1 /* $NetBSD: n_j0.c,v 1.7 2011/11/02 02:34:56 christos Exp $ */
2 /*-
3 * Copyright (c) 1992, 1993
4 * The Regents of the University of California. All rights reserved.
5 *
6 * Redistribution and use in source and binary forms, with or without
7 * modification, are permitted provided that the following conditions
8 * are met:
9 * 1. Redistributions of source code must retain the above copyright
10 * notice, this list of conditions and the following disclaimer.
11 * 2. Redistributions in binary form must reproduce the above copyright
12 * notice, this list of conditions and the following disclaimer in the
13 * documentation and/or other materials provided with the distribution.
14 * 3. Neither the name of the University nor the names of its contributors
15 * may be used to endorse or promote products derived from this software
16 * without specific prior written permission.
17 *
18 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
19 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
20 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
21 * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
22 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
23 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
24 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
25 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
26 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
27 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
28 * SUCH DAMAGE.
29 */
30
31 #ifndef lint
32 #if 0
33 static char sccsid[] = "@(#)j0.c 8.2 (Berkeley) 11/30/93";
34 #endif
35 #endif /* not lint */
36
37 /*
38 * 16 December 1992
39 * Minor modifications by Peter McIlroy to adapt non-IEEE architecture.
40 */
41
42 /*
43 * ====================================================
44 * Copyright (C) 1992 by Sun Microsystems, Inc.
45 *
46 * Developed at SunPro, a Sun Microsystems, Inc. business.
47 * Permission to use, copy, modify, and distribute this
48 * software is freely granted, provided that this notice
49 * is preserved.
50 * ====================================================
51 *
52 * ******************* WARNING ********************
53 * This is an alpha version of SunPro's FDLIBM (Freely
54 * Distributable Math Library) for IEEE double precision
55 * arithmetic. FDLIBM is a basic math library written
56 * in C that runs on machines that conform to IEEE
57 * Standard 754/854. This alpha version is distributed
58 * for testing purpose. Those who use this software
59 * should report any bugs to
60 *
61 * fdlibm-comments@sunpro.eng.sun.com
62 *
63 * -- K.C. Ng, Oct 12, 1992
64 * ************************************************
65 */
66
67 /* double j0(double x), y0(double x)
68 * Bessel function of the first and second kinds of order zero.
69 * Method -- j0(x):
70 * 1. For tiny x, we use j0(x) = 1 - x^2/4 + x^4/64 - ...
71 * 2. Reduce x to |x| since j0(x)=j0(-x), and
72 * for x in (0,2)
73 * j0(x) = 1-z/4+ z^2*R0/S0, where z = x*x;
74 * (precision: |j0-1+z/4-z^2R0/S0 |<2**-63.67 )
75 * for x in (2,inf)
76 * j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)-q0(x)*sin(x0))
77 * where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
78 * as follow:
79 * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
80 * = 1/sqrt(2) * (cos(x) + sin(x))
81 * sin(x0) = sin(x)cos(pi/4)-cos(x)sin(pi/4)
82 * = 1/sqrt(2) * (sin(x) - cos(x))
83 * (To avoid cancellation, use
84 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
85 * to compute the worse one.)
86 *
87 * 3 Special cases
88 * j0(nan)= nan
89 * j0(0) = 1
90 * j0(inf) = 0
91 *
92 * Method -- y0(x):
93 * 1. For x<2.
94 * Since
95 * y0(x) = 2/pi*(j0(x)*(ln(x/2)+Euler) + x^2/4 - ...)
96 * therefore y0(x)-2/pi*j0(x)*ln(x) is an even function.
97 * We use the following function to approximate y0,
98 * y0(x) = U(z)/V(z) + (2/pi)*(j0(x)*ln(x)), z= x^2
99 * where
100 * U(z) = u0 + u1*z + ... + u6*z^6
101 * V(z) = 1 + v1*z + ... + v4*z^4
102 * with absolute approximation error bounded by 2**-72.
103 * Note: For tiny x, U/V = u0 and j0(x)~1, hence
104 * y0(tiny) = u0 + (2/pi)*ln(tiny), (choose tiny<2**-27)
105 * 2. For x>=2.
106 * y0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)+q0(x)*sin(x0))
107 * where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
108 * by the method mentioned above.
109 * 3. Special cases: y0(0)=-inf, y0(x<0)=NaN, y0(inf)=0.
