2 * Copyright (c) 1992, 1993
3 * The Regents of the University of California. All rights reserved.
5 * Redistribution and use in source and binary forms, with or without
6 * modification, are permitted provided that the following conditions
8 * 1. Redistributions of source code must retain the above copyright
9 * notice, this list of conditions and the following disclaimer.
10 * 2. Redistributions in binary form must reproduce the above copyright
11 * notice, this list of conditions and the following disclaimer in the
12 * documentation and/or other materials provided with the distribution.
13 * 3. All advertising materials mentioning features or use of this software
14 * must display the following acknowledgement:
15 * This product includes software developed by the University of
16 * California, Berkeley and its contributors.
17 * 4. Neither the name of the University nor the names of its contributors
18 * may be used to endorse or promote products derived from this software
19 * without specific prior written permission.
21 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
22 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
23 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
24 * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
25 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
26 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
27 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
28 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
29 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
30 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
35 static char sccsid[] = "@(#)j0.c 8.2 (Berkeley) 11/30/93";
40 * Minor modifications by Peter McIlroy to adapt non-IEEE architecture.
44 * ====================================================
45 * Copyright (C) 1992 by Sun Microsystems, Inc.
47 * Developed at SunPro, a Sun Microsystems, Inc. business.
48 * Permission to use, copy, modify, and distribute this
49 * software is freely granted, provided that this notice
51 * ====================================================
53 * ******************* WARNING ********************
54 * This is an alpha version of SunPro's FDLIBM (Freely
55 * Distributable Math Library) for IEEE double precision
56 * arithmetic. FDLIBM is a basic math library written
57 * in C that runs on machines that conform to IEEE
58 * Standard 754/854. This alpha version is distributed
59 * for testing purpose. Those who use this software
60 * should report any bugs to
62 * fdlibm-comments@sunpro.eng.sun.com
64 * -- K.C. Ng, Oct 12, 1992
65 * ************************************************
68 /* double j0(double x), y0(double x)
69 * Bessel function of the first and second kinds of order zero.
71 * 1. For tiny x, we use j0(x) = 1 - x^2/4 + x^4/64 - ...
72 * 2. Reduce x to |x| since j0(x)=j0(-x), and
74 * j0(x) = 1-z/4+ z^2*R0/S0, where z = x*x;
75 * (precision: |j0-1+z/4-z^2R0/S0 |<2**-63.67 )
77 * j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)-q0(x)*sin(x0))
78 * where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
80 * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
81 * = 1/sqrt(2) * (cos(x) + sin(x))
82 * sin(x0) = sin(x)cos(pi/4)-cos(x)sin(pi/4)
83 * = 1/sqrt(2) * (sin(x) - cos(x))
84 * (To avoid cancellation, use
85 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
86 * to compute the worse one.)
96 * y0(x) = 2/pi*(j0(x)*(ln(x/2)+Euler) + x^2/4 - ...)
97 * therefore y0(x)-2/pi*j0(x)*ln(x) is an even function.
98 * We use the following function to approximate y0,
99 * y0(x) = U(z)/V(z) + (2/pi)*(j0(x)*ln(x)), z= x^2
101 * U(z) = u0 + u1*z + ... + u6*z^6
102 * V(z) = 1 + v1*z + ... + v4*z^4
103 * with absolute approximation error bounded by 2**-72.
104 * Note: For tiny x, U/V = u0 and j0(x)~1, hence
105 * y0(tiny) = u0 + (2/pi)*ln(tiny), (choose tiny<2**-27)
107 * y0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)+q0(x)*sin(x0))
108 * where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
109 * by the method mentioned above.
110 * 3. Special cases: y0(0)=-inf, y0(x<0)=NaN, y0(inf)=0.
