1*181254a7Smrg /* Quad-precision floating point sine and cosine on <-pi/4,pi/4>.
2*181254a7Smrg Copyright (C) 1999-2018 Free Software Foundation, Inc.
3*181254a7Smrg This file is part of the GNU C Library.
4*181254a7Smrg Contributed by Jakub Jelinek <jj@ultra.linux.cz>
5*181254a7Smrg
6*181254a7Smrg The GNU C Library is free software; you can redistribute it and/or
7*181254a7Smrg modify it under the terms of the GNU Lesser General Public
8*181254a7Smrg License as published by the Free Software Foundation; either
9*181254a7Smrg version 2.1 of the License, or (at your option) any later version.
10*181254a7Smrg
11*181254a7Smrg The GNU C Library is distributed in the hope that it will be useful,
12*181254a7Smrg but WITHOUT ANY WARRANTY; without even the implied warranty of
13*181254a7Smrg MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
14*181254a7Smrg Lesser General Public License for more details.
15*181254a7Smrg
16*181254a7Smrg You should have received a copy of the GNU Lesser General Public
17*181254a7Smrg License along with the GNU C Library; if not, see
18*181254a7Smrg <http://www.gnu.org/licenses/>. */
19*181254a7Smrg
20*181254a7Smrg #include "quadmath-imp.h"
21*181254a7Smrg
22*181254a7Smrg static const __float128 c[] = {
23*181254a7Smrg #define ONE c[0]
24*181254a7Smrg 1.00000000000000000000000000000000000E+00Q, /* 3fff0000000000000000000000000000 */
25*181254a7Smrg
26*181254a7Smrg /* cos x ~ ONE + x^2 ( SCOS1 + SCOS2 * x^2 + ... + SCOS4 * x^6 + SCOS5 * x^8 )
27*181254a7Smrg x in <0,1/256> */
28*181254a7Smrg #define SCOS1 c[1]
29*181254a7Smrg #define SCOS2 c[2]
30*181254a7Smrg #define SCOS3 c[3]
31*181254a7Smrg #define SCOS4 c[4]
32*181254a7Smrg #define SCOS5 c[5]
33*181254a7Smrg -5.00000000000000000000000000000000000E-01Q, /* bffe0000000000000000000000000000 */
34*181254a7Smrg 4.16666666666666666666666666556146073E-02Q, /* 3ffa5555555555555555555555395023 */
35*181254a7Smrg -1.38888888888888888888309442601939728E-03Q, /* bff56c16c16c16c16c16a566e42c0375 */
36*181254a7Smrg 2.48015873015862382987049502531095061E-05Q, /* 3fefa01a01a019ee02dcf7da2d6d5444 */
37*181254a7Smrg -2.75573112601362126593516899592158083E-07Q, /* bfe927e4f5dce637cb0b54908754bde0 */
38*181254a7Smrg
39*181254a7Smrg /* cos x ~ ONE + x^2 ( COS1 + COS2 * x^2 + ... + COS7 * x^12 + COS8 * x^14 )
40*181254a7Smrg x in <0,0.1484375> */
41*181254a7Smrg #define COS1 c[6]
42*181254a7Smrg #define COS2 c[7]
43*181254a7Smrg #define COS3 c[8]
44*181254a7Smrg #define COS4 c[9]
45*181254a7Smrg #define COS5 c[10]
46*181254a7Smrg #define COS6 c[11]
47*181254a7Smrg #define COS7 c[12]
48*181254a7Smrg #define COS8 c[13]
49*181254a7Smrg -4.99999999999999999999999999999999759E-01Q, /* bffdfffffffffffffffffffffffffffb */
50*181254a7Smrg 4.16666666666666666666666666651287795E-02Q, /* 3ffa5555555555555555555555516f30 */
51*181254a7Smrg -1.38888888888888888888888742314300284E-03Q, /* bff56c16c16c16c16c16c16a463dfd0d */
52*181254a7Smrg 2.48015873015873015867694002851118210E-05Q, /* 3fefa01a01a01a01a0195cebe6f3d3a5 */
