n_j1.c revision 1.4
1 /* $NetBSD: n_j1.c,v 1.4 1999/07/02 15:37:37 simonb 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. All advertising materials mentioning features or use of this software 15 * must display the following acknowledgement: 16 * This product includes software developed by the University of 17 * California, Berkeley and its contributors. 18 * 4. Neither the name of the University nor the names of its contributors 19 * may be used to endorse or promote products derived from this software 20 * without specific prior written permission. 21 * 22 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND 23 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 24 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 25 * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE 26 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL 27 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS 28 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 29 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT 30 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY 31 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF 32 * SUCH DAMAGE. 33 */ 34 35 #ifndef lint 36 #if 0 37 static char sccsid[] = "@(#)j1.c 8.2 (Berkeley) 11/30/93"; 38 #endif 39 #endif /* not lint */ 40 41 /* 42 * 16 December 1992 43 * Minor modifications by Peter McIlroy to adapt non-IEEE architecture. 44 */ 45 46 /* 47 * ==================================================== 48 * Copyright (C) 1992 by Sun Microsystems, Inc. 49 * 50 * Developed at SunPro, a Sun Microsystems, Inc. business. 51 * Permission to use, copy, modify, and distribute this 52 * software is freely granted, provided that this notice 53 * is preserved. 54 * ==================================================== 55 * 56 * ******************* WARNING ******************** 57 * This is an alpha version of SunPro's FDLIBM (Freely 58 * Distributable Math Library) for IEEE double precision 59 * arithmetic. FDLIBM is a basic math library written 60 * in C that runs on machines that conform to IEEE 61 * Standard 754/854. This alpha version is distributed 62 * for testing purpose. Those who use this software 63 * should report any bugs to 64 * 65 * fdlibm-comments (at) sunpro.eng.sun.com 66 * 67 * -- K.C. Ng, Oct 12, 1992 68 * ************************************************ 69 */ 70 71 /* double j1(double x), y1(double x) 72 * Bessel function of the first and second kinds of order zero. 73 * Method -- j1(x): 74 * 1. For tiny x, we use j1(x) = x/2 - x^3/16 + x^5/384 - ... 75 * 2. Reduce x to |x| since j1(x)=-j1(-x), and 76 * for x in (0,2) 77 * j1(x) = x/2 + x*z*R0/S0, where z = x*x; 78 * (precision: |j1/x - 1/2 - R0/S0 |<2**-61.51 ) 79 * for x in (2,inf) 80 * j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x1)-q1(x)*sin(x1)) 81 * y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1)) 82 * where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1) 83 * as follows: 84 * cos(x1) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4) 85 * = 1/sqrt(2) * (sin(x) - cos(x)) 86 * sin(x1) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4) 87 * = -1/sqrt(2) * (sin(x) + cos(x)) 88 * (To avoid cancellation, use 89 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) 90 * to compute the worse one.) 91 * 92 * 3 Special cases 93 * j1(nan)= nan 94 * j1(0) = 0 95 * j1(inf) = 0 96 * 97 * Method -- y1(x): 98 * 1. screen out x<=0 cases: y1(0)=-inf, y1(x<0)=NaN 99 * 2. For x<2. 100 * Since 101 * y1(x) = 2/pi*(j1(x)*(ln(x/2)+Euler)-1/x-x/2+5/64*x^3-...) 102 * therefore y1(x)-2/pi*j1(x)*ln(x)-1/x is an odd function. 103 * We use the following function to approximate y1, 104 * y1(x) = x*U(z)/V(z) + (2/pi)*(j1(x)*ln(x)-1/x), z= x^2 105 * where for x in [0,2] (abs err less than 2**-65.89) 106 * U(z) = u0 + u1*z + ... + u4*z^4 107 * V(z) = 1 + v1*z + ... + v5*z^5 108 * Note: For tiny x, 1/x dominate y1 and hence 109 * y1(tiny) = -2/pi/tiny, (choose tiny<2**-54) 110 * 3. For x>=2. 