115 #if defined(vax) || defined(tahoe)
119 #define infnan(x) (0.0)
122 static double pzero __P((double)), qzero __P((double));
128 invsqrtpi= 5.641895835477562869480794515607725858441e-0001,
129 tpi = 0.636619772367581343075535053490057448,
130 /* R0/S0 on [0, 2.00] */
131 r02 = 1.562499999999999408594634421055018003102e-0002,
132 r03 = -1.899792942388547334476601771991800712355e-0004,
133 r04 = 1.829540495327006565964161150603950916854e-0006,
134 r05 = -4.618326885321032060803075217804816988758e-0009,
135 s01 = 1.561910294648900170180789369288114642057e-0002,
136 s02 = 1.169267846633374484918570613449245536323e-0004,
137 s03 = 5.135465502073181376284426245689510134134e-0007,
138 s04 = 1.166140033337900097836930825478674320464e-0009;
144 double z, s,c,ss,cc,r,u,v;
147 if (_IEEE) return one/(x*x);
150 if (x >= 2.0) { /* |x| >= 2.0 */
155 if (x < .5 * DBL_MAX) { /* make sure x+x not overflow */
157 if ((s*c)<zero) cc = z/ss;
161 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
162 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
164 if (_IEEE && x> 6.80564733841876927e+38) /* 2^129 */
165 z = (invsqrtpi*cc)/sqrt(x);
167 u = pzero(x); v = qzero(x);
168 z = invsqrtpi*(u*cc-v*ss)/sqrt(x);
172 if (x < 1.220703125e-004) { /* |x| < 2**-13 */
173 if (huge+x > one) { /* raise inexact if x != 0 */
174 if (x < 7.450580596923828125e-009) /* |x|<2**-27 */
176 else return (one - 0.25*x*x);
180 r = z*(r02+z*(r03+z*(r04+z*r05)));
181 s = one+z*(s01+z*(s02+z*(s03+z*s04)));
182 if (x < one) { /* |x| < 1.00 */
183 return (one + z*(-0.25+(r/s)));
186 return ((one+u)*(one-u)+z*(r/s));
191 u00 = -7.380429510868722527422411862872999615628e-0002,
192 u01 = 1.766664525091811069896442906220827182707e-0001,
193 u02 = -1.381856719455968955440002438182885835344e-0002,
194 u03 = 3.474534320936836562092566861515617053954e-0004,
195 u04 = -3.814070537243641752631729276103284491172e-0006,
196 u05 = 1.955901370350229170025509706510038090009e-0008,
197 u06 = -3.982051941321034108350630097330144576337e-0011,
198 v01 = 1.273048348341237002944554656529224780561e-0002,
199 v02 = 7.600686273503532807462101309675806839635e-0005,
200 v03 = 2.591508518404578033173189144579208685163e-0007,
201 v04 = 4.411103113326754838596529339004302243157e-0010;
207 double z, s, c, ss, cc, u, v;
208 /* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0 */
211 return (one/(x+x*x));
215 if (_IEEE) return (-one/zero);
216 else return(infnan(-ERANGE));
218 if (_IEEE) return (zero/zero);
219 else return (infnan(EDOM));
220 if (x >= 2.00) { /* |x| >= 2.0 */
221 /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
224 * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
225 * = 1/sqrt(2) * (sin(x) + cos(x))
226 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
227 * = 1/sqrt(2) * (sin(x) - cos(x))
228 * To avoid cancellation, use
229 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
230 * to compute the worse one.
237 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
238 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
240 if (x < .5 * DBL_MAX) { /* make sure x+x not overflow */
242 if ((s*c)<zero) cc = z/ss;
245 if (_IEEE && x > 6.80564733841876927e+38) /* > 2^129 */
246 z = (invsqrtpi*ss)/sqrt(x);
248 u = pzero(x); v = qzero(x);
249 z = invsqrtpi*(u*ss+v*cc)/sqrt(x);
253 if (x <= 7.450580596923828125e-009) { /* x < 2**-27 */
254 return (u00 + tpi*log(x));
257 u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06)))));
258 v = one+z*(v01+z*(v02+z*(v03+z*v04)));
259 return (u/v + tpi*(j0(x)*log(x)));
262 /* The asymptotic expansions of pzero is
263 * 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x.
264 * For x >= 2, We approximate pzero by
265 * pzero(x) = 1 + (R/S)
266 * where R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10
267 * S = 1 + ps0*s^2 + ... + ps4*s^10
269 * | pzero(x)-1-R/S | <= 2 ** ( -60.26)
271 static double pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
273 -7.031249999999003994151563066182798210142e-0002,
274 -8.081670412753498508883963849859423939871e+0000,
275 -2.570631056797048755890526455854482662510e+0002,
276 -2.485216410094288379417154382189125598962e+0003,
277 -5.253043804907295692946647153614119665649e+0003,
279 static double ps8[5] = {
280 1.165343646196681758075176077627332052048e+0002,
281 3.833744753641218451213253490882686307027e+0003,
282 4.059785726484725470626341023967186966531e+0004,
283 1.167529725643759169416844015694440325519e+0005,
284 4.762772841467309430100106254805711722972e+0004,
287 static double pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
288 -1.141254646918944974922813501362824060117e-0011,
289 -7.031249408735992804117367183001996028304e-0002,
290 -4.159610644705877925119684455252125760478e+0000,
291 -6.767476522651671942610538094335912346253e+0001,
292 -3.312312996491729755731871867397057689078e+0002,
293 -3.464333883656048910814187305901796723256e+0002,
295 static double ps5[5] = {
296 6.075393826923003305967637195319271932944e+0001,
297 1.051252305957045869801410979087427910437e+0003,
298 5.978970943338558182743915287887408780344e+0003,
299 9.625445143577745335793221135208591603029e+0003,
300 2.406058159229391070820491174867406875471e+0003,
303 static double pr3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
304 -2.547046017719519317420607587742992297519e-0009,
305 -7.031196163814817199050629727406231152464e-0002,
306 -2.409032215495295917537157371488126555072e+0000,
307 -2.196597747348830936268718293366935843223e+0001,
308 -5.807917047017375458527187341817239891940e+0001,
309 -3.144794705948885090518775074177485744176e+0001,
311 static double ps3[5] = {
312 3.585603380552097167919946472266854507059e+0001,
313 3.615139830503038919981567245265266294189e+0002,
314 1.193607837921115243628631691509851364715e+0003,
315 1.127996798569074250675414186814529958010e+0003,
316 1.735809308133357510239737333055228118910e+0002,
319 static double pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
320 -8.875343330325263874525704514800809730145e-0008,
321 -7.030309954836247756556445443331044338352e-0002,
322 -1.450738467809529910662233622603401167409e+0000,
323 -7.635696138235277739186371273434739292491e+0000,
324 -1.119316688603567398846655082201614524650e+0001,
325 -3.233645793513353260006821113608134669030e+0000,
327 static double ps2[5] = {
328 2.222029975320888079364901247548798910952e+0001,
329 1.362067942182152109590340823043813120940e+0002,
330 2.704702786580835044524562897256790293238e+0002,
331 1.538753942083203315263554770476850028583e+0002,
332 1.465761769482561965099880599279699314477e+0001,
335 static double pzero(x)
339 if (x >= 8.00) {p = pr8; q= ps8;}
340 else if (x >= 4.54545211791992188) {p = pr5; q= ps5;}
341 else if (x >= 2.85714149475097656) {p = pr3; q= ps3;}
342 else if (x >= 2.00) {p = pr2; q= ps2;}
344 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
345 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
350 /* For x >= 8, the asymptotic expansions of qzero is
351 * -1/8 s + 75/1024 s^3 - ..., where s = 1/x.
