overby.me
this repo has no description
1/* @(#)e_j0.c 5.1 93/09/24 */
2/*
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5 *
6 * Developed at SunPro, a Sun Microsystems, Inc. business.
7 * Permission to use, copy, modify, and distribute this
8 * software is freely granted, provided that this notice
9 * is preserved.
10 * ====================================================
11 */
12
13
14#include <sys/cdefs.h>
15#if defined(LIBM_SCCS) && !defined(lint)
16__RCSID("$NetBSD: e_j0.c,v 1.10 1999/07/02 15:37:39 simonb Exp $");
17#endif
18
19/* __ieee754_j0(x), __ieee754_y0(x)
20 * Bessel function of the first and second kinds of order zero.
21 * Method -- j0(x):
22 * 1. For tiny x, we use j0(x) = 1 - x^2/4 + x^4/64 - ...
23 * 2. Reduce x to |x| since j0(x)=j0(-x), and
24 * for x in (0,2)
25 * j0(x) = 1-z/4+ z^2*R0/S0, where z = x*x;
26 * (precision: |j0-1+z/4-z^2R0/S0 |<2**-63.67 )
27 * for x in (2,inf)
28 * j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)-q0(x)*sin(x0))
29 * where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
30 * as follow:
31 * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
32 * = 1/sqrt(2) * (cos(x) + sin(x))
33 * sin(x0) = sin(x)cos(pi/4)-cos(x)sin(pi/4)
34 * = 1/sqrt(2) * (sin(x) - cos(x))
35 * (To avoid cancellation, use
36 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
37 * to compute the worse one.)
38 *
39 * 3 Special cases
40 * j0(nan)= nan
41 * j0(0) = 1
42 * j0(inf) = 0
43 *
44 * Method -- y0(x):
45 * 1. For x<2.
46 * Since
47 * y0(x) = 2/pi*(j0(x)*(ln(x/2)+Euler) + x^2/4 - ...)
48 * therefore y0(x)-2/pi*j0(x)*ln(x) is an even function.
49 * We use the following function to approximate y0,
50 * y0(x) = U(z)/V(z) + (2/pi)*(j0(x)*ln(x)), z= x^2
51 * where
52 * U(z) = u00 + u01*z + ... + u06*z^6
53 * V(z) = 1 + v01*z + ... + v04*z^4
54 * with absolute approximation error bounded by 2**-72.
55 * Note: For tiny x, U/V = u0 and j0(x)~1, hence
56 * y0(tiny) = u0 + (2/pi)*ln(tiny), (choose tiny<2**-27)
57 * 2. For x>=2.
58 * y0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)+q0(x)*sin(x0))
59 * where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
60 * by the method mentioned above.
61 * 3. Special cases: y0(0)=-inf, y0(x<0)=NaN, y0(inf)=0.
62 */
63
64#include "math.h"
65#include "math_private.h"
66
67#ifdef __STDC__
68static double pzero(double), qzero(double);
69#else
70static double pzero(), qzero();
71#endif
72
73static const double
74huge = 1e300,
75one = 1.0,
76invsqrtpi= 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
77tpi = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
78 /* R0/S0 on [0, 2.00] */
79R02 = 1.56249999999999947958e-02, /* 0x3F8FFFFF, 0xFFFFFFFD */
80R03 = -1.89979294238854721751e-04, /* 0xBF28E6A5, 0xB61AC6E9 */
81R04 = 1.82954049532700665670e-06, /* 0x3EBEB1D1, 0x0C503919 */
82R05 = -4.61832688532103189199e-09, /* 0xBE33D5E7, 0x73D63FCE */
83S01 = 1.56191029464890010492e-02, /* 0x3F8FFCE8, 0x82C8C2A4 */
84S02 = 1.16926784663337450260e-04, /* 0x3F1EA6D2, 0xDD57DBF4 */
85S03 = 5.13546550207318111446e-07, /* 0x3EA13B54, 0xCE84D5A9 */
86S04 = 1.16614003333790000205e-09; /* 0x3E1408BC, 0xF4745D8F */
87
88static const double zero = 0.0;
89
90double j0(double x)
91{
92 double z, s,c,ss,cc,r,u,v;
93 int32_t hx,ix;
94
95 GET_HIGH_WORD(hx,x);
96 ix = hx&0x7fffffff;
97 if(ix>=0x7ff00000) return one/(x*x);
98 x = fabs(x);
99 if(ix >= 0x40000000) { /* |x| >= 2.0 */
