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[] = "@(#)j1.c 8.2 (Berkeley) 11/30/93"; 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 j1(double x), y1(double x) 43 * Bessel function of the first and second kinds of order zero. 44 * Method -- j1(x): 45 * 1. For tiny x, we use j1(x) = x/2 - x^3/16 + x^5/384 - ... 46 * 2. Reduce x to |x| since j1(x)=-j1(-x), and 47 * for x in (0,2) 48 * j1(x) = x/2 + x*z*R0/S0, where z = x*x; 49 * (precision: |j1/x - 1/2 - R0/S0 |<2**-61.51 ) 50 * for x in (2,inf) 51 * j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x1)-q1(x)*sin(x1)) 52 * y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1)) 53 * where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1) 54 * as follows: 55 * cos(x1) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4) 56 * = 1/sqrt(2) * (sin(x) - cos(x)) 57 * sin(x1) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4) 58 * = -1/sqrt(2) * (sin(x) + cos(x)) 59 * (To avoid cancellation, use 60 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) 61 * to compute the worse one.) 62 * 63 * 3 Special cases 64 * j1(nan)= nan 65 * j1(0) = 0 66 * j1(inf) = 0 67 * 68 * Method -- y1(x): 69 * 1. screen out x<=0 cases: y1(0)=-inf, y1(x<0)=NaN 70 * 2. For x<2. 71 * Since 72 * y1(x) = 2/pi*(j1(x)*(ln(x/2)+Euler)-1/x-x/2+5/64*x^3-...) 73 * therefore y1(x)-2/pi*j1(x)*ln(x)-1/x is an odd function. 74 * We use the following function to approximate y1, 75 * y1(x) = x*U(z)/V(z) + (2/pi)*(j1(x)*ln(x)-1/x), z= x^2 76 * where for x in [0,2] (abs err less than 2**-65.89) 77 * U(z) = u0 + u1*z + ... + u4*z^4 78 * V(z) = 1 + v1*z + ... + v5*z^5 79 * Note: For tiny x, 1/x dominate y1 and hence 80 * y1(tiny) = -2/pi/tiny, (choose tiny<2**-54) 81 * 3. For x>=2. 82 * y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1)) 83 * where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1) 84 * by method mentioned above. 85 */ 86 87 #include <math.h> 88 #include <float.h> 89 90 #if defined(vax) || defined(tahoe) 91 #define _IEEE 0 92 #else 93 #define _IEEE 1 94 #define infnan(x) (0.0) 95 #endif 96 97 static double pone(), qone(); 98 99 static double 100 huge = 1e300, 101 zero = 0.0, 102 one = 1.0, 103 invsqrtpi= 5.641895835477562869480794515607725858441e-0001, 104 tpi = 0.636619772367581343075535053490057448, 105 106 /* R0/S0 on [0,2] */ 107 r00 = -6.250000000000000020842322918309200910191e-0002, 108 r01 = 1.407056669551897148204830386691427791200e-0003, 109 r02 = -1.599556310840356073980727783817809847071e-0005, 110 r03 = 4.967279996095844750387702652791615403527e-0008, 111 s01 = 1.915375995383634614394860200531091839635e-0002, 112 s02 = 1.859467855886309024045655476348872850396e-0004, 113 s03 = 1.177184640426236767593432585906758230822e-0006, 114 s04 = 5.046362570762170559046714468225101016915e-0009, 115 s05 = 1.235422744261379203512624973117299248281e-0011; 116 117 #define two_129 6.80564733841876926e+038 /* 2^129 */ 118 #define two_m54 5.55111512312578270e-017 /* 2^-54 */ 119 double j1(x) 120 double x; 121 { 122 double z, s,c,ss,cc,r,u,v,y; 123 y = fabs(x); 124 if (!finite(x)) /* Inf or NaN */ 125 if (_IEEE && x != x) 126 return(x); 127 else 128 return (copysign(x, zero)); 129 y = fabs(x); 130 if (y >= 2) /* |x| >= 2.0 */ 131 { 132 s = sin(y); 133 c = cos(y); 134 ss = -s-c; 135 cc = s-c; 136 if (y < .5*DBL_MAX) { /* make sure y+y not overflow */ 137 z = cos(y+y); 138 if ((s*c)<zero) cc = z/ss; 139 else ss = z/cc; 