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 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 = 1e300, 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 142 j0(double x) 143 { 144 double z, s,c,ss,cc,r,u,v; 145 146 if (!finite(x)) { 147 if (_IEEE) return one/(x*x); 148 else return (0); 149 } 150 x = fabs(x); 151 if (x >= 2.0) { /* |x| >= 2.0 */ 152 s = sin(x); 153 c = cos(x); 154 ss = s-c; 155 cc = s+c; 156 if (x < .5 * DBL_MAX) { /* make sure x+x not overflow */ 157 z = -cos(x+x); 158 if ((s*c)<zero) cc = z/ss; 159 else ss = z/cc; 160 } 161 /* 162 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x) 163 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x) 164 */ 165 if (_IEEE && x> 6.80564733841876927e+38) /* 2^129 */ 166 z = (invsqrtpi*cc)/sqrt(x); 167 else { 168 u = pzero(x); v = qzero(x); 169 z = invsqrtpi*(u*cc-v*ss)/sqrt(x); 170 } 171 return z; 172 } 173 if (x < 1.220703125e-004) { /* |x| < 2**-13 */ 174 if (huge+x > one) { /* raise inexact if x != 0 */ 175 if (x < 7.450580596923828125e-009) /* |x|<2**-27 */ 176 return one; 177 else return (one - 0.25*x*x); 178 } 179 } 180 z = x*x; 181 r = z*(r02+z*(r03+z*(r04+z*r05))); 182 s = one+z*(s01+z*(s02+z*(s03+z*s04))); 183 if (x < one) { /* |x| < 1.00 */ 184 return (one + z*(-0.25+(r/s))); 185 } else { 186 u = 0.5*x; 187 return ((one+u)*(one-u)+z*(r/s)); 188 } 189 } 190 191 static const double 192 u00 = -7.380429510868722527422411862872999615628e-0002, 193 u01 = 1.766664525091811069896442906220827182707e-0001, 194 u02 = -1.381856719455968955440002438182885835344e-0002, 195 u03 = 3.474534320936836562092566861515617053954e-0004, 196 u04 = -3.814070537243641752631729276103284491172e-0006, 197 u05 = 1.955901370350229170025509706510038090009e-0008, 198 u06 = -3.982051941321034108350630097330144576337e-0011, 199 v01 = 1.273048348341237002944554656529224780561e-0002, 200 v02 = 7.600686273503532807462101309675806839635e-0005, 201 v03 = 2.591508518404578033173189144579208685163e-0007, 202 v04 = 4.411103113326754838596529339004302243157e-0010; 203 204 double 205 y0(double x) 206 { 207 double z, s, c, ss, cc, u, v; 208 /* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0 */ 209 if (!finite(x)) { 210 if (_IEEE) 211 return (one/(x+x*x)); 212 else 213 return (0); 214 } 215 if (x == 0) { 216 if (_IEEE) return (-one/zero); 217 else return(infnan(-ERANGE)); 218 } 219 if (x<0) { 220 if (_IEEE) return (zero/zero); 221 else return (infnan(EDOM)); 222 } 223 if (x >= 2.00) { /* |x| >= 2.0 */ 224 /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0)) 225 * where x0 = x-pi/4 226 * Better formula: 227 * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4) 228 * = 1/sqrt(2) * (sin(x) + cos(x)) 229 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4) 230 * = 1/sqrt(2) * (sin(x) - cos(x)) 231 * To avoid cancellation, use 232 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) 233 * to compute the worse one. 234 */ 235 s = sin(x); 236 c = cos(x); 237 ss = s-c; 238 cc = s+c; 239 /* 240 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x) 241 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x) 242 */ 243 if (x < .5 * DBL_MAX) { /* make sure x+x not overflow */ 244 z = -cos(x+x); 245 if ((s*c)<zero) cc = z/ss; 246 else ss = z/cc; 247 } 248 if (_IEEE && x > 6.80564733841876927e+38) /* > 2^129 */ 249 z = (invsqrtpi*ss)/sqrt(x); 250 else { 251 u = pzero(x); v = qzero(x); 252 z = invsqrtpi*(u*ss+v*cc)/sqrt(x); 253 } 254 return z; 255 } 256 if (x <= 7.450580596923828125e-009) { /* x < 2**-27 */ 257 return (u00 + tpi*log(x)); 258 } 259 z = x*x; 260 u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06))))); 261 v = one+z*(v01+z*(v02+z*(v03+z*v04))); 262 return (u/v + tpi*(j0(x)*log(x))); 263 } 264 265 /* The asymptotic