100 s = sin(x);
101 c = cos(x);
102 ss = s-c;
103 cc = s+c;
104 if(ix<0x7fe00000) { /* make sure x+x not overflow */
105 z = -cos(x+x);
106 if ((s*c)<zero) cc = z/ss;
107 else ss = z/cc;
108 }
109 /*
110 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
111 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
112 */
113 if(ix>0x48000000) z = (invsqrtpi*cc)/sqrt(x);
114 else {
115 u = pzero(x); v = qzero(x);
116 z = invsqrtpi*(u*cc-v*ss)/sqrt(x);
117 }
118 return z;
119 }
120 if(ix<0x3f200000) { /* |x| < 2**-13 */
121 if(huge+x>one) { /* raise inexact if x != 0 */
122 if(ix<0x3e400000) return one; /* |x|<2**-27 */
123 else return one - 0.25*x*x;
124 }
125 }
126 z = x*x;
127 r = z*(R02+z*(R03+z*(R04+z*R05)));
128 s = one+z*(S01+z*(S02+z*(S03+z*S04)));
129 if(ix < 0x3FF00000) { /* |x| < 1.00 */
130 return one + z*(-0.25+(r/s));
131 } else {
132 u = 0.5*x;
133 return((one+u)*(one-u)+z*(r/s));
134 }
135}
136
137static const double
138u00 = -7.38042951086872317523e-02, /* 0xBFB2E4D6, 0x99CBD01F */
139u01 = 1.76666452509181115538e-01, /* 0x3FC69D01, 0x9DE9E3FC */
140u02 = -1.38185671945596898896e-02, /* 0xBF8C4CE8, 0xB16CFA97 */
141u03 = 3.47453432093683650238e-04, /* 0x3F36C54D, 0x20B29B6B */
142u04 = -3.81407053724364161125e-06, /* 0xBECFFEA7, 0x73D25CAD */
143u05 = 1.95590137035022920206e-08, /* 0x3E550057, 0x3B4EABD4 */
144u06 = -3.98205194132103398453e-11, /* 0xBDC5E43D, 0x693FB3C8 */
145v01 = 1.27304834834123699328e-02, /* 0x3F8A1270, 0x91C9C71A */
146v02 = 7.60068627350353253702e-05, /* 0x3F13ECBB, 0xF578C6C1 */
147v03 = 2.59150851840457805467e-07, /* 0x3E91642D, 0x7FF202FD */
148v04 = 4.41110311332675467403e-10; /* 0x3DFE5018, 0x3BD6D9EF */
149
150#define __ieee754_j0 j0
151#define __ieee754_log log
152
153double y0(double x)
154{
155 double z, s,c,ss,cc,u,v;
156 int32_t hx,ix,lx;
157
158 EXTRACT_WORDS(hx,lx,x);
159 ix = 0x7fffffff&hx;
160 /* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0 */
161 if(ix>=0x7ff00000) return one/(x+x*x);
162 if((ix|lx)==0) return -one/zero;
163 if(hx<0) return zero/zero;
164 if(ix >= 0x40000000) { /* |x| >= 2.0 */
165 /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
166 * where x0 = x-pi/4
167 * Better formula:
168 * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
169 * = 1/sqrt(2) * (sin(x) + cos(x))
170 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
171 * = 1/sqrt(2) * (sin(x) - cos(x))
172 * To avoid cancellation, use
173 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
174 * to compute the worse one.
175 */
176 s = sin(x);
177 c = cos(x);
178 ss = s-c;
179 cc = s+c;
180 /*
181 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
182 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
183 */
184 if(ix<0x7fe00000) { /* make sure x+x not overflow */
185 z = -cos(x+x);
186 if ((s*c)<zero) cc = z/ss;
187 else ss = z/cc;
188 }
189 if(ix>0x48000000) z = (invsqrtpi*ss)/sqrt(x);
190 else {
191 u = pzero(x); v = qzero(x);
192 z = invsqrtpi*(u*ss+v*cc)/sqrt(x);
193 }
194 return z;
195 }
196 if(ix<=0x3e400000) { /* x < 2**-27 */
197 return(u00 + tpi*__ieee754_log(x));
198 }
199 z = x*x;
200 u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06)))));
201 v = one+z*(v01+z*(v02+z*(v03+z*v04)));
202 return(u/v + tpi*(__ieee754_j0(x)*__ieee754_log(x)));
203}
204
205/* The asymptotic expansions of pzero is
206 * 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x.
