POK
e_j1.c
1 /*
2  * POK header
3  *
4  * The following file is a part of the POK project. Any modification should
5  * made according to the POK licence. You CANNOT use this file or a part of
6  * this file is this part of a file for your own project
7  *
8  * For more information on the POK licence, please see our LICENCE FILE
9  *
10  * Please follow the coding guidelines described in doc/CODING_GUIDELINES
11  *
12  * Copyright (c) 2007-2009 POK team
13  *
14  * Created by julien on Fri Jan 30 14:41:34 2009
15  */
16 
17 /* @(#)e_j1.c 5.1 93/09/24 */
18 /*
19  * ====================================================
20  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
21  *
22  * Developed at SunPro, a Sun Microsystems, Inc. business.
23  * Permission to use, copy, modify, and distribute this
24  * software is freely granted, provided that this notice
25  * is preserved.
26  * ====================================================
27  */
28 
29 /* __ieee754_j1(x), __ieee754_y1(x)
30  * Bessel function of the first and second kinds of order zero.
31  * Method -- j1(x):
32  * 1. For tiny x, we use j1(x) = x/2 - x^3/16 + x^5/384 - ...
33  * 2. Reduce x to |x| since j1(x)=-j1(-x), and
34  * for x in (0,2)
35  * j1(x) = x/2 + x*z*R0/S0, where z = x*x;
36  * (precision: |j1/x - 1/2 - R0/S0 |<2**-61.51 )
37  * for x in (2,inf)
38  * j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x1)-q1(x)*sin(x1))
39  * y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1))
40  * where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1)
41  * as follow:
42  * cos(x1) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
43  * = 1/sqrt(2) * (sin(x) - cos(x))
44  * sin(x1) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
45  * = -1/sqrt(2) * (sin(x) + cos(x))
46  * (To avoid cancellation, use
47  * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
48  * to compute the worse one.)
49  *
50  * 3 Special cases
51  * j1(nan)= nan
52  * j1(0) = 0
53  * j1(inf) = 0
54  *
55  * Method -- y1(x):
56  * 1. screen out x<=0 cases: y1(0)=-inf, y1(x<0)=NaN
57  * 2. For x<2.
58  * Since
59  * y1(x) = 2/pi*(j1(x)*(ln(x/2)+Euler)-1/x-x/2+5/64*x^3-...)
60  * therefore y1(x)-2/pi*j1(x)*ln(x)-1/x is an odd function.
61  * We use the following function to approximate y1,
62  * y1(x) = x*U(z)/V(z) + (2/pi)*(j1(x)*ln(x)-1/x), z= x^2
63  * where for x in [0,2] (abs err less than 2**-65.89)
64  * U(z) = U0[0] + U0[1]*z + ... + U0[4]*z^4
65  * V(z) = 1 + v0[0]*z + ... + v0[4]*z^5
66  * Note: For tiny x, 1/x dominate y1 and hence
67  * y1(tiny) = -2/pi/tiny, (choose tiny<2**-54)
68  * 3. For x>=2.
69  * y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1))
70  * where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1)
71  * by method mentioned above.
