1/* e_j1f.c -- float version of e_j1.c.
2 * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
3 */
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#include <errno.h>
17#include <float.h>
18#include <math.h>
19#include <math-narrow-eval.h>
20#include <math_private.h>
21#include <math-underflow.h>
22
23static float ponef(float), qonef(float);
24
25static const float
26huge = 1e30,
27one = 1.0,
28invsqrtpi= 5.6418961287e-01, /* 0x3f106ebb */
29tpi = 6.3661974669e-01, /* 0x3f22f983 */
30 /* R0/S0 on [0,2] */
31r00 = -6.2500000000e-02, /* 0xbd800000 */
32r01 = 1.4070566976e-03, /* 0x3ab86cfd */
33r02 = -1.5995563444e-05, /* 0xb7862e36 */
34r03 = 4.9672799207e-08, /* 0x335557d2 */
35s01 = 1.9153760746e-02, /* 0x3c9ce859 */
36s02 = 1.8594678841e-04, /* 0x3942fab6 */
37s03 = 1.1771846857e-06, /* 0x359dffc2 */
38s04 = 5.0463624390e-09, /* 0x31ad6446 */
39s05 = 1.2354227016e-11; /* 0x2d59567e */
40
41static const float zero = 0.0;
42
43float
44__ieee754_j1f(float x)
45{
46 float z, s,c,ss,cc,r,u,v,y;
47 int32_t hx,ix;
48
49 GET_FLOAT_WORD(hx,x);
50 ix = hx&0x7fffffff;
51 if(__builtin_expect(ix>=0x7f800000, 0)) return one/x;
52 y = fabsf(x);
53 if(ix >= 0x40000000) { /* |x| >= 2.0 */
54 __sincosf (y, &s, &c);
55 ss = -s-c;
56 cc = s-c;
57 if(ix<0x7f000000) { /* make sure y+y not overflow */
58 z = __cosf(y+y);
59 if ((s*c)>zero) cc = z/ss;
60 else ss = z/cc;
61 }
62 /*
63 * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x)
64 * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x)
65 */
66 if(ix>0x48000000) z = (invsqrtpi*cc)/sqrtf(y);
67 else {
68 u = ponef(y); v = qonef(y);
69 z = invsqrtpi*(u*cc-v*ss)/sqrtf(y);
70 }
71 if(hx<0) return -z;
72 else return z;
73 }
74 if(__builtin_expect(ix<0x32000000, 0)) { /* |x|<2**-27 */
75 if(huge+x>one) { /* inexact if x!=0 necessary */
76 float ret = math_narrow_eval ((float) 0.5 * x);
77 math_check_force_underflow (ret);
78 if (ret == 0 && x != 0)
79 __set_errno (ERANGE);
80 return ret;
81 }
82 }
83 z = x*x;
84 r = z*(r00+z*(r01+z*(r02+z*r03)));
85 s = one+z*(s01+z*(s02+z*(s03+z*(s04+z*s05))));
86 r *= x;
87 return(x*(float)0.5+r/s);
88}
89strong_alias (__ieee754_j1f, __j1f_finite)
90
91static const float U0[5] = {
92 -1.9605709612e-01, /* 0xbe48c331 */
93 5.0443872809e-02, /* 0x3d4e9e3c */
94 -1.9125689287e-03, /* 0xbafaaf2a */
95 2.3525259166e-05, /* 0x37c5581c */
96 -9.1909917899e-08, /* 0xb3c56003 */
97};
98static const float V0[5] = {
99 1.9916731864e-02, /* 0x3ca3286a */
100 2.0255257550e-04, /* 0x3954644b */
101 1.3560879779e-06, /* 0x35b602d4 */
102 6.2274145840e-09, /* 0x31d5f8eb */
103 1.6655924903e-11, /* 0x2d9281cf */
104};
105
106float
107__ieee754_y1f(float x)
108{
109 float z, s,c,ss,cc,u,v;
110 int32_t hx,ix;
111
112 GET_FLOAT_WORD(hx,x);
113 ix = 0x7fffffff&hx;
114 /* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */
115 if(__builtin_expect(ix>=0x7f800000, 0)) return one/(x+x*x);
116 if(__builtin_expect(ix==0, 0))
117 return -1/zero; /* -inf and divide by zero exception. */
118 if(__builtin_expect(hx<0, 0)) return zero/(zero*x);
119 if(ix >= 0x40000000) { /* |x| >= 2.0 */
120 SET_RESTORE_ROUNDF (FE_TONEAREST);
121 __sincosf (x, &s, &c);
122 ss = -s-c;
123 cc = s-c;
124 if(ix<0x7f000000) { /* make sure x+x not overflow */
125 z = __cosf(x+x);
126 if ((s*c)>zero) cc = z/ss;
127 else ss = z/cc;
128 }
129 /* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0))
130 * where x0 = x-3pi/4
131 * Better formula:
132 * cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
133 * = 1/sqrt(2) * (sin(x) - cos(x))
134 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
135 * = -1/sqrt(2) * (cos(x) + sin(x))
136 * To avoid cancellation, use
137 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
138 * to compute the worse one.