207 * For x >= 2, We approximate pzero by
208 * pzero(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 * | pzero(x)-1-R/S | <= 2 ** ( -60.26)
213 */
214#ifdef __STDC__
215static const double pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
216#else
217static double pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
218#endif
219 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
220 -7.03124999999900357484e-02, /* 0xBFB1FFFF, 0xFFFFFD32 */
221 -8.08167041275349795626e+00, /* 0xC02029D0, 0xB44FA779 */
222 -2.57063105679704847262e+02, /* 0xC0701102, 0x7B19E863 */
223 -2.48521641009428822144e+03, /* 0xC0A36A6E, 0xCD4DCAFC */
224 -5.25304380490729545272e+03, /* 0xC0B4850B, 0x36CC643D */
225};
226#ifdef __STDC__
227static const double pS8[5] = {
228#else
229static double pS8[5] = {
230#endif
231 1.16534364619668181717e+02, /* 0x405D2233, 0x07A96751 */
232 3.83374475364121826715e+03, /* 0x40ADF37D, 0x50596938 */
233 4.05978572648472545552e+04, /* 0x40E3D2BB, 0x6EB6B05F */
234 1.16752972564375915681e+05, /* 0x40FC810F, 0x8F9FA9BD */
235 4.76277284146730962675e+04, /* 0x40E74177, 0x4F2C49DC */
236};
237
238#ifdef __STDC__
239static const double pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
240#else
241static double pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
242#endif
243 -1.14125464691894502584e-11, /* 0xBDA918B1, 0x47E495CC */
244 -7.03124940873599280078e-02, /* 0xBFB1FFFF, 0xE69AFBC6 */
245 -4.15961064470587782438e+00, /* 0xC010A370, 0xF90C6BBF */
246 -6.76747652265167261021e+01, /* 0xC050EB2F, 0x5A7D1783 */
247 -3.31231299649172967747e+02, /* 0xC074B3B3, 0x6742CC63 */
248 -3.46433388365604912451e+02, /* 0xC075A6EF, 0x28A38BD7 */
249};
250#ifdef __STDC__
251static const double pS5[5] = {
252#else
253static double pS5[5] = {
254#endif
255 6.07539382692300335975e+01, /* 0x404E6081, 0x0C98C5DE */
256 1.05125230595704579173e+03, /* 0x40906D02, 0x5C7E2864 */
257 5.97897094333855784498e+03, /* 0x40B75AF8, 0x8FBE1D60 */
258 9.62544514357774460223e+03, /* 0x40C2CCB8, 0xFA76FA38 */
259 2.40605815922939109441e+03, /* 0x40A2CC1D, 0xC70BE864 */
260};
261
262#ifdef __STDC__
263static const double pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
264#else
265static double pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
266#endif
267 -2.54704601771951915620e-09, /* 0xBE25E103, 0x6FE1AA86 */
268 -7.03119616381481654654e-02, /* 0xBFB1FFF6, 0xF7C0E24B */
269 -2.40903221549529611423e+00, /* 0xC00345B2, 0xAEA48074 */
270 -2.19659774734883086467e+01, /* 0xC035F74A, 0x4CB94E14 */
271 -5.80791704701737572236e+01, /* 0xC04D0A22, 0x420A1A45 */
272 -3.14479470594888503854e+01, /* 0xC03F72AC, 0xA892D80F */
273};
274#ifdef __STDC__
275static const double pS3[5] = {
276#else
277static double pS3[5] = {
278#endif
279 3.58560338055209726349e+01, /* 0x4041ED92, 0x84077DD3 */
280 3.61513983050303863820e+02, /* 0x40769839, 0x464A7C0E */
281 1.19360783792111533330e+03, /* 0x4092A66E, 0x6D1061D6 */
282 1.12799679856907414432e+03, /* 0x40919FFC, 0xB8C39B7E */
283 1.73580930813335754692e+02, /* 0x4065B296, 0xFC379081 */
284};
285
286#ifdef __STDC__
287static const double pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
288#else
289static double pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
290#endif
