1 /* e_j1f.c -- float version of e_j1.c.
5 * ====================================================
6 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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
12 * ====================================================
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>
25 static float ponef(float), qonef(float);
30 invsqrtpi
= 5.6418961287e-01, /* 0x3f106ebb */
31 tpi
= 6.3661974669e-01, /* 0x3f22f983 */
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 */
43 static const float zero
= 0.0;
45 /* This is the nearest approximation of the first positive zero of j1. */
46 #define FIRST_ZERO_J1 0x3.d4eabp+0f
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.
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.
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 */
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). */
202 float cst
= 0xc.c422ap
-4; /* sqrt(2/pi) rounded to nearest */
208 double y
= 1.0 / (double) x
;
210 double beta1
= 1.0f
+ y2
* (0x3p
-4 - 0x3.18p
-4 * y2
);
212 alpha1
= y
* (-0x6p
-4 + y2
* (0x2.ap
-4 - 0x5.ef33333333334p
-4 * y2
));
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
;
220 float t
= cst
/ sqrtf (x
) * (float) beta1
;
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
);
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). */
236 j1f_near_root (float x
, float z
)
238 float index_f
, sign
= 1.0f
;
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
];
253 /* If not in the interval [x0,x1] around xmid, return the value z. */
254 if (! (x0
<= x
&& x
<= x1
))
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
)));
263 __ieee754_j1f(float x
)
265 float z
, s
,c
,ss
,cc
,r
,u
,v
,y
;
268 GET_FLOAT_WORD(hx
,x
);
270 if(__builtin_expect(ix
>=0x7f800000, 0)) return one
/x
;
272 if(ix
>= 0x40000000) { /* |x| >= 2.0 */
273 SET_RESTORE_ROUNDF (FE_TONEAREST
);
274 __sincosf (y
, &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. */
282 if ((s
*c
)>zero
) cc
= z
/ss
;
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)
288 if (ix
<= 0x5c000000)
290 u
= ponef(y
); v
= qonef(y
);
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
)
303 return j1f_near_root (x
, z
);
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
);
315 r
= z
*(r00
+z
*(r01
+z
*(r02
+z
*r03
)));
316 s
= one
+z
*(s01
+z
*(s02
+z
*(s03
+z
*(s04
+z
*s05
))));
318 return(x
*(float)0.5+r
/s
);
320 libm_alias_finite (__ieee754_j1f
, __j1f
)
322 static 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 */
329 static 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 */
337 /* This is the nearest approximation of the first zero of y1. */
338 #define FIRST_ZERO_Y1 0x2.3277dcp+0f
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.
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.
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 */
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). */
494 float cst
= 0xc.c422ap
-4; /* sqrt(2/pi) rounded to nearest */
495 double y
= 1.0 / (double) x
;
497 double beta1
= 1.0f
+ y2
* (0x3p
-4 - 0x3.18p
-4 * y2
);
499 alpha1
= y
* (-0x6p
-4 + y2
* (0x2.ap
-4 - 0x5.ef33333333334p
-4 * y2
));
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
;
507 float t
= cst
/ sqrtf (x
) * (float) beta1
;
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
);
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). */
523 y1f_near_root (float x
, float z
)
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
];
535 /* If not in the interval [x0,x1] around xmid, return the value z. */
536 if (! (x0
<= x
&& x
<= x1
))
539 float y
= x
- xmid
, p6
;
541 p6
= p
[6] + y
* (-0x1.28043p
-8 + y
* 0x2.50e83p
-8);
543 p6
= p
[6] + y
* -0xf.ff6b8p
-12;
546 return p
[3] + y
* (p
[4] + y
* (p
[5] + y
* p6
));
550 __ieee754_y1f(float x
)
552 float z
, s
,c
,ss
,cc
,u
,v
;
555 GET_FLOAT_WORD(hx
,x
);
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
);
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. */
572 if ((s
*c
)>zero
) cc
= z
/ss
;
574 /* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0))
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.
585 if (ix
<= 0x5c000000)
587 u
= ponef(x
); v
= qonef(x
);
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
)
598 return y1f_near_root (x
, z
);
600 if(__builtin_expect(ix
<=0x33000000, 0)) { /* x < 2**-25 */
603 __set_errno (ERANGE
);
606 /* Now 2**-25 <= x < 0x1.c1bf78p+0. */
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
));
612 libm_alias_finite (__ieee754_y1f
, __y1f
)
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
621 * | pone(x)-1-R/S | <= 2 ** ( -60.06)
624 static 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 */
632 static 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 */
640 static 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 */
648 static 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 */
656 static 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 */
664 static 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 */
672 static 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 */
680 static 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 */
694 GET_FLOAT_WORD(ix
,x
);
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
;}
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]))));
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
714 * | qone(x)/s -0.375-R/S | <= 2 ** ( -61.13)
717 static 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 */
725 static 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 */
734 static 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 */
742 static 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 */
751 static 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 */
759 static 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 */
768 static 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 */
776 static 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 */
791 GET_FLOAT_WORD(ix
,x
);
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 */
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
;