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Commit | Line | Data |
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f7eac6eb | 1 | /* e_j1f.c -- float version of e_j1.c. |
f7eac6eb RM |
2 | */ |
3 | ||
4 | /* | |
5 | * ==================================================== | |
6 | * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. | |
7 | * | |
8 | * Developed at SunPro, a Sun Microsystems, Inc. business. | |
9 | * Permission to use, copy, modify, and distribute this | |
10659981 | 10 | * software is freely granted, provided that this notice |
f7eac6eb RM |
11 | * is preserved. |
12 | * ==================================================== | |
13 | */ | |
14 | ||
46a3d3c7 | 15 | #include <errno.h> |
63dbe5f3 | 16 | #include <float.h> |
1ed0291c | 17 | #include <math.h> |
aaee3cd8 | 18 | #include <math-narrow-eval.h> |
1ed0291c | 19 | #include <math_private.h> |
70e2ba33 | 20 | #include <fenv_private.h> |
8f5b00d3 | 21 | #include <math-underflow.h> |
220622dd | 22 | #include <libm-alias-finite.h> |
9acda61d | 23 | #include <reduce_aux.h> |
f7eac6eb | 24 | |
f7eac6eb | 25 | static float ponef(float), qonef(float); |
f7eac6eb | 26 | |
10659981 | 27 | static const float |
f7eac6eb RM |
28 | huge = 1e30, |
29 | one = 1.0, | |
30 | invsqrtpi= 5.6418961287e-01, /* 0x3f106ebb */ | |
31 | tpi = 6.3661974669e-01, /* 0x3f22f983 */ | |
32 | /* R0/S0 on [0,2] */ | |
33 | r00 = -6.2500000000e-02, /* 0xbd800000 */ | |
34 | r01 = 1.4070566976e-03, /* 0x3ab86cfd */ | |
35 | r02 = -1.5995563444e-05, /* 0xb7862e36 */ | |
36 | r03 = 4.9672799207e-08, /* 0x335557d2 */ | |
37 | s01 = 1.9153760746e-02, /* 0x3c9ce859 */ | |
38 | s02 = 1.8594678841e-04, /* 0x3942fab6 */ | |
39 | s03 = 1.1771846857e-06, /* 0x359dffc2 */ | |
40 | s04 = 5.0463624390e-09, /* 0x31ad6446 */ | |
41 | s05 = 1.2354227016e-11; /* 0x2d59567e */ | |
42 | ||
f7eac6eb | 43 | static const float zero = 0.0; |
f7eac6eb | 44 | |
9acda61d PZ |
45 | /* This is the nearest approximation of the first positive zero of j1. */ |
46 | #define FIRST_ZERO_J1 0x3.d4eabp+0f | |
47 | ||
48 | #define SMALL_SIZE 64 | |
49 | ||
50 | /* The following table contains successive zeros of j1 and degree-3 | |
51 | polynomial approximations of j1 around these zeros: Pj[0] for the first | |
52 | positive zero (3.831705), Pj[1] for the second one (7.015586), and so on. | |
53 | Each line contains: | |
54 | {x0, xmid, x1, p0, p1, p2, p3} | |
55 | where [x0,x1] is the interval around the zero, xmid is the binary32 number | |
56 | closest to the zero, and p0+p1*x+p2*x^2+p3*x^3 is the approximation | |
57 | polynomial. Each polynomial was generated using Sollya on the interval | |
58 | [x0,x1] around the corresponding zero where the error exceeds 9 ulps | |
59 | for the alternate code. Degree 3 is enough to get an error at most | |
60 | 9 ulps, except around the first zero. | |
61 | */ | |
62 | static const float Pj[SMALL_SIZE][7] = { | |
63 | /* For index 0, we use a degree-4 polynomial generated by Sollya, with the | |
64 | coefficient of degree 4 hard-coded in j1f_near_root(). */ | |
65 | { 0x1.e09e5ep+1, 0x1.ea7558p+1, 0x1.ef7352p+1, -0x8.4f069p-28, | |
66 | -0x6.71b3d8p-4, 0xd.744a2p-8, 0xd.acd9p-8/*, -0x1.3e51aap-8*/ }, /* 0 */ | |
67 | { 0x1.bdb4c2p+2, 0x1.c0ff6p+2, 0x1.c27a8cp+2, 0xe.c2858p-28, | |
68 | 0x4.cd464p-4, -0x5.79b71p-8, -0xc.11124p-8 }, /* 1 */ | |
69 | { 0x1.43b214p+3, 0x1.458d0ep+3, 0x1.460ccep+3, -0x1.e7acecp-24, | |
70 | -0x3.feca9p-4, 0x3.2470f8p-8, 0xa.625b7p-8 }, /* 2 */ | |
71 | { 0x1.a9c98p+3, 0x1.aa5bbp+3, 0x1.aaa4d8p+3, 0x1.698158p-24, | |
72 | 0x3.7e666cp-4, -0x2.1900ap-8, -0x9.2755p-8 }, /* 3 */ | |
73 | { 0x1.073be4p+4, 0x1.0787b4p+4, 0x1.07aed8p+4, -0x1.f5f658p-24, | |
74 | -0x3.24b8ep-4, 0x1.86e35cp-8, 0x8.4e4bbp-8 }, /* 4 */ | |
75 | { 0x1.39ae2ap+4, 0x1.39da8ep+4, 0x1.39f3dap+4, -0x1.4e744p-24, | |
76 | 0x2.e18a24p-4, -0x1.2ccd16p-8, -0x7.a27ep-8 }, /* 5 */ | |
77 | { 0x1.6bfa46p+4, 0x1.6c294ep+4, 0x1.6c412p+4, 0xa.3fb7fp-28, | |
78 | -0x2.acc9c4p-4, 0xf.0b783p-12, 0x7.1c0d3p-8 }, /* 6 */ | |
79 | { 0x1.9e42bep+4, 0x1.9e757p+4, 0x1.9e876ep+4, -0x2.29f6f4p-24, | |
80 | 0x2.81f21p-4, -0xc.641bp-12, -0x6.a7ea58p-8 }, /* 7 */ | |
81 | { 0x1.d08a3ep+4, 0x1.d0bfdp+4, 0x1.d0cd3cp+4, -0x1.b5d196p-24, | |
82 | -0x2.5e40e4p-4, 0xa.7059fp-12, 0x6.4d6bfp-8 }, /* 8 */ | |
83 | { 0x1.017794p+5, 0x1.018476p+5, 0x1.018b8cp+5, -0x4.0e001p-24, | |
84 | 0x2.3febep-4, -0x8.f23aap-12, -0x6.0102cp-8 }, /* 9 */ | |
85 | { 0x1.1a9e78p+5, 0x1.1aa89p+5, 0x1.1aaf88p+5, 0x3.b26f2p-24, | |
86 | -0x2.25babp-4, 0x7.c6d948p-12, 0x5.a1d988p-8 }, /* 10 */ | |
87 | { 0x1.33bddep+5, 0x1.33cc52p+5, 0x1.33d2e4p+5, -0xf.c8cdap-28, | |
88 | 0x2.0ed05p-4, -0x6.d97dbp-12, -0x5.8da498p-8 }, /* 11 */ | |
89 | { 0x1.4ce7cp+5, 0x1.4cefdp+5, 0x1.4cf7d4p+5, -0x3.9940e4p-24, | |
90 | -0x1.fa8b4p-4, 0x6.16108p-12, 0x5.4355e8p-8 }, /* 12 */ | |
91 | { 0x1.6603e8p+5, 0x1.661316p+5, 0x1.66173ap+5, 0x9.da15dp-28, | |
92 | 0x1.e8727ep-4, -0x5.742468p-12, -0x5.117c28p-8 }, /* 13 */ | |
93 | { 0x1.7f2ebcp+5, 0x1.7f3632p+5, 0x1.7f3a7ep+5, -0x3.39b218p-24, | |
94 | -0x1.d8293ap-4, 0x4.ee3348p-12, 0x4.f9bep-8 }, /* 14 */ | |
