1/* $NetBSD: n_j0.c,v 1.6 2003/08/07 16:44:51 agc 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 33static 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 123static double pzero (double), qzero (double); 124 125static const double 126huge = _HUGE, 127zero = 0.0, 128one = 1.0, 129invsqrtpi= 5.641895835477562869480794515607725858441e-0001, 130tpi = 0.636619772367581343075535053490057448, 131 /* R0/S0 on [0, 2.00] */ 132r02 = 1.562499999999999408594634421055018003102e-0002, 133r03 = -1.899792942388547334476601771991800712355e-0004, 134r04 = 1.829540495327006565964161150603950916854e-0006, 135r05 = -4.618326885321032060803075217804816988758e-0009, 136s01 = 1.561910294648900170180789369288114642057e-0002, 137s02 = 1.169267846633374484918570613449245536323e-0004, 138s03 = 5.135465502073181376284426245689510134134e-0007, 139s04 = 1.166140033337900097836930825478674320464e-0009; 140 141double 142j0(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 197static const double 198u00 = -7.380429510868722527422411862872999615628e-0002, 199u01 = 1.766664525091811069896442906220827182707e-0001, 200u02 = -1.381856719455968955440002438182885835344e-0002, 201u03 = 3.474534320936836562092566861515617053954e-0004, 202u04 = -3.814070537243641752631729276103284491172e-0006, 203u05 = 1.955901370350229170025509706510038090009e-0008, 204u06 = -3.982051941321034108350630097330144576337e-0011, 205v01 = 1.273048348341237002944554656529224780561e-0002, 206v02 = 7.600686273503532807462101309675806839635e-0005, 207v03 = 2.591508518404578033173189144579208685163e-0007, 208v04 = 4.411103113326754838596529339004302243157e-0010; 209 210double 211y0(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 */ 290static 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}; 298static 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 306static 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}; 314static 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 322static 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}; 330static 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 338static 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}; 346static 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 354static double 355pzero(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 */ 379static 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}; 387static 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 396static 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}; 404static 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 413static 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}; 421static 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 430static 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}; 438static 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 447static double 448qzero(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