110 */
111
112 #include "mathimpl.h"
113 #include <float.h>
114 #include <errno.h>
115
116 #if defined(__vax__) || defined(tahoe)
117 #define _IEEE 0
118 #else
119 #define _IEEE 1
120 #define infnan(x) (0.0)
121 #endif
122
123 static double pzero (double), qzero (double);
124
125 static const double
126 huge = _HUGE,
127 zero = 0.0,
128 one = 1.0,
129 invsqrtpi= 5.641895835477562869480794515607725858441e-0001,
130 tpi = 0.636619772367581343075535053490057448,
131 /* R0/S0 on [0, 2.00] */
132 r02 = 1.562499999999999408594634421055018003102e-0002,
133 r03 = -1.899792942388547334476601771991800712355e-0004,
134 r04 = 1.829540495327006565964161150603950916854e-0006,
135 r05 = -4.618326885321032060803075217804816988758e-0009,
136 s01 = 1.561910294648900170180789369288114642057e-0002,
137 s02 = 1.169267846633374484918570613449245536323e-0004,
138 s03 = 5.135465502073181376284426245689510134134e-0007,
139 s04 = 1.166140033337900097836930825478674320464e-0009;
140
141 double
j0(double x)142 j0(double x)
143 {
144 double z, s,c,ss,cc,r,u,v;
145
146 if (!finite(x)) {
147 #if _IEEE
148 return one/(x*x);
149 #else
150 return (0);
151 #endif
152 }
153 x = fabs(x);
154 if (x >= 2.0) { /* |x| >= 2.0 */
155 s = sin(x);
156 c = cos(x);
157 ss = s-c;
158 cc = s+c;
159 if (x < .5 * DBL_MAX) { /* make sure x+x not overflow */
160 z = -cos(x+x);
161 if ((s*c)<zero) cc = z/ss;
162 else ss = z/cc;
163 }
164 /*
165 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
166 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
167 */
168 #if _IEEE
169 if (x > 6.80564733841876927e+38) /* 2^129 */
170 z = (invsqrtpi*cc)/sqrt(x);
171 else
172 #endif
173 {
174 u = pzero(x); v = qzero(x);
175 z = invsqrtpi*(u*cc-v*ss)/sqrt(x);
176 }
177 return z;
178 }
179 if (x < 1.220703125e-004) { /* |x| < 2**-13 */
180 if (huge+x > one) { /* raise inexact if x != 0 */
181 if (x < 7.450580596923828125e-009) /* |x|<2**-27 */
182 return one;
183 else return (one - 0.25*x*x);
184 }
185 }
186 z = x*x;
187 r = z*(r02+z*(r03+z*(r04+z*r05)));
188 s = one+z*(s01+z*(s02+z*(s03+z*s04)));
189 if (x < one) { /* |x| < 1.00 */
190 return (one + z*(-0.25+(r/s)));
191 } else {
192 u = 0.5*x;
193 return ((one+u)*(one-u)+z*(r/s));
194 }
195 }
196
197 static const double
198 u00 = -7.380429510868722527422411862872999615628e-0002,
199 u01 = 1.766664525091811069896442906220827182707e-0001,
200 u02 = -1.381856719455968955440002438182885835344e-0002,
201 u03 = 3.474534320936836562092566861515617053954e-0004,
202 u04 = -3.814070537243641752631729276103284491172e-0006,
203 u05 = 1.955901370350229170025509706510038090009e-0008,
204 u06 = -3.982051941321034108350630097330144576337e-0011,
205 v01 = 1.273048348341237002944554656529224780561e-0002,
206 v02 = 7.600686273503532807462101309675806839635e-0005,
207 v03 = 2.591508518404578033173189144579208685163e-0007,
208 v04 = 4.411103113326754838596529339004302243157e-0010;
209
210 double
y0(double x)211 y0(double x)
212 {
213 double z, s, c, ss, cc, u, v;
214 /* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0 */
215 if (!finite(x)) {
216 #if _IEEE
217 return (one/(x+x*x));
218 #else
219 return (0);
220 #endif
221 }
222 if (x == 0) {
223 #if _IEEE
224 return (-one/zero);
225 #else
226 return(infnan(-ERANGE));
227 #endif
228 }
229 if (x<0) {
230 #if _IEEE
231 return (zero/zero);
232 #else
233 return (infnan(EDOM));
234 #endif
235 }
236 if (x >= 2.00) { /* |x| >= 2.0 */
237 /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
238 * where x0 = x-pi/4
239 * Better formula:
240 * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
241 * = 1/sqrt(2) * (sin(x) + cos(x))
242 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
243 * = 1/sqrt(2) * (sin(x) - cos(x))
244 * To avoid cancellation, use
245 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
246 * to compute the worse one.
