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/* 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
#ifndef lint
17
static char rcsid[] = "$\Id: e_j1f.c,v 1.2 1995/05/30 05:48:23 rgrimes Exp $";
18
#endif
19
 
20
#include "math.h"
21
#include "math_private.h"
22
 
23
#ifdef __STDC__
24
static float ponef(float), qonef(float);
25
#else
26
static float ponef(), qonef();
27
#endif
28
 
29
#ifdef __STDC__
30
static const float
31
#else
32
static float
33
#endif
34
huge    = 1e30,
35
one     = 1.0,
36
invsqrtpi=  5.6418961287e-01, /* 0x3f106ebb */
37
tpi      =  6.3661974669e-01, /* 0x3f22f983 */
38
        /* R0/S0 on [0,2] */
39
r00  = -6.2500000000e-02, /* 0xbd800000 */
40
r01  =  1.4070566976e-03, /* 0x3ab86cfd */
41
r02  = -1.5995563444e-05, /* 0xb7862e36 */
42
r03  =  4.9672799207e-08, /* 0x335557d2 */
43
s01  =  1.9153760746e-02, /* 0x3c9ce859 */
44
s02  =  1.8594678841e-04, /* 0x3942fab6 */
45
s03  =  1.1771846857e-06, /* 0x359dffc2 */
46
s04  =  5.0463624390e-09, /* 0x31ad6446 */
47
s05  =  1.2354227016e-11; /* 0x2d59567e */
48
 
49
#ifdef __STDC__
50
static const float zero    = 0.0;
51
#else
52
static float zero    = 0.0;
53
#endif
54
 
55
#ifdef __STDC__
56
        float __ieee754_j1f(float x)
57
#else
58
        float __ieee754_j1f(x)
59
        float x;
60
#endif
61
{
62
        float z, s,c,ss,cc,r,u,v,y;
63
        int32_t hx,ix;
64
 
65
        GET_FLOAT_WORD(hx,x);
66
        ix = hx&0x7fffffff;
67
        if(ix>=0x7f800000) return one/x;
68
        y = fabsf(x);
69
        if(ix >= 0x40000000) {  /* |x| >= 2.0 */
70
                s = sinf(y);
71
                c = cosf(y);
72
                ss = -s-c;
73
                cc = s-c;
74
                if(ix<0x7f000000) {  /* make sure y+y not overflow */
75
                    z = cosf(y+y);
76
                    if ((s*c)>zero) cc = z/ss;
77
                    else            ss = z/cc;
78
                }
79
        /*
80
         * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x)
81
         * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x)
82
         */
83
                if(ix>0x80000000) z = (invsqrtpi*cc)/sqrtf(y);
84
                else {
85
                    u = ponef(y); v = qonef(y);
86
                    z = invsqrtpi*(u*cc-v*ss)/sqrtf(y);
87
                }
88
                if(hx<0) return -z;
89
                else     return  z;
90
        }
91
        if(ix<0x32000000) {     /* |x|<2**-27 */
92
            if(huge+x>one) return (float)0.5*x;/* inexact if x!=0 necessary */
93
        }
94
        z = x*x;
95
        r =  z*(r00+z*(r01+z*(r02+z*r03)));
96
        s =  one+z*(s01+z*(s02+z*(s03+z*(s04+z*s05))));
97
        r *= x;
98
        return(x*(float)0.5+r/s);
99
}
100
 
101
#ifdef __STDC__
102
static const float U0[5] = {
103
#else
104
static float U0[5] = {
105
#endif
106
 -1.9605709612e-01, /* 0xbe48c331 */
107
  5.0443872809e-02, /* 0x3d4e9e3c */
108
 -1.9125689287e-03, /* 0xbafaaf2a */
109
  2.3525259166e-05, /* 0x37c5581c */
110
 -9.1909917899e-08, /* 0xb3c56003 */
111
};
112
#ifdef __STDC__
113
static const float V0[5] = {
114
#else
115
static float V0[5] = {
116
#endif
117
  1.9916731864e-02, /* 0x3ca3286a */
118
  2.0255257550e-04, /* 0x3954644b */
119
  1.3560879779e-06, /* 0x35b602d4 */
120
  6.2274145840e-09, /* 0x31d5f8eb */
121
  1.6655924903e-11, /* 0x2d9281cf */
122
};
123
 