95 | { 0x1.9850e6p+5, 0x1.985928p+5, 0x1.985d9ep+5, -0x3.7b5108p-24, | |
96 | 0x1.c96702p-4, -0x4.7b0d08p-12, -0x4.c784a8p-8 }, /* 15 */ | |
97 | { 0x1.b172e8p+5, 0x1.b17c04p+5, 0x1.b1805cp+5, -0x1.91e43ep-24, | |
98 | -0x1.bbf246p-4, 0x4.18ad78p-12, 0x4.9bfae8p-8 }, /* 16 */ | |
99 | { 0x1.ca955ap+5, 0x1.ca9ec6p+5, 0x1.caa2a4p+5, 0x1.28453cp-24, | |
100 | 0x1.af9cb4p-4, -0x3.c3a494p-12, -0x4.78b69p-8 }, /* 17 */ | |
101 | { 0x1.e3bc94p+5, 0x1.e3c174p+5, 0x1.e3c64p+5, -0x2.e7fef4p-24, | |
102 | -0x1.a4407ep-4, 0x3.79b228p-12, 0x4.874f7p-8 }, /* 18 */ | |
103 | { 0x1.fcdf16p+5, 0x1.fce40ep+5, 0x1.fce71p+5, -0x3.23b2fcp-24, | |
104 | 0x1.99be76p-4, -0x3.39ad7cp-12, -0x4.92a55p-8 }, /* 19 */ | |
105 | { 0x1.0afe34p+6, 0x1.0b034ep+6, 0x1.0b054ap+6, -0xd.19e93p-28, | |
106 | -0x1.8ffc9cp-4, 0x2.fee7f8p-12, 0x4.2d33b8p-8 }, /* 20 */ | |
107 | { 0x1.179344p+6, 0x1.17948ep+6, 0x1.1795bp+6, 0x1.c2ac48p-24, | |
108 | 0x1.86e51cp-4, -0x2.cc5abp-12, -0x4.866d08p-8 }, /* 21 */ | |
109 | { 0x1.24231ep+6, 0x1.2425c8p+6, 0x1.2426e2p+6, -0xd.31027p-28, | |
110 | -0x1.7e656ep-4, 0x2.9db23cp-12, 0x3.cc63c8p-8 }, /* 22 */ | |
111 | { 0x1.30b5a8p+6, 0x1.30b6fep+6, 0x1.30b84ep+6, 0x5.b5e53p-24, | |
112 | 0x1.766dc2p-4, -0x2.754cfcp-12, -0x3.c39bb4p-8 }, /* 23 */ | |
113 | { 0x1.3d46ccp+6, 0x1.3d482ep+6, 0x1.3d495ep+6, -0x1.340a8ap-24, | |
114 | -0x1.6ef07ep-4, 0x2.4ff9d4p-12, 0x3.9b36e4p-8 }, /* 24 */ | |
115 | { 0x1.49d688p+6, 0x1.49d95ap+6, 0x1.49dabep+6, -0x3.ba66p-24, | |
116 | 0x1.67e1dcp-4, -0x2.2f32b8p-12, -0x3.e2aaf4p-8 }, /* 25 */ | |
117 | { 0x1.566916p+6, 0x1.566a84p+6, 0x1.566bcp+6, 0x6.47ca5p-28, | |
118 | -0x1.61379ap-4, 0x2.1096acp-12, 0x4.2d0968p-8 }, /* 26 */ | |
119 | { 0x1.62f8dap+6, 0x1.62fbaap+6, 0x1.62fc9cp+6, -0x2.12affp-24, | |
120 | 0x1.5ae8c4p-4, -0x1.f32444p-12, -0x3.9e592p-8 }, /* 27 */ | |
121 | { 0x1.6f89e6p+6, 0x1.6f8ccep+6, 0x1.6f8e34p+6, -0x7.4853ap-28, | |
122 | -0x1.54ed76p-4, 0x1.db004ap-12, 0x3.907034p-8 }, /* 28 */ | |
123 | { 0x1.7c1c6ap+6, 0x1.7c1deep+6, 0x1.7c1f4cp+6, -0x4.f0a998p-24, | |
124 | 0x1.4f3ebcp-4, -0x1.c26808p-12, -0x2.da8df8p-8 }, /* 29 */ | |
125 | { 0x1.88adaep+6, 0x1.88af0ep+6, 0x1.88afc4p+6, -0x1.80c246p-24, | |
126 | -0x1.49d668p-4, 0x1.aebc26p-12, 0x3.af7b5cp-8 }, /* 30 */ | |
127 | { 0x1.953d7p+6, 0x1.95402ap+6, 0x1.9540ep+6, -0x2.22aff8p-24, | |
128 | 0x1.44aefap-4, -0x1.99f25p-12, -0x3.5e9198p-8 }, /* 31 */ | |
129 | { 0x1.a1d01ep+6, 0x1.a1d146p+6, 0x1.a1d20ap+6, -0x3.aad6d4p-24, | |
130 | -0x1.3fc386p-4, 0x1.892858p-12, 0x3.fe0184p-8 }, /* 32 */ | |
131 | { 0x1.ae60ecp+6, 0x1.ae625ep+6, 0x1.ae6326p+6, -0x2.010be4p-24, | |
132 | 0x1.3b0fa4p-4, -0x1.7539ap-12, -0x2.b2c9bp-8 }, /* 33 */ | |
133 | { 0x1.baf234p+6, 0x1.baf376p+6, 0x1.baf442p+6, -0xd.4fd17p-32, | |
134 | -0x1.368f5cp-4, 0x1.6734e4p-12, 0x3.59f514p-8 }, /* 34 */ | |
135 | { 0x1.c782e6p+6, 0x1.c7848cp+6, 0x1.c78516p+6, -0xa.d662dp-28, | |
136 | 0x1.323f18p-4, -0x1.571c02p-12, -0x3.2be5bp-8 }, /* 35 */ | |
137 | { 0x1.d4144ep+6, 0x1.d415ap+6, 0x1.d41622p+6, 0x4.9f217p-24, | |
138 | -0x1.2e1b9ap-4, 0x1.4a2edap-12, 0x3.a4e96p-8 }, /* 36 */ | |
139 | { 0x1.e0a5ep+6, 0x1.e0a6b4p+6, 0x1.e0a788p+6, -0x2.d167p-24, | |
140 | 0x1.2a21eep-4, -0x1.3c4b46p-12, -0x4.9e0978p-8 }, /* 37 */ | |
141 | { 0x1.ed36eep+6, 0x1.ed37c8p+6, 0x1.ed3892p+6, -0x4.15a83p-24, | |
142 | -0x1.264f66p-4, 0x1.31dea4p-12, 0x3.d125ecp-8 }, /* 38 */ | |
143 | { 0x1.f9c77p+6, 0x1.f9c8d8p+6, 0x1.f9c9acp+6, -0x2.a5bbbp-24, | |
144 | 0x1.22a192p-4, -0x1.25e59ep-12, -0x2.ef6934p-8 }, /* 39 */ | |
145 | { 0x1.032c54p+7, 0x1.032cf4p+7, 0x1.032d66p+7, 0x4.e828bp-24, | |
146 | -0x1.1f1634p-4, 0x1.1c2394p-12, 0x3.6d744cp-8 }, /* 40 */ | |
147 | { 0x1.09750cp+7, 0x1.09757cp+7, 0x1.0975b6p+7, -0x3.28a3bcp-24, | |
148 | 0x1.1bab3ep-4, -0x1.1569cep-12, -0x5.84da7p-8 }, /* 41 */ | |
149 | { 0x1.0fbd9ap+7, 0x1.0fbe04p+7, 0x1.0fbe5ep+7, -0x2.2f667p-24, | |
150 | -0x1.185eccp-4, 0x1.07f42cp-12, 0x2.87896cp-8 }, /* 42 */ | |
151 | { 0x1.160628p+7, 0x1.16068ap+7, 0x1.1606cep+7, -0x6.9097dp-24, | |
152 | 0x1.152f28p-4, -0x1.0227fep-12, -0x5.da6e6p-8 }, /* 43 */ | |
153 | { 0x1.1c4e9ap+7, 0x1.1c4f12p+7, 0x1.1c4f7cp+7, -0x5.1b408p-24, | |
154 | -0x1.121abp-4, 0xf.6efcp-16, 0x2.c5e954p-8 }, /* 44 */ | |
155 | { 0x1.2296aap+7, 0x1.229798p+7, 0x1.2297d4p+7, 0x2.70d0dp-24, | |
156 | 0x1.0f1ffp-4, -0xf.523f5p-16, -0x3.5c0568p-8 }, /* 45 */ | |
157 | { 0x1.28dfa4p+7, 0x1.28e01ep+7, 0x1.28e054p+7, -0x2.7c176p-24, | |
158 | -0x1.0c3d8ap-4, 0xe.8329ap-16, 0x3.5eb34p-8 }, /* 46 */ | |
159 | { 0x1.2f282ap+7, 0x1.2f28a4p+7, 0x1.2f28dep+7, 0x4.fd6368p-24, | |
160 | 0x1.097236p-4, -0xe.17299p-16, -0x3.73a2e4p-8 }, /* 47 */ | |
161 | { 0x1.3570bp+7, 0x1.357128p+7, 0x1.35716p+7, 0x6.b05f68p-24, | |
162 | -0x1.06bccap-4, 0xd.527b8p-16, 0x2.b8bf9cp-8 }, /* 48 */ | |
163 | { 0x1.3bb932p+7, 0x1.3bb9aep+7, 0x1.3bb9eap+7, 0x4.0d622p-28, | |
164 | 0x1.041c28p-4, -0xd.0ac11p-16, -0x1.65f2ccp-8 }, /* 49 */ | |
165 | { 0x1.4201b6p+7, 0x1.420232p+7, 0x1.42027p+7, 0x7.0d98cp-24, | |
166 | -0x1.018f52p-4, 0xc.c4d8ep-16, 0x2.8f250cp-8 }, /* 50 */ | |
167 | { 0x1.484a78p+7, 0x1.484ab8p+7, 0x1.484af4p+7, 0x3.93d10cp-24, | |
