1 /*- 2 * Copyright (c) 1992, 1993 3 * The Regents of the University of California. All rights reserved. 4 * 5 * %sccs.include.redist.c% 6 */ 7 8 #ifndef lint 9 static char sccsid[] = "@(#)j0.c 8.1 (Berkeley) 06/04/93"; 10 #endif /* not lint */ 11 12 /* 13 * 16 December 1992 14 * Minor modifications by Peter McIlroy to adapt non-IEEE architecture. 15 */ 16 17 /* 18 * ==================================================== 19 * Copyright (C) 1992 by Sun Microsystems, Inc. 20 * 21 * Developed at SunPro, a Sun Microsystems, Inc. business. 22 * Permission to use, copy, modify, and distribute this 23 * software is freely granted, provided that this notice 24 * is preserved. 25 * ==================================================== 26 * 27 * ******************* WARNING ******************** 28 * This is an alpha version of SunPro's FDLIBM (Freely 29 * Distributable Math Library) for IEEE double precision 30 * arithmetic. FDLIBM is a basic math library written 31 * in C that runs on machines that conform to IEEE 32 * Standard 754/854. This alpha version is distributed 33 * for testing purpose. Those who use this software 34 * should report any bugs to 35 * 36 * fdlibm-comments@sunpro.eng.sun.com 37 * 38 * -- K.C. Ng, Oct 12, 1992 39 * ************************************************ 40 */ 41 42 /* double j0(double x), y0(double x) 43 * Bessel function of the first and second kinds of order zero. 44 * Method -- j0(x): 45 * 1. For tiny x, we use j0(x) = 1 - x^2/4 + x^4/64 - ... 46 * 2. Reduce x to |x| since j0(x)=j0(-x), and 47 * for x in (0,2) 48 * j0(x) = 1-z/4+ z^2*R0/S0, where z = x*x; 49 * (precision: |j0-1+z/4-z^2R0/S0 |<2**-63.67 ) 50 * for x in (2,inf) 51 * j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)-q0(x)*sin(x0)) 52 * where x0 = x-pi/4. It is better to compute sin(x0),cos(x0) 53 * as follow: 54 * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4) 55 * = 1/sqrt(2) * (cos(x) + sin(x)) 56 * sin(x0) = sin(x)cos(pi/4)-cos(x)sin(pi/4) 57 * = 1/sqrt(2) * (sin(x) - cos(x)) 58 * (To avoid cancellation, use 59 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) 60 * to compute the worse one.) 61 * 62 * 3 Special cases 63 * j0(nan)= nan 64 * j0(0) = 1 65 * j0(inf) = 0 66 * 67 * Method -- y0(x): 68 * 1. For x<2. 69 * Since 70 * y0(x) = 2/pi*(j0(x)*(ln(x/2)+Euler) + x^2/4 - ...) 71 * therefore y0(x)-2/pi*j0(x)*ln(x) is an even function. 72 * We use the following function to approximate y0, 73 * y0(x) = U(z)/V(z) + (2/pi)*(j0(x)*ln(x)), z= x^2 74 * where 75 * U(z) = u0 + u1*z + ... + u6*z^6 76 * V(z) = 1 + v1*z + ... + v4*z^4 77 * with absolute approximation error bounded by 2**-72. 78 * Note: For tiny x, U/V = u0 and j0(x)~1, hence 79 * y0(tiny) = u0 + (2/pi)*ln(tiny), (choose tiny<2**-27) 80 * 2. For x>=2. 81 * y0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)+q0(x)*sin(x0)) 82 * where x0 = x-pi/4. It is better to compute sin(x0),cos(x0) 83 * by the method mentioned above. 84 * 3. Special cases: y0(0)=-inf, y0(x<0)=NaN, y0(inf)=0. 85 */ 86 87 #include <math.h> 88 #include <float.h> 89 #if defined(vax) || defined(tahoe) 90 #define _IEEE 0 91 #else 92 #define _IEEE 1 93 #define infnan(x) (0.0) 94 #endif 95 96 static double pzero __P((double)), qzero __P((double)); 97 98 static double 99 huge = 1e300, 100 zero = 0.0, 101 one = 1.0, 102 invsqrtpi= 5.641895835477562869480794515607725858441e-0001, 103 tpi = 0.636619772367581343075535053490057448, 104 /* R0/S0 on [0, 2.00] */ 105 r02 = 1.562499999999999408594634421055018003102e-0002, 106 r03 = -1.899792942388547334476601771991800712355e-0004, 107 r04 = 1.829540495327006565964161150603950916854e-0006, 108 r05 = -4.618326885321032060803075217804816988758e-0009, 109 s01 = 