS4 c[4]
32#define SCOS5 c[5]
33-5.00000000000000000000000000000000000E-01L, /* bffe0000000000000000000000000000 */
34 4.16666666666666666666666666556146073E-02L, /* 3ffa5555555555555555555555395023 */
35-1.38888888888888888888309442601939728E-03L, /* bff56c16c16c16c16c16a566e42c0375 */
36 2.48015873015862382987049502531095061E-05L, /* 3fefa01a01a019ee02dcf7da2d6d5444 */
37-2.75573112601362126593516899592158083E-07L, /* bfe927e4f5dce637cb0b54908754bde0 */
38
39/* cos x ~ ONE + x^2 ( COS1 + COS2 * x^2 + ... + COS7 * x^12 + COS8 * x^14 )
40 x in <0,0.1484375> */
41#define COS1 c[6]
42#define COS2 c[7]
43#define COS3 c[8]
44#define COS4 c[9]
45#define COS5 c[10]
46#define COS6 c[11]
47#define COS7 c[12]
48#define COS8 c[13]
49-4.99999999999999999999999999999999759E-01L, /* bffdfffffffffffffffffffffffffffb */
50 4.16666666666666666666666666651287795E-02L, /* 3ffa5555555555555555555555516f30 */
51-1.38888888888888888888888742314300284E-03L, /* bff56c16c16c16c16c16c16a463dfd0d */
52 2.48015873015873015867694002851118210E-05L, /* 3fefa01a01a01a01a0195cebe6f3d3a5 */
53-2.75573192239858811636614709689300351E-07L, /* bfe927e4fb7789f5aa8142a22044b51f */
54 2.08767569877762248667431926878073669E-09L, /* 3fe21eed8eff881d1e9262d7adff4373 */
55-1.14707451049343817400420280514614892E-11L, /* bfda9397496922a9601ed3d4ca48944b */
56 4.77810092804389587579843296923533297E-14L, /* 3fd2ae5f8197cbcdcaf7c3fb4523414c */
57
58/* sin x ~ ONE * x + x^3 ( SSIN1 + SSIN2 * x^2 + ... + SSIN4 * x^6 + SSIN5 * x^8 )
59 x in <0,1/256> */
60#define SSIN1 c[14]
61#define SSIN2 c[15]
62#define SSIN3 c[16]
63#define SSIN4 c[17]
64#define SSIN5 c[18]
65-1.66666666666666666666666666666666659E-01L, /* bffc5555555555555555555555555555 */
66 8.33333333333333333333333333146298442E-03L, /* 3ff81111111111111111111110fe195d */
67-1.98412698412698412697726277416810661E-04L, /* bff2a01a01a01a01a019e7121e080d88 */
68 2.75573192239848624174178393552189149E-06L, /* 3fec71de3a556c640c6aaa51aa02ab41 */
69-2.50521016467996193495359189395805639E-08L, /* bfe5ae644ee90c47dc71839de75b2787 */
70};
71
72#define SINCOSL_COS_HI 0
73#define SINCOSL_COS_LO 1
74#define SINCOSL_SIN_HI 2
75#define SINCOSL_SIN_LO 3
76extern const long double __sincosl_table[];
77
78long double
79__kernel_cosl(long double x, long double y)
80{
81 long double h, l, z, sin_l, cos_l_m1;
82 int64_t ix;
83 uint32_t tix, hix, index;
84 double xhi, hhi;
85
86 xhi = ldbl_high (x);
87 EXTRACT_WORDS64 (ix, xhi);
88 tix = ((uint64_t)ix) >> 32;
89 tix &= ~0x80000000; /* tix = |x|'s high 32 bits */
90 if (tix < 0x3fc30000) /* |x| < 0.1484375 */
91 {
92 /* Argument is small enough to approximate it by a Chebyshev
93 polynomial of degree 16. */
94 if (tix < 0x3c600000) /* |x| < 2^-57 */
95 if (!((int)x)) return ONE; /* generate inexact */
96 z = x * x;
97 return ONE + (z*(COS1+z*(COS2+z*(COS3+z*(COS4+
98 z*(COS5+z*(COS6+z*(COS7+z*COS8))))))));
99 }
100 else
101 {
102 /* So that we don't have to use too large polynomial, we find
103 l and h such that x = l + h, where fabsl(l) <= 1.0/256 with 83
104 possible values for h. We look up cosl(h) and sinl(h) in
105 pre-computed tables, compute cosl(l) and sinl(l) using a
106 Chebyshev polynomial of degree 10(11) and compute
107 cosl(h+l) = cosl(h)cosl(l) - sinl(h)sinl(l). */
108 int six = tix;
109 tix = ((six - 0x3ff00000) >> 4) + 0x3fff0000;
110 index = 0x3ffe - (tix >> 16);
111 hix = (tix + (0x200 << index)) & (0xfffffc00 << index);
112 x = fabsl (x: x);
113 switch (index)
114 {
115 case 0: index = ((45 << 10) + hix - 0x3ffe0000) >> 8; break;
116 case 1: index = ((13 << 11) + hix - 0x3ffd0000) >> 9; break;
117 default:
118 case 2: index = (hix - 0x3ffc3000) >> 10; break;
119 }
120 hix = (hix << 4) & 0x3fffffff;
121/*
122 The following should work for double but generates the wrong index.
123 For now the code above converts double to ieee extended to compute
124 the index back to double for the h value.
125
126 index = 0x3fe - (tix >> 20);
127 hix = (tix + (0x200 << index)) & (0xfffffc00 << index);
128 if (signbit (x))
129 {
130 x = -x;
131 y = -y;
132 }
133 switch (index)
134 {
135 case 0: index = ((45 << 14) + hix - 0x3fe00000) >> 12; break;
136 case 1: index = ((13 << 15) + hix - 0x3fd00000) >> 13; break;
137 default:
138 case 2: index = (hix - 0x3fc30000) >> 14; break;
139 }
140*/
141 INSERT_WORDS64 (hhi, ((uint64_t)hix) << 32);
142 h = hhi;
143 l = y - (h - x);
144 z = l * l;
145 sin_l = l*(ONE+z*(SSIN1+z*(SSIN2+z*(SSIN3+z*(SSIN4+z*SSIN5)))));
146 cos_l_m1 = z*(SCOS1+z*(SCOS2+z*(SCOS3+z*(SCOS4+z*SCOS5))));
147 return __sincosl_table [index + SINCOSL_COS_HI]
148 + (__sincosl_table [index + SINCOSL_COS_LO]
149 - (__sincosl_table [index + SINCOSL_SIN_HI] * sin_l
150 - __sincosl_table [index + SINCOSL_COS_HI] * cos_l_m1));
151 }
152}
153

source code of glibc/sysdeps/ieee754/ldbl-128ibm/k_cosl.c