168 | 0xf.f154fp-8, -0xc.7b7fep-16, -0x3.6b6e4cp-8 }, /* 51 */ | |
169 | { 0x1.4e92c8p+7, 0x1.4e933cp+7, 0x1.4e9368p+7, 0xd.88185p-32, | |
170 | -0xf.cad3fp-8, 0xc.1462p-16, 0x2.bd66p-8 }, /* 52 */ | |
171 | { 0x1.54db84p+7, 0x1.54dbcp+7, 0x1.54dbf4p+7, -0x1.fe6b92p-24, | |
172 | 0xf.a564cp-8, -0xb.c4e6cp-16, -0x3.d51decp-8 }, /* 53 */ | |
173 | { 0x1.5b23c4p+7, 0x1.5b2444p+7, 0x1.5b2486p+7, 0x2.6137f4p-24, | |
174 | -0xf.80faep-8, 0xb.5199ep-16, 0x1.9ca85ap-8 }, /* 54 */ | |
175 | { 0x1.616c62p+7, 0x1.616cc8p+7, 0x1.616d0ap+7, -0x1.55468p-24, | |
176 | 0xf.5d8acp-8, -0xb.21d16p-16, -0x1.b8809ap-8 }, /* 55 */ | |
177 | { 0x1.67b4fp+7, 0x1.67b54cp+7, 0x1.67b588p+7, -0x1.08c6bep-24, | |
178 | -0xf.3b096p-8, 0xa.e65efp-16, 0x3.642304p-8 }, /* 56 */ | |
179 | { 0x1.6dfd8ep+7, 0x1.6dfddp+7, 0x1.6dfe0ap+7, 0x4.9ebfa8p-24, | |
180 | 0xf.196c7p-8, -0xa.ba8c8p-16, -0x5.ad6a2p-8 }, /* 57 */ | |
181 | { 0x1.74461p+7, 0x1.744652p+7, 0x1.744692p+7, 0x5.a4017p-24, | |
182 | -0xe.f8aa5p-8, 0xa.49748p-16, 0x2.a86498p-8 }, /* 58 */ | |
183 | { 0x1.7a8e5ep+7, 0x1.7a8ed6p+7, 0x1.7a8ef8p+7, 0x3.bcb2a8p-28, | |
184 | 0xe.d8b9dp-8, -0x9.c43bep-16, -0x1.e7124ap-8 }, /* 59 */ | |
185 | { 0x1.80d6cep+7, 0x1.80d75ap+7, 0x1.80d78ap+7, -0x7.1091fp-24, | |
186 | -0xe.b9925p-8, 0x9.c43dap-16, 0x1.aba86p-8 }, /* 60 */ | |
187 | { 0x1.871f58p+7, 0x1.871fdcp+7, 0x1.87201ep+7, 0x2.ca1cf4p-28, | |
188 | 0xe.9b2bep-8, -0x9.843b3p-16, -0x2.093e68p-8 }, /* 61 */ | |
189 | { 0x1.8d67e8p+7, 0x1.8d685ep+7, 0x1.8d688ep+7, 0x5.aa8908p-24, | |
190 | -0xe.7d7ecp-8, 0x9.501a8p-16, 0x2.54a754p-8 }, /* 62 */ | |
191 | { 0x1.93b09cp+7, 0x1.93b0e2p+7, 0x1.93b10ep+7, 0x3.d9cd9cp-24, | |
192 | 0xe.6083ap-8, -0x9.45dadp-16, -0x5.112908p-8 }, /* 63 */ | |
193 | }; | |
194 | ||
195 | /* Formula page 5 of https://www.cl.cam.ac.uk/~jrh13/papers/bessel.pdf: | |
196 | j1f(x) ~ sqrt(2/(pi*x))*beta1(x)*cos(x-3pi/4-alpha1(x)) | |
197 | where beta1(x) = 1 + 3/(16*x^2) - 99/(512*x^4) | |
198 | and alpha1(x) = -3/(8*x) + 21/(128*x^3) - 1899/(5120*x^5). */ | |
199 | static float | |
200 | j1f_asympt (float x) | |
201 | { | |
202 | float cst = 0xc.c422ap-4; /* sqrt(2/pi) rounded to nearest */ | |
203 | if (x < 0) | |
204 | { | |
205 | x = -x; | |
206 | cst = -cst; | |
207 | } | |
208 | double y = 1.0 / (double) x; | |
209 | double y2 = y * y; | |
210 | double beta1 = 1.0f + y2 * (0x3p-4 - 0x3.18p-4 * y2); | |
211 | double alpha1; | |
212 | alpha1 = y * (-0x6p-4 + y2 * (0x2.ap-4 - 0x5.ef33333333334p-4 * y2)); | |
213 | double h; | |
214 | int n; | |
215 | h = reduce_aux (x, &n, alpha1); | |
216 | n--; /* Subtract pi/2. */ | |
217 | /* Now x - 3pi/4 - alpha1 = h + n*pi/2 mod (2*pi). */ | |
218 | float xr = (float) h; | |
219 | n = n & 3; | |
220 | float t = cst / sqrtf (x) * (float) beta1; | |
221 | if (n == 0) | |
222 | return t * __cosf (xr); | |
223 | else if (n == 2) /* cos(x+pi) = -cos(x) */ | |
224 | return -t * __cosf (xr); | |
225 | else if (n == 1) /* cos(x+pi/2) = -sin(x) */ | |
226 | return -t * __sinf (xr); | |
227 | else /* cos(x+3pi/2) = sin(x) */ | |
228 | return t * __sinf (xr); | |
229 | } | |
230 | ||
231 | /* Special code for x near a root of j1. | |
232 | z is the value computed by the generic code. | |
233 | For small x, we use a polynomial approximating j1 around its root. | |
234 | For large x, we use an asymptotic formula (j1f_asympt). */ | |
235 | static float | |
236 | j1f_near_root (float x, float z) | |
237 | { | |
238 | float index_f, sign = 1.0f; | |
239 | int index; | |
240 | ||
241 | if (x < 0) | |
242 | { | |
243 | x = -x; | |
244 | sign = -1.0f; | |
245 | } | |
347a5b59 | 246 | index_f = roundf ((x - FIRST_ZERO_J1) / M_PIf); |
9acda61d PZ |
247 | if (index_f >= SMALL_SIZE) |
248 | return sign * j1f_asympt (x); | |
249 | index = (int) index_f; | |
250 | const float *p = Pj[index]; | |
251 | float x0 = p[0]; | |
252 | float x1 = p[2]; | |
253 | /* If not in the interval [x0,x1] around xmid, return the value z. */ | |
254 | if (! (x0 <= x && x <= x1)) | |
255 | return z; | |
256 | float xmid = p[1]; | |
257 | float y = x - xmid; | |
258 | float p6 = (index > 0) ? p[6] : p[6] + y * -0x1.3e51aap-8f; | |
259 | return sign * (p[3] + y * (p[4] + y * (p[5] + y * p6))); | |
260 | } | |
261 | ||
0ac5ae23 UD |
262 | float |
263 | __ieee754_j1f(float x) | |
f7eac6eb RM |
264 | { |
265 | float z, s,c,ss,cc,r,u,v,y; | |
266 | int32_t hx,ix; | |
267 | ||
268 | GET_FLOAT_WORD(hx,x); | |
269 | ix = hx&0x7fffffff; | |
0ac5ae23 | 270 | if(__builtin_expect(ix>=0x7f800000, 0)) return one/x; |
f7eac6eb RM |
271 | y = fabsf(x); |
272 | if(ix >= 0x40000000) { /* |x| >= 2.0 */ | |
9acda61d | 273 | SET_RESTORE_ROUNDF (FE_TONEAREST); |
5d2a88a7 | 274 | __sincosf (y, &s, &c); |
f7eac6eb RM |
275 | ss = -s-c; |
276 | cc = s-c; | |
9acda61d PZ |
277 | if (ix >= 0x7f000000) |
278 | /* x >= 2^127: use asymptotic expansion. */ | |
279 | return j1f_asympt (x); | |
280 | /* Now we are sure that x+x cannot overflow. */ | |
281 | z = __cosf(y+y); | |
282 | if ((s*c)>zero) cc = z/ss; | |
283 | else ss = z/cc; | |
f7eac6eb RM |
284 | /* |
285 | * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x) | |