1.561910294648900170180789369288114642057e-0002, 110 s02 = 1.169267846633374484918570613449245536323e-0004, 111 s03 = 5.135465502073181376284426245689510134134e-0007, 112 s04 = 1.166140033337900097836930825478674320464e-0009; 113 114 double 115 j0(x) 116 double x; 117 { 118 double z, s,c,ss,cc,r,u,v; 119 120 if (!finite(x)) 121 if (_IEEE) return one/(x*x); 122 else return (0); 123 x = fabs(x); 124 if (x >= 2.0) { /* |x| >= 2.0 */ 125 s = sin(x); 126 c = cos(x); 127 ss = s-c; 128 cc = s+c; 129 if (x < .5 * DBL_MAX) { /* make sure x+x not overflow */ 130 z = -cos(x+x); 131 if ((s*c)<zero) cc = z/ss; 132 else ss = z/cc; 133 } 134 /* 135 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x) 136 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x) 137 */ 138 if (_IEEE && x> 6.80564733841876927e+38) /* 2^129 */ 139 z = (invsqrtpi*cc)/sqrt(x); 140 else { 141 u = pzero(x); v = qzero(x); 142 z = invsqrtpi*(u*cc-v*ss)/sqrt(x); 143 } 144 return z; 145 } 146 if (x < 1.220703125e-004) { /* |x| < 2**-13 */ 147 if (huge+x > one) { /* raise inexact if x != 0 */ 148 if (x < 7.450580596923828125e-009) /* |x|<2**-27 */ 149 return one; 150 else return (one - 0.25*x*x); 151 } 152 } 153 z = x*x; 154 r = z*(r02+z*(r03+z*(r04+z*r05))); 155 s = one+z*(s01+z*(s02+z*(s03+z*s04))); 156 if (x < one) { /* |x| < 1.00 */ 157 return (one + z*(-0.25+(r/s))); 158 } else { 159 u = 0.5*x; 160 return ((one+u)*(one-u)+z*(r/s)); 161 } 162 } 163 164 static double 165 u00 = -7.380429510868722527422411862872999615628e-0002, 166 u01 = 1.766664525091811069896442906220827182707e-0001, 167 u02 = -1.381856719455968955440002438182885835344e-0002, 168 u03 = 3.474534320936836562092566861515617053954e-0004, 169 u04 = -3.814070537243641752631729276103284491172e-0006, 170 u05 = 1.955901370350229170025509706510038090009e-0008, 171 u06 = -3.982051941321034108350630097330144576337e-0011, 172 v01 = 1.273048348341237002944554656529224780561e-0002, 173 v02 = 7.600686273503532807462101309675806839635e-0005, 174 v03 = 2.591508518404578033173189144579208685163e-0007, 175 v04 = 4.411103113326754838596529339004302243157e-0010; 176 177 double 178 y0(x) 179 double x; 180 { 181 double z, s,c,ss,cc,u,v,j0(); 182 /* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0 */ 183 if (!finite(x)) 184 if (_IEEE) 185 return (one/(x+x*x)); 186 else 187 return (0); 188 if (x == 0) 189 if (_IEEE) return (-one/zero); 190 else return(infnan(-ERANGE)); 191 if (x<0) 192 if (_IEEE) return (zero/zero); 193 else return (infnan(EDOM)); 194 if (x >= 2.00) { /* |x| >= 2.0 */ 195 /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0)) 196 * where x0 = x-pi/4 197 * Better formula: 198 * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4) 199 * = 1/sqrt(2) * (sin(x) + cos(x)) 200 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4) 201 * = 1/sqrt(2) * (sin(x) - cos(x)) 202 * To avoid cancellation, use 203 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) 204 * to compute the worse one. 205 */ 206 s = sin(x); 207 c = cos(x); 208 ss = s-c; 209 cc = s+c; 210 /* 211 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x) 212 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x) 213 */ 214 if (x < .5 * DBL_MAX) { /* make sure x+x not overflow */ 215 z = -cos(x+x); 216 if ((s*c)<zero) cc = z/ss; 217 else ss = z/cc; 218 } 219 if (_IEEE && x > 6.80564733841876927e+38) /* > 2^129 */ 220 z = (invsqrtpi*ss)/sqrt(x); 221 else { 222 u = pzero(x); v = qzero(x); 223 z = invsqrtpi*(u*ss+v*cc)/sqrt(x); 224 } 225 return z; 226 } 227 if (x <= 7.450580596923828125e-009) { /* x < 2**-27 */ 228 return (u00 + tpi*log(x)); 229 } 230 z = x*x; 231 u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06))))); 232 v = one+z*(v01+z*(v02+z*(v03+z*v04))); 