111 * y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1)) 112 * where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1) 113 * by method mentioned above. 114 */ 115 116 #include "mathimpl.h" 117 #include <float.h> 118 #include <errno.h> 119 120 #if defined(__vax__) || defined(tahoe) 121 #define _IEEE 0 122 #else 123 #define _IEEE 1 124 #define infnan(x) (0.0) 125 #endif 126 127 static double pone __P((double)), qone __P((double)); 128 129 static double 130 huge = 1e300, 131 zero = 0.0, 132 one = 1.0, 133 invsqrtpi= 5.641895835477562869480794515607725858441e-0001, 134 tpi = 0.636619772367581343075535053490057448, 135 136 /* R0/S0 on [0,2] */ 137 r00 = -6.250000000000000020842322918309200910191e-0002, 138 r01 = 1.407056669551897148204830386691427791200e-0003, 139 r02 = -1.599556310840356073980727783817809847071e-0005, 140 r03 = 4.967279996095844750387702652791615403527e-0008, 141 s01 = 1.915375995383634614394860200531091839635e-0002, 142 s02 = 1.859467855886309024045655476348872850396e-0004, 143 s03 = 1.177184640426236767593432585906758230822e-0006, 144 s04 = 5.046362570762170559046714468225101016915e-0009, 145 s05 = 1.235422744261379203512624973117299248281e-0011; 146 147 #define two_129 6.80564733841876926e+038 /* 2^129 */ 148 #define two_m54 5.55111512312578270e-017 /* 2^-54 */ 149 double j1(x) 150 double x; 151 { 152 double z, s,c,ss,cc,r,u,v,y; 153 y = fabs(x); 154 if (!finite(x)) { /* Inf or NaN */ 155 if (_IEEE && x != x) 156 return(x); 157 else 158 return (copysign(x, zero)); 159 } 160 y = fabs(x); 161 if (y >= 2) /* |x| >= 2.0 */ 162 { 163 s = sin(y); 164 c = cos(y); 165 ss = -s-c; 166 cc = s-c; 167 if (y < .5*DBL_MAX) { /* make sure y+y not overflow */ 168 z = cos(y+y); 169 if ((s*c)<zero) cc = z/ss; 170 else ss = z/cc; 171 } 172 /* 173 * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x) 174 * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x) 175 */ 176 #if !defined(__vax__) && !defined(tahoe) 177 if (y > two_129) /* x > 2^129 */ 178 z = (invsqrtpi*cc)/sqrt(y); 179 else 180 #endif /* defined(__vax__) || defined(tahoe) */ 181 { 182 u = pone(y); v = qone(y); 183 z = invsqrtpi*(u*cc-v*ss)/sqrt(y); 184 } 185 if (x < 0) return -z; 186 else return z; 187 } 188 if (y < 7.450580596923828125e-009) { /* |x|<2**-27 */ 189 if(huge+x>one) return 0.5*x;/* inexact if x!=0 necessary */ 190 } 191 z = x*x; 192 r = z*(r00+z*(r01+z*(r02+z*r03))); 193 s = one+z*(s01+z*(s02+z*(s03+z*(s04+z*s05)))); 194 r *= x; 195 return (x*0.5+r/s); 196 } 197 198 static double u0[5] = { 199 -1.960570906462389484206891092512047539632e-0001, 200 5.044387166398112572026169863174882070274e-0002, 201 -1.912568958757635383926261729464141209569e-0003, 202 2.352526005616105109577368905595045204577e-0005, 203 -9.190991580398788465315411784276789663849e-0008, 204 }; 205 static double v0[5] = { 206 1.991673182366499064031901734535479833387e-0002, 207 2.025525810251351806268483867032781294682e-0004, 208 1.356088010975162198085369545564475416398e-0006, 209 6.227414523646214811803898435084697863445e-0009, 210 1.665592462079920695971450872592458916421e-0011, 211 }; 212 213 double y1(x) 214 double x; 215 { 216 double z, s, c, ss, cc, u, v; 217 /* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */ 218 if (!finite(x)) { 219 if (!_IEEE) return (infnan(EDOM)); 220 else if (x < 0) 221 return(zero/zero); 222 else if (x > 0) 223 return (0); 224 else 225 return(x); 226 } 227 if (x <= 0) { 228 if (_IEEE && x == 0) return -one/zero; 229 else if(x == 0) return(infnan(-ERANGE)); 230 else if(_IEEE) return (zero/zero); 231 else return(infnan(EDOM)); 232 } 233 if (x >= 2) /* |x| >= 2.0 */ 234 { 235 s = sin(x); 236 c = cos(x); 237 ss = -s-c; 238 cc = s-c; 239 if (x < .5 * DBL_MAX) /* make sure x+x not overflow */ 240 { 241 z = cos(x+x); 242 if ((s*c)>zero) cc = z/ss; 243 else ss = z/cc; 244 } 245 /* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0)) 246 * where x0 = x-3pi/4 247 * Better formula: 248 * cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4) 249 * = 1/sqrt(2) * (sin(x) - cos(x)) 250 