53*181254a7Smrg -2.75573192239858811636614709689300351E-07Q, /* bfe927e4fb7789f5aa8142a22044b51f */
54*181254a7Smrg 2.08767569877762248667431926878073669E-09Q, /* 3fe21eed8eff881d1e9262d7adff4373 */
55*181254a7Smrg -1.14707451049343817400420280514614892E-11Q, /* bfda9397496922a9601ed3d4ca48944b */
56*181254a7Smrg 4.77810092804389587579843296923533297E-14Q, /* 3fd2ae5f8197cbcdcaf7c3fb4523414c */
57*181254a7Smrg
58*181254a7Smrg /* sin x ~ ONE * x + x^3 ( SSIN1 + SSIN2 * x^2 + ... + SSIN4 * x^6 + SSIN5 * x^8 )
59*181254a7Smrg x in <0,1/256> */
60*181254a7Smrg #define SSIN1 c[14]
61*181254a7Smrg #define SSIN2 c[15]
62*181254a7Smrg #define SSIN3 c[16]
63*181254a7Smrg #define SSIN4 c[17]
64*181254a7Smrg #define SSIN5 c[18]
65*181254a7Smrg -1.66666666666666666666666666666666659E-01Q, /* bffc5555555555555555555555555555 */
66*181254a7Smrg 8.33333333333333333333333333146298442E-03Q, /* 3ff81111111111111111111110fe195d */
67*181254a7Smrg -1.98412698412698412697726277416810661E-04Q, /* bff2a01a01a01a01a019e7121e080d88 */
68*181254a7Smrg 2.75573192239848624174178393552189149E-06Q, /* 3fec71de3a556c640c6aaa51aa02ab41 */
69*181254a7Smrg -2.50521016467996193495359189395805639E-08Q, /* bfe5ae644ee90c47dc71839de75b2787 */
70*181254a7Smrg
71*181254a7Smrg /* sin x ~ ONE * x + x^3 ( SIN1 + SIN2 * x^2 + ... + SIN7 * x^12 + SIN8 * x^14 )
72*181254a7Smrg x in <0,0.1484375> */
73*181254a7Smrg #define SIN1 c[19]
74*181254a7Smrg #define SIN2 c[20]
75*181254a7Smrg #define SIN3 c[21]
76*181254a7Smrg #define SIN4 c[22]
77*181254a7Smrg #define SIN5 c[23]
78*181254a7Smrg #define SIN6 c[24]
79*181254a7Smrg #define SIN7 c[25]
80*181254a7Smrg #define SIN8 c[26]
81*181254a7Smrg -1.66666666666666666666666666666666538e-01Q, /* bffc5555555555555555555555555550 */
82*181254a7Smrg 8.33333333333333333333333333307532934e-03Q, /* 3ff811111111111111111111110e7340 */
83*181254a7Smrg -1.98412698412698412698412534478712057e-04Q, /* bff2a01a01a01a01a01a019e7a626296 */
84*181254a7Smrg 2.75573192239858906520896496653095890e-06Q, /* 3fec71de3a556c7338fa38527474b8f5 */
85*181254a7Smrg -2.50521083854417116999224301266655662e-08Q, /* bfe5ae64567f544e16c7de65c2ea551f */
86*181254a7Smrg 1.60590438367608957516841576404938118e-10Q, /* 3fde6124613a811480538a9a41957115 */
87*181254a7Smrg -7.64716343504264506714019494041582610e-13Q, /* bfd6ae7f3d5aef30c7bc660b060ef365 */
88*181254a7Smrg 2.81068754939739570236322404393398135e-15Q, /* 3fce9510115aabf87aceb2022a9a9180 */
89*181254a7Smrg };
90*181254a7Smrg
91*181254a7Smrg #define SINCOSL_COS_HI 0
92*181254a7Smrg #define SINCOSL_COS_LO 1
93*181254a7Smrg #define SINCOSL_SIN_HI 2
94*181254a7Smrg #define SINCOSL_SIN_LO 3
95*181254a7Smrg extern const __float128 __sincosq_table[];
96*181254a7Smrg
97*181254a7Smrg void
__quadmath_kernel_sincosq(__float128 x,__float128 y,__float128 * sinx,__float128 * cosx,int iy)98*181254a7Smrg __quadmath_kernel_sincosq(__float128 x, __float128 y, __float128 *sinx, __float128 *cosx, int iy)
99*181254a7Smrg {
100*181254a7Smrg __float128 h, l, z, sin_l, cos_l_m1;
101*181254a7Smrg int64_t ix;
102*181254a7Smrg uint32_t tix, hix, index;