286 | * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x) | |
287 | */ | |
9acda61d PZ |
288 | if (ix <= 0x5c000000) |
289 | { | |
f7eac6eb | 290 | u = ponef(y); v = qonef(y); |
9acda61d PZ |
291 | cc = u*cc-v*ss; |
292 | } | |
293 | z = (invsqrtpi * cc) / sqrtf(y); | |
294 | /* Adjust sign of z. */ | |
295 | z = (hx < 0) ? -z : z; | |
296 | /* The following threshold is optimal: for x=0x1.e09e5ep+1 | |
297 | and rounding upwards, cc=0x1.b79638p-4 and z is 10 ulps | |
298 | far from the correctly rounded value. */ | |
299 | float threshold = 0x1.b79638p-4; | |
300 | if (fabsf (cc) > threshold) | |
301 | return z; | |
302 | else | |
303 | return j1f_near_root (x, z); | |
f7eac6eb | 304 | } |
0ac5ae23 | 305 | if(__builtin_expect(ix<0x32000000, 0)) { /* |x|<2**-27 */ |
63dbe5f3 | 306 | if(huge+x>one) { /* inexact if x!=0 necessary */ |
c643db87 | 307 | float ret = math_narrow_eval ((float) 0.5 * x); |
d96164c3 | 308 | math_check_force_underflow (ret); |
c643db87 JM |
309 | if (ret == 0 && x != 0) |
310 | __set_errno (ERANGE); | |
63dbe5f3 JM |
311 | return ret; |
312 | } | |
f7eac6eb RM |
313 | } |
314 | z = x*x; | |
315 | r = z*(r00+z*(r01+z*(r02+z*r03))); | |
316 | s = one+z*(s01+z*(s02+z*(s03+z*(s04+z*s05)))); | |
317 | r *= x; | |
318 | return(x*(float)0.5+r/s); | |
319 | } | |
220622dd | 320 | libm_alias_finite (__ieee754_j1f, __j1f) |
f7eac6eb | 321 | |
f7eac6eb | 322 | static const float U0[5] = { |
f7eac6eb RM |
323 | -1.9605709612e-01, /* 0xbe48c331 */ |
324 | 5.0443872809e-02, /* 0x3d4e9e3c */ | |
325 | -1.9125689287e-03, /* 0xbafaaf2a */ | |
326 | 2.3525259166e-05, /* 0x37c5581c */ | |
327 | -9.1909917899e-08, /* 0xb3c56003 */ | |
328 | }; | |
f7eac6eb | 329 | static const float V0[5] = { |
f7eac6eb RM |
330 | 1.9916731864e-02, /* 0x3ca3286a */ |
331 | 2.0255257550e-04, /* 0x3954644b */ | |
332 | 1.3560879779e-06, /* 0x35b602d4 */ | |
333 | 6.2274145840e-09, /* 0x31d5f8eb */ | |
334 | 1.6655924903e-11, /* 0x2d9281cf */ | |
335 | }; | |
336 | ||
9acda61d PZ |
337 | /* This is the nearest approximation of the first zero of y1. */ |
338 | #define FIRST_ZERO_Y1 0x2.3277dcp+0f | |
339 | ||
340 | /* The following table contains successive zeros of y1 and degree-3 | |
341 | polynomial approximations of y1 around these zeros: Py[0] for the first | |
342 | positive zero (2.197141), Py[1] for the second one (5.429681), and so on. | |
343 | Each line contains: | |
344 | {x0, xmid, x1, p0, p1, p2, p3} | |
345 | where [x0,x1] is the interval around the zero, xmid is the binary32 number | |
346 | closest to the zero, and p0+p1*x+p2*x^2+p3*x^3 is the approximation | |
347 | polynomial. Each polynomial was generated using Sollya on the interval | |
348 | [x0,x1] around the corresponding zero where the error exceeds 9 ulps | |
349 | for the alternate code. Degree 3 is enough, except for the first roots. | |
350 | */ | |
351 | static const float Py[SMALL_SIZE][7] = { | |
352 | /* For index 0, we use a degree-5 polynomial generated by Sollya, with the | |
353 | coefficients of degree 4 and 5 hard-coded in y1f_near_root(). */ | |
354 | { 0x1.f7f16ap+0, 0x1.193beep+1, 0x1.2105dcp+1, 0xb.96749p-28, | |
355 | 0x8.55241p-4, -0x1.e570bp-4, -0x8.68b61p-8 | |
356 | /*, -0x1.28043p-8, 0x2.50e83p-8*/ }, /* 0 */ | |
357 | /* For index 1, we use a degree-4 polynomial generated by Sollya, with the | |
358 | coefficient of degree 4 hard-coded in y1f_near_root(). */ | |
359 | { 0x1.55c6d2p+2, 0x1.5b7fe4p+2, 0x1.5cf8cap+2, 0x1.3c7822p-24, | |
360 | -0x5.71f158p-4, 0x8.05cb4p-8, 0xd.0b15p-8/*, -0xf.ff6b8p-12*/ }, /* 1 */ | |
361 | { 0x1.113c6p+3, 0x1.13127ap+3, 0x1.1387dcp+3, -0x1.f3ad8ep-24, | |
362 | 0x4.57e66p-4, -0x4.0afb58p-8, -0xb.29207p-8 }, /* 2 */ | |
363 | { 0x1.76e7dep+3, 0x1.77f914p+3, 0x1.786a6ap+3, -0xd.5608fp-28, | |
364 | -0x3.b829d4p-4, 0x2.8852cp-8, 0x9.b70e3p-8 }, /* 3 */ | |
365 | { 0x1.dc2794p+3, 0x1.dcb7d8p+3, 0x1.dd032p+3, -0xe.a7c04p-28, | |
366 | 0x3.4e0458p-4, -0x1.c64b18p-8, -0x8.b0e7fp-8 }, /* 4 */ | |
367 | { 0x1.20874p+4, 0x1.20b1c6p+4, 0x1.20c71p+4, 0x1.c2626p-24, | |
368 | -0x3.00f03cp-4, 0x1.54f806p-8, 0x7.f9cf9p-8 }, /* 5 */ | |
369 | { 0x1.52d848p+4, 0x1.530254p+4, 0x1.531962p+4, -0x1.9503ecp-24, | |
370 | 0x2.c5b29cp-4, -0x1.0bf28p-8, -0x7.562e58p-8 }, /* 6 */ | |
371 | { 0x1.851e64p+4, 0x1.854fa4p+4, 0x1.85679p+4, -0x2.8d40fcp-24, | |
372 | -0x2.96547p-4, 0xd.9c38bp-12, 0x6.dcbf8p-8 }, /* 7 */ | |
373 | { 0x1.b7808ep+4, 0x1.b79acep+4, 0x1.b7b2a8p+4, -0x2.36df5cp-24, | |
374 | 0x2.6f55ap-4, -0xb.57f9fp-12, -0x6.82569p-8 }, /* 8 */ | |
375 | { 0x1.e9c8fp+4, 0x1.e9e48p+4, 0x1.e9f24p+4, 0xd.e2eb7p-28, | |
376 | -0x2.4e8104p-4, 0x9.a4be2p-12, 0x6.2541fp-8 }, /* 9 */ | |
377 | { 0x1.0e0808p+5, 0x1.0e169p+5, 0x1.0e1d92p+5, -0x2.3070f4p-24, | |
378 | 0x2.325e4cp-4, -0x8.53604p-12, -0x5.ca03a8p-8 }, /* 10 */ | |
379 | { 0x1.272e08p+5, 0x1.273a7cp+5, 0x1.2741fcp+5, -0x3.525508p-24, | |
380 | -0x2.19e7dcp-4, 0x7.49d1dp-12, 0x5.9cb02p-8 }, /* 11 */ | |
381 | { 0x1.404ec6p+5, 0x1.405e18p+5, 0x1.4065cep+5, -0xe.6e158p-28, | |
382 | 0x2.046174p-4, -0x6.71b3dp-12, -0x5.4c3c8p-8 }, /* 12 */ | |