124
#ifdef __STDC__
125
        float __ieee754_y1f(float x)
126
#else
127
        float __ieee754_y1f(x)
128
        float x;
129
#endif
130
{
131
        float z, s,c,ss,cc,u,v;
132
        int32_t hx,ix;
133
 
134
        GET_FLOAT_WORD(hx,x);
135
        ix = 0x7fffffff&hx;
136
    /* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */
137
        if(ix>=0x7f800000) return  one/(x+x*x);
138
        if(ix==0) return -one/zero;
139
        if(hx<0) return zero/zero;
140
        if(ix >= 0x40000000) {  /* |x| >= 2.0 */
141
                s = sinf(x);
142
                c = cosf(x);
143
                ss = -s-c;
144
                cc = s-c;
145
                if(ix<0x7f000000) {  /* make sure x+x not overflow */
146
                    z = cosf(x+x);
147
                    if ((s*c)>zero) cc = z/ss;
148
                    else            ss = z/cc;
149
                }
150
        /* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0))
151
         * where x0 = x-3pi/4
152
         *      Better formula:
153
         *              cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
154
         *                      =  1/sqrt(2) * (sin(x) - cos(x))
155
         *              sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
156
         *                      = -1/sqrt(2) * (cos(x) + sin(x))
157
         * To avoid cancellation, use
158
         *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
159
         * to compute the worse one.
160
         */
161
                if(ix>0x48000000) z = (invsqrtpi*ss)/sqrtf(x);
162
                else {
163
                    u = ponef(x); v = qonef(x);
164
                    z = invsqrtpi*(u*ss+v*cc)/sqrtf(x);
165
                }
166
                return z;
167
        }
168
        if(ix<=0x24800000) {    /* x < 2**-54 */
169
            return(-tpi/x);
170
        }
171
        z = x*x;
172
        u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4])));
173
        v = one+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4]))));
174
        return(x*(u/v) + tpi*(__ieee754_j1f(x)*__ieee754_logf(x)-one/x));
175
}
176
 
177
/* For x >= 8, the asymptotic expansions of pone is
178
 *      1 + 15/128 s^2 - 4725/2^15 s^4 - ...,   where s = 1/x.
179
 * We approximate pone by
180
 *      pone(x) = 1 + (R/S)
181
 * where  R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10
182
 *        S = 1 + ps0*s^2 + ... + ps4*s^10
183
 * and
184
 *      | pone(x)-1-R/S | <= 2  ** ( -60.06)
185
 */
186
 
187
#ifdef __STDC__
188
static const float pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
189
#else
190
static float pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
191
#endif
192
  0.0000000000e+00, /* 0x00000000 */
193
  1.1718750000e-01, /* 0x3df00000 */
194
  1.3239480972e+01, /* 0x4153d4ea */
195
  4.1205184937e+02, /* 0x43ce06a3 */
196
  3.8747453613e+03, /* 0x45722bed */
197
  7.9144794922e+03, /* 0x45f753d6 */
198
};
199
#ifdef __STDC__
200
static const float ps8[5] = {
201
#else
202
static float ps8[5] = {
203
#endif
204
  1.1420736694e+02, /* 0x42e46a2c */
205
  3.6509309082e+03, /* 0x45642ee5 */
206
  3.6956207031e+04, /* 0x47105c35 */
207
  9.7602796875e+04, /* 0x47bea166 */
208
  3.0804271484e+04, /* 0x46f0a88b */
209
};
210
 
211
#ifdef __STDC__
212
static const float pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
213
#else
214
static float pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
215
#endif
216
  1.3199052094e-11, /* 0x2d68333f */
217
  1.1718749255e-01, /* 0x3defffff */
218
  6.8027510643e+00, /* 0x40d9b023 */
219
  1.0830818176e+02, /* 0x42d89dca */
220
  5.1763616943e+02, /* 0x440168b7 */
221
  5.2871520996e+02, /* 0x44042dc6 */
222
};
223
#ifdef __STDC__
224
static const float ps5[5] = {
225
#else
226
static float ps5[5] = {
227
#endif
228
  5.9280597687e+01, /* 0x426d1f55 */
229
  9.9140142822e+02, /* 0x4477d9b1 */
230
  5.3532670898e+03, /* 0x45a74a23 */
231
  7.8446904297e+03, /* 0x45f52586 */
232
  1.5040468750e+03, /* 0x44bc0180 */
233
};
234
 
