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e_j1.c
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1 
2 /* @(#)e_j1.c 1.3 95/01/18 */
3 /*
4  * ====================================================
5  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
6  *
7  * Developed at SunSoft, a Sun Microsystems, Inc. business.
8  * Permission to use, copy, modify, and distribute this
9  * software is freely granted, provided that this notice
10  * is preserved.
11  * ====================================================
12  */
13 
14 /* __ieee754_j1(x), __ieee754_y1(x)
15  * Bessel function of the first and second kinds of order zero.
16  * Method -- j1(x):
17  * 1. For tiny x, we use j1(x) = x/2 - x^3/16 + x^5/384 - ...
18  * 2. Reduce x to |x| since j1(x)=-j1(-x), and
19  * for x in (0,2)
20  * j1(x) = x/2 + x*z*R0/S0, where z = x*x;
21  * (precision: |j1/x - 1/2 - R0/S0 |<2**-61.51 )
22  * for x in (2,inf)
23  * j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x1)-q1(x)*sin(x1))
24  * y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1))
25  * where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1)
26  * as follow:
27  * cos(x1) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
28  * = 1/sqrt(2) * (sin(x) - cos(x))
29  * sin(x1) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
30  * = -1/sqrt(2) * (sin(x) + cos(x))
31  * (To avoid cancellation, use
32  * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
33  * to compute the worse one.)
34  *
35  * 3 Special cases
36  * j1(nan)= nan
37  * j1(0) = 0
38  * j1(inf) = 0
39  *
40  * Method -- y1(x):
41  * 1. screen out x<=0 cases: y1(0)=-inf, y1(x<0)=NaN
42  * 2. For x<2.
43  * Since
44  * y1(x) = 2/pi*(j1(x)*(ln(x/2)+Euler)-1/x-x/2+5/64*x^3-...)
45  * therefore y1(x)-2/pi*j1(x)*ln(x)-1/x is an odd function.
46  * We use the following function to approximate y1,
47  * y1(x) = x*U(z)/V(z) + (2/pi)*(j1(x)*ln(x)-1/x), z= x^2
48  * where for x in [0,2] (abs err less than 2**-65.89)
49  * U(z) = U0[0] + U0[1]*z + ... + U0[4]*z^4
50  * V(z) = 1 + v0[0]*z + ... + v0[4]*z^5
51  * Note: For tiny x, 1/x dominate y1 and hence
52  * y1(tiny) = -2/pi/tiny, (choose tiny<2**-54)
53  * 3. For x>=2.
54  * y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1))
55  * where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1)
56  * by method mentioned above.
57  */
58 
59 #include "fdlibm.h"
60 
61 #ifdef __STDC__
62 static double pone(double), qone(double);
63 #else
64 static double pone(), qone();
65 #endif
66 
67 #ifdef __STDC__
68 static const double
69 #else
70 static double
71 #endif
72 huge = 1e300,
73 one = 1.0,
74 invsqrtpi= 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
75 tpi = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
76  /* R0/S0 on [0,2] */
77 r00 = -6.25000000000000000000e-02, /* 0xBFB00000, 0x00000000 */
78 r01 = 1.40705666955189706048e-03, /* 0x3F570D9F, 0x98472C61 */
79 r02 = -1.59955631084035597520e-05, /* 0xBEF0C5C6, 0xBA169668 */
80 r03 = 4.96727999609584448412e-08, /* 0x3E6AAAFA, 0x46CA0BD9 */
81 s01 = 1.91537599538363460805e-02, /* 0x3F939D0B, 0x12637E53 */
82 s02 = 1.85946785588630915560e-04, /* 0x3F285F56, 0xB9CDF664 */
83 s03 = 1.17718464042623683263e-06, /* 0x3EB3BFF8, 0x333F8498 */
84 s04 = 5.04636257076217042715e-09, /* 0x3E35AC88, 0xC97DFF2C */
85 s05 = 1.23542274426137913908e-11; /* 0x3DAB2ACF, 0xCFB97ED8 */
86 
87 static double zero = 0.0;
88 
89 #ifdef __STDC__
90  double __ieee754_j1(double x)
91 #else
92  double __ieee754_j1(x)
93  double x;
94 #endif
95 {