140 } 141 /* 142 * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x) 143 * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x) 144 */ 145 #if !defined(vax) && !defined(tahoe) 146 if (y > two_129) /* x > 2^129 */ 147 z = (invsqrtpi*cc)/sqrt(y); 148 else 149 #endif /* defined(vax) || defined(tahoe) */ 150 { 151 u = pone(y); v = qone(y); 152 z = invsqrtpi*(u*cc-v*ss)/sqrt(y); 153 } 154 if (x < 0) return -z; 155 else return z; 156 } 157 if (y < 7.450580596923828125e-009) { /* |x|<2**-27 */ 158 if(huge+x>one) return 0.5*x;/* inexact if x!=0 necessary */ 159 } 160 z = x*x; 161 r = z*(r00+z*(r01+z*(r02+z*r03))); 162 s = one+z*(s01+z*(s02+z*(s03+z*(s04+z*s05)))); 163 r *= x; 164 return (x*0.5+r/s); 165 } 166 167 static double u0[5] = { 168 -1.960570906462389484206891092512047539632e-0001, 169 5.044387166398112572026169863174882070274e-0002, 170 -1.912568958757635383926261729464141209569e-0003, 171 2.352526005616105109577368905595045204577e-0005, 172 -9.190991580398788465315411784276789663849e-0008, 173 }; 174 static double v0[5] = { 175 1.991673182366499064031901734535479833387e-0002, 176 2.025525810251351806268483867032781294682e-0004, 177 1.356088010975162198085369545564475416398e-0006, 178 6.227414523646214811803898435084697863445e-0009, 179 1.665592462079920695971450872592458916421e-0011, 180 }; 181 182 double y1(x) 183 double x; 184 { 185 double z, s, c, ss, cc, u, v; 186 /* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */ 187 if (!finite(x)) 188 if (!_IEEE) return (infnan(EDOM)); 189 else if (x < 0) 190 return(zero/zero); 191 else if (x > 0) 192 return (0); 193 else 194 return(x); 195 if (x <= 0) { 196 if (_IEEE && x == 0) return -one/zero; 197 else if(x == 0) return(infnan(-ERANGE)); 198 else if(_IEEE) return (zero/zero); 199 else return(infnan(EDOM)); 200 } 201 if (x >= 2) /* |x| >= 2.0 */ 202 { 203 s = sin(x); 204 c = cos(x); 205 ss = -s-c; 206 cc = s-c; 207 if (x < .5 * DBL_MAX) /* make sure x+x not overflow */ 208 { 209 z = cos(x+x); 210 if ((s*c)>zero) cc = z/ss; 211 else ss = z/cc; 212 } 213 /* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0)) 214 * where x0 = x-3pi/4 215 * Better formula: 216 * cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4) 217 * = 1/sqrt(2) * (sin(x) - cos(x)) 218 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4) 219 * = -1/sqrt(2) * (cos(x) + sin(x)) 220 * To avoid cancellation, use 221 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) 222 * to compute the worse one. 223 */ 224 if (_IEEE && x>two_129) 225 z = (invsqrtpi*ss)/sqrt(x); 226 else { 227 u = pone(x); v = qone(x); 228 z = invsqrtpi*(u*ss+v*cc)/sqrt(x); 229 } 230 return z; 231 } 232 if (x <= two_m54) { /* x < 2**-54 */ 233 return (-tpi/x); 234 } 235 z = x*x; 236 u = u0[0]+z*(u0[1]+z*(u0[2]+z*(u0[3]+z*u0[4]))); 237 v = one+z*(v0[0]+z*(v0[1]+z*(v0[2]+z*(v0[3]+z*v0[4])))); 238 return (x*(u/v) + tpi*(j1(x)*log(x)-one/x)); 239 } 240 241 /* For x >= 8, the asymptotic expansions of pone is 242 * 1 + 15/128 s^2 - 4725/2^15 s^4 - ..., where s = 1/x. 243 * We approximate pone by 244 * pone(x) = 1 + (R/S) 245 * where R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10 246 * S = 1 + ps0*s^2 + ... + ps4*s^10 247 * and 248 * | pone(x)-1-R/S | <= 2 ** ( -60.06) 249 */ 250 251 static double pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 252 0.0, 253 1.171874999999886486643746274751925399540e-0001, 254 1.323948065930735690925827997575471527252e+0001, 