247 */
248 s = sin(x);
249 c = cos(x);
250 ss = s-c;
251 cc = s+c;
252 /*
253 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
254 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
255 */
256 if (x < .5 * DBL_MAX) { /* make sure x+x not overflow */
257 z = -cos(x+x);
258 if ((s*c)<zero) cc = z/ss;
259 else ss = z/cc;
260 }
261 #if _IEEE
262 if (x > 6.80564733841876927e+38) /* > 2^129 */
263 z = (invsqrtpi*ss)/sqrt(x);
264 else
265 #endif
266 {
267 u = pzero(x); v = qzero(x);
268 z = invsqrtpi*(u*ss+v*cc)/sqrt(x);
269 }
270 return z;
271 }
272 if (x <= 7.450580596923828125e-009) { /* x < 2**-27 */
273 return (u00 + tpi*log(x));
274 }
275 z = x*x;
276 u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06)))));
277 v = one+z*(v01+z*(v02+z*(v03+z*v04)));
278 return (u/v + tpi*(j0(x)*log(x)));
279 }
280
281 /* The asymptotic expansions of pzero is
282 * 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x.
283 * For x >= 2, We approximate pzero by
284 * pzero(x) = 1 + (R/S)
285 * where R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10
286 * S = 1 + ps0*s^2 + ... + ps4*s^10
287 * and
288 * | pzero(x)-1-R/S | <= 2 ** ( -60.26)
289 */
290 static const double pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
291 0.0,
292 -7.031249999999003994151563066182798210142e-0002,
293 -8.081670412753498508883963849859423939871e+0000,
294 -2.570631056797048755890526455854482662510e+0002,
295 -2.485216410094288379417154382189125598962e+0003,
296 -5.253043804907295692946647153614119665649e+0003,
297 };
298 static const double ps8[5] = {
299 1.165343646196681758075176077627332052048e+0002,
300 3.833744753641218451213253490882686307027e+0003,
301 4.059785726484725470626341023967186966531e+0004,
302 1.167529725643759169416844015694440325519e+0005,
303 4.762772841467309430100106254805711722972e+0004,
304 };
305
306 static const double pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
307 -1.141254646918944974922813501362824060117e-0011,
308 -7.031249408735992804117367183001996028304e-0002,
309 -4.159610644705877925119684455252125760478e+0000,
310 -6.767476522651671942610538094335912346253e+0001,
311 -3.312312996491729755731871867397057689078e+0002,
312 -3.464333883656048910814187305901796723256e+0002,
313 };
314 static const double ps5[5] = {
315 6.075393826923003305967637195319271932944e+0001,
316 1.051252305957045869801410979087427910437e+0003,
317 5.978970943338558182743915287887408780344e+0003,
318 9.625445143577745335793221135208591603029e+0003,
319 2.406058159229391070820491174867406875471e+0003,
320 };
321
322 static const double pr3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
323 -2.547046017719519317420607587742992297519e-0009,
324 -7.031196163814817199050629727406231152464e-0002,
325 -2.409032215495295917537157371488126555072e+0000,
326 -2.196597747348830936268718293366935843223e+0001,
327 -5.807917047017375458527187341817239891940e+0001,
328 -3.144794705948885090518775074177485744176e+0001,
329 };
330 static const double ps3[5] = {
331 3.585603380552097167919946472266854507059e+0001,
332 3.615139830503038919981567245265266294189e+0002,
333 1.193607837921115243628631691509851364715e+0003,
334 1.127996798569074250675414186814529958010e+0003,
335 1.735809308133357510239737333055228118910e+0002,
336 };
337
338 static const double pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
339 -8.875343330325263874525704514800809730145e-0008,
340 -7.030309954836247756556445443331044338352e-0002,
341 -1.450738467809529910662233622603401167409e+0000,
342 -7.635696138235277739186371273434739292491e+0000,
343 -1.119316688603567398846655082201614524650e+0001,
344 -3.233645793513353260006821113608134669030e+0000,
345 };
346 static const double ps2[5] = {
347 2.222029975320888079364901247548798910952e+0001,
348 1.362067942182152109590340823043813120940e+0002,
349 2.704702786580835044524562897256790293238e+0002,
350 1.538753942083203315263554770476850028583e+0002,
351 1.465761769482561965099880599279699314477e+0001,
352 };
353
354 static double
pzero(double x)355 pzero(double x)
356 {
357 const double *p,*q;
358 double z,r,s;
359 if (x >= 8.00) {p = pr8; q= ps8;}
360 else if (x >= 4.54545211791992188) {p = pr5; q= ps5;}
361 else if (x >= 2.85714149475097656) {p = pr3; q= ps3;}
362 else /* if (x >= 2.00) */ {p = pr2; q= ps2;}
363 z = one/(x*x);
364 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
365 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
366 return one+ r/s;
367 }
368
369
370 /* For x >= 8, the asymptotic expansions of qzero is
371 * -1/8 s + 75/1024 s^3 - ..., where s = 1/x.
