1/* e_j1f.c -- float version of e_j1.c.
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
10 * software is freely granted, provided that this notice
11 * is preserved.
12 * ====================================================
13 */
14
15#include <errno.h>
16#include <float.h>
17#include <math.h>
18#include <math-narrow-eval.h>
19#include <math_private.h>
20#include <fenv_private.h>
21#include <math-underflow.h>
22#include <libm-alias-finite.h>
23#include <reduce_aux.h>
24
25static float ponef(float), qonef(float);
26
27static const float
28huge = 1e30,
29one = 1.0,
30invsqrtpi= 5.6418961287e-01, /* 0x3f106ebb */
31tpi = 6.3661974669e-01, /* 0x3f22f983 */
32 /* R0/S0 on [0,2] */
33r00 = -6.2500000000e-02, /* 0xbd800000 */
34r01 = 1.4070566976e-03, /* 0x3ab86cfd */
35r02 = -1.5995563444e-05, /* 0xb7862e36 */
36r03 = 4.9672799207e-08, /* 0x335557d2 */
37s01 = 1.9153760746e-02, /* 0x3c9ce859 */
38s02 = 1.8594678841e-04, /* 0x3942fab6 */
39s03 = 1.1771846857e-06, /* 0x359dffc2 */
40s04 = 5.0463624390e-09, /* 0x31ad6446 */
41s05 = 1.2354227016e-11; /* 0x2d59567e */
42
43static const float zero = 0.0;
44
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*/
62static 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). */
199static float
200j1f_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). */
235static float
236j1f_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 }
246 index_f = roundf ((x - FIRST_ZERO_J1) / M_PIf);
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
262float
263__ieee754_j1f(float x)
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;
270 if(__builtin_expect(ix>=0x7f800000, 0)) return one/x;
271 y = fabsf(x);
272 if(ix >= 0x40000000) { /* |x| >= 2.0 */
273 SET_RESTORE_ROUNDF (FE_TONEAREST);
274 __sincosf (y, &s, &c);
275 ss = -s-c;
276 cc = s-c;
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;
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 */
288 if (ix <= 0x5c000000)
289 {
290 u = ponef(y); v = qonef(y);
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);
304 }
305 if(__builtin_expect(ix<0x32000000, 0)) { /* |x|<2**-27 */
306 if(huge+x>one) { /* inexact if x!=0 necessary */
307 float ret = math_narrow_eval ((float) 0.5 * x);
308 math_check_force_underflow (ret);
309 if (ret == 0 && x != 0)
310 __set_errno (ERANGE);
311 return ret;
312 }
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}
320libm_alias_finite (__ieee754_j1f, __j1f)
321
322static const float U0[5] = {
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};
329static const float V0[5] = {
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
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*/
351static 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). */
491static float
492y1f_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). */
522static float
523y1f_near_root (float x, float z)
524{
525 float index_f;
526 int index;
527
528 index_f = roundf ((x - FIRST_ZERO_Y1) / M_PIf);
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
549float
550__ieee754_y1f(float x)
551{
552 float z, s,c,ss,cc,u,v;
553 int32_t hx,ix;
554
555 GET_FLOAT_WORD(hx,x);
556 ix = 0x7fffffff&hx;
557 /* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */
558 if(__builtin_expect(ix>=0x7f800000, 0)) return one/(x+x*x);
559 if(__builtin_expect(ix==0, 0))
560 return -1/zero; /* -inf and divide by zero exception. */
561 if(__builtin_expect(hx<0, 0)) return zero/(zero*x);
562 if (ix >= 0x3fe0dfbc) { /* |x| >= 0x1.c1bf78p+0 */
563 SET_RESTORE_ROUNDF (FE_TONEAREST);
564 __sincosf (x, &s, &c);
565 ss = -s-c;
566 cc = s-c;
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;
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 */
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);
599 }
600 if(__builtin_expect(ix<=0x33000000, 0)) { /* x < 2**-25 */
601 z = -tpi / x;
602 if (isinf (z))
603 __set_errno (ERANGE);
604 return z;
605 }
606 /* Now 2**-25 <= x < 0x1.c1bf78p+0. */
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));
611}
612libm_alias_finite (__ieee754_y1f, __y1f)
613
614/* For x >= 8, the asymptotic expansion of pone is
615 * 1 + 15/128 s^2 - 4725/2^15 s^4 - ..., where s = 1/x.
616 * We approximate pone by
617 * pone(x) = 1 + (R/S)
618 * where R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10
619 * S = 1 + ps0*s^2 + ... + ps4*s^10
620 * and
621 * | pone(x)-1-R/S | <= 2 ** ( -60.06)
622 */
623
624static const float pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
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};
632static const float ps8[5] = {
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
640static const float pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
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};
648static const float ps5[5] = {
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
656static const float pr3[6] = {
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};
664static const float ps3[5] = {
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
672static const float pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
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};
680static const float ps2[5] = {
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
688static float
689ponef(float x)
690{
691 const float *p,*q;
692 float z,r,s;
693 int32_t ix;
694 GET_FLOAT_WORD(ix,x);
695 ix &= 0x7fffffff;
696 /* ix >= 0x40000000 for all calls to this function. */
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;}
700 else {p = pr2; q= ps2;}
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;
705}
706
707/* For x >= 8, the asymptotic expansion of qone is
708 * 3/8 s - 105/1024 s^3 - ..., where s = 1/x.
709 * We approximate pone by
710 * qone(x) = s*(0.375 + (R/S))
711 * where R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10
712 * S = 1 + qs1*s^2 + ... + qs6*s^12
713 * and
714 * | qone(x)/s -0.375-R/S | <= 2 ** ( -61.13)
715 */
716
717static const float qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
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};
725static const float qs8[6] = {
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
734static const float qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
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};
742static const float qs5[6] = {
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
751static const float qr3[6] = {
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};
759static const float qs3[6] = {
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
768static const float qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
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};
776static const float qs2[6] = {
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
785static float
786qonef(float x)
787{
788 const float *p,*q;
789 float s,r,z;
790 int32_t ix;
791 GET_FLOAT_WORD(ix,x);
792 ix &= 0x7fffffff;
793 /* ix >= 0x40000000 for all calls to this function. */
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 */
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}
803