summaryrefslogtreecommitdiffstats
path: root/lib/msun/src/e_j1f.c
blob: 2d4b93aec31dfa581d34b13b8ce63b3d3889798d (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
/* e_j1f.c -- float version of e_j1.c.
 * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
 */

/*
 * ====================================================
 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
 *
 * Developed at SunPro, a Sun Microsystems, Inc. business.
 * Permission to use, copy, modify, and distribute this
 * software is freely granted, provided that this notice
 * is preserved.
 * ====================================================
 */

#ifndef lint
static char rcsid[] = "$FreeBSD$";
#endif

#include "math.h"
#include "math_private.h"

#ifdef __STDC__
static float ponef(float), qonef(float);
#else
static float ponef(), qonef();
#endif

#ifdef __STDC__
static const float
#else
static float
#endif
huge    = 1e30,
one	= 1.0,
invsqrtpi=  5.6418961287e-01, /* 0x3f106ebb */
tpi      =  6.3661974669e-01, /* 0x3f22f983 */
	/* R0/S0 on [0,2] */
r00  = -6.2500000000e-02, /* 0xbd800000 */
r01  =  1.4070566976e-03, /* 0x3ab86cfd */
r02  = -1.5995563444e-05, /* 0xb7862e36 */
r03  =  4.9672799207e-08, /* 0x335557d2 */
s01  =  1.9153760746e-02, /* 0x3c9ce859 */
s02  =  1.8594678841e-04, /* 0x3942fab6 */
s03  =  1.1771846857e-06, /* 0x359dffc2 */
s04  =  5.0463624390e-09, /* 0x31ad6446 */
s05  =  1.2354227016e-11; /* 0x2d59567e */

#ifdef __STDC__
static const float zero    = 0.0;
#else
static float zero    = 0.0;
#endif

#ifdef __STDC__
	float __ieee754_j1f(float x)
#else
	float __ieee754_j1f(x)
	float x;
#endif
{
	float z, s,c,ss,cc,r,u,v,y;
	int32_t hx,ix;

	GET_FLOAT_WORD(hx,x);
	ix = hx&0x7fffffff;
	if(ix>=0x7f800000) return one/x;
	y = fabsf(x);
	if(ix >= 0x40000000) {	/* |x| >= 2.0 */
		s = sinf(y);
		c = cosf(y);
		ss = -s-c;
		cc = s-c;
		if(ix<0x7f000000) {  /* make sure y+y not overflow */
		    z = cosf(y+y);
		    if ((s*c)>zero) cc = z/ss;
		    else 	    ss = z/cc;
		}
	/*
	 * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x)
	 * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x)
	 */
		if(ix>0x80000000) z = (invsqrtpi*cc)/sqrtf(y);
		else {
		    u = ponef(y); v = qonef(y);
		    z = invsqrtpi*(u*cc-v*ss)/sqrtf(y);
		}
		if(hx<0) return -z;
		else  	 return  z;
	}
	if(ix<0x32000000) {	/* |x|<2**-27 */
	    if(huge+x>one) return (float)0.5*x;/* inexact if x!=0 necessary */
	}
	z = x*x;
	r =  z*(r00+z*(r01+z*(r02+z*r03)));
	s =  one+z*(s01+z*(s02+z*(s03+z*(s04+z*s05))));
	r *= x;
	return(x*(float)0.5+r/s);
}

#ifdef __STDC__
static const float U0[5] = {
#else
static float U0[5] = {
#endif
 -1.9605709612e-01, /* 0xbe48c331 */
  5.0443872809e-02, /* 0x3d4e9e3c */
 -1.9125689287e-03, /* 0xbafaaf2a */
  2.3525259166e-05, /* 0x37c5581c */
 -9.1909917899e-08, /* 0xb3c56003 */
};
#ifdef __STDC__
static const float V0[5] = {
#else
static float V0[5] = {
#endif
  1.9916731864e-02, /* 0x3ca3286a */
  2.0255257550e-04, /* 0x3954644b */
  1.3560879779e-06, /* 0x35b602d4 */
  6.2274145840e-09, /* 0x31d5f8eb */
  1.6655924903e-11, /* 0x2d9281cf */
};

#ifdef __STDC__
	float __ieee754_y1f(float x)
#else
	float __ieee754_y1f(x)
	float x;
#endif
{
	float z, s,c,ss,cc,u,v;
	int32_t hx,ix;