291 -8.87534333032526411254e-08, /* 0xBE77D316, 0xE927026D */
292 -7.03030995483624743247e-02, /* 0xBFB1FF62, 0x495E1E42 */
293 -1.45073846780952986357e+00, /* 0xBFF73639, 0x8A24A843 */
294 -7.63569613823527770791e+00, /* 0xC01E8AF3, 0xEDAFA7F3 */
295 -1.11931668860356747786e+01, /* 0xC02662E6, 0xC5246303 */
296 -3.23364579351335335033e+00, /* 0xC009DE81, 0xAF8FE70F */
297};
298#ifdef __STDC__
299static const double pS2[5] = {
300#else
301static double pS2[5] = {
302#endif
303 2.22202997532088808441e+01, /* 0x40363865, 0x908B5959 */
304 1.36206794218215208048e+02, /* 0x4061069E, 0x0EE8878F */
305 2.70470278658083486789e+02, /* 0x4070E786, 0x42EA079B */
306 1.53875394208320329881e+02, /* 0x40633C03, 0x3AB6FAFF */
307 1.46576176948256193810e+01, /* 0x402D50B3, 0x44391809 */
308};
309
310#ifdef __STDC__
311 static double pzero(double x)
312#else
313 static double pzero(x)
314 double x;
315#endif
316{
317#ifdef __STDC__
318 const double *p,*q;
319#else
320 double *p,*q;
321#endif
322 double z,r,s;
323 int32_t ix;
324
325 p = q = 0;
326 GET_HIGH_WORD(ix,x);
327 ix &= 0x7fffffff;
328 if(ix>=0x40200000) {p = pR8; q= pS8;}
329 else if(ix>=0x40122E8B){p = pR5; q= pS5;}
330 else if(ix>=0x4006DB6D){p = pR3; q= pS3;}
331 else if(ix>=0x40000000){p = pR2; q= pS2;}
332 z = one/(x*x);
333 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
334 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
335 return one+ r/s;
336}
337
338
339/* For x >= 8, the asymptotic expansions of qzero is
340 * -1/8 s + 75/1024 s^3 - ..., where s = 1/x.
341 * We approximate pzero by
342 * qzero(x) = s*(-1.25 + (R/S))
343 * where R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10
344 * S = 1 + qS0*s^2 + ... + qS5*s^12
345 * and
346 * | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22)
347 */
348#ifdef __STDC__
349static const double qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
350#else
351static double qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
352#endif
353 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
354 7.32421874999935051953e-02, /* 0x3FB2BFFF, 0xFFFFFE2C */
355 1.17682064682252693899e+01, /* 0x40278952, 0x5BB334D6 */
356 5.57673380256401856059e+02, /* 0x40816D63, 0x15301825 */
357 8.85919720756468632317e+03, /* 0x40C14D99, 0x3E18F46D */
358 3.70146267776887834771e+04, /* 0x40E212D4, 0x0E901566 */
359};
360#ifdef __STDC__
361static const double qS8[6] = {
362#else
363static double qS8[6] = {
364#endif
365 1.63776026895689824414e+02, /* 0x406478D5, 0x365B39BC */
366 8.09834494656449805916e+03, /* 0x40BFA258, 0x4E6B0563 */
367 1.42538291419120476348e+05, /* 0x41016652, 0x54D38C3F */
368 8.03309257119514397345e+05, /* 0x412883DA, 0x83A52B43 */
369 8.40501579819060512818e+05, /* 0x4129A66B, 0x28DE0B3D */
370 -3.43899293537866615225e+05, /* 0xC114FD6D, 0x2C9530C5 */
371};
372
373#ifdef __STDC__
374static const double qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
375#else
376static double qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
377#endif
378 1.84085963594515531381e-11, /* 0x3DB43D8F, 0x29CC8CD9 */
379 7.32421766612684765896e-02, /* 0x3FB2BFFF, 0xD172B04C */
380 5.83563508962056953777e+00, /* 0x401757B0, 0xB9953DD3 */
381 1.35111577286449829671e+02, /* 0x4060E392, 0x0A8788E9 */
382 1.02724376596164097464e+03, /* 0x40900CF9, 0x9DC8C481 */