expansions of pzero is 266 * 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x. 267 * For x >= 2, We approximate pzero by 268 * pzero(x) = 1 + (R/S) 269 * where R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10 270 * S = 1 + ps0*s^2 + ... + ps4*s^10 271 * and 272 * | pzero(x)-1-R/S | <= 2 ** ( -60.26) 273 */ 274 static const double pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 275 0.0, 276 -7.031249999999003994151563066182798210142e-0002, 277 -8.081670412753498508883963849859423939871e+0000, 278 -2.570631056797048755890526455854482662510e+0002, 279 -2.485216410094288379417154382189125598962e+0003, 280 -5.253043804907295692946647153614119665649e+0003, 281 }; 282 static const double ps8[5] = { 283 1.165343646196681758075176077627332052048e+0002, 284 3.833744753641218451213253490882686307027e+0003, 285 4.059785726484725470626341023967186966531e+0004, 286 1.167529725643759169416844015694440325519e+0005, 287 4.762772841467309430100106254805711722972e+0004, 288 }; 289 290 static const double pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 291 -1.141254646918944974922813501362824060117e-0011, 292 -7.031249408735992804117367183001996028304e-0002, 293 -4.159610644705877925119684455252125760478e+0000, 294 -6.767476522651671942610538094335912346253e+0001, 295 -3.312312996491729755731871867397057689078e+0002, 296 -3.464333883656048910814187305901796723256e+0002, 297 }; 298 static const double ps5[5] = { 299 6.075393826923003305967637195319271932944e+0001, 300 1.051252305957045869801410979087427910437e+0003, 301 5.978970943338558182743915287887408780344e+0003, 302 9.625445143577745335793221135208591603029e+0003, 303 2.406058159229391070820491174867406875471e+0003, 304 }; 305 306 static const double pr3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ 307 -2.547046017719519317420607587742992297519e-0009, 308 -7.031196163814817199050629727406231152464e-0002, 309 -2.409032215495295917537157371488126555072e+0000, 310 -2.196597747348830936268718293366935843223e+0001, 311 -5.807917047017375458527187341817239891940e+0001, 312 -3.144794705948885090518775074177485744176e+0001, 313 }; 314 static const double ps3[5] = { 315 3.585603380552097167919946472266854507059e+0001, 316 3.615139830503038919981567245265266294189e+0002, 317 1.193607837921115243628631691509851364715e+0003, 318 1.127996798569074250675414186814529958010e+0003, 319 1.735809308133357510239737333055228118910e+0002, 320 }; 321 322 static const double pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 323 -8.875343330325263874525704514800809730145e-0008, 324 -7.030309954836247756556445443331044338352e-0002, 325 -1.450738467809529910662233622603401167409e+0000, 326 -7.635696138235277739186371273434739292491e+0000, 327 -1.119316688603567398846655082201614524650e+0001, 328 -3.233645793513353260006821113608134669030e+0000, 329 }; 330 static const double ps2[5] = { 331 2.222029975320888079364901247548798910952e+0001, 332 1.362067942182152109590340823043813120940e+0002, 333 2.704702786580835044524562897256790293238e+0002, 334 1.538753942083203315263554770476850028583e+0002, 335 1.465761769482561965099880599279699314477e+0001, 336 }; 337 338 static double 339 pzero(double x) 340 { 341 const double *p,*q; 342 double z,r,s; 343 if (x >= 8.00) {p = pr8; q= ps8;} 344 else if (x >= 4.54545211791992188) {p = pr5; q= ps5;} 345 else if (x >= 2.85714149475097656) {p = pr3; q= ps3;} 346 else /* if (x >= 2.00) */ {p = pr2; q= ps2;} 347 z = one/(x*x); 348 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); 349 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4])))); 350 return one+ r/s; 351 } 352 353 354 /* For x >= 8, the asymptotic expansions of qzero is 355 * -1/8 s + 75/1024 s^3 - ..., where s = 1/x. 356 * We approximate pzero by 357 * qzero(x) = s*(-1.25 + (R/S)) 358 * where R = qr0 + qr1*s^2 + qr2*s^4 + ... + qr5*s^10 359 * S = 1 + qs0*s^2 + ... + qs5*s^12 360 * and 361 * | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22) 362 */ 363 static const double qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 364 0.0, 365 7.324218749999350414479738504551775297096e-0002, 366 1.176820646822526933903301695932765232456e+0001, 367 5.576733802564018422407734683549251364365e+0002, 368 8.859197207564685717547076568608235802317e+0003, 369 3.701462677768878501173055581933725704809e+0004, 370 }; 371 static const double qs8[6] = { 372 1.637760268956898345680262381842235272369e+0002, 373 8.098344946564498460163123708054674227492e+0003, 374 1.425382914191204905277585267143216379136e+0005, 375 8.033092571195144136565231198526081387047e+0005, 376 8.405015798190605130722042369969184811488e+0005, 377 -3.438992935378666373204500729736454421006e+0005, 378 }; 379 380 static const double qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 381 1.840859635945155400568380711372759921179e-0011, 382 7.324217666126847411304688081129741939255e-0002, 383 5.835635089620569401157245917610984757296e+0000, 384 1.351115772864498375785526599119895942361e+0002, 385 1.027243765961641042977177679021711341529e+0003, 386 1.989977858646053872589042328678602481924e+0003, 387 }; 388 static const double qs5[6] = { 389 8.277661022365377058749454444343415524509e+0001, 390 2.077814164213929827140178285401017305309e+0003, 391 1.884728877857180787101956800212453218179e+0004, 392 5.675111228949473657576693406600265778689e+0004, 393 3.597675384251145011342454247417399490174e+0004, 394 -5.354342756019447546671440667961399442388e+0003, 395 }; 396 397 static const double qr3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ 398 4.377410140897386263955149197672576223054e-0009, 399 7.324111800429115152536250525131924283018e-0002, 400 3.344231375161707158666412987337679317358e+0000, 401 4.262184407454126175974453269277100206290e+0001, 402 1.708080913405656078640701512007621675724e+0002, 403 1.667339486966511691019925923456050558293e+0002, 404 }; 405 static const double qs3[6] = { 406 4.875887297245871932865584382810260676713e+0001, 407 7.096892210566060535416958362640184894280e+0002, 408 3.704148226201113687434290319905207398682e+0003, 409 6.460425167525689088321109036469797462086e+0003, 410 2.516333689203689683999196167394889715078e+0003, 411 -1.492474518361563818275130131510339371048e+0002, 412 }; 413 414 static const double qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 415 1.504444448869832780257436041633206366087e-0007, 416 7.322342659630792930894554535717104926902e-0002, 417 1.998191740938159956838594407540292600331e+0000, 418 1.449560293478857407645853071687125850962e+0001, 419 3.166623175047815297062638132537957315395e+0001, 420 1.625270757109292688799540258329430963726e+0001, 421 }; 422 static const double qs2[6] = { 423 3.036558483552191922522729838478169383969e+0001, 424 2.693481186080498724211751445725708524507e+0002, 425 8.447837575953201460013136756723746023736e+0002, 426 8.829358451124885811233995083187666981299e+0002, 427 2.126663885117988324180482985363624996652e+0002, 428 -5.310954938826669402431816125780738924463e+0000, 429 }; 430 431 static double 432 qzero(double x) 433 { 434 const double *p,*q; 435 double s,r,z; 436 if (x >= 8.00) {p = qr8; q= qs8;} 437 else if (x >= 4.54545211791992188) {p = qr5; q= qs5;} 438 else if (x >= 2.85714149475097656) {p = qr3; q= qs3;} 439 else /* if (x >= 2.00) */ {p = qr2; q= qs2;} 440 z = one/(x*x); 441 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); 442 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5]))))); 443 return (-.125 + r/s)/x; 444 } 445