1 /* origin: FreeBSD /usr/src/lib/msun/src/e_j1f.c */
2 /*
3 * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
4 */
5 /*
6 * ====================================================
7 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
8 *
9 * Developed at SunPro, a Sun Microsystems, Inc. business.
10 * Permission to use, copy, modify, and distribute this
11 * software is freely granted, provided that this notice
12 * is preserved.
13 * ====================================================
14 */
15
16 #define _GNU_SOURCE
17 #include "libm.h"
18
19 static float ponef(float), qonef(float);
20
21 static const float
22 invsqrtpi = 5.6418961287e-01, /* 0x3f106ebb */
23 tpi = 6.3661974669e-01; /* 0x3f22f983 */
24
25 static float common(uint32_t ix, float x, int y1, int sign)
26 {
27 double z,s,c,ss,cc;
28
29 s = sinf(x);
30 if (y1)
31 s = -s;
32 c = cosf(x);
33 cc = s-c;
34 if (ix < 0x7f000000) {
35 ss = -s-c;
36 z = cosf(2*x);
37 if (s*c > 0)
38 cc = z/ss;
39 else
40 ss = z/cc;
41 if (ix < 0x58800000) {
42 if (y1)
43 ss = -ss;
44 cc = ponef(x)*cc-qonef(x)*ss;
45 }
46 }
47 if (sign)
48 cc = -cc;
49 return invsqrtpi*cc/sqrtf(x);
50 }
51
52 /* R0/S0 on [0,2] */
53 static const float
54 r00 = -6.2500000000e-02, /* 0xbd800000 */
55 r01 = 1.4070566976e-03, /* 0x3ab86cfd */
56 r02 = -1.5995563444e-05, /* 0xb7862e36 */
57 r03 = 4.9672799207e-08, /* 0x335557d2 */
58 s01 = 1.9153760746e-02, /* 0x3c9ce859 */
59 s02 = 1.8594678841e-04, /* 0x3942fab6 */
60 s03 = 1.1771846857e-06, /* 0x359dffc2 */
61 s04 = 5.0463624390e-09, /* 0x31ad6446 */
62 s05 = 1.2354227016e-11; /* 0x2d59567e */
63
64 float j1f(float x)
65 {
66 float z,r,s;
67 uint32_t ix;
68 int sign;
69
70 GET_FLOAT_WORD(ix, x);
71 sign = ix>>31;
72 ix &= 0x7fffffff;
73 if (ix >= 0x7f800000)
74 return 1/(x*x);
75 if (ix >= 0x40000000) /* |x| >= 2 */
76 return common(ix, fabsf(x), 0, sign);
77 if (ix >= 0x39000000) { /* |x| >= 2**-13 */
78 z = x*x;
79 r = z*(r00+z*(r01+z*(r02+z*r03)));
80 s = 1+z*(s01+z*(s02+z*(s03+z*(s04+z*s05))));
81 z = 0.5f + r/s;
82 } else
83 z = 0.5f;
84 return z*x;
85 }
86
87 static const float U0[5] = {
88 -1.9605709612e-01, /* 0xbe48c331 */
89 5.0443872809e-02, /* 0x3d4e9e3c */
90 -1.9125689287e-03, /* 0xbafaaf2a */
91 2.3525259166e-05, /* 0x37c5581c */
92 -9.1909917899e-08, /* 0xb3c56003 */
93 };
94 static const float V0[5] = {
95 1.9916731864e-02, /* 0x3ca3286a */
96 2.0255257550e-04, /* 0x3954644b */
97 1.3560879779e-06, /* 0x35b602d4 */
98 6.2274145840e-09, /* 0x31d5f8eb */
99 1.6655924903e-11, /* 0x2d9281cf */
100 };
101
102 float y1f(float x)
103 {
104 float z,u,v;
105 uint32_t ix;
106
107 GET_FLOAT_WORD(ix, x);
108 if ((ix & 0x7fffffff) == 0)
109 return -1/0.0f;
110 if (ix>>31)
111 return 0/0.0f;
112 if (ix >= 0x7f800000)
113 return 1/x;
114 if (ix >= 0x40000000) /* |x| >= 2.0 */
115 return common(ix,x,1,0);
116 if (ix < 0x33000000) /* x < 2**-25 */
117 return -tpi/x;
118 z = x*x;
119 u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4])));
120 v = 1.0f+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4]))));
121 return x*(u/v) + tpi*(j1f(x)*logf(x)-1.0f/x);
122 }
123
124 /* For x >= 8, the asymptotic expansions of pone is
125 * 1 + 15/128 s^2 - 4725/2^15 s^4 - ..., where s = 1/x.