96  double z, s,c,ss,cc,r,u,v,y;
97  int hx,ix;
98 
99  hx = __HI(x);
100  ix = hx&0x7fffffff;
101  if(ix>=0x7ff00000) return one/x;
102  y = fabs(x);
103  if(ix >= 0x40000000) { /* |x| >= 2.0 */
104  s = sin(y);
105  c = cos(y);
106  ss = -s-c;
107  cc = s-c;
108  if(ix<0x7fe00000) { /* make sure y+y not overflow */
109  z = cos(y+y);
110  if ((s*c)>zero) cc = z/ss;
111  else ss = z/cc;
112  }
113  /*
114  * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x)
115  * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x)
116  */
117  if(ix>0x48000000) z = (invsqrtpi*cc)/sqrt(y);
118  else {
119  u = pone(y); v = qone(y);
120  z = invsqrtpi*(u*cc-v*ss)/sqrt(y);
121  }
122  if(hx<0) return -z;
123  else return z;
124  }
125  if(ix<0x3e400000) { /* |x|<2**-27 */
126  if(huge+x>one) return 0.5*x;/* inexact if x!=0 necessary */
127  }
128  z = x*x;
129  r = z*(r00+z*(r01+z*(r02+z*r03)));
130  s = one+z*(s01+z*(s02+z*(s03+z*(s04+z*s05))));
131  r *= x;
132  return(x*0.5+r/s);
133 }
134 
135 #ifdef __STDC__
136 static const double U0[5] = {
137 #else
138 static double U0[5] = {
139 #endif
140  -1.96057090646238940668e-01, /* 0xBFC91866, 0x143CBC8A */
141  5.04438716639811282616e-02, /* 0x3FA9D3C7, 0x76292CD1 */
142  -1.91256895875763547298e-03, /* 0xBF5F55E5, 0x4844F50F */
143  2.35252600561610495928e-05, /* 0x3EF8AB03, 0x8FA6B88E */
144  -9.19099158039878874504e-08, /* 0xBE78AC00, 0x569105B8 */
145 };
146 #ifdef __STDC__
147 static const double V0[5] = {
148 #else
149 static double V0[5] = {
150 #endif
151  1.99167318236649903973e-02, /* 0x3F94650D, 0x3F4DA9F0 */
152  2.02552581025135171496e-04, /* 0x3F2A8C89, 0x6C257764 */
153  1.35608801097516229404e-06, /* 0x3EB6C05A, 0x894E8CA6 */
154  6.22741452364621501295e-09, /* 0x3E3ABF1D, 0x5BA69A86 */
155  1.66559246207992079114e-11, /* 0x3DB25039, 0xDACA772A */
156 };
157 
158 #ifdef __STDC__
159  double __ieee754_y1(double x)
160 #else
161  double __ieee754_y1(x)
162  double x;
163 #endif
164 {
165  double z, s,c,ss,cc,u,v;
166  int hx,ix,lx;
167 
168  hx = __HI(x);
169  ix = 0x7fffffff&hx;
170  lx = __LO(x);
171  /* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */
172  if(ix>=0x7ff00000) return one/(x+x*x);
173  if((ix|lx)==0) return -one/zero;
174  if(hx<0) return zero/zero;
175  if(ix >= 0x40000000) { /* |x| >= 2.0 */
176  s = sin(x);
177  c = cos(x);
178  ss = -s-c;
179  cc = s-c;
180  if(ix<0x7fe00000) { /* make sure x+x not overflow */
181  z = cos(x+x);
182  if ((s*c)>zero) cc = z/ss;
183  else ss = z/cc;
184  }
185  /* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0))
186  * where x0 = x-3pi/4
187  * Better formula:
188  * cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
189  * = 1/sqrt(2) * (sin(x) - cos(x))
190  * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
191  * = -1/sqrt(2) * (cos(x) + sin(x))
192  * To avoid cancellation, use
193  * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
194  * to compute the worse one.
195  */
196  if(ix>0x48000000) z = (invsqrtpi*ss)/sqrt(x);
197  else {
198  u = pone(x); v = qone(x);
199  z = invsqrtpi*(u*ss+v*cc)/sqrt(x);
200  }
201  return z;
202  }
203  if(ix<=0x3c900000) { /* x < 2**-54 */
204  return(-tpi/x);
205  }
206  z = x*x;
207  u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4])));
208  v = one+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4]))));
209  return(x*(u/v) + tpi*(__ieee754_j1(x)*__ieee754_log(x)-one/x));
210 }
211 
212 /* For x >= 8, the asymptotic expansions of pone is
213  * 1 + 15/128 s^2 - 4725/2^15 s^4 - ..., where s = 1/x.