383 | { 0x1.5971dcp+5, 0x1.598178p+5, 0x1.598592p+5, 0x1.e72698p-24, | |
384 | -0x1.f13fb2p-4, 0x5.c0f938p-12, 0x5.28ca78p-8 }, /* 13 */ | |
385 | { 0x1.729c4ep+5, 0x1.72a4a8p+5, 0x1.72a8eap+5, -0x1.5bed9cp-24, | |
386 | 0x1.e018dcp-4, -0x5.2f11e8p-12, -0x5.16ce48p-8 }, /* 14 */ | |
387 | { 0x1.8bbf4ep+5, 0x1.8bc7b2p+5, 0x1.8bcc1p+5, -0x3.6b654cp-24, | |
388 | -0x1.d09b2p-4, 0x4.b1747p-12, 0x4.bd22fp-8 }, /* 15 */ | |
389 | { 0x1.a4e272p+5, 0x1.a4ea9ap+5, 0x1.a4eef4p+5, 0x1.6f11bp-24, | |
390 | 0x1.c28612p-4, -0x4.47462p-12, -0x4.947c5p-8 }, /* 16 */ | |
391 | { 0x1.be08bep+5, 0x1.be0d68p+5, 0x1.be1088p+5, -0x2.0bc074p-24, | |
392 | -0x1.b5a622p-4, 0x3.ed52d4p-12, 0x4.b76fc8p-8 }, /* 17 */ | |
393 | { 0x1.d7272ap+5, 0x1.d7301ep+5, 0x1.d734aep+5, -0x2.87dd4p-24, | |
394 | 0x1.a9d184p-4, -0x3.9cf494p-12, -0x4.6303ep-8 }, /* 18 */ | |
395 | { 0x1.f0499ap+5, 0x1.f052c4p+5, 0x1.f05758p+5, -0x2.fb964p-24, | |
396 | -0x1.9ee5eep-4, 0x3.5800dp-12, 0x4.4e9f9p-8 }, /* 19 */ | |
397 | { 0x1.04b63ap+6, 0x1.04baacp+6, 0x1.04bc92p+6, 0x2.cf5adp-24, | |
398 | 0x1.94c6f4p-4, -0x3.1a83e4p-12, -0x4.2311fp-8 }, /* 20 */ | |
399 | { 0x1.1146dp+6, 0x1.114beep+6, 0x1.114e12p+6, 0x3.6766fp-24, | |
400 | -0x1.8b5cccp-4, 0x2.e4a4e4p-12, 0x4.20bf9p-8 }, /* 21 */ | |
401 | { 0x1.1dda8cp+6, 0x1.1ddd2cp+6, 0x1.1dde7ap+6, 0x3.501424p-24, | |
402 | 0x1.829356p-4, -0x2.b47524p-12, -0x4.04bf18p-8 }, /* 22 */ | |
403 | { 0x1.2a6bcp+6, 0x1.2a6e64p+6, 0x1.2a6faap+6, -0x5.c05808p-24, | |
404 | -0x1.7a597ep-4, 0x2.8a0498p-12, 0x4.187258p-8 }, /* 23 */ | |
405 | { 0x1.36fcd6p+6, 0x1.36ff96p+6, 0x1.3700f6p+6, 0x7.1e1478p-28, | |
406 | 0x1.72a09ap-4, -0x2.61a7fp-12, -0x3.c0b54p-8 }, /* 24 */ | |
407 | { 0x1.438f46p+6, 0x1.4390c4p+6, 0x1.4392p+6, 0x3.e36e6cp-24, | |
408 | -0x1.6b5c06p-4, 0x2.3f612p-12, 0x4.18f868p-8 }, /* 25 */ | |
409 | { 0x1.501f4cp+6, 0x1.5021fp+6, 0x1.50235p+6, 0x1.3f9e5ap-24, | |
410 | 0x1.6480c4p-4, -0x2.1f28fcp-12, -0x3.bb4e3cp-8 }, /* 26 */ | |
411 | { 0x1.5cb07cp+6, 0x1.5cb318p+6, 0x1.5cb464p+6, -0x2.39e41cp-24, | |
412 | -0x1.5e0544p-4, 0x2.0189f4p-12, 0x3.8b55acp-8 }, /* 27 */ | |
413 | { 0x1.694166p+6, 0x1.69443cp+6, 0x1.694594p+6, -0x2.912f84p-24, | |
414 | 0x1.57e12p-4, -0x1.e6fabep-12, -0x3.850174p-8 }, /* 28 */ | |
415 | { 0x1.75d27cp+6, 0x1.75d55ep+6, 0x1.75d67ep+6, 0x3.d5b00cp-24, | |
416 | -0x1.520ceep-4, 0x1.d0286ep-12, 0x3.8e7d1p-8 }, /* 29 */ | |
417 | { 0x1.82653ep+6, 0x1.82667ep+6, 0x1.82674p+6, -0x3.1726ecp-24, | |
418 | 0x1.4c8222p-4, -0x1.b98206p-12, -0x3.f34978p-8 }, /* 30 */ | |
419 | { 0x1.8ef4b4p+6, 0x1.8ef79cp+6, 0x1.8ef888p+6, 0x1.949e22p-24, | |
420 | -0x1.473ae6p-4, 0x1.a47388p-12, 0x3.69eefcp-8 }, /* 31 */ | |
421 | { 0x1.9b8728p+6, 0x1.9b88b8p+6, 0x1.9b896cp+6, -0x5.5553bp-28, | |
422 | 0x1.42320ap-4, -0x1.90f0b8p-12, -0x3.6565p-8 }, /* 32 */ | |
423 | { 0x1.a8183cp+6, 0x1.a819d2p+6, 0x1.a81aecp+6, 0x3.2df7ecp-28, | |
424 | -0x1.3d62e4p-4, 0x1.7dae28p-12, 0x2.9eb128p-8 }, /* 33 */ | |
425 | { 0x1.b4aa1cp+6, 0x1.b4aaeap+6, 0x1.b4abb8p+6, -0x1.e13fcep-24, | |
426 | 0x1.38c948p-4, -0x1.6eb0ecp-12, -0x1.f9ddf8p-8 }, /* 34 */ | |
427 | { 0x1.c13a7ap+6, 0x1.c13c02p+6, 0x1.c13cbp+6, -0x3.ad9974p-24, | |
428 | -0x1.34616ep-4, 0x1.5e36ecp-12, 0x2.a9fc5p-8 }, /* 35 */ | |
429 | { 0x1.cdcb76p+6, 0x1.cdcd16p+6, 0x1.cdcde4p+6, -0x3.6977e8p-24, | |
430 | 0x1.3027fp-4, -0x1.4f703p-12, -0x2.9817d4p-8 }, /* 36 */ | |
431 | { 0x1.da5cdep+6, 0x1.da5e2ap+6, 0x1.da5efp+6, 0x4.654cbp-24, | |
432 | -0x1.2c19b6p-4, 0x1.455982p-12, 0x3.f1c564p-8 }, /* 37 */ | |
433 | { 0x1.e6edccp+6, 0x1.e6ef3ep+6, 0x1.e6f00ap+6, 0x8.825c8p-32, | |
434 | 0x1.2833eep-4, -0x1.39097p-12, -0x3.b2646p-8 }, /* 38 */ | |
435 | { 0x1.f37f72p+6, 0x1.f3805p+6, 0x1.f3812ap+6, -0x2.0d11d8p-28, | |
436 | -0x1.24740ap-4, 0x1.2c16p-12, 0x1.fc3804p-8 }, /* 39 */ | |
437 | { 0x1.000842p+7, 0x1.0008bp+7, 0x1.000908p+7, -0x4.4e495p-24, | |
438 | 0x1.20d7b6p-4, -0x1.20816p-12, -0x2.d1ebe8p-8 }, /* 40 */ | |
439 | { 0x1.06505cp+7, 0x1.065138p+7, 0x1.06518p+7, 0x4.81c1c8p-24, | |
440 | -0x1.1d5ccap-4, 0x1.17ad5ap-12, 0x2.fda33p-8 }, /* 41 */ | |
441 | { 0x1.0c98dap+7, 0x1.0c99cp+7, 0x1.0c9a28p+7, -0xe.99386p-28, | |
442 | 0x1.1a015p-4, -0x1.0bd50ap-12, -0x2.9dfb68p-8 }, /* 42 */ | |
443 | { 0x1.12e212p+7, 0x1.12e248p+7, 0x1.12e29p+7, -0x6.16f1c8p-24, | |
444 | -0x1.16c37ap-4, 0x1.0303dcp-12, 0x4.34316p-8 }, /* 43 */ | |
445 | { 0x1.192a68p+7, 0x1.192acep+7, 0x1.192b02p+7, -0x1.129336p-24, | |
446 | 0x1.13a19ep-4, -0xf.bd247p-16, -0x3.851d18p-8 }, /* 44 */ | |
447 | { 0x1.1f727p+7, 0x1.1f7354p+7, 0x1.1f73ap+7, 0x5.19c09p-24, | |
448 | -0x1.109a32p-4, 0xf.09644p-16, 0x2.d78194p-8 }, /* 45 */ | |
449 | { 0x1.25bb8p+7, 0x1.25bbdap+7, 0x1.25bc12p+7, -0x6.497dp-24, | |
450 | 0x1.0dabc8p-4, -0xe.a1d25p-16, -0x2.3378bp-8 }, /* 46 */ | |
451 | { 0x1.2c04p+7, 0x1.2c046p+7, 0x1.2c04ap+7, 0x4.e4f338p-24, | |
452 | -0x1.0ad512p-4, 0xe.52d84p-16, 0x4.3bfa08p-8 }, /* 47 */ | |
453 | { 0x1.324cbp+7, 0x1.324ce6p+7, 0x1.324d4p+7, -0x1.287c58p-24, | |
454 | 0x1.0814d4p-4, -0xe.03a95p-16, 0x3.9930ap-12 }, /* 48 */ | |
455 | { 0x1.3894f6p+7, 0x1.38956cp+7, 0x1.3895ap+7, -0x4.b594ep-24, | |
456 | -0x1.0569fp-4, 0xd.6787ep-16, 0x4.0a5148p-8 }, /* 49 */ | |