	GET_FLOAT_WORD(hx,x);
        ix = 0x7fffffff&hx;
    /* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */
	if(ix>=0x7f800000) return  one/(x+x*x);
        if(ix==0) return -one/zero;
        if(hx<0) return zero/zero;
        if(ix >= 0x40000000) {  /* |x| >= 2.0 */
                s = sinf(x);
                c = cosf(x);
                ss = -s-c;
                cc = s-c;
                if(ix<0x7f000000) {  /* make sure x+x not overflow */
                    z = cosf(x+x);
                    if ((s*c)>zero) cc = z/ss;
                    else            ss = z/cc;
                }
        /* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0))
         * where x0 = x-3pi/4
         *      Better formula:
         *              cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
         *                      =  1/sqrt(2) * (sin(x) - cos(x))
         *              sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
         *                      = -1/sqrt(2) * (cos(x) + sin(x))
         * To avoid cancellation, use
         *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
         * to compute the worse one.
         */
                if(ix>0x48000000) z = (invsqrtpi*ss)/sqrtf(x);
                else {
                    u = ponef(x); v = qonef(x);
                    z = invsqrtpi*(u*ss+v*cc)/sqrtf(x);
                }
                return z;
        }
        if(ix<=0x24800000) {    /* x < 2**-54 */
            return(-tpi/x);
        }
        z = x*x;
        u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4])));
        v = one+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4]))));
        return(x*(u/v) + tpi*(__ieee754_j1f(x)*__ieee754_logf(x)-one/x));
}

/* For x >= 8, the asymptotic expansions of pone is
 *	1 + 15/128 s^2 - 4725/2^15 s^4 - ...,	where s = 1/x.
 * We approximate pone by
 * 	pone(x) = 1 + (R/S)
 * where  R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10
 * 	  S = 1 + ps0*s^2 + ... + ps4*s^10
 * and
 *	| pone(x)-1-R/S | <= 2  ** ( -60.06)
 */

#ifdef __STDC__
static const float pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
#else
static float pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
#endif
  0.0000000000e+00, /* 0x00000000 */
  1.1718750000e-01, /* 0x3df00000 */
  1.3239480972e+01, /* 0x4153d4ea */
  4.1205184937e+02, /* 0x43ce06a3 */
  3.8747453613e+03, /* 0x45722bed */
  7.9144794922e+03, /* 0x45f753d6 */
};
#ifdef __STDC__
static const float ps8[5] = {
#else
static float ps8[5] = {
#endif
  1.1420736694e+02, /* 0x42e46a2c */
  3.6509309082e+03, /* 0x45642ee5 */
  3.6956207031e+04, /* 0x47105c35 */
  9.7602796875e+04, /* 0x47bea166 */
  3.0804271484e+04, /* 0x46f0a88b */
};

#ifdef __STDC__
static const float pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
#else
static float pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
#endif
  1.3199052094e-11, /* 0x2d68333f */
  1.1718749255e-01, /* 0x3defffff */
  6.8027510643e+00, /* 0x40d9b023 */
  1.0830818176e+02, /* 0x42d89dca */
  5.1763616943e+02, /* 0x440168b7 */
  5.2871520996e+02, /* 0x44042dc6 */
};
#ifdef __STDC__
static const float ps5[5] = {
#else
static float ps5[5] = {
#endif
  5.9280597687e+01, /* 0x426d1f55 */
  9.9140142822e+02, /* 0x4477d9b1 */
  5.3532670898e+03, /* 0x45a74a23 */
  7.8446904297e+03, /* 0x45f52586 */
  1.5040468750e+03, /* 0x44bc0180 */
};

#ifdef __STDC__
static const float pr3[6] = {
#else
static float pr3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
#endif
  3.0250391081e-09, /* 0x314fe10d */
  1.1718686670e-01, /* 0x3defffab */
  3.9329774380e+00, /* 0x407bb5e7 */
  3.5119403839e+01, /* 0x420c7a45 */
  9.1055007935e+01, /* 0x42b61c2a */
  4.8559066772e+01, /* 0x42423c7c */
};
#ifdef __STDC__
static const float ps3[5] = {
#else
static float ps3[5] = {
#endif
  3.4791309357e+01, /* 0x420b2a4d */
  3.3676245117e+02, /* 0x43a86198 */
  1.0468714600e+03, /* 0x4482dbe3 */
  8.9081134033e+02, /* 0x445eb3ed */
  1.0378793335e+02, /* 0x42cf936c */
};

#ifdef __STDC__
static const float pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
#else
static float pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
#endif
  1.0771083225e-07, /* 0x33e74ea8 */
  1.1717621982e-01, /* 0x3deffa16 */
  2.3685150146e+00, /* 0x401795c0 */
  1.2242610931e+01, /* 0x4143e1bc */
  1.7693971634e+01, /* 0x418d8d41 */
  5.0735230446e+00, /* 0x40a25a4d */
};
#ifdef __STDC__
static const float ps2[5] = {
#else
static float ps2[5] = {
#endif
  2.1436485291e+01, /* 0x41ab7dec */
  1.2529022980e+02, /* 0x42fa9499 */
  2.3227647400e+02, /* 0x436846c7 */
  1.1767937469e+02, /* 0x42eb5bd7 */
  8.3646392822e+00, /* 0x4105d590 */
};

#ifdef __STDC__
	static float ponef(float x)
#else
	static float ponef(x)
	float x;
#endif
{
#ifdef __STDC__
	const float *p,*q;
#else
	float *p,*q;
#endif
	float z,r,s;
        int32_t ix;
	GET_FLOAT_WORD(ix,x);
	ix &= 0x7fffffff;
        if(ix>=0x41000000)     {p = pr8; q= ps8;}
        else if(ix>=0x40f71c58){p = pr5; q= ps5;}
        else if(ix>=0x4036db68){p = pr3; q= ps3;}
        else if(ix>=0x40000000){p = pr2; q= ps2;}
        z = one/(x*x);
        r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
        s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
        return one+ r/s;
}