72  */
73 
74 #ifdef POK_NEEDS_LIBMATH
75 
76 #include <libm.h>
77 #include "math_private.h"
78 
79 static double pone(double), qone(double);
80 
81 static const double
82 huge = 1e300,
83 one = 1.0,
84 invsqrtpi= 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
85 tpi = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
86  /* R0/S0 on [0,2] */
87 r00 = -6.25000000000000000000e-02, /* 0xBFB00000, 0x00000000 */
88 r01 = 1.40705666955189706048e-03, /* 0x3F570D9F, 0x98472C61 */
89 r02 = -1.59955631084035597520e-05, /* 0xBEF0C5C6, 0xBA169668 */
90 r03 = 4.96727999609584448412e-08, /* 0x3E6AAAFA, 0x46CA0BD9 */
91 s01 = 1.91537599538363460805e-02, /* 0x3F939D0B, 0x12637E53 */
92 s02 = 1.85946785588630915560e-04, /* 0x3F285F56, 0xB9CDF664 */
93 s03 = 1.17718464042623683263e-06, /* 0x3EB3BFF8, 0x333F8498 */
94 s04 = 5.04636257076217042715e-09, /* 0x3E35AC88, 0xC97DFF2C */
95 s05 = 1.23542274426137913908e-11; /* 0x3DAB2ACF, 0xCFB97ED8 */
96 
97 static const double zero = 0.0;
98 
99 double
100 __ieee754_j1(double x)
101 {
102  double z, s,c,ss,cc,r,u,v,y;
103  int32_t hx,ix;
104 
105  GET_HIGH_WORD(hx,x);
106  ix = hx&0x7fffffff;
107  if(ix>=0x7ff00000) return one/x;
108  y = fabs(x);
109  if(ix >= 0x40000000) { /* |x| >= 2.0 */
110  s = sin(y);
111  c = cos(y);
112  ss = -s-c;
113  cc = s-c;
114  if(ix<0x7fe00000) { /* make sure y+y not overflow */
115  z = cos(y+y);
116  if ((s*c)>zero) cc = z/ss;
117  else ss = z/cc;
118  }
119  /*
120  * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x)
121  * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x)
122  */
123  if(ix>0x48000000) z = (invsqrtpi*cc)/sqrt(y);
124  else {
125  u = pone(y); v = qone(y);
126  z = invsqrtpi*(u*cc-v*ss)/sqrt(y);
127  }
128  if(hx<0) return -z;
129  else return z;
130  }
131  if(ix<0x3e400000) { /* |x|<2**-27 */
132  if(huge+x>one) return 0.5*x;/* inexact if x!=0 necessary */
133  }
134  z = x*x;
135  r = z*(r00+z*(r01+z*(r02+z*r03)));
136  s = one+z*(s01+z*(s02+z*(s03+z*(s04+z*s05))));
137  r *= x;
138  return(x*0.5+r/s);
139 }
140 
141 static const double U0[5] = {
142  -1.96057090646238940668e-01, /* 0xBFC91866, 0x143CBC8A */
143  5.04438716639811282616e-02, /* 0x3FA9D3C7, 0x76292CD1 */
144  -1.91256895875763547298e-03, /* 0xBF5F55E5, 0x4844F50F */
145  2.35252600561610495928e-05, /* 0x3EF8AB03, 0x8FA6B88E */
146  -9.19099158039878874504e-08, /* 0xBE78AC00, 0x569105B8 */
147 };
148 static const double V0[5] = {
149  1.99167318236649903973e-02, /* 0x3F94650D, 0x3F4DA9F0 */
150  2.02552581025135171496e-04, /* 0x3F2A8C89, 0x6C257764 */
151  1.35608801097516229404e-06, /* 0x3EB6C05A, 0x894E8CA6 */
152  6.22741452364621501295e-09, /* 0x3E3ABF1D, 0x5BA69A86 */
153  1.66559246207992079114e-11, /* 0x3DB25039, 0xDACA772A */
154 };
155 
156 double
157 __ieee754_y1(double x)
158 {
159  double z, s,c,ss,cc,u,v;
160  int32_t hx,ix,lx;
161 
162  EXTRACT_WORDS(hx,lx,x);
163  ix = 0x7fffffff&hx;
164  /* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */
165  if(ix>=0x7ff00000) return one/(x+x*x);
166  if((ix|lx)==0) return -one/zero;
167  if(hx<0) return zero/zero;
168  if(ix >= 0x40000000) { /* |x| >= 2.0 */
169  s = sin(x);
170  c = cos(x);
171  ss = -s-c;
172  cc = s-c;
173  if(ix<0x7fe00000) { /* make sure x+x not overflow */
174  z = cos(x+x);
175  if ((s*c)>zero) cc = z/ss;
176  else ss = z/cc;
177  }
178  /* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0))
179  * where x0 = x-3pi/4
180  * Better formula:
181  * cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
182  * = 1/sqrt(2) * (sin(x) - cos(x))
183  * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
184  * = -1/sqrt(2) * (cos(x) + sin(x))
185  * To avoid cancellation, use
186  * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
187  * to compute the worse one.