457 | { 0x1.3edd98p+7, 0x1.3eddfp+7, 0x1.3ede2ap+7, -0x3.a8f164p-24, | |
458 | 0x1.02d354p-4, -0xd.0309dp-16, -0x3.2ebfb4p-8 }, /* 50 */ | |
459 | { 0x1.452638p+7, 0x1.452676p+7, 0x1.4526b4p+7, -0x6.12505p-24, | |
460 | -0x1.005004p-4, 0xc.a0045p-16, 0x4.87c67p-8 }, /* 51 */ | |
461 | { 0x1.4b6e8p+7, 0x1.4b6efap+7, 0x1.4b6f34p+7, 0x1.8acf4ep-24, | |
462 | 0xf.ddf16p-8, -0xc.2d207p-16, -0x1.da6c36p-8 }, /* 52 */ | |
463 | { 0x1.51b742p+7, 0x1.51b77ep+7, 0x1.51b7b2p+7, 0x1.39cf86p-24, | |
464 | -0xf.b7faep-8, 0xb.db598p-16, -0x8.945b1p-12 }, /* 53 */ | |
465 | { 0x1.57ffc4p+7, 0x1.580002p+7, 0x1.58003cp+7, -0x2.5f8de8p-24, | |
466 | 0xf.930fep-8, -0xb.91889p-16, -0xa.30df9p-12 }, /* 54 */ | |
467 | { 0x1.5e483p+7, 0x1.5e4886p+7, 0x1.5e48c8p+7, 0x2.073d64p-24, | |
468 | -0xf.6f245p-8, 0xb.4085fp-16, 0x2.128188p-8 }, /* 55 */ | |
469 | { 0x1.64908cp+7, 0x1.64910ap+7, 0x1.64912ap+7, -0x4.ed26ep-28, | |
470 | 0xf.4c2cep-8, -0xa.fe719p-16, -0x2.9374b8p-8 }, /* 56 */ | |
471 | { 0x1.6ad91ep+7, 0x1.6ad98ep+7, 0x1.6ad9cep+7, -0x2.ae5204p-24, | |
472 | -0xf.2a1efp-8, 0xa.aa585p-16, 0x2.1c0834p-8 }, /* 57 */ | |
473 | { 0x1.7121cep+7, 0x1.712212p+7, 0x1.712238p+7, 0x6.d72168p-24, | |
474 | 0xf.08f09p-8, -0xa.7da49p-16, -0x3.4f5f1cp-8 }, /* 58 */ | |
475 | { 0x1.776a0cp+7, 0x1.776a94p+7, 0x1.776accp+7, 0x2.d3f294p-24, | |
476 | -0xe.e8986p-8, 0xa.23ccdp-16, 0x2.2a6678p-8 }, /* 59 */ | |
477 | { 0x1.7db2e8p+7, 0x1.7db318p+7, 0x1.7db35ap+7, 0x3.88c0fp-24, | |
478 | 0xe.c90d7p-8, -0x9.eaeap-16, -0x2.86438cp-8 }, /* 60 */ | |
479 | { 0x1.83fb56p+7, 0x1.83fb9ap+7, 0x1.83fbep+7, 0x3.d94d34p-24, | |
480 | -0xe.aa478p-8, 0x9.abac7p-16, 0x1.ac2d84p-8 }, /* 61 */ | |
481 | { 0x1.8a43e8p+7, 0x1.8a441ep+7, 0x1.8a446p+7, 0x4.66b7ep-24, | |
482 | 0xe.8c3e9p-8, -0x9.87682p-16, -0x7.9ab4a8p-12 }, /* 62 */ | |
483 | { 0x1.908c6p+7, 0x1.908cap+7, 0x1.908ce6p+7, 0xf.f7ac9p-28, | |
484 | -0xe.6eeb6p-8, 0x9.4423p-16, 0x4.54c4d8p-8 }, /* 63 */ | |
485 | }; | |
486 | ||
487 | /* Formula page 5 of https://www.cl.cam.ac.uk/~jrh13/papers/bessel.pdf: | |
488 | y1f(x) ~ sqrt(2/(pi*x))*beta1(x)*sin(x-3pi/4-alpha1(x)) | |
489 | where beta1(x) = 1 + 3/(16*x^2) - 99/(512*x^4) | |
490 | and alpha1(x) = -3/(8*x) + 21/(128*x^3) - 1899/(5120*x^5). */ | |
491 | static float | |
492 | y1f_asympt (float x) | |
493 | { | |
494 | float cst = 0xc.c422ap-4; /* sqrt(2/pi) rounded to nearest */ | |
495 | double y = 1.0 / (double) x; | |
496 | double y2 = y * y; | |
497 | double beta1 = 1.0f + y2 * (0x3p-4 - 0x3.18p-4 * y2); | |
498 | double alpha1; | |
499 | alpha1 = y * (-0x6p-4 + y2 * (0x2.ap-4 - 0x5.ef33333333334p-4 * y2)); | |
500 | double h; | |
501 | int n; | |
502 | h = reduce_aux (x, &n, alpha1); | |
503 | n--; /* Subtract pi/2. */ | |
504 | /* Now x - 3pi/4 - alpha1 = h + n*pi/2 mod (2*pi). */ | |
505 | float xr = (float) h; | |
506 | n = n & 3; | |
507 | float t = cst / sqrtf (x) * (float) beta1; | |
508 | if (n == 0) | |
509 | return t * __sinf (xr); | |
510 | else if (n == 2) /* sin(x+pi) = -sin(x) */ | |
511 | return -t * __sinf (xr); | |
512 | else if (n == 1) /* sin(x+pi/2) = cos(x) */ | |
513 | return t * __cosf (xr); | |
514 | else /* sin(x+3pi/2) = -cos(x) */ | |
515 | return -t * __cosf (xr); | |
516 | } | |
517 | ||
518 | /* Special code for x near a root of y1. | |
519 | z is the value computed by the generic code. | |
520 | For small x, we use a polynomial approximating y1 around its root. | |
521 | For large x, we use an asymptotic formula (y1f_asympt). */ | |
522 | static float | |
523 | y1f_near_root (float x, float z) | |
524 | { | |
525 | float index_f; | |
526 | int index; | |
527 | ||
347a5b59 | 528 | index_f = roundf ((x - FIRST_ZERO_Y1) / M_PIf); |
9acda61d PZ |
529 | if (index_f >= SMALL_SIZE) |
530 | return y1f_asympt (x); | |
531 | index = (int) index_f; | |
532 | const float *p = Py[index]; | |
533 | float x0 = p[0]; | |
534 | float x1 = p[2]; | |
535 | /* If not in the interval [x0,x1] around xmid, return the value z. */ | |
536 | if (! (x0 <= x && x <= x1)) | |
537 | return z; | |
538 | float xmid = p[1]; | |
539 | float y = x - xmid, p6; | |
540 | if (index == 0) | |
541 | p6 = p[6] + y * (-0x1.28043p-8 + y * 0x2.50e83p-8); | |
542 | else if (index == 1) | |
543 | p6 = p[6] + y * -0xf.ff6b8p-12; | |
544 | else | |
545 | p6 = p[6]; | |
546 | return p[3] + y * (p[4] + y * (p[5] + y * p6)); | |
547 | } | |
548 | ||
0ac5ae23 UD |
549 | float |
550 | __ieee754_y1f(float x) | |
f7eac6eb RM |
551 | { |
552 | float z, s,c,ss,cc,u,v; | |
553 | int32_t hx,ix; | |
554 | ||
555 | GET_FLOAT_WORD(hx,x); | |
0ac5ae23 | 556 | ix = 0x7fffffff&hx; |
f7eac6eb | 557 | /* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */ |
0ac5ae23 UD |
558 | if(__builtin_expect(ix>=0x7f800000, 0)) return one/(x+x*x); |
559 | if(__builtin_expect(ix==0, 0)) | |
4918e5f4 | 560 | return -1/zero; /* -inf and divide by zero exception. */ |
0ac5ae23 | 561 | if(__builtin_expect(hx<0, 0)) return zero/(zero*x); |
9acda61d | 562 | if (ix >= 0x3fe0dfbc) { /* |x| >= 0x1.c1bf78p+0 */ |
30f3d07c | 563 | SET_RESTORE_ROUNDF (FE_TONEAREST); |
5d2a88a7 | 564 | __sincosf (x, &s, &c); |