383 1.98997785864605384631e+03, /* 0x409F17E9, 0x53C6E3A6 */
384};
385#ifdef __STDC__
386static const double qS5[6] = {
387#else
388static double qS5[6] = {
389#endif
390 8.27766102236537761883e+01, /* 0x4054B1B3, 0xFB5E1543 */
391 2.07781416421392987104e+03, /* 0x40A03BA0, 0xDA21C0CE */
392 1.88472887785718085070e+04, /* 0x40D267D2, 0x7B591E6D */
393 5.67511122894947329769e+04, /* 0x40EBB5E3, 0x97E02372 */
394 3.59767538425114471465e+04, /* 0x40E19118, 0x1F7A54A0 */
395 -5.35434275601944773371e+03, /* 0xC0B4EA57, 0xBEDBC609 */
396};
397
398#ifdef __STDC__
399static const double qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
400#else
401static double qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
402#endif
403 4.37741014089738620906e-09, /* 0x3E32CD03, 0x6ADECB82 */
404 7.32411180042911447163e-02, /* 0x3FB2BFEE, 0x0E8D0842 */
405 3.34423137516170720929e+00, /* 0x400AC0FC, 0x61149CF5 */
406 4.26218440745412650017e+01, /* 0x40454F98, 0x962DAEDD */
407 1.70808091340565596283e+02, /* 0x406559DB, 0xE25EFD1F */
408 1.66733948696651168575e+02, /* 0x4064D77C, 0x81FA21E0 */
409};
410#ifdef __STDC__
411static const double qS3[6] = {
412#else
413static double qS3[6] = {
414#endif
415 4.87588729724587182091e+01, /* 0x40486122, 0xBFE343A6 */
416 7.09689221056606015736e+02, /* 0x40862D83, 0x86544EB3 */
417 3.70414822620111362994e+03, /* 0x40ACF04B, 0xE44DFC63 */
418 6.46042516752568917582e+03, /* 0x40B93C6C, 0xD7C76A28 */
419 2.51633368920368957333e+03, /* 0x40A3A8AA, 0xD94FB1C0 */
420 -1.49247451836156386662e+02, /* 0xC062A7EB, 0x201CF40F */
421};
422
423#ifdef __STDC__
424static const double qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
425#else
426static double qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
427#endif
428 1.50444444886983272379e-07, /* 0x3E84313B, 0x54F76BDB */
429 7.32234265963079278272e-02, /* 0x3FB2BEC5, 0x3E883E34 */
430 1.99819174093815998816e+00, /* 0x3FFFF897, 0xE727779C */
431 1.44956029347885735348e+01, /* 0x402CFDBF, 0xAAF96FE5 */
432 3.16662317504781540833e+01, /* 0x403FAA8E, 0x29FBDC4A */
433 1.62527075710929267416e+01, /* 0x403040B1, 0x71814BB4 */
434};
435#ifdef __STDC__
436static const double qS2[6] = {
437#else
438static double qS2[6] = {
439#endif
440 3.03655848355219184498e+01, /* 0x403E5D96, 0xF7C07AED */
441 2.69348118608049844624e+02, /* 0x4070D591, 0xE4D14B40 */
442 8.44783757595320139444e+02, /* 0x408A6645, 0x22B3BF22 */
443 8.82935845112488550512e+02, /* 0x408B977C, 0x9C5CC214 */
444 2.12666388511798828631e+02, /* 0x406A9553, 0x0E001365 */
445 -5.31095493882666946917e+00, /* 0xC0153E6A, 0xF8B32931 */
446};
447
448#ifdef __STDC__
449 static double qzero(double x)
450#else
451 static double qzero(x)
452 double x;
453#endif
454{
455#ifdef __STDC__
456 const double *p,*q;
457#else
458 double *p,*q;
459#endif
460 double s,r,z;
461 int32_t ix;
462
463 p = q = 0;
464 GET_HIGH_WORD(ix,x);
465 ix &= 0x7fffffff;
466 if(ix>=0x40200000) {p = qR8; q= qS8;}
467 else if(ix>=0x40122E8B){p = qR5; q= qS5;}
468 else if(ix>=0x4006DB6D){p = qR3; q= qS3;}
469 else if(ix>=0x40000000){p = qR2; q= qS2;}
470 z = one/(x*x);
471 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
472 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
473 return (-.125 + r/s)/x;
474}