139 */
140 if(ix>0x48000000) z = (invsqrtpi*ss)/sqrtf(x);
141 else {
142 u = ponef(x); v = qonef(x);
143 z = invsqrtpi*(u*ss+v*cc)/sqrtf(x);
144 }
145 return z;
146 }
147 if(__builtin_expect(ix<=0x33000000, 0)) { /* x < 2**-25 */
148 z = -tpi / x;
149 if (isinf (z))
150 __set_errno (ERANGE);
151 return z;
152 }
153 z = x*x;
154 u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4])));
155 v = one+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4]))));
156 return(x*(u/v) + tpi*(__ieee754_j1f(x)*__ieee754_logf(x)-one/x));
157}
158strong_alias (__ieee754_y1f, __y1f_finite)
159
160/* For x >= 8, the asymptotic expansions of pone is
161 * 1 + 15/128 s^2 - 4725/2^15 s^4 - ..., where s = 1/x.
162 * We approximate pone by
163 * pone(x) = 1 + (R/S)
164 * where R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10
165 * S = 1 + ps0*s^2 + ... + ps4*s^10
166 * and
167 * | pone(x)-1-R/S | <= 2 ** ( -60.06)
168 */
169
170static const float pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
171 0.0000000000e+00, /* 0x00000000 */
172 1.1718750000e-01, /* 0x3df00000 */
173 1.3239480972e+01, /* 0x4153d4ea */
174 4.1205184937e+02, /* 0x43ce06a3 */
175 3.8747453613e+03, /* 0x45722bed */
176 7.9144794922e+03, /* 0x45f753d6 */
177};
178static const float ps8[5] = {
179 1.1420736694e+02, /* 0x42e46a2c */
180 3.6509309082e+03, /* 0x45642ee5 */
181 3.6956207031e+04, /* 0x47105c35 */
182 9.7602796875e+04, /* 0x47bea166 */
183 3.0804271484e+04, /* 0x46f0a88b */
184};
185
186static const float pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
187 1.3199052094e-11, /* 0x2d68333f */
188 1.1718749255e-01, /* 0x3defffff */
189 6.8027510643e+00, /* 0x40d9b023 */
190 1.0830818176e+02, /* 0x42d89dca */
191 5.1763616943e+02, /* 0x440168b7 */
192 5.2871520996e+02, /* 0x44042dc6 */
193};
194static const float ps5[5] = {
195 5.9280597687e+01, /* 0x426d1f55 */
196 9.9140142822e+02, /* 0x4477d9b1 */
197 5.3532670898e+03, /* 0x45a74a23 */
198 7.8446904297e+03, /* 0x45f52586 */
199 1.5040468750e+03, /* 0x44bc0180 */
200};
201
202static const float pr3[6] = {
203 3.0250391081e-09, /* 0x314fe10d */
204 1.1718686670e-01, /* 0x3defffab */
205 3.9329774380e+00, /* 0x407bb5e7 */
206 3.5119403839e+01, /* 0x420c7a45 */
207 9.1055007935e+01, /* 0x42b61c2a */
208 4.8559066772e+01, /* 0x42423c7c */
209};
210static const float ps3[5] = {
211 3.4791309357e+01, /* 0x420b2a4d */
212 3.3676245117e+02, /* 0x43a86198 */
213 1.0468714600e+03, /* 0x4482dbe3 */
214 8.9081134033e+02, /* 0x445eb3ed */
215 1.0378793335e+02, /* 0x42cf936c */
216};
217
218static const float pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
219 1.0771083225e-07, /* 0x33e74ea8 */
220 1.1717621982e-01, /* 0x3deffa16 */
221 2.3685150146e+00, /* 0x401795c0 */
222 1.2242610931e+01, /* 0x4143e1bc */
223 1.7693971634e+01, /* 0x418d8d41 */
224 5.0735230446e+00, /* 0x40a25a4d */
225};
226static const float ps2[5] = {
227 2.1436485291e+01, /* 0x41ab7dec */
228 1.2529022980e+02, /* 0x42fa9499 */
229 2.3227647400e+02, /* 0x436846c7 */
230 1.1767937469e+02, /* 0x42eb5bd7 */
231 8.3646392822e+00, /* 0x4105d590 */
232};
233
234static float
235ponef(float x)
236{
237 const float *p,*q;
238 float z,r,s;
239 int32_t ix;
240 GET_FLOAT_WORD(ix,x);
241 ix &= 0x7fffffff;
242 /* ix >= 0x40000000 for all calls to this function. */
243 if(ix>=0x41000000) {p = pr8; q= ps8;}
244 else if(ix>=0x40f71c58){p = pr5; q= ps5;}
245 else if(ix>=0x4036db68){p = pr3; q= ps3;}
246 else {p = pr2; q= ps2;}
247 z = one/(x*x);
248 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
249 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
250 return one+ r/s;
251}
252
253
254/* For x >= 8, the asymptotic expansions of qone is
255 * 3/8 s - 105/1024 s^3 - ..., where s = 1/x.