126 * We approximate pone by
127 * pone(x) = 1 + (R/S)
128 * where R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10
129 * S = 1 + ps0*s^2 + ... + ps4*s^10
130 * and
131 * | pone(x)-1-R/S | <= 2 ** ( -60.06)
132 */
133
134 static const float pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
135 0.0000000000e+00, /* 0x00000000 */
136 1.1718750000e-01, /* 0x3df00000 */
137 1.3239480972e+01, /* 0x4153d4ea */
138 4.1205184937e+02, /* 0x43ce06a3 */
139 3.8747453613e+03, /* 0x45722bed */
140 7.9144794922e+03, /* 0x45f753d6 */
141 };
142 static const float ps8[5] = {
143 1.1420736694e+02, /* 0x42e46a2c */
144 3.6509309082e+03, /* 0x45642ee5 */
145 3.6956207031e+04, /* 0x47105c35 */
146 9.7602796875e+04, /* 0x47bea166 */
147 3.0804271484e+04, /* 0x46f0a88b */
148 };
149
150 static const float pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
151 1.3199052094e-11, /* 0x2d68333f */
152 1.1718749255e-01, /* 0x3defffff */
153 6.8027510643e+00, /* 0x40d9b023 */
154 1.0830818176e+02, /* 0x42d89dca */
155 5.1763616943e+02, /* 0x440168b7 */
156 5.2871520996e+02, /* 0x44042dc6 */
157 };
158 static const float ps5[5] = {
159 5.9280597687e+01, /* 0x426d1f55 */
160 9.9140142822e+02, /* 0x4477d9b1 */
161 5.3532670898e+03, /* 0x45a74a23 */
162 7.8446904297e+03, /* 0x45f52586 */
163 1.5040468750e+03, /* 0x44bc0180 */
164 };
165
166 static const float pr3[6] = {
167 3.0250391081e-09, /* 0x314fe10d */
168 1.1718686670e-01, /* 0x3defffab */
169 3.9329774380e+00, /* 0x407bb5e7 */
170 3.5119403839e+01, /* 0x420c7a45 */
171 9.1055007935e+01, /* 0x42b61c2a */
172 4.8559066772e+01, /* 0x42423c7c */
173 };
174 static const float ps3[5] = {
175 3.4791309357e+01, /* 0x420b2a4d */
176 3.3676245117e+02, /* 0x43a86198 */
177 1.0468714600e+03, /* 0x4482dbe3 */
178 8.9081134033e+02, /* 0x445eb3ed */
179 1.0378793335e+02, /* 0x42cf936c */
180 };
181
182 static const float pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
183 1.0771083225e-07, /* 0x33e74ea8 */
184 1.1717621982e-01, /* 0x3deffa16 */
185 2.3685150146e+00, /* 0x401795c0 */
186 1.2242610931e+01, /* 0x4143e1bc */
187 1.7693971634e+01, /* 0x418d8d41 */
188 5.0735230446e+00, /* 0x40a25a4d */
189 };
190 static const float ps2[5] = {
191 2.1436485291e+01, /* 0x41ab7dec */
192 1.2529022980e+02, /* 0x42fa9499 */
193 2.3227647400e+02, /* 0x436846c7 */
194 1.1767937469e+02, /* 0x42eb5bd7 */
195 8.3646392822e+00, /* 0x4105d590 */
196 };
197
198 static float ponef(float x)
199 {
200 const float *p,*q;
201 float_t z,r,s;
202 uint32_t ix;
203
204 GET_FLOAT_WORD(ix, x);
205 ix &= 0x7fffffff;
206 if (ix >= 0x41000000){p = pr8; q = ps8;}
207 else if (ix >= 0x409173eb){p = pr5; q = ps5;}
208 else if (ix >= 0x4036d917){p = pr3; q = ps3;}
209 else /*ix >= 0x40000000*/ {p = pr2; q = ps2;}
210 z = 1.0f/(x*x);
211 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
212 s = 1.0f+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
213 return 1.0f + r/s;
214 }
215
216 /* For x >= 8, the asymptotic expansions of qone is
217 * 3/8 s - 105/1024 s^3 - ..., where s = 1/x.