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4) 251 * = -1/sqrt(2) * (cos(x) + sin(x)) 252 * To avoid cancellation, use 253 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) 254 * to compute the worse one. 255 */ 256 if (_IEEE && x>two_129) 257 z = (invsqrtpi*ss)/sqrt(x); 258 else { 259 u = pone(x); v = qone(x); 260 z = invsqrtpi*(u*ss+v*cc)/sqrt(x); 261 } 262 return z; 263 } 264 if (x <= two_m54) { /* x < 2**-54 */ 265 return (-tpi/x); 266 } 267 z = x*x; 268 u = u0[0]+z*(u0[1]+z*(u0[2]+z*(u0[3]+z*u0[4]))); 269 v = one+z*(v0[0]+z*(v0[1]+z*(v0[2]+z*(v0[3]+z*v0[4])))); 270 return (x*(u/v) + tpi*(j1(x)*log(x)-one/x)); 271 } 272 273 /* For x >= 8, the asymptotic expansions of pone is 274 * 1 + 15/128 s^2 - 4725/2^15 s^4 - ..., where s = 1/x. 275 * We approximate pone by 276 * pone(x) = 1 + (R/S) 277 * where R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10 278 * S = 1 + ps0*s^2 + ... + ps4*s^10 279 * and 280 * | pone(x)-1-R/S | <= 2 ** ( -60.06) 281 */ 282 283 static double pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 284 0.0, 285 1.171874999999886486643746274751925399540e-0001, 286 1.323948065930735690925827997575471527252e+0001, 287 4.120518543073785433325860184116512799375e+0002, 288 3.874745389139605254931106878336700275601e+0003, 289 7.914479540318917214253998253147871806507e+0003, 290 }; 291 static double ps8[5] = { 292 1.142073703756784104235066368252692471887e+0002, 293 3.650930834208534511135396060708677099382e+0003, 294 3.695620602690334708579444954937638371808e+0004, 295 9.760279359349508334916300080109196824151e+0004, 296 3.080427206278887984185421142572315054499e+0004, 297 }; 298 299 static double pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 300 1.319905195562435287967533851581013807103e-0011, 301 1.171874931906140985709584817065144884218e-0001, 302 6.802751278684328781830052995333841452280e+0000, 303 1.083081829901891089952869437126160568246e+0002, 304 5.176361395331997166796512844100442096318e+0002, 305 5.287152013633375676874794230748055786553e+0002, 306 }; 307 static double ps5[5] = { 308 5.928059872211313557747989128353699746120e+0001, 309 9.914014187336144114070148769222018425781e+0002, 310 5.353266952914879348427003712029704477451e+0003, 311 7.844690317495512717451367787640014588422e+0003, 312 1.504046888103610723953792002716816255382e+0003, 313 }; 314 315 static double pr3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ 316 3.025039161373736032825049903408701962756e-0009, 317 1.171868655672535980750284752227495879921e-0001, 318 3.932977500333156527232725812363183251138e+0000, 319 3.511940355916369600741054592597098912682e+0001, 320 9.105501107507812029367749771053045219094e+0001, 321 4.855906851973649494139275085628195457113e+0001, 322 }; 323 static double ps3[5] = { 324 3.479130950012515114598605916318694946754e+0001, 325 3.367624587478257581844639171605788622549e+0002, 326 1.046871399757751279180649307467612538415e+0003, 327 8.908113463982564638443204408234739237639e+0002, 328 1.037879324396392739952487012284401031859e+0002, 329 }; 330 331 static double pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 332 1.077108301068737449490056513753865482831e-0007, 333 1.171762194626833490512746348050035171545e-0001, 334 2.368514966676087902251125130227221462134e+0000, 335 1.224261091482612280835153832574115951447e+0001, 336 1.769397112716877301904532320376586509782e+0001, 337 5.073523125888185399030700509321145995160e+0000, 338 }; 339 static double ps2[5] = { 340 2.143648593638214170243114358933327983793e+0001, 341 1.252902271684027493309211410842525120355e+0002, 342 2.322764690571628159027850677565128301361e+0002, 343 1.176793732871470939654351793502076106651e+0002, 344 8.364638933716182492500902115164881195742e+0000, 345 }; 346 347 static double pone(x) 348 double x; 349 { 350 double *p,*q,z,r,s; 351 if (x >= 8.0) {p = pr8; q= ps8;} 352 else if (x >= 4.54545211791992188) {p = pr5; q= ps5;} 353 else if (x >= 2.85714149475097656) {p = pr3; q= ps3;} 354 else /* if (x >= 2.0) */ {p = pr2; q= ps2;} 355 z = one/(x*x); 