256 * We approximate pone by
257 * qone(x) = s*(0.375 + (R/S))
258 * where R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10
259 * S = 1 + qs1*s^2 + ... + qs6*s^12
260 * and
261 * | qone(x)/s -0.375-R/S | <= 2 ** ( -61.13)
262 */
263
264static const float qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
265 0.0000000000e+00, /* 0x00000000 */
266 -1.0253906250e-01, /* 0xbdd20000 */
267 -1.6271753311e+01, /* 0xc1822c8d */
268 -7.5960174561e+02, /* 0xc43de683 */
269 -1.1849806641e+04, /* 0xc639273a */
270 -4.8438511719e+04, /* 0xc73d3683 */
271};
272static const float qs8[6] = {
273 1.6139537048e+02, /* 0x43216537 */
274 7.8253862305e+03, /* 0x45f48b17 */
275 1.3387534375e+05, /* 0x4802bcd6 */
276 7.1965775000e+05, /* 0x492fb29c */
277 6.6660125000e+05, /* 0x4922be94 */
278 -2.9449025000e+05, /* 0xc88fcb48 */
279};
280
281static const float qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
282 -2.0897993405e-11, /* 0xadb7d219 */
283 -1.0253904760e-01, /* 0xbdd1fffe */
284 -8.0564479828e+00, /* 0xc100e736 */
285 -1.8366960144e+02, /* 0xc337ab6b */
286 -1.3731937256e+03, /* 0xc4aba633 */
287 -2.6124443359e+03, /* 0xc523471c */
288};
289static const float qs5[6] = {
290 8.1276550293e+01, /* 0x42a28d98 */
291 1.9917987061e+03, /* 0x44f8f98f */
292 1.7468484375e+04, /* 0x468878f8 */
293 4.9851425781e+04, /* 0x4742bb6d */
294 2.7948074219e+04, /* 0x46da5826 */
295 -4.7191835938e+03, /* 0xc5937978 */
296};
297
298static const float qr3[6] = {
299 -5.0783124372e-09, /* 0xb1ae7d4f */
300 -1.0253783315e-01, /* 0xbdd1ff5b */
301 -4.6101160049e+00, /* 0xc0938612 */
302 -5.7847221375e+01, /* 0xc267638e */
303 -2.2824453735e+02, /* 0xc3643e9a */
304 -2.1921012878e+02, /* 0xc35b35cb */
305};
306static const float qs3[6] = {
307 4.7665153503e+01, /* 0x423ea91e */
308 6.7386511230e+02, /* 0x4428775e */
309 3.3801528320e+03, /* 0x45534272 */
310 5.5477290039e+03, /* 0x45ad5dd5 */
311 1.9031191406e+03, /* 0x44ede3d0 */
312 -1.3520118713e+02, /* 0xc3073381 */
313};
314
315static const float qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
316 -1.7838172539e-07, /* 0xb43f8932 */
317 -1.0251704603e-01, /* 0xbdd1f475 */
318 -2.7522056103e+00, /* 0xc0302423 */
319 -1.9663616180e+01, /* 0xc19d4f16 */
320 -4.2325313568e+01, /* 0xc2294d1f */
321 -2.1371921539e+01, /* 0xc1aaf9b2 */
322};
323static const float qs2[6] = {
324 2.9533363342e+01, /* 0x41ec4454 */
325 2.5298155212e+02, /* 0x437cfb47 */
326 7.5750280762e+02, /* 0x443d602e */
327 7.3939318848e+02, /* 0x4438d92a */
328 1.5594900513e+02, /* 0x431bf2f2 */
329 -4.9594988823e+00, /* 0xc09eb437 */
330};
331
332static float
333qonef(float x)
334{
335 const float *p,*q;
336 float s,r,z;
337 int32_t ix;
338 GET_FLOAT_WORD(ix,x);
339 ix &= 0x7fffffff;
340 /* ix >= 0x40000000 for all calls to this function. */
341 if(ix>=0x40200000) {p = qr8; q= qs8;}
342 else if(ix>=0x40f71c58){p = qr5; q= qs5;}
343 else if(ix>=0x4036db68){p = qr3; q= qs3;}
344 else {p = qr2; q= qs2;}
345 z = one/(x*x);
346 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
347 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
348 return ((float).375 + r/s)/x;
349}
350