214  * We approximate pone by
215  * pone(x) = 1 + (R/S)
216  * where R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10
217  * S = 1 + ps0*s^2 + ... + ps4*s^10
218  * and
219  * | pone(x)-1-R/S | <= 2 ** ( -60.06)
220  */
221 
222 #ifdef __STDC__
223 static const double pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
224 #else
225 static double pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
226 #endif
227  0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
228  1.17187499999988647970e-01, /* 0x3FBDFFFF, 0xFFFFFCCE */
229  1.32394806593073575129e+01, /* 0x402A7A9D, 0x357F7FCE */
230  4.12051854307378562225e+02, /* 0x4079C0D4, 0x652EA590 */
231  3.87474538913960532227e+03, /* 0x40AE457D, 0xA3A532CC */
232  7.91447954031891731574e+03, /* 0x40BEEA7A, 0xC32782DD */
233 };
234 #ifdef __STDC__
235 static const double ps8[5] = {
236 #else
237 static double ps8[5] = {
238 #endif
239  1.14207370375678408436e+02, /* 0x405C8D45, 0x8E656CAC */
240  3.65093083420853463394e+03, /* 0x40AC85DC, 0x964D274F */
241  3.69562060269033463555e+04, /* 0x40E20B86, 0x97C5BB7F */
242  9.76027935934950801311e+04, /* 0x40F7D42C, 0xB28F17BB */
243  3.08042720627888811578e+04, /* 0x40DE1511, 0x697A0B2D */
244 };
245 
246 #ifdef __STDC__
247 static const double pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
248 #else
249 static double pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
250 #endif
251  1.31990519556243522749e-11, /* 0x3DAD0667, 0xDAE1CA7D */
252  1.17187493190614097638e-01, /* 0x3FBDFFFF, 0xE2C10043 */
253  6.80275127868432871736e+00, /* 0x401B3604, 0x6E6315E3 */
254  1.08308182990189109773e+02, /* 0x405B13B9, 0x452602ED */
255  5.17636139533199752805e+02, /* 0x40802D16, 0xD052D649 */
256  5.28715201363337541807e+02, /* 0x408085B8, 0xBB7E0CB7 */
257 };
258 #ifdef __STDC__
259 static const double ps5[5] = {
260 #else
261 static double ps5[5] = {
262 #endif
263  5.92805987221131331921e+01, /* 0x404DA3EA, 0xA8AF633D */
264  9.91401418733614377743e+02, /* 0x408EFB36, 0x1B066701 */
265  5.35326695291487976647e+03, /* 0x40B4E944, 0x5706B6FB */
266  7.84469031749551231769e+03, /* 0x40BEA4B0, 0xB8A5BB15 */
267  1.50404688810361062679e+03, /* 0x40978030, 0x036F5E51 */
268 };
269 
270 #ifdef __STDC__
271 static const double pr3[6] = {
272 #else
273 static double pr3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
274 #endif
275  3.02503916137373618024e-09, /* 0x3E29FC21, 0xA7AD9EDD */
276  1.17186865567253592491e-01, /* 0x3FBDFFF5, 0x5B21D17B */
277  3.93297750033315640650e+00, /* 0x400F76BC, 0xE85EAD8A */
278  3.51194035591636932736e+01, /* 0x40418F48, 0x9DA6D129 */
279  9.10550110750781271918e+01, /* 0x4056C385, 0x4D2C1837 */
280  4.85590685197364919645e+01, /* 0x4048478F, 0x8EA83EE5 */
281 };
282 #ifdef __STDC__
283 static const double ps3[5] = {
284 #else
285 static double ps3[5] = {
286 #endif
287  3.47913095001251519989e+01, /* 0x40416549, 0xA134069C */
288  3.36762458747825746741e+02, /* 0x40750C33, 0x07F1A75F */
289  1.04687139975775130551e+03, /* 0x40905B7C, 0x5037D523 */
290  8.90811346398256432622e+02, /* 0x408BD67D, 0xA32E31E9 */
291  1.03787932439639277504e+02, /* 0x4059F26D, 0x7C2EED53 */
292 };
293 
294 #ifdef __STDC__
295 static const double pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
296 #else
297 static double pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