356 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); 357 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4])))); 358 return (one + r/s); 359 } 360 361 362 /* For x >= 8, the asymptotic expansions of qone is 363 * 3/8 s - 105/1024 s^3 - ..., where s = 1/x. 364 * We approximate pone by 365 * qone(x) = s*(0.375 + (R/S)) 366 * where R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10 367 * S = 1 + qs1*s^2 + ... + qs6*s^12 368 * and 369 * | qone(x)/s -0.375-R/S | <= 2 ** ( -61.13) 370 */ 371 372 static double qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 373 0.0, 374 -1.025390624999927207385863635575804210817e-0001, 375 -1.627175345445899724355852152103771510209e+0001, 376 -7.596017225139501519843072766973047217159e+0002, 377 -1.184980667024295901645301570813228628541e+0004, 378 -4.843851242857503225866761992518949647041e+0004, 379 }; 380 static double qs8[6] = { 381 1.613953697007229231029079421446916397904e+0002, 382 7.825385999233484705298782500926834217525e+0003, 383 1.338753362872495800748094112937868089032e+0005, 384 7.196577236832409151461363171617204036929e+0005, 385 6.666012326177764020898162762642290294625e+0005, 386 -2.944902643038346618211973470809456636830e+0005, 387 }; 388 389 static double qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 390 -2.089799311417640889742251585097264715678e-0011, 391 -1.025390502413754195402736294609692303708e-0001, 392 -8.056448281239359746193011295417408828404e+0000, 393 -1.836696074748883785606784430098756513222e+0002, 394 -1.373193760655081612991329358017247355921e+0003, 395 -2.612444404532156676659706427295870995743e+0003, 396 }; 397 static double qs5[6] = { 398 8.127655013843357670881559763225310973118e+0001, 399 1.991798734604859732508048816860471197220e+0003, 400 1.746848519249089131627491835267411777366e+0004, 401 4.985142709103522808438758919150738000353e+0004, 402 2.794807516389181249227113445299675335543e+0004, 403 -4.719183547951285076111596613593553911065e+0003, 404 }; 405 406 static double qr3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ 407 -5.078312264617665927595954813341838734288e-0009, 408 -1.025378298208370901410560259001035577681e-0001, 409 -4.610115811394734131557983832055607679242e+0000, 410 -5.784722165627836421815348508816936196402e+0001, 411 -2.282445407376317023842545937526967035712e+0002, 412 -2.192101284789093123936441805496580237676e+0002, 413 }; 414 static double qs3[6] = { 415 4.766515503237295155392317984171640809318e+0001, 416 6.738651126766996691330687210949984203167e+0002, 417 3.380152866795263466426219644231687474174e+0003, 418 5.547729097207227642358288160210745890345e+0003, 419 1.903119193388108072238947732674639066045e+0003, 420 -1.352011914443073322978097159157678748982e+0002, 421 }; 422 423 static double qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 424 -1.783817275109588656126772316921194887979e-0007, 425 -1.025170426079855506812435356168903694433e-0001, 426 -2.752205682781874520495702498875020485552e+0000, 427 -1.966361626437037351076756351268110418862e+0001, 428 -4.232531333728305108194363846333841480336e+0001, 429 -2.137192117037040574661406572497288723430e+0001, 430 }; 431 static double qs2[6] = { 432 2.953336290605238495019307530224241335502e+0001, 433 2.529815499821905343698811319455305266409e+0002, 434 7.575028348686454070022561120722815892346e+0002, 435 7.393932053204672479746835719678434981599e+0002, 436 1.559490033366661142496448853793707126179e+0002, 437 -4.959498988226281813825263003231704397158e+0000, 438 }; 439 440 static double qone(x) 441 double x; 442 { 443 double *p,*q, s,r,z; 444 if (x >= 8.0) {p = qr8; q= qs8;} 445 else if (x >= 4.54545211791992188) {p = qr5; q= qs5;} 446 else if (x >= 2.85714149475097656) {p = qr3; q= qs3;} 447 else /* if (x >= 2.0) */ {p = qr2; q= qs2;} 448 z = one/(x*x); 449 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); 450 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5]))))); 451 return (.375 + r/s)/x; 452 } 453
Indexes created Tue Sep 23 14:10:03 GMT 2025