235
#ifdef __STDC__
236
static const float pr3[6] = {
237
#else
238
static float pr3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
239
#endif
240
  3.0250391081e-09, /* 0x314fe10d */
241
  1.1718686670e-01, /* 0x3defffab */
242
  3.9329774380e+00, /* 0x407bb5e7 */
243
  3.5119403839e+01, /* 0x420c7a45 */
244
  9.1055007935e+01, /* 0x42b61c2a */
245
  4.8559066772e+01, /* 0x42423c7c */
246
};
247
#ifdef __STDC__
248
static const float ps3[5] = {
249
#else
250
static float ps3[5] = {
251
#endif
252
  3.4791309357e+01, /* 0x420b2a4d */
253
  3.3676245117e+02, /* 0x43a86198 */
254
  1.0468714600e+03, /* 0x4482dbe3 */
255
  8.9081134033e+02, /* 0x445eb3ed */
256
  1.0378793335e+02, /* 0x42cf936c */
257
};
258
 
259
#ifdef __STDC__
260
static const float pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
261
#else
262
static float pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
263
#endif
264
  1.0771083225e-07, /* 0x33e74ea8 */
265
  1.1717621982e-01, /* 0x3deffa16 */
266
  2.3685150146e+00, /* 0x401795c0 */
267
  1.2242610931e+01, /* 0x4143e1bc */
268
  1.7693971634e+01, /* 0x418d8d41 */
269
  5.0735230446e+00, /* 0x40a25a4d */
270
};
271
#ifdef __STDC__
272
static const float ps2[5] = {
273
#else
274
static float ps2[5] = {
275
#endif
276
  2.1436485291e+01, /* 0x41ab7dec */
277
  1.2529022980e+02, /* 0x42fa9499 */
278
  2.3227647400e+02, /* 0x436846c7 */
279
  1.1767937469e+02, /* 0x42eb5bd7 */
280
  8.3646392822e+00, /* 0x4105d590 */
281
};
282
 
283
#ifdef __STDC__
284
        static float ponef(float x)
285
#else
286
        static float ponef(x)
287
        float x;
288
#endif
289
{
290
#ifdef __STDC__
291
        const float *p,*q;
292
#else
293
        float *p,*q;
294
#endif
295
        float z,r,s;
296
        int32_t ix;
297
        GET_FLOAT_WORD(ix,x);
298
        ix &= 0x7fffffff;
299
        if(ix>=0x41000000)     {p = pr8; q= ps8;}
300
        else if(ix>=0x40f71c58){p = pr5; q= ps5;}
301
        else if(ix>=0x4036db68){p = pr3; q= ps3;}
302
        else if(ix>=0x40000000){p = pr2; q= ps2;}
303
        z = one/(x*x);
304
        r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
305
        s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
306
        return one+ r/s;
307
}
308
 
309
 
310
/* For x >= 8, the asymptotic expansions of qone is
311
 *      3/8 s - 105/1024 s^3 - ..., where s = 1/x.
312
 * We approximate pone by
313
 *      qone(x) = s*(0.375 + (R/S))
314
 * where  R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10
315
 *        S = 1 + qs1*s^2 + ... + qs6*s^12
316
 * and
317
 *      | qone(x)/s -0.375-R/S | <= 2  ** ( -61.13)
318
 */
319
 
320
#ifdef __STDC__
321
static const float qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
322
#else
323
static float qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
324
#endif
325
  0.0000000000e+00, /* 0x00000000 */
326
 -1.0253906250e-01, /* 0xbdd20000 */
327
 -1.6271753311e+01, /* 0xc1822c8d */
328
 -7.5960174561e+02, /* 0xc43de683 */
329
 -1.1849806641e+04, /* 0xc639273a */
330
 -4.8438511719e+04, /* 0xc73d3683 */
331
};
332
#ifdef __STDC__
333
static const float qs8[6] = {
334
#else
335
static float qs8[6] = {
336
#endif
337
  1.6139537048e+02, /* 0x43216537 */
338
  7.8253862305e+03, /* 0x45f48b17 */
339
  1.3387534375e+05, /* 0x4802bcd6 */
340
  7.1965775000e+05, /* 0x492fb29c */
341
  6.6660125000e+05, /* 0x4922be94 */
342
 -2.9449025000e+05, /* 0xc88fcb48 */
343
};
344
 