298 #endif
299  1.07710830106873743082e-07, /* 0x3E7CE9D4, 0xF65544F4 */
300  1.17176219462683348094e-01, /* 0x3FBDFF42, 0xBE760D83 */
301  2.36851496667608785174e+00, /* 0x4002F2B7, 0xF98FAEC0 */
302  1.22426109148261232917e+01, /* 0x40287C37, 0x7F71A964 */
303  1.76939711271687727390e+01, /* 0x4031B1A8, 0x177F8EE2 */
304  5.07352312588818499250e+00, /* 0x40144B49, 0xA574C1FE */
305 };
306 #ifdef __STDC__
307 static const double ps2[5] = {
308 #else
309 static double ps2[5] = {
310 #endif
311  2.14364859363821409488e+01, /* 0x40356FBD, 0x8AD5ECDC */
312  1.25290227168402751090e+02, /* 0x405F5293, 0x14F92CD5 */
313  2.32276469057162813669e+02, /* 0x406D08D8, 0xD5A2DBD9 */
314  1.17679373287147100768e+02, /* 0x405D6B7A, 0xDA1884A9 */
315  8.36463893371618283368e+00, /* 0x4020BAB1, 0xF44E5192 */
316 };
317 
318 #ifdef __STDC__
319  static double pone(double x)
320 #else
321  static double pone(x)
322  double x;
323 #endif
324 {
325 #ifdef __STDC__
326  const double *p=NULL,*q=NULL;
327 #else
328  double *p,*q;
329 #endif
330  double z,r,s;
331  int ix;
332  ix = 0x7fffffff&__HI(x);
333  if(ix>=0x40200000) {p = pr8; q= ps8;}
334  else if(ix>=0x40122E8B){p = pr5; q= ps5;}
335  else if(ix>=0x4006DB6D){p = pr3; q= ps3;}
336  else if(ix>=0x40000000){p = pr2; q= ps2;}
337  z = one/(x*x);
338  r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
339  s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
340  return one+ r/s;
341 }
342 
343 
344 /* For x >= 8, the asymptotic expansions of qone is
345  * 3/8 s - 105/1024 s^3 - ..., where s = 1/x.
346  * We approximate pone by
347  * qone(x) = s*(0.375 + (R/S))
348  * where R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10
349  * S = 1 + qs1*s^2 + ... + qs6*s^12
350  * and
351  * | qone(x)/s -0.375-R/S | <= 2 ** ( -61.13)
352  */
353 
354 #ifdef __STDC__
355 static const double qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
356 #else
357 static double qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
358 #endif
359  0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
360  -1.02539062499992714161e-01, /* 0xBFBA3FFF, 0xFFFFFDF3 */
361  -1.62717534544589987888e+01, /* 0xC0304591, 0xA26779F7 */
362  -7.59601722513950107896e+02, /* 0xC087BCD0, 0x53E4B576 */
363  -1.18498066702429587167e+04, /* 0xC0C724E7, 0x40F87415 */
364  -4.84385124285750353010e+04, /* 0xC0E7A6D0, 0x65D09C6A */
365 };
366 #ifdef __STDC__
367 static const double qs8[6] = {
368 #else
369 static double qs8[6] = {
370 #endif
371  1.61395369700722909556e+02, /* 0x40642CA6, 0xDE5BCDE5 */
372  7.82538599923348465381e+03, /* 0x40BE9162, 0xD0D88419 */
373  1.33875336287249578163e+05, /* 0x4100579A, 0xB0B75E98 */
374  7.19657723683240939863e+05, /* 0x4125F653, 0x72869C19 */
375  6.66601232617776375264e+05, /* 0x412457D2, 0x7719AD5C */
376  -2.94490264303834643215e+05, /* 0xC111F969, 0x0EA5AA18 */
377 };
378 
379 #ifdef __STDC__
380 static const double qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
381 #else
382 static double qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
383 #endif
384  -2.08979931141764104297e-11, /* 0xBDB6FA43, 0x1AA1A098 */
385  -1.02539050241375426231e-01, /* 0xBFBA3FFF, 0xCB597FEF */
386  -8.05644828123936029840e+00, /* 0xC0201CE6, 0xCA03AD4B */
387  -1.83669607474888380239e+02, /* 0xC066F56D, 0x6CA7B9B0 */