372 * We approximate pzero by
373 * qzero(x) = s*(-1.25 + (R/S))
374 * where R = qr0 + qr1*s^2 + qr2*s^4 + ... + qr5*s^10
375 * S = 1 + qs0*s^2 + ... + qs5*s^12
376 * and
377 * | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22)
378 */
379 static const double qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
380 0.0,
381 7.324218749999350414479738504551775297096e-0002,
382 1.176820646822526933903301695932765232456e+0001,
383 5.576733802564018422407734683549251364365e+0002,
384 8.859197207564685717547076568608235802317e+0003,
385 3.701462677768878501173055581933725704809e+0004,
386 };
387 static const double qs8[6] = {
388 1.637760268956898345680262381842235272369e+0002,
389 8.098344946564498460163123708054674227492e+0003,
390 1.425382914191204905277585267143216379136e+0005,
391 8.033092571195144136565231198526081387047e+0005,
392 8.405015798190605130722042369969184811488e+0005,
393 -3.438992935378666373204500729736454421006e+0005,
394 };
395
396 static const double qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
397 1.840859635945155400568380711372759921179e-0011,
398 7.324217666126847411304688081129741939255e-0002,
399 5.835635089620569401157245917610984757296e+0000,
400 1.351115772864498375785526599119895942361e+0002,
401 1.027243765961641042977177679021711341529e+0003,
402 1.989977858646053872589042328678602481924e+0003,
403 };
404 static const double qs5[6] = {
405 8.277661022365377058749454444343415524509e+0001,
406 2.077814164213929827140178285401017305309e+0003,
407 1.884728877857180787101956800212453218179e+0004,
408 5.675111228949473657576693406600265778689e+0004,
409 3.597675384251145011342454247417399490174e+0004,
410 -5.354342756019447546671440667961399442388e+0003,
411 };
412
413 static const double qr3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
414 4.377410140897386263955149197672576223054e-0009,
415 7.324111800429115152536250525131924283018e-0002,
416 3.344231375161707158666412987337679317358e+0000,
417 4.262184407454126175974453269277100206290e+0001,
418 1.708080913405656078640701512007621675724e+0002,
419 1.667339486966511691019925923456050558293e+0002,
420 };
421 static const double qs3[6] = {
422 4.875887297245871932865584382810260676713e+0001,
423 7.096892210566060535416958362640184894280e+0002,
424 3.704148226201113687434290319905207398682e+0003,
425 6.460425167525689088321109036469797462086e+0003,
426 2.516333689203689683999196167394889715078e+0003,
427 -1.492474518361563818275130131510339371048e+0002,
428 };
429
430 static const double qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
431 1.504444448869832780257436041633206366087e-0007,
432 7.322342659630792930894554535717104926902e-0002,
433 1.998191740938159956838594407540292600331e+0000,
434 1.449560293478857407645853071687125850962e+0001,
435 3.166623175047815297062638132537957315395e+0001,
436 1.625270757109292688799540258329430963726e+0001,
437 };
438 static const double qs2[6] = {
439 3.036558483552191922522729838478169383969e+0001,
440 2.693481186080498724211751445725708524507e+0002,
441 8.447837575953201460013136756723746023736e+0002,
442 8.829358451124885811233995083187666981299e+0002,
443 2.126663885117988324180482985363624996652e+0002,
444 -5.310954938826669402431816125780738924463e+0000,
445 };
446
447 static double
qzero(double x)448 qzero(double x)
449 {
450 const double *p,*q;
451 double s,r,z;
452 if (x >= 8.00) {p = qr8; q= qs8;}
453 else if (x >= 4.54545211791992188) {p = qr5; q= qs5;}
454 else if (x >= 2.85714149475097656) {p = qr3; q= qs3;}
455 else /* if (x >= 2.00) */ {p = qr2; q= qs2;}
456 z = one/(x*x);
457 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
458 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
459 return (-.125 + r/s)/x;
460 }
461