218 * We approximate pone by
219 * qone(x) = s*(0.375 + (R/S))
220 * where R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10
221 * S = 1 + qs1*s^2 + ... + qs6*s^12
222 * and
223 * | qone(x)/s -0.375-R/S | <= 2 ** ( -61.13)
224 */
225
226 static const float qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
227 0.0000000000e+00, /* 0x00000000 */
228 -1.0253906250e-01, /* 0xbdd20000 */
229 -1.6271753311e+01, /* 0xc1822c8d */
230 -7.5960174561e+02, /* 0xc43de683 */
231 -1.1849806641e+04, /* 0xc639273a */
232 -4.8438511719e+04, /* 0xc73d3683 */
233 };
234 static const float qs8[6] = {
235 1.6139537048e+02, /* 0x43216537 */
236 7.8253862305e+03, /* 0x45f48b17 */
237 1.3387534375e+05, /* 0x4802bcd6 */
238 7.1965775000e+05, /* 0x492fb29c */
239 6.6660125000e+05, /* 0x4922be94 */
240 -2.9449025000e+05, /* 0xc88fcb48 */
241 };
242
243 static const float qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
244 -2.0897993405e-11, /* 0xadb7d219 */
245 -1.0253904760e-01, /* 0xbdd1fffe */
246 -8.0564479828e+00, /* 0xc100e736 */
247 -1.8366960144e+02, /* 0xc337ab6b */
248 -1.3731937256e+03, /* 0xc4aba633 */
249 -2.6124443359e+03, /* 0xc523471c */
250 };
251 static const float qs5[6] = {
252 8.1276550293e+01, /* 0x42a28d98 */
253 1.9917987061e+03, /* 0x44f8f98f */
254 1.7468484375e+04, /* 0x468878f8 */
255 4.9851425781e+04, /* 0x4742bb6d */
256 2.7948074219e+04, /* 0x46da5826 */
257 -4.7191835938e+03, /* 0xc5937978 */
258 };
259
260 static const float qr3[6] = {
261 -5.0783124372e-09, /* 0xb1ae7d4f */
262 -1.0253783315e-01, /* 0xbdd1ff5b */
263 -4.6101160049e+00, /* 0xc0938612 */
264 -5.7847221375e+01, /* 0xc267638e */
265 -2.2824453735e+02, /* 0xc3643e9a */
266 -2.1921012878e+02, /* 0xc35b35cb */
267 };
268 static const float qs3[6] = {
269 4.7665153503e+01, /* 0x423ea91e */
270 6.7386511230e+02, /* 0x4428775e */
271 3.3801528320e+03, /* 0x45534272 */
272 5.5477290039e+03, /* 0x45ad5dd5 */
273 1.9031191406e+03, /* 0x44ede3d0 */
274 -1.3520118713e+02, /* 0xc3073381 */
275 };
276
277 static const float qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
278 -1.7838172539e-07, /* 0xb43f8932 */
279 -1.0251704603e-01, /* 0xbdd1f475 */
280 -2.7522056103e+00, /* 0xc0302423 */
281 -1.9663616180e+01, /* 0xc19d4f16 */
282 -4.2325313568e+01, /* 0xc2294d1f */
283 -2.1371921539e+01, /* 0xc1aaf9b2 */
284 };
285 static const float qs2[6] = {
286 2.9533363342e+01, /* 0x41ec4454 */
287 2.5298155212e+02, /* 0x437cfb47 */
288 7.5750280762e+02, /* 0x443d602e */
289 7.3939318848e+02, /* 0x4438d92a */
290 1.5594900513e+02, /* 0x431bf2f2 */
291 -4.9594988823e+00, /* 0xc09eb437 */
292 };
293
294 static float qonef(float x)
295 {
296 const float *p,*q;
297 float_t s,r,z;
298 uint32_t ix;
299
300 GET_FLOAT_WORD(ix, x);
301 ix &= 0x7fffffff;
302 if (ix >= 0x41000000){p = qr8; q = qs8;}
303 else if (ix >= 0x409173eb){p = qr5; q = qs5;}
304 else if (ix >= 0x4036d917){p = qr3; q = qs3;}
305 else /*ix >= 0x40000000*/ {p = qr2; q = qs2;}
306 z = 1.0f/(x*x);
307 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
308 s = 1.0f+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
309 return (.375f + r/s)/x;
310 }
311