388  -1.37319376065508163265e+03, /* 0xC09574C6, 0x6931734F */
389  -2.61244440453215656817e+03, /* 0xC0A468E3, 0x88FDA79D */
390 };
391 #ifdef __STDC__
392 static const double qs5[6] = {
393 #else
394 static double qs5[6] = {
395 #endif
396  8.12765501384335777857e+01, /* 0x405451B2, 0xFF5A11B2 */
397  1.99179873460485964642e+03, /* 0x409F1F31, 0xE77BF839 */
398  1.74684851924908907677e+04, /* 0x40D10F1F, 0x0D64CE29 */
399  4.98514270910352279316e+04, /* 0x40E8576D, 0xAABAD197 */
400  2.79480751638918118260e+04, /* 0x40DB4B04, 0xCF7C364B */
401  -4.71918354795128470869e+03, /* 0xC0B26F2E, 0xFCFFA004 */
402 };
403 
404 #ifdef __STDC__
405 static const double qr3[6] = {
406 #else
407 static double qr3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
408 #endif
409  -5.07831226461766561369e-09, /* 0xBE35CFA9, 0xD38FC84F */
410  -1.02537829820837089745e-01, /* 0xBFBA3FEB, 0x51AEED54 */
411  -4.61011581139473403113e+00, /* 0xC01270C2, 0x3302D9FF */
412  -5.78472216562783643212e+01, /* 0xC04CEC71, 0xC25D16DA */
413  -2.28244540737631695038e+02, /* 0xC06C87D3, 0x4718D55F */
414  -2.19210128478909325622e+02, /* 0xC06B66B9, 0x5F5C1BF6 */
415 };
416 #ifdef __STDC__
417 static const double qs3[6] = {
418 #else
419 static double qs3[6] = {
420 #endif
421  4.76651550323729509273e+01, /* 0x4047D523, 0xCCD367E4 */
422  6.73865112676699709482e+02, /* 0x40850EEB, 0xC031EE3E */
423  3.38015286679526343505e+03, /* 0x40AA684E, 0x448E7C9A */
424  5.54772909720722782367e+03, /* 0x40B5ABBA, 0xA61D54A6 */
425  1.90311919338810798763e+03, /* 0x409DBC7A, 0x0DD4DF4B */
426  -1.35201191444307340817e+02, /* 0xC060E670, 0x290A311F */
427 };
428 
429 #ifdef __STDC__
430 static const double qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
431 #else
432 static double qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
433 #endif
434  -1.78381727510958865572e-07, /* 0xBE87F126, 0x44C626D2 */
435  -1.02517042607985553460e-01, /* 0xBFBA3E8E, 0x9148B010 */
436  -2.75220568278187460720e+00, /* 0xC0060484, 0x69BB4EDA */
437  -1.96636162643703720221e+01, /* 0xC033A9E2, 0xC168907F */
438  -4.23253133372830490089e+01, /* 0xC04529A3, 0xDE104AAA */
439  -2.13719211703704061733e+01, /* 0xC0355F36, 0x39CF6E52 */
440 };
441 #ifdef __STDC__
442 static const double qs2[6] = {
443 #else
444 static double qs2[6] = {
445 #endif
446  2.95333629060523854548e+01, /* 0x403D888A, 0x78AE64FF */
447  2.52981549982190529136e+02, /* 0x406F9F68, 0xDB821CBA */
448  7.57502834868645436472e+02, /* 0x4087AC05, 0xCE49A0F7 */
449  7.39393205320467245656e+02, /* 0x40871B25, 0x48D4C029 */
450  1.55949003336666123687e+02, /* 0x40637E5E, 0x3C3ED8D4 */
451  -4.95949898822628210127e+00, /* 0xC013D686, 0xE71BE86B */
452 };
453 
454 #ifdef __STDC__
455  static double qone(double x)
456 #else
457  static double qone(x)
458  double x;
459 #endif
460 {
461 #ifdef __STDC__
462  const double *p=NULL,*q=NULL;
463 #else
464  double *p,*q;
465 #endif
466  double s,r,z;
467  int ix;
468  ix = 0x7fffffff&__HI(x);
469  if(ix>=0x40200000) {p = qr8; q= qs8;}
470  else if(ix>=0x40122E8B){p = qr5; q= qs5;}
471  else if(ix>=0x4006DB6D){p = qr3; q= qs3;}
472  else if(ix>=0x40000000){p = qr2; q= qs2;}
473  z = one/(x*x);
474  r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
475  s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
476  return (.375 + r/s)/x;
477 }