352 * We approximate pzero by
353 * qzero(x) = s*(-1.25 + (R/S))
354 * where R = qr0 + qr1*s^2 + qr2*s^4 + ... + qr5*s^10
355 * S = 1 + qs0*s^2 + ... + qs5*s^12
357 * | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22)
359 static double qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
361 7.324218749999350414479738504551775297096e-0002,
362 1.176820646822526933903301695932765232456e+0001,
363 5.576733802564018422407734683549251364365e+0002,
364 8.859197207564685717547076568608235802317e+0003,
365 3.701462677768878501173055581933725704809e+0004,
367 static double qs8[6] = {
368 1.637760268956898345680262381842235272369e+0002,
369 8.098344946564498460163123708054674227492e+0003,
370 1.425382914191204905277585267143216379136e+0005,
371 8.033092571195144136565231198526081387047e+0005,
372 8.405015798190605130722042369969184811488e+0005,
373 -3.438992935378666373204500729736454421006e+0005,
376 static double qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
377 1.840859635945155400568380711372759921179e-0011,
378 7.324217666126847411304688081129741939255e-0002,
379 5.835635089620569401157245917610984757296e+0000,
380 1.351115772864498375785526599119895942361e+0002,
381 1.027243765961641042977177679021711341529e+0003,
382 1.989977858646053872589042328678602481924e+0003,
384 static double qs5[6] = {
385 8.277661022365377058749454444343415524509e+0001,
386 2.077814164213929827140178285401017305309e+0003,
387 1.884728877857180787101956800212453218179e+0004,
388 5.675111228949473657576693406600265778689e+0004,
389 3.597675384251145011342454247417399490174e+0004,
390 -5.354342756019447546671440667961399442388e+0003,
393 static double qr3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
394 4.377410140897386263955149197672576223054e-0009,
395 7.324111800429115152536250525131924283018e-0002,
396 3.344231375161707158666412987337679317358e+0000,
397 4.262184407454126175974453269277100206290e+0001,
398 1.708080913405656078640701512007621675724e+0002,
399 1.667339486966511691019925923456050558293e+0002,
401 static double qs3[6] = {
402 4.875887297245871932865584382810260676713e+0001,
403 7.096892210566060535416958362640184894280e+0002,
404 3.704148226201113687434290319905207398682e+0003,
405 6.460425167525689088321109036469797462086e+0003,
406 2.516333689203689683999196167394889715078e+0003,
407 -1.492474518361563818275130131510339371048e+0002,
410 static double qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
411 1.504444448869832780257436041633206366087e-0007,
412 7.322342659630792930894554535717104926902e-0002,
413 1.998191740938159956838594407540292600331e+0000,
414 1.449560293478857407645853071687125850962e+0001,
415 3.166623175047815297062638132537957315395e+0001,
416 1.625270757109292688799540258329430963726e+0001,
418 static double qs2[6] = {
419 3.036558483552191922522729838478169383969e+0001,
420 2.693481186080498724211751445725708524507e+0002,
421 8.447837575953201460013136756723746023736e+0002,
422 8.829358451124885811233995083187666981299e+0002,
423 2.126663885117988324180482985363624996652e+0002,
424 -5.310954938826669402431816125780738924463e+0000,
427 static double qzero(x)
431 if (x >= 8.00) {p = qr8; q= qs8;}
432 else if (x >= 4.54545211791992188) {p = qr5; q= qs5;}
433 else if (x >= 2.85714149475097656) {p = qr3; q= qs3;}
434 else if (x >= 2.00) {p = qr2; q= qs2;}
436 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
437 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
438 return (-.125 + r/s)/x;