103*181254a7Smrg GET_FLT128_MSW64 (ix, x);
104*181254a7Smrg tix = ((uint64_t)ix) >> 32;
105*181254a7Smrg tix &= ~0x80000000; /* tix = |x|'s high 32 bits */
106*181254a7Smrg if (tix < 0x3ffc3000) /* |x| < 0.1484375 */
107*181254a7Smrg {
108*181254a7Smrg /* Argument is small enough to approximate it by a Chebyshev
109*181254a7Smrg polynomial of degree 16(17). */
110*181254a7Smrg if (tix < 0x3fc60000) /* |x| < 2^-57 */
111*181254a7Smrg {
112*181254a7Smrg math_check_force_underflow (x);
113*181254a7Smrg if (!((int)x)) /* generate inexact */
114*181254a7Smrg {
115*181254a7Smrg *sinx = x;
116*181254a7Smrg *cosx = ONE;
117*181254a7Smrg return;
118*181254a7Smrg }
119*181254a7Smrg }
120*181254a7Smrg z = x * x;
121*181254a7Smrg *sinx = x + (x * (z*(SIN1+z*(SIN2+z*(SIN3+z*(SIN4+
122*181254a7Smrg z*(SIN5+z*(SIN6+z*(SIN7+z*SIN8)))))))));
123*181254a7Smrg *cosx = ONE + (z*(COS1+z*(COS2+z*(COS3+z*(COS4+
124*181254a7Smrg z*(COS5+z*(COS6+z*(COS7+z*COS8))))))));
125*181254a7Smrg }
126*181254a7Smrg else
127*181254a7Smrg {
128*181254a7Smrg /* So that we don't have to use too large polynomial, we find
129*181254a7Smrg l and h such that x = l + h, where fabsq(l) <= 1.0/256 with 83
130*181254a7Smrg possible values for h. We look up cosq(h) and sinq(h) in
131*181254a7Smrg pre-computed tables, compute cosq(l) and sinq(l) using a
132*181254a7Smrg Chebyshev polynomial of degree 10(11) and compute
133*181254a7Smrg sinq(h+l) = sinq(h)cosq(l) + cosq(h)sinq(l) and
134*181254a7Smrg cosq(h+l) = cosq(h)cosq(l) - sinq(h)sinq(l). */
135*181254a7Smrg index = 0x3ffe - (tix >> 16);
136*181254a7Smrg hix = (tix + (0x200 << index)) & (0xfffffc00 << index);
137*181254a7Smrg if (signbitq (x))
138*181254a7Smrg {
139*181254a7Smrg x = -x;
140*181254a7Smrg y = -y;
141*181254a7Smrg }
142*181254a7Smrg switch (index)
143*181254a7Smrg {
144*181254a7Smrg case 0: index = ((45 << 10) + hix - 0x3ffe0000) >> 8; break;
145*181254a7Smrg case 1: index = ((13 << 11) + hix - 0x3ffd0000) >> 9; break;
146*181254a7Smrg default:
147*181254a7Smrg case 2: index = (hix - 0x3ffc3000) >> 10; break;
148*181254a7Smrg }
149*181254a7Smrg
150*181254a7Smrg SET_FLT128_WORDS64(h, ((uint64_t)hix) << 32, 0);
151*181254a7Smrg if (iy)
152*181254a7Smrg l = y - (h - x);
153*181254a7Smrg else
154*181254a7Smrg l = x - h;
155*181254a7Smrg z = l * l;
156*181254a7Smrg sin_l = l*(ONE+z*(SSIN1+z*(SSIN2+z*(SSIN3+z*(SSIN4+z*SSIN5)))));
157*181254a7Smrg cos_l_m1 = z*(SCOS1+z*(SCOS2+z*(SCOS3+z*(SCOS4+z*SCOS5))));
158*181254a7Smrg z = __sincosq_table [index + SINCOSL_SIN_HI]
159*181254a7Smrg + (__sincosq_table [index + SINCOSL_SIN_LO]
160*181254a7Smrg + (__sincosq_table [index + SINCOSL_SIN_HI] * cos_l_m1)
161*181254a7Smrg + (__sincosq_table [index + SINCOSL_COS_HI] * sin_l));
162*181254a7Smrg *sinx = (ix < 0) ? -z : z;
163*181254a7Smrg *cosx = __sincosq_table [index + SINCOSL_COS_HI]
164*181254a7Smrg + (__sincosq_table [index + SINCOSL_COS_LO]
165*181254a7Smrg - (__sincosq_table [index + SINCOSL_SIN_HI] * sin_l
166*181254a7Smrg - __sincosq_table [index + SINCOSL_COS_HI] * cos_l_m1));
167*181254a7Smrg }
168*181254a7Smrg }
169