188  */
189  if(ix>0x48000000) z = (invsqrtpi*ss)/sqrt(x);
190  else {
191  u = pone(x); v = qone(x);
192  z = invsqrtpi*(u*ss+v*cc)/sqrt(x);
193  }
194  return z;
195  }
196  if(ix<=0x3c900000) { /* x < 2**-54 */
197  return(-tpi/x);
198  }
199  z = x*x;
200  u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4])));
201  v = one+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4]))));
202  return(x*(u/v) + tpi*(__ieee754_j1(x)*__ieee754_log(x)-one/x));
203 }
204 
205 /* For x >= 8, the asymptotic expansions of pone is
206  * 1 + 15/128 s^2 - 4725/2^15 s^4 - ..., where s = 1/x.
207  * We approximate pone by
208  * pone(x) = 1 + (R/S)
209  * where R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10
210  * S = 1 + ps0*s^2 + ... + ps4*s^10
211  * and
212  * | pone(x)-1-R/S | <= 2 ** ( -60.06)
213  */
214 
215 static const double pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
216  0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
217  1.17187499999988647970e-01, /* 0x3FBDFFFF, 0xFFFFFCCE */
218  1.32394806593073575129e+01, /* 0x402A7A9D, 0x357F7FCE */
219  4.12051854307378562225e+02, /* 0x4079C0D4, 0x652EA590 */
220  3.87474538913960532227e+03, /* 0x40AE457D, 0xA3A532CC */
221  7.91447954031891731574e+03, /* 0x40BEEA7A, 0xC32782DD */
222 };
223 static const double ps8[5] = {
224  1.14207370375678408436e+02, /* 0x405C8D45, 0x8E656CAC */
225  3.65093083420853463394e+03, /* 0x40AC85DC, 0x964D274F */
226  3.69562060269033463555e+04, /* 0x40E20B86, 0x97C5BB7F */
227  9.76027935934950801311e+04, /* 0x40F7D42C, 0xB28F17BB */
228  3.08042720627888811578e+04, /* 0x40DE1511, 0x697A0B2D */
229 };
230 
231 static const double pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
232  1.31990519556243522749e-11, /* 0x3DAD0667, 0xDAE1CA7D */
233  1.17187493190614097638e-01, /* 0x3FBDFFFF, 0xE2C10043 */
234  6.80275127868432871736e+00, /* 0x401B3604, 0x6E6315E3 */
235  1.08308182990189109773e+02, /* 0x405B13B9, 0x452602ED */
236  5.17636139533199752805e+02, /* 0x40802D16, 0xD052D649 */
237  5.28715201363337541807e+02, /* 0x408085B8, 0xBB7E0CB7 */
238 };
239 static const double ps5[5] = {
240  5.92805987221131331921e+01, /* 0x404DA3EA, 0xA8AF633D */
241  9.91401418733614377743e+02, /* 0x408EFB36, 0x1B066701 */
242  5.35326695291487976647e+03, /* 0x40B4E944, 0x5706B6FB */
243  7.84469031749551231769e+03, /* 0x40BEA4B0, 0xB8A5BB15 */
244  1.50404688810361062679e+03, /* 0x40978030, 0x036F5E51 */
245 };
246 
247 static const double pr3[6] = {
248  3.02503916137373618024e-09, /* 0x3E29FC21, 0xA7AD9EDD */
249  1.17186865567253592491e-01, /* 0x3FBDFFF5, 0x5B21D17B */
250  3.93297750033315640650e+00, /* 0x400F76BC, 0xE85EAD8A */
251  3.51194035591636932736e+01, /* 0x40418F48, 0x9DA6D129 */
252  9.10550110750781271918e+01, /* 0x4056C385, 0x4D2C1837 */
253  4.85590685197364919645e+01, /* 0x4048478F, 0x8EA83EE5 */
254 };
255 static const double ps3[5] = {
256  3.47913095001251519989e+01, /* 0x40416549, 0xA134069C */