0ac5ae23 UD |
565 | ss = -s-c; |
566 | cc = s-c; | |
9acda61d PZ |
567 | if (ix >= 0x7f000000) |
568 | /* x >= 2^127: use asymptotic expansion. */ | |
569 | return y1f_asympt (x); | |
570 | /* Now we are sure that x+x cannot overflow. */ | |
571 | z = __cosf(x+x); | |
572 | if ((s*c)>zero) cc = z/ss; | |
573 | else ss = z/cc; | |
0ac5ae23 UD |
574 | /* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0)) |
575 | * where x0 = x-3pi/4 | |
576 | * Better formula: | |
577 | * cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4) | |
578 | * = 1/sqrt(2) * (sin(x) - cos(x)) | |
579 | * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4) | |
580 | * = -1/sqrt(2) * (cos(x) + sin(x)) | |
581 | * To avoid cancellation, use | |
582 | * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) | |
583 | * to compute the worse one. | |
584 | */ | |
9acda61d PZ |
585 | if (ix <= 0x5c000000) |
586 | { | |
587 | u = ponef(x); v = qonef(x); | |
588 | ss = u*ss+v*cc; | |
589 | } | |
590 | z = (invsqrtpi * ss) / sqrtf(x); | |
591 | float threshold = 0x1.3e014cp-2; | |
592 | /* The following threshold is optimal: for x=0x1.f7f16ap+0 | |
593 | and rounding upwards, |ss|=-0x1.3e014cp-2 and z is 11 ulps | |
594 | far from the correctly rounded value. */ | |
595 | if (fabsf (ss) > threshold) | |
596 | return z; | |
597 | else | |
598 | return y1f_near_root (x, z); | |
0ac5ae23 | 599 | } |
05b227bd | 600 | if(__builtin_expect(ix<=0x33000000, 0)) { /* x < 2**-25 */ |
46a3d3c7 | 601 | z = -tpi / x; |
d81f90cc | 602 | if (isinf (z)) |
46a3d3c7 JM |
603 | __set_errno (ERANGE); |
604 | return z; | |
0ac5ae23 | 605 | } |
9acda61d | 606 | /* Now 2**-25 <= x < 0x1.c1bf78p+0. */ |
0ac5ae23 UD |
607 | z = x*x; |
608 | u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4]))); | |
609 | v = one+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4])))); | |
610 | return(x*(u/v) + tpi*(__ieee754_j1f(x)*__ieee754_logf(x)-one/x)); | |
f7eac6eb | 611 | } |
220622dd | 612 | libm_alias_finite (__ieee754_y1f, __y1f) |
f7eac6eb | 613 | |
9acda61d | 614 | /* For x >= 8, the asymptotic expansion of pone is |
f7eac6eb RM |
615 | * 1 + 15/128 s^2 - 4725/2^15 s^4 - ..., where s = 1/x. |
616 | * We approximate pone by | |
0ac5ae23 | 617 | * pone(x) = 1 + (R/S) |
f7eac6eb | 618 | * where R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10 |
0ac5ae23 | 619 | * S = 1 + ps0*s^2 + ... + ps4*s^10 |
f7eac6eb RM |
620 | * and |
621 | * | pone(x)-1-R/S | <= 2 ** ( -60.06) | |
622 | */ | |
623 | ||
f7eac6eb | 624 | static const float pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ |
f7eac6eb RM |
625 | 0.0000000000e+00, /* 0x00000000 */ |
626 | 1.1718750000e-01, /* 0x3df00000 */ | |
627 | 1.3239480972e+01, /* 0x4153d4ea */ | |
628 | 4.1205184937e+02, /* 0x43ce06a3 */ | |
629 | 3.8747453613e+03, /* 0x45722bed */ | |
630 | 7.9144794922e+03, /* 0x45f753d6 */ | |
631 | }; | |
f7eac6eb | 632 | static const float ps8[5] = { |
f7eac6eb RM |
633 | 1.1420736694e+02, /* 0x42e46a2c */ |
634 | 3.6509309082e+03, /* 0x45642ee5 */ | |
635 | 3.6956207031e+04, /* 0x47105c35 */ | |
636 | 9.7602796875e+04, /* 0x47bea166 */ | |
637 | 3.0804271484e+04, /* 0x46f0a88b */ | |
638 | }; | |
639 | ||
f7eac6eb | 640 | static const float pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ |
f7eac6eb RM |
641 | 1.3199052094e-11, /* 0x2d68333f */ |
642 | 1.1718749255e-01, /* 0x3defffff */ | |
643 | 6.8027510643e+00, /* 0x40d9b023 */ | |
644 | 1.0830818176e+02, /* 0x42d89dca */ | |
645 | 5.1763616943e+02, /* 0x440168b7 */ | |
646 | 5.2871520996e+02, /* 0x44042dc6 */ | |
647 | }; | |
f7eac6eb | 648 | static const float ps5[5] = { |
f7eac6eb RM |
649 | 5.9280597687e+01, /* 0x426d1f55 */ |
650 | 9.9140142822e+02, /* 0x4477d9b1 */ | |
651 | 5.3532670898e+03, /* 0x45a74a23 */ | |
652 | 7.8446904297e+03, /* 0x45f52586 */ | |
653 | 1.5040468750e+03, /* 0x44bc0180 */ | |
654 | }; | |
655 | ||
f7eac6eb | 656 | static const float pr3[6] = { |
f7eac6eb RM |
657 | 3.0250391081e-09, /* 0x314fe10d */ |
658 | 1.1718686670e-01, /* 0x3defffab */ | |
659 | 3.9329774380e+00, /* 0x407bb5e7 */ | |
660 | 3.5119403839e+01, /* 0x420c7a45 */ | |
661 | 9.1055007935e+01, /* 0x42b61c2a */ | |
662 | 4.8559066772e+01, /* 0x42423c7c */ | |
663 | }; | |
f7eac6eb | 664 | static const float ps3[5] = { |
f7eac6eb RM |
665 | 3.4791309357e+01, /* 0x420b2a4d */ |
666 | 3.3676245117e+02, /* 0x43a86198 */ | |
667 | 1.0468714600e+03, /* 0x4482dbe3 */ | |
668 | 8.9081134033e+02, /* 0x445eb3ed */ | |
669 | 1.0378793335e+02, /* 0x42cf936c */ | |
670 | }; | |
671 | ||
f7eac6eb | 672 | static const float pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ |
f7eac6eb RM |
673 | 1.0771083225e-07, /* 0x33e74ea8 */ |
674 | 1.1717621982e-01, /* 0x3deffa16 */ | |
675 | 2.3685150146e+00, /* 0x401795c0 */ | |
676 | 1.2242610931e+01, /* 0x4143e1bc */ | |
677 | 1.7693971634e+01, /* 0x418d8d41 */ | |
678 | 5.0735230446e+00, /* 0x40a25a4d */ | |
679 | }; | |
f7eac6eb | 680 | static const float ps2[5] = { |
f7eac6eb RM |
681 | 2.1436485291e+01, /* 0x41ab7dec */ |
682 | 1.2529022980e+02, /* 0x42fa9499 */ | |
683 | 2.3227647400e+02, /* 0x436846c7 */ | |