233 return (u/v + tpi*(j0(x)*log(x))); 234 } 235 236 /* The asymptotic expansions of pzero is 237 * 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x. 238 * For x >= 2, We approximate pzero by 239 * pzero(x) = 1 + (R/S) 240 * where R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10 241 * S = 1 + ps0*s^2 + ... + ps4*s^10 242 * and 243 * | pzero(x)-1-R/S | <= 2 ** ( -60.26) 244 */ 245 static double pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 246 0.0, 247 -7.031249999999003994151563066182798210142e-0002, 248 -8.081670412753498508883963849859423939871e+0000, 249 -2.570631056797048755890526455854482662510e+0002, 250 -2.485216410094288379417154382189125598962e+0003, 251 -5.253043804907295692946647153614119665649e+0003, 252 }; 253 static double ps8[5] = { 254 1.165343646196681758075176077627332052048e+0002, 255 3.833744753641218451213253490882686307027e+0003, 256 4.059785726484725470626341023967186966531e+0004, 257 1.167529725643759169416844015694440325519e+0005, 258 4.762772841467309430100106254805711722972e+0004, 259 }; 260 261 static double pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 262 -1.141254646918944974922813501362824060117e-0011, 263 -7.031249408735992804117367183001996028304e-0002, 264 -4.159610644705877925119684455252125760478e+0000, 265 -6.767476522651671942610538094335912346253e+0001, 266 -3.312312996491729755731871867397057689078e+0002, 267 -3.464333883656048910814187305901796723256e+0002, 268 }; 269 static double ps5[5] = { 270 6.075393826923003305967637195319271932944e+0001, 271 1.051252305957045869801410979087427910437e+0003, 272 5.978970943338558182743915287887408780344e+0003, 273 9.625445143577745335793221135208591603029e+0003, 274 2.406058159229391070820491174867406875471e+0003, 275 }; 276 277 static double pr3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ 278 -2.547046017719519317420607587742992297519e-0009, 279 -7.031196163814817199050629727406231152464e-0002, 280 -2.409032215495295917537157371488126555072e+0000, 281 -2.196597747348830936268718293366935843223e+0001, 282 -5.807917047017375458527187341817239891940e+0001, 283 -3.144794705948885090518775074177485744176e+0001, 284 }; 285 static double ps3[5] = { 286 3.585603380552097167919946472266854507059e+0001, 287 3.615139830503038919981567245265266294189e+0002, 288 1.193607837921115243628631691509851364715e+0003, 289 1.127996798569074250675414186814529958010e+0003, 290 1.735809308133357510239737333055228118910e+0002, 291 }; 292 293 static double pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 294 -8.875343330325263874525704514800809730145e-0008, 295 -7.030309954836247756556445443331044338352e-0002, 296 -1.450738467809529910662233622603401167409e+0000, 297 -7.635696138235277739186371273434739292491e+0000, 298 -1.119316688603567398846655082201614524650e+0001, 299 -3.233645793513353260006821113608134669030e+0000, 300 }; 301 static double ps2[5] = { 302 2.222029975320888079364901247548798910952e+0001, 303 1.362067942182152109590340823043813120940e+0002, 304 2.704702786580835044524562897256790293238e+0002, 305 1.538753942083203315263554770476850028583e+0002, 306 1.465761769482561965099880599279699314477e+0001, 307 }; 308 309 static double pzero(x) 310 double x; 311 { 312 double *p,*q,z,r,s; 313 if (x >= 8.00) {p = pr8; q= ps8;} 314 else if (x >= 4.54545211791992188) {p = pr5; q= ps5;} 315 else if (x >= 2.85714149475097656) {p = pr3; q= ps3;} 316 else if (x >= 2.00) {p = pr2; q= ps2;} 317 z = one/(x*x); 318 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); 319 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4])))); 320 return one+ r/s; 321 } 322 323 324 /* For x >= 8, the asymptotic expansions of qzero is 325 * -1/8 s + 75/1024 