257  3.36762458747825746741e+02, /* 0x40750C33, 0x07F1A75F */
258  1.04687139975775130551e+03, /* 0x40905B7C, 0x5037D523 */
259  8.90811346398256432622e+02, /* 0x408BD67D, 0xA32E31E9 */
260  1.03787932439639277504e+02, /* 0x4059F26D, 0x7C2EED53 */
261 };
262 
263 static const double pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
264  1.07710830106873743082e-07, /* 0x3E7CE9D4, 0xF65544F4 */
265  1.17176219462683348094e-01, /* 0x3FBDFF42, 0xBE760D83 */
266  2.36851496667608785174e+00, /* 0x4002F2B7, 0xF98FAEC0 */
267  1.22426109148261232917e+01, /* 0x40287C37, 0x7F71A964 */
268  1.76939711271687727390e+01, /* 0x4031B1A8, 0x177F8EE2 */
269  5.07352312588818499250e+00, /* 0x40144B49, 0xA574C1FE */
270 };
271 static const double ps2[5] = {
272  2.14364859363821409488e+01, /* 0x40356FBD, 0x8AD5ECDC */
273  1.25290227168402751090e+02, /* 0x405F5293, 0x14F92CD5 */
274  2.32276469057162813669e+02, /* 0x406D08D8, 0xD5A2DBD9 */
275  1.17679373287147100768e+02, /* 0x405D6B7A, 0xDA1884A9 */
276  8.36463893371618283368e+00, /* 0x4020BAB1, 0xF44E5192 */
277 };
278 
279 static double
280 pone(double x)
281 {
282  const double *p,*q;
283  double z,r,s;
284  int32_t ix;
285 
286  p = q = 0;
287  GET_HIGH_WORD(ix,x);
288  ix &= 0x7fffffff;
289  if(ix>=0x40200000) {p = pr8; q= ps8;}
290  else if(ix>=0x40122E8B){p = pr5; q= ps5;}
291  else if(ix>=0x4006DB6D){p = pr3; q= ps3;}
292  else if(ix>=0x40000000){p = pr2; q= ps2;}
293  z = one/(x*x);
294  r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
295  s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
296  return one+ r/s;
297 }
298 
299 
300 /* For x >= 8, the asymptotic expansions of qone is
301  * 3/8 s - 105/1024 s^3 - ..., where s = 1/x.
302  * We approximate pone by
303  * qone(x) = s*(0.375 + (R/S))
304  * where R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10
305  * S = 1 + qs1*s^2 + ... + qs6*s^12
306  * and
307  * | qone(x)/s -0.375-R/S | <= 2 ** ( -61.13)
308  */
309 
310 static const double qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
311  0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
312  -1.02539062499992714161e-01, /* 0xBFBA3FFF, 0xFFFFFDF3 */
313  -1.62717534544589987888e+01, /* 0xC0304591, 0xA26779F7 */
314  -7.59601722513950107896e+02, /* 0xC087BCD0, 0x53E4B576 */
315  -1.18498066702429587167e+04, /* 0xC0C724E7, 0x40F87415 */
316  -4.84385124285750353010e+04, /* 0xC0E7A6D0, 0x65D09C6A */
317 };
318 static const double qs8[6] = {
319  1.61395369700722909556e+02, /* 0x40642CA6, 0xDE5BCDE5 */
320  7.82538599923348465381e+03, /* 0x40BE9162, 0xD0D88419 */
321  1.33875336287249578163e+05, /* 0x4100579A, 0xB0B75E98 */
322  7.19657723683240939863e+05, /* 0x4125F653, 0x72869C19 */
323  6.66601232617776375264e+05, /* 0x412457D2, 0x7719AD5C */
324  -2.94490264303834643215e+05, /* 0xC111F969, 0x0EA5AA18 */
325 };
326 
327 static const double qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
328  -2.08979931141764104297e-11, /* 0xBDB6FA43, 0x1AA1A098 */