/* For x >= 8, the asymptotic expansions of qone is
 *	3/8 s - 105/1024 s^3 - ..., where s = 1/x.
 * We approximate pone by
 * 	qone(x) = s*(0.375 + (R/S))
 * where  R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10
 * 	  S = 1 + qs1*s^2 + ... + qs6*s^12
 * and
 *	| qone(x)/s -0.375-R/S | <= 2  ** ( -61.13)
 */

#ifdef __STDC__
static const float qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
#else
static float qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
#endif
  0.0000000000e+00, /* 0x00000000 */
 -1.0253906250e-01, /* 0xbdd20000 */
 -1.6271753311e+01, /* 0xc1822c8d */
 -7.5960174561e+02, /* 0xc43de683 */
 -1.1849806641e+04, /* 0xc639273a */
 -4.8438511719e+04, /* 0xc73d3683 */
};
#ifdef __STDC__
static const float qs8[6] = {
#else
static float qs8[6] = {
#endif
  1.6139537048e+02, /* 0x43216537 */
  7.8253862305e+03, /* 0x45f48b17 */
  1.3387534375e+05, /* 0x4802bcd6 */
  7.1965775000e+05, /* 0x492fb29c */
  6.6660125000e+05, /* 0x4922be94 */
 -2.9449025000e+05, /* 0xc88fcb48 */
};

#ifdef __STDC__
static const float qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
#else
static float qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
#endif
 -2.0897993405e-11, /* 0xadb7d219 */
 -1.0253904760e-01, /* 0xbdd1fffe */
 -8.0564479828e+00, /* 0xc100e736 */
 -1.8366960144e+02, /* 0xc337ab6b */
 -1.3731937256e+03, /* 0xc4aba633 */
 -2.6124443359e+03, /* 0xc523471c */
};
#ifdef __STDC__
static const float qs5[6] = {
#else
static float qs5[6] = {
#endif
  8.1276550293e+01, /* 0x42a28d98 */
  1.9917987061e+03, /* 0x44f8f98f */
  1.7468484375e+04, /* 0x468878f8 */
  4.9851425781e+04, /* 0x4742bb6d */
  2.7948074219e+04, /* 0x46da5826 */
 -4.7191835938e+03, /* 0xc5937978 */
};

#ifdef __STDC__
static const float qr3[6] = {
#else
static float qr3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
#endif
 -5.0783124372e-09, /* 0xb1ae7d4f */
 -1.0253783315e-01, /* 0xbdd1ff5b */
 -4.6101160049e+00, /* 0xc0938612 */
 -5.7847221375e+01, /* 0xc267638e */
 -2.2824453735e+02, /* 0xc3643e9a */
 -2.1921012878e+02, /* 0xc35b35cb */
};
#ifdef __STDC__
static const float qs3[6] = {
#else
static float qs3[6] = {
#endif
  4.7665153503e+01, /* 0x423ea91e */
  6.7386511230e+02, /* 0x4428775e */
  3.3801528320e+03, /* 0x45534272 */
  5.5477290039e+03, /* 0x45ad5dd5 */
  1.9031191406e+03, /* 0x44ede3d0 */
 -1.3520118713e+02, /* 0xc3073381 */
};

#ifdef __STDC__
static const float qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
#else
static float qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
#endif
 -1.7838172539e-07, /* 0xb43f8932 */
 -1.0251704603e-01, /* 0xbdd1f475 */
 -2.7522056103e+00, /* 0xc0302423 */
 -1.9663616180e+01, /* 0xc19d4f16 */
 -4.2325313568e+01, /* 0xc2294d1f */
 -2.1371921539e+01, /* 0xc1aaf9b2 */
};
#ifdef __STDC__
static const float qs2[6] = {
#else
static float qs2[6] = {
#endif
  2.9533363342e+01, /* 0x41ec4454 */
  2.5298155212e+02, /* 0x437cfb47 */
  7.5750280762e+02, /* 0x443d602e */
  7.3939318848e+02, /* 0x4438d92a */
  1.5594900513e+02, /* 0x431bf2f2 */
 -4.9594988823e+00, /* 0xc09eb437 */
};

#ifdef __STDC__
	static float qonef(float x)
#else
	static float qonef(x)
	float x;
#endif
{
#ifdef __STDC__
	const float *p,*q;
#else
	float *p,*q;
#endif
	float  s,r,z;
	int32_t ix;
	GET_FLOAT_WORD(ix,x);
	ix &= 0x7fffffff;
	if(ix>=0x40200000)     {p = qr8; q= qs8;}
	else if(ix>=0x40f71c58){p = qr5; q= qs5;}
	else if(ix>=0x4036db68){p = qr3; q= qs3;}
	else if(ix>=0x40000000){p = qr2; q= qs2;}
	z = one/(x*x);
	r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
	s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
	return ((float).375 + r/s)/x;
}
OpenPOWER on IntegriCloud