s^3 - ..., where s = 1/x. 326 * We approximate pzero by 327 * qzero(x) = s*(-1.25 + (R/S)) 328 * where R = qr0 + qr1*s^2 + qr2*s^4 + ... + qr5*s^10 329 * S = 1 + qs0*s^2 + ... + qs5*s^12 330 * and 331 * | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22) 332 */ 333 static double qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 334 0.0, 335 7.324218749999350414479738504551775297096e-0002, 336 1.176820646822526933903301695932765232456e+0001, 337 5.576733802564018422407734683549251364365e+0002, 338 8.859197207564685717547076568608235802317e+0003, 339 3.701462677768878501173055581933725704809e+0004, 340 }; 341 static double qs8[6] = { 342 1.637760268956898345680262381842235272369e+0002, 343 8.098344946564498460163123708054674227492e+0003, 344 1.425382914191204905277585267143216379136e+0005, 345 8.033092571195144136565231198526081387047e+0005, 346 8.405015798190605130722042369969184811488e+0005, 347 -3.438992935378666373204500729736454421006e+0005, 348 }; 349 350 static double qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 351 1.840859635945155400568380711372759921179e-0011, 352 7.324217666126847411304688081129741939255e-0002, 353 5.835635089620569401157245917610984757296e+0000, 354 1.351115772864498375785526599119895942361e+0002, 355 1.027243765961641042977177679021711341529e+0003, 356 1.989977858646053872589042328678602481924e+0003, 357 }; 358 static double qs5[6] = { 359 8.277661022365377058749454444343415524509e+0001, 360 2.077814164213929827140178285401017305309e+0003, 361 1.884728877857180787101956800212453218179e+0004, 362 5.675111228949473657576693406600265778689e+0004, 363 3.597675384251145011342454247417399490174e+0004, 364 -5.354342756019447546671440667961399442388e+0003, 365 }; 366 367 static double qr3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ 368 4.377410140897386263955149197672576223054e-0009, 369 7.324111800429115152536250525131924283018e-0002, 370 3.344231375161707158666412987337679317358e+0000, 371 4.262184407454126175974453269277100206290e+0001, 372 1.708080913405656078640701512007621675724e+0002, 373 1.667339486966511691019925923456050558293e+0002, 374 }; 375 static double qs3[6] = { 376 4.875887297245871932865584382810260676713e+0001, 377 7.096892210566060535416958362640184894280e+0002, 378 3.704148226201113687434290319905207398682e+0003, 379 6.460425167525689088321109036469797462086e+0003, 380 2.516333689203689683999196167394889715078e+0003, 381 -1.492474518361563818275130131510339371048e+0002, 382 }; 383 384 static double qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 385 1.504444448869832780257436041633206366087e-0007, 386 7.322342659630792930894554535717104926902e-0002, 387 1.998191740938159956838594407540292600331e+0000, 388 1.449560293478857407645853071687125850962e+0001, 389 3.166623175047815297062638132537957315395e+0001, 390 1.625270757109292688799540258329430963726e+0001, 391 }; 392 static double qs2[6] = { 393 3.036558483552191922522729838478169383969e+0001, 394 2.693481186080498724211751445725708524507e+0002, 395 8.447837575953201460013136756723746023736e+0002, 396 8.829358451124885811233995083187666981299e+0002, 397 2.126663885117988324180482985363624996652e+0002, 398 -5.310954938826669402431816125780738924463e+0000, 399 }; 400 401 static double qzero(x) 402 double x; 403 { 404 double *p,*q, s,r,z; 405 if (x >= 8.00) {p = qr8; q= qs8;} 406 else if (x >= 4.54545211791992188) {p = qr5; q= qs5;} 407 else if (x >= 2.85714149475097656) {p = qr3; q= qs3;} 408 else if (x >= 2.00) {p = qr2; q= qs2;} 409 z = one/(x*x); 410 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); 411 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5]))))); 412 return (-.125 + r/s)/x; 413 } 414