345
#ifdef __STDC__
346
static const float qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
347
#else
348
static float qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
349
#endif
350
 -2.0897993405e-11, /* 0xadb7d219 */
351
 -1.0253904760e-01, /* 0xbdd1fffe */
352
 -8.0564479828e+00, /* 0xc100e736 */
353
 -1.8366960144e+02, /* 0xc337ab6b */
354
 -1.3731937256e+03, /* 0xc4aba633 */
355
 -2.6124443359e+03, /* 0xc523471c */
356
};
357
#ifdef __STDC__
358
static const float qs5[6] = {
359
#else
360
static float qs5[6] = {
361
#endif
362
  8.1276550293e+01, /* 0x42a28d98 */
363
  1.9917987061e+03, /* 0x44f8f98f */
364
  1.7468484375e+04, /* 0x468878f8 */
365
  4.9851425781e+04, /* 0x4742bb6d */
366
  2.7948074219e+04, /* 0x46da5826 */
367
 -4.7191835938e+03, /* 0xc5937978 */
368
};
369
 
370
#ifdef __STDC__
371
static const float qr3[6] = {
372
#else
373
static float qr3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
374
#endif
375
 -5.0783124372e-09, /* 0xb1ae7d4f */
376
 -1.0253783315e-01, /* 0xbdd1ff5b */
377
 -4.6101160049e+00, /* 0xc0938612 */
378
 -5.7847221375e+01, /* 0xc267638e */
379
 -2.2824453735e+02, /* 0xc3643e9a */
380
 -2.1921012878e+02, /* 0xc35b35cb */
381
};
382
#ifdef __STDC__
383
static const float qs3[6] = {
384
#else
385
static float qs3[6] = {
386
#endif
387
  4.7665153503e+01, /* 0x423ea91e */
388
  6.7386511230e+02, /* 0x4428775e */
389
  3.3801528320e+03, /* 0x45534272 */
390
  5.5477290039e+03, /* 0x45ad5dd5 */
391
  1.9031191406e+03, /* 0x44ede3d0 */
392
 -1.3520118713e+02, /* 0xc3073381 */
393
};
394
 
395
#ifdef __STDC__
396
static const float qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
397
#else
398
static float qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
399
#endif
400
 -1.7838172539e-07, /* 0xb43f8932 */
401
 -1.0251704603e-01, /* 0xbdd1f475 */
402
 -2.7522056103e+00, /* 0xc0302423 */
403
 -1.9663616180e+01, /* 0xc19d4f16 */
404
 -4.2325313568e+01, /* 0xc2294d1f */
405
 -2.1371921539e+01, /* 0xc1aaf9b2 */
406
};
407
#ifdef __STDC__
408
static const float qs2[6] = {
409
#else
410
static float qs2[6] = {
411
#endif
412
  2.9533363342e+01, /* 0x41ec4454 */
413
  2.5298155212e+02, /* 0x437cfb47 */
414
  7.5750280762e+02, /* 0x443d602e */
415
  7.3939318848e+02, /* 0x4438d92a */
416
  1.5594900513e+02, /* 0x431bf2f2 */
417
 -4.9594988823e+00, /* 0xc09eb437 */
418
};
419
 
420
#ifdef __STDC__
421
        static float qonef(float x)
422
#else
423
        static float qonef(x)
424
        float x;
425
#endif
426
{
427
#ifdef __STDC__
428
        const float *p,*q;
429
#else
430
        float *p,*q;
431
#endif
432
        float  s,r,z;
433
        int32_t ix;
434
        GET_FLOAT_WORD(ix,x);
435
        ix &= 0x7fffffff;
436
        if(ix>=0x40200000)     {p = qr8; q= qs8;}
437
        else if(ix>=0x40f71c58){p = qr5; q= qs5;}
438
        else if(ix>=0x4036db68){p = qr3; q= qs3;}
439
        else if(ix>=0x40000000){p = qr2; q= qs2;}
440
        z = one/(x*x);
441
        r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
442
        s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
443
        return ((float).375 + r/s)/x;
444
}