329  -1.02539050241375426231e-01, /* 0xBFBA3FFF, 0xCB597FEF */
330  -8.05644828123936029840e+00, /* 0xC0201CE6, 0xCA03AD4B */
331  -1.83669607474888380239e+02, /* 0xC066F56D, 0x6CA7B9B0 */
332  -1.37319376065508163265e+03, /* 0xC09574C6, 0x6931734F */
333  -2.61244440453215656817e+03, /* 0xC0A468E3, 0x88FDA79D */
334 };
335 static const double qs5[6] = {
336  8.12765501384335777857e+01, /* 0x405451B2, 0xFF5A11B2 */
337  1.99179873460485964642e+03, /* 0x409F1F31, 0xE77BF839 */
338  1.74684851924908907677e+04, /* 0x40D10F1F, 0x0D64CE29 */
339  4.98514270910352279316e+04, /* 0x40E8576D, 0xAABAD197 */
340  2.79480751638918118260e+04, /* 0x40DB4B04, 0xCF7C364B */
341  -4.71918354795128470869e+03, /* 0xC0B26F2E, 0xFCFFA004 */
342 };
343 
344 static const double qr3[6] = {
345  -5.07831226461766561369e-09, /* 0xBE35CFA9, 0xD38FC84F */
346  -1.02537829820837089745e-01, /* 0xBFBA3FEB, 0x51AEED54 */
347  -4.61011581139473403113e+00, /* 0xC01270C2, 0x3302D9FF */
348  -5.78472216562783643212e+01, /* 0xC04CEC71, 0xC25D16DA */
349  -2.28244540737631695038e+02, /* 0xC06C87D3, 0x4718D55F */
350  -2.19210128478909325622e+02, /* 0xC06B66B9, 0x5F5C1BF6 */
351 };
352 static const double qs3[6] = {
353  4.76651550323729509273e+01, /* 0x4047D523, 0xCCD367E4 */
354  6.73865112676699709482e+02, /* 0x40850EEB, 0xC031EE3E */
355  3.38015286679526343505e+03, /* 0x40AA684E, 0x448E7C9A */
356  5.54772909720722782367e+03, /* 0x40B5ABBA, 0xA61D54A6 */
357  1.90311919338810798763e+03, /* 0x409DBC7A, 0x0DD4DF4B */
358  -1.35201191444307340817e+02, /* 0xC060E670, 0x290A311F */
359 };
360 
361 static const double qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
362  -1.78381727510958865572e-07, /* 0xBE87F126, 0x44C626D2 */
363  -1.02517042607985553460e-01, /* 0xBFBA3E8E, 0x9148B010 */
364  -2.75220568278187460720e+00, /* 0xC0060484, 0x69BB4EDA */
365  -1.96636162643703720221e+01, /* 0xC033A9E2, 0xC168907F */
366  -4.23253133372830490089e+01, /* 0xC04529A3, 0xDE104AAA */
367  -2.13719211703704061733e+01, /* 0xC0355F36, 0x39CF6E52 */
368 };
369 static const double qs2[6] = {
370  2.95333629060523854548e+01, /* 0x403D888A, 0x78AE64FF */
371  2.52981549982190529136e+02, /* 0x406F9F68, 0xDB821CBA */
372  7.57502834868645436472e+02, /* 0x4087AC05, 0xCE49A0F7 */
373  7.39393205320467245656e+02, /* 0x40871B25, 0x48D4C029 */
374  1.55949003336666123687e+02, /* 0x40637E5E, 0x3C3ED8D4 */
375  -4.95949898822628210127e+00, /* 0xC013D686, 0xE71BE86B */
376 };
377 
378 static double
379 qone(double x)
380 {
381  const double *p,*q;
382  double s,r,z;
383  int32_t ix;
384 
385  p = q = 0;
386  GET_HIGH_WORD(ix,x);
387  ix &= 0x7fffffff;
388  if(ix>=0x40200000) {p = qr8; q= qs8;}
389  else if(ix>=0x40122E8B){p = qr5; q= qs5;}
390  else if(ix>=0x4006DB6D){p = qr3; q= qs3;}
391  else if(ix>=0x40000000){p = qr2; q= qs2;}
392  z = one/(x*x);
393  r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
394  s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
395  return (.375 + r/s)/x;
396 }
397 #endif
398