684 | 1.1767937469e+02, /* 0x42eb5bd7 */ | |
685 | 8.3646392822e+00, /* 0x4105d590 */ | |
686 | }; | |
687 | ||
0ac5ae23 UD |
688 | static float |
689 | ponef(float x) | |
f7eac6eb | 690 | { |
f7eac6eb | 691 | const float *p,*q; |
f7eac6eb | 692 | float z,r,s; |
0ac5ae23 | 693 | int32_t ix; |
f7eac6eb RM |
694 | GET_FLOAT_WORD(ix,x); |
695 | ix &= 0x7fffffff; | |
09220e66 | 696 | /* ix >= 0x40000000 for all calls to this function. */ |
0ac5ae23 UD |
697 | if(ix>=0x41000000) {p = pr8; q= ps8;} |
698 | else if(ix>=0x40f71c58){p = pr5; q= ps5;} | |
699 | else if(ix>=0x4036db68){p = pr3; q= ps3;} | |
09220e66 | 700 | else {p = pr2; q= ps2;} |
0ac5ae23 UD |
701 | z = one/(x*x); |
702 | r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); | |
703 | s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4])))); | |
704 | return one+ r/s; | |
f7eac6eb | 705 | } |
10659981 | 706 | |
9acda61d | 707 | /* For x >= 8, the asymptotic expansion of qone is |
f7eac6eb RM |
708 | * 3/8 s - 105/1024 s^3 - ..., where s = 1/x. |
709 | * We approximate pone by | |
0ac5ae23 | 710 | * qone(x) = s*(0.375 + (R/S)) |
f7eac6eb | 711 | * where R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10 |
0ac5ae23 | 712 | * S = 1 + qs1*s^2 + ... + qs6*s^12 |
f7eac6eb RM |
713 | * and |
714 | * | qone(x)/s -0.375-R/S | <= 2 ** ( -61.13) | |
715 | */ | |
716 | ||
f7eac6eb | 717 | static const float qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ |
f7eac6eb RM |
718 | 0.0000000000e+00, /* 0x00000000 */ |
719 | -1.0253906250e-01, /* 0xbdd20000 */ | |
720 | -1.6271753311e+01, /* 0xc1822c8d */ | |
721 | -7.5960174561e+02, /* 0xc43de683 */ | |
722 | -1.1849806641e+04, /* 0xc639273a */ | |
723 | -4.8438511719e+04, /* 0xc73d3683 */ | |
724 | }; | |
f7eac6eb | 725 | static const float qs8[6] = { |
f7eac6eb RM |
726 | 1.6139537048e+02, /* 0x43216537 */ |
727 | 7.8253862305e+03, /* 0x45f48b17 */ | |
728 | 1.3387534375e+05, /* 0x4802bcd6 */ | |
729 | 7.1965775000e+05, /* 0x492fb29c */ | |
730 | 6.6660125000e+05, /* 0x4922be94 */ | |
731 | -2.9449025000e+05, /* 0xc88fcb48 */ | |
732 | }; | |
733 | ||
f7eac6eb | 734 | static const float qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ |
f7eac6eb RM |
735 | -2.0897993405e-11, /* 0xadb7d219 */ |
736 | -1.0253904760e-01, /* 0xbdd1fffe */ | |
737 | -8.0564479828e+00, /* 0xc100e736 */ | |
738 | -1.8366960144e+02, /* 0xc337ab6b */ | |
739 | -1.3731937256e+03, /* 0xc4aba633 */ | |
740 | -2.6124443359e+03, /* 0xc523471c */ | |
741 | }; | |
f7eac6eb | 742 | static const float qs5[6] = { |
f7eac6eb RM |
743 | 8.1276550293e+01, /* 0x42a28d98 */ |
744 | 1.9917987061e+03, /* 0x44f8f98f */ | |
745 | 1.7468484375e+04, /* 0x468878f8 */ | |
746 | 4.9851425781e+04, /* 0x4742bb6d */ | |
747 | 2.7948074219e+04, /* 0x46da5826 */ | |
748 | -4.7191835938e+03, /* 0xc5937978 */ | |
749 | }; | |
750 | ||
f7eac6eb | 751 | static const float qr3[6] = { |
f7eac6eb RM |
752 | -5.0783124372e-09, /* 0xb1ae7d4f */ |
753 | -1.0253783315e-01, /* 0xbdd1ff5b */ | |
754 | -4.6101160049e+00, /* 0xc0938612 */ | |
755 | -5.7847221375e+01, /* 0xc267638e */ | |
756 | -2.2824453735e+02, /* 0xc3643e9a */ | |
757 | -2.1921012878e+02, /* 0xc35b35cb */ | |
758 | }; | |
f7eac6eb | 759 | static const float qs3[6] = { |
f7eac6eb RM |
760 | 4.7665153503e+01, /* 0x423ea91e */ |
761 | 6.7386511230e+02, /* 0x4428775e */ | |
762 | 3.3801528320e+03, /* 0x45534272 */ | |
763 | 5.5477290039e+03, /* 0x45ad5dd5 */ | |
764 | 1.9031191406e+03, /* 0x44ede3d0 */ | |
765 | -1.3520118713e+02, /* 0xc3073381 */ | |
766 | }; | |
767 | ||
f7eac6eb | 768 | static const float qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ |
f7eac6eb RM |
769 | -1.7838172539e-07, /* 0xb43f8932 */ |
770 | -1.0251704603e-01, /* 0xbdd1f475 */ | |
771 | -2.7522056103e+00, /* 0xc0302423 */ | |
772 | -1.9663616180e+01, /* 0xc19d4f16 */ | |
773 | -4.2325313568e+01, /* 0xc2294d1f */ | |
774 | -2.1371921539e+01, /* 0xc1aaf9b2 */ | |
775 | }; | |
f7eac6eb | 776 | static const float qs2[6] = { |
f7eac6eb RM |
777 | 2.9533363342e+01, /* 0x41ec4454 */ |
778 | 2.5298155212e+02, /* 0x437cfb47 */ | |
779 | 7.5750280762e+02, /* 0x443d602e */ | |
780 | 7.3939318848e+02, /* 0x4438d92a */ | |
781 | 1.5594900513e+02, /* 0x431bf2f2 */ | |
782 | -4.9594988823e+00, /* 0xc09eb437 */ | |
783 | }; | |
784 | ||
0ac5ae23 UD |
785 | static float |
786 | qonef(float x) | |
f7eac6eb | 787 | { |
f7eac6eb | 788 | const float *p,*q; |
f7eac6eb RM |
789 | float s,r,z; |
790 | int32_t ix; | |
791 | GET_FLOAT_WORD(ix,x); | |
792 | ix &= 0x7fffffff; | |
09220e66 | 793 | /* ix >= 0x40000000 for all calls to this function. */ |
9acda61d PZ |
794 | if(ix>=0x41000000) {p = qr8; q= qs8;} /* x >= 8 */ |
795 | else if(ix>=0x40f71c58){p = qr5; q= qs5;} /* x >= 7.722209930e+00 */ | |
796 | else if(ix>=0x4036db68){p = qr3; q= qs3;} /* x >= 2.857141495e+00 */ | |
797 | else {p = qr2; q= qs2;} /* x >= 2 */ | |
f7eac6eb RM |
798 | z = one/(x*x); |
799 | r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); | |
800 | s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5]))))); | |
801 | return ((float).375 + r/s)/x; | |
802 | } |