255 4.120518543073785433325860184116512799375e+0002, 256 3.874745389139605254931106878336700275601e+0003, 257 7.914479540318917214253998253147871806507e+0003, 258 }; 259 static double ps8[5] = { 260 1.142073703756784104235066368252692471887e+0002, 261 3.650930834208534511135396060708677099382e+0003, 262 3.695620602690334708579444954937638371808e+0004, 263 9.760279359349508334916300080109196824151e+0004, 264 3.080427206278887984185421142572315054499e+0004, 265 }; 266 267 static double pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 268 1.319905195562435287967533851581013807103e-0011, 269 1.171874931906140985709584817065144884218e-0001, 270 6.802751278684328781830052995333841452280e+0000, 271 1.083081829901891089952869437126160568246e+0002, 272 5.176361395331997166796512844100442096318e+0002, 273 5.287152013633375676874794230748055786553e+0002, 274 }; 275 static double ps5[5] = { 276 5.928059872211313557747989128353699746120e+0001, 277 9.914014187336144114070148769222018425781e+0002, 278 5.353266952914879348427003712029704477451e+0003, 279 7.844690317495512717451367787640014588422e+0003, 280 1.504046888103610723953792002716816255382e+0003, 281 }; 282 283 static double pr3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ 284 3.025039161373736032825049903408701962756e-0009, 285 1.171868655672535980750284752227495879921e-0001, 286 3.932977500333156527232725812363183251138e+0000, 287 3.511940355916369600741054592597098912682e+0001, 288 9.105501107507812029367749771053045219094e+0001, 289 4.855906851973649494139275085628195457113e+0001, 290 }; 291 static double ps3[5] = { 292 3.479130950012515114598605916318694946754e+0001, 293 3.367624587478257581844639171605788622549e+0002, 294 1.046871399757751279180649307467612538415e+0003, 295 8.908113463982564638443204408234739237639e+0002, 296 1.037879324396392739952487012284401031859e+0002, 297 }; 298 299 static double pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 300 1.077108301068737449490056513753865482831e-0007, 301 1.171762194626833490512746348050035171545e-0001, 302 2.368514966676087902251125130227221462134e+0000, 303 1.224261091482612280835153832574115951447e+0001, 304 1.769397112716877301904532320376586509782e+0001, 305 5.073523125888185399030700509321145995160e+0000, 306 }; 307 static double ps2[5] = { 308 2.143648593638214170243114358933327983793e+0001, 309 1.252902271684027493309211410842525120355e+0002, 310 2.322764690571628159027850677565128301361e+0002, 311 1.176793732871470939654351793502076106651e+0002, 312 8.364638933716182492500902115164881195742e+0000, 313 }; 314 315 static double pone(x) 316 double x; 317 { 318 double *p,*q,z,r,s; 319 if (x >= 8.0) {p = pr8; q= ps8;} 320 else if (x >= 4.54545211791992188) {p = pr5; q= ps5;} 321 else if (x >= 2.85714149475097656) {p = pr3; q= ps3;} 322 else /* if (x >= 2.0) */ {p = pr2; q= ps2;} 323 z = one/(x*x); 324 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); 325 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4])))); 326 return (one + r/s); 327 } 328 329 330 /* For x >= 8, the asymptotic expansions of qone is 331 * 3/8 s - 105/1024 s^3 - ..., where s = 1/x. 332 * We approximate pone by 333 * qone(x) = s*(0.375 + (R/S)) 334 * where R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10 335 * S = 1 + qs1*s^2 + ... + qs6*s^12 336 * and 337 * | qone(x)/s -0.375-R/S | <= 2 ** ( -61.13) 338 */ 339 340 static double qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 341 0.0, 342 -1.025390624999927207385863635575804210817e-0001, 343 -1.627175345445899724355852152103771510209e+0001, 344 -7.596017225139501519843072766973047217159e+0002, 345 -1.184980667024295901645301570813228628541e+0004, 346 -4.843851242857503225866761992518949647041e+0004, 347 }; 348 static double qs8[6] = { 349 1.613953697007229231029079421446916397904e+0002, 350 7.825385999233484705298782500926834217525e+0003, 351 1.338753362872495800748094112937868089032e+0005, 352 7.196577236832409151461363171617204036929e+0005, 353 6.666012326177764020898162762642290294625e+0005, 354 -2.944902643038346618211973470809456636830e+0005, 355 }; 356 357 static double qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 358 -2.089799311417640889742251585097264715678e-0011, 359 -1.025390502413754195402736294609692303708e-0001, 360 -8.056448281239359746193011295417408828404e+0000, 361 -1.836696074748883785606784430098756513222e+0002, 362 -1.373193760655081612991329358017247355921e+0003, 363 -2.612444404532156676659706427295870995743e+0003, 364 }; 365 static double qs5[6] = { 366 8.127655013843357670881559763225310973118e+0001, 367 1.991798734604859732508048816860471197220e+0003, 368 1.746848519249089131627491835267411777366e+0004, 369 4.985142709103522808438758919150738000353e+0004, 370 2.794807516389181249227113445299675335543e+0004, 371 -4.719183547951285076111596613593553911065e+0003, 372 }; 373 374 static double qr3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ 375 -5.078312264617665927595954813341838734288e-0009, 376 -1.025378298208370901410560259001035577681e-0001, 377 -4.610115811394734131557983832055607679242e+0000, 378 -5.784722165627836421815348508816936196402e+0001, 379 -2.282445407376317023842545937526967035712e+0002, 380 -2.192101284789093123936441805496580237676e+0002, 381 }; 382 static double qs3[6] = { 383 4.766515503237295155392317984171640809318e+0001, 384 6.738651126766996691330687210949984203167e+0002, 385 3.380152866795263466426219644231687474174e+0003, 386 5.547729097207227642358288160210745890345e+0003, 387 1.903119193388108072238947732674639066045e+0003, 388 -1.352011914443073322978097159157678748982e+0002, 389 }; 390 391 static double qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 392 -1.783817275109588656126772316921194887979e-0007, 393 -1.025170426079855506812435356168903694433e-0001, 394 -2.752205682781874520495702498875020485552e+0000, 395 -1.966361626437037351076756351268110418862e+0001, 396 -4.232531333728305108194363846333841480336e+0001, 397 -2.137192117037040574661406572497288723430e+0001, 398 }; 399 static double qs2[6] = { 400 2.953336290605238495019307530224241335502e+0001, 401 2.529815499821905343698811319455305266409e+0002, 402 7.575028348686454070022561120722815892346e+0002, 403 7.393932053204672479746835719678434981599e+0002, 404 1.559490033366661142496448853793707126179e+0002, 405 -4.959498988226281813825263003231704397158e+0000, 406 }; 407 408 static double qone(x) 409 double x; 410 { 411 double *p,*q, s,r,z; 412 if (x >= 8.0) {p = qr8; q= qs8;} 413 else if (x >= 4.54545211791992188) {p = qr5; q= qs5;} 414 else if (x >= 2.85714149475097656) {p = qr3; q= qs3;} 415 else /* if (x >= 2.0) */ {p = qr2; q= qs2;} 416 z = one/(x*x); 417 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); 418 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5]))))); 419 return (.375 + r/s)/x; 420 } 421