1 /* 2 * CDDL HEADER START 3 * 4 * The contents of this file are subject to the terms of the 5 * Common Development and Distribution License (the "License"). 6 * You may not use this file except in compliance with the License. 7 * 8 * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE 9 * or http://www.opensolaris.org/os/licensing. 10 * See the License for the specific language governing permissions 11 * and limitations under the License. 12 * 13 * When distributing Covered Code, include this CDDL HEADER in each 14 * file and include the License file at usr/src/OPENSOLARIS.LICENSE. 15 * If applicable, add the following below this CDDL HEADER, with the 16 * fields enclosed by brackets "[]" replaced with your own identifying 17 * information: Portions Copyright [yyyy] [name of copyright owner] 18 * 19 * CDDL HEADER END 20 */ 21 22 /* 23 * Copyright 2011 Nexenta Systems, Inc. All rights reserved. 24 */ 25 /* 26 * Copyright 2006 Sun Microsystems, Inc. All rights reserved. 27 * Use is subject to license terms. 28 */ 29 30 /* 31 * Floating point Bessel's function of the first and second kinds 32 * of order zero: j0(x),y0(x); 33 * 34 * Special cases: 35 * y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal; 36 * y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal. 37 */ 38 39 #pragma weak __j0 = j0 40 #pragma weak __y0 = y0 41 42 #include "libm.h" 43 #include "libm_protos.h" 44 #include <math.h> 45 #include <values.h> 46 47 #define GENERIC double 48 static const GENERIC 49 zero = 0.0, 50 small = 1.0e-5, 51 tiny = 1.0e-18, 52 one = 1.0, 53 eight = 8.0, 54 invsqrtpi = 5.641895835477562869480794515607725858441e-0001, 55 tpi = 0.636619772367581343075535053490057448; 56 57 static GENERIC pzero(GENERIC), qzero(GENERIC); 58 static const GENERIC r0[4] = { /* [1.e-5, 1.28] */ 59 -2.500000000000003622131880894830476755537e-0001, 60 1.095597547334830263234433855932375353303e-0002, 61 -1.819734750463320921799187258987098087697e-0004, 62 9.977001946806131657544212501069893930846e-0007, 63 }; 64 static const GENERIC s0[4] = { /* [1.e-5, 1.28] */ 65 1.0, 66 1.867609810662950169966782360588199673741e-0002, 67 1.590389206181565490878430827706972074208e-0004, 68 6.520867386742583632375520147714499522721e-0007, 69 }; 70 static const GENERIC r1[9] = { /* [1.28,8] */ 71 9.999999999999999942156495584397047660949e-0001, 72 -2.389887722731319130476839836908143731281e-0001, 73 1.293359476138939027791270393439493640570e-0002, 74 -2.770985642343140122168852400228563364082e-0004, 75 2.905241575772067678086738389169625218912e-0006, 76 -1.636846356264052597969042009265043251279e-0008, 77 5.072306160724884775085431059052611737827e-0011, 78 -8.187060730684066824228914775146536139112e-0014, 79 5.422219326959949863954297860723723423842e-0017, 80 }; 81 static const GENERIC s1[9] = { /* [1.28,8] */ 82 1.0, 83 1.101122772686807702762104741932076228349e-0002, 84 6.140169310641649223411427764669143978228e-0005, 85 2.292035877515152097976946119293215705250e-0007, 86 6.356910426504644334558832036362219583789e-0010, 87 1.366626326900219555045096999553948891401e-0012, 88 2.280399586866739522891837985560481180088e-0015, 89 2.801559820648939665270492520004836611187e-0018, 90 2.073101088320349159764410261466350732968e-0021, 91 }; 92 93 GENERIC 94 j0(GENERIC x) { 95 GENERIC z, s, c, ss, cc, r, u, v, ox; 96 int i; 97 98 if (isnan(x)) 99 return (x*x); /* + -> * for Cheetah */ 100 ox = x; 101 x = fabs(x); 102 if (x > 8.0) { 103 if (!finite(x)) 104 return (zero); 105 s = sin(x); 106 c = cos(x); 107 /* 108 * j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)-q0(x)*sin(x0)) 109 * where x0 = x-pi/4 110 * Better formula: 111 * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4) 112 * = 1/sqrt(2) * (cos(x) + sin(x)) 113 * sin(x0) = sin(x)cos(pi/4)-cos(x)sin(pi/4) 114 * = 1/sqrt(2) * (sin(x) - cos(x)) 115 * To avoid cancellation, use 116 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) 117 * to compute the worse one. 118 */ 119 if (x > 8.9e307) { /* x+x may overflow */ 120 ss = s-c; 121 cc = s+c; 122 } else if (signbit(s) != signbit(c)) { 123 ss = s - c; 124 cc = -cos(x+x)/ss; 125 } else { 126 cc = s + c; 127 ss = -cos(x+x)/cc; 128 } 129 /* 130 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x) 131 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x) 132 */ 133 if (x > 1.0e40) z = (invsqrtpi*cc)/sqrt(x); 134 else { 135 u = pzero(x); v = qzero(x); 136 z = invsqrtpi*(u*cc-v*ss)/sqrt(x); 137 } 138 /* force to pass SVR4 even the result is wrong (sign) */ 139 if (x > X_TLOSS) 140 return (_SVID_libm_err(ox, z, 34)); 141 else 142 return (z); 143 } 144 if (x <= small) { 145 if (x <= tiny) 146 return (one-x); 147 else 148 return (one-x*x*0.25); 149 } 150 z = x*x; 151 if (x <= 1.28) { 152 r = r0[0]+z*(r0[1]+z*(r0[2]+z*r0[3])); 153 s = s0[0]+z*(s0[1]+z*(s0[2]+z*s0[3])); 154 return (one + z*(r/s)); 155 } else { 156 for (r = r1[8], s = s1[8], i = 7; i >= 0; i--) { 157 r = r*z + r1[i]; 158 s = s*z + s1[i]; 159 } 160 return (r/s); 161 } 162 } 163 164 static const GENERIC u0[13] = { 165 -7.380429510868722526754723020704317641941e-0002, 166 1.772607102684869924301459663049874294814e-0001, 167 -1.524370666542713828604078090970799356306e-0002, 168 4.650819100693891757143771557629924591915e-0004, 169 -7.125768872339528975036316108718239946022e-0006, 170 6.411017001656104598327565004771515257146e-0008, 171 -3.694275157433032553021246812379258781665e-0010, 172 1.434364544206266624252820889648445263842e-0012, 173 -3.852064731859936455895036286874139896861e-0015, 174 7.182052899726138381739945881914874579696e-0018, 175 -9.060556574619677567323741194079797987200e-0021, 176 7.124435467408860515265552217131230511455e-0024, 177 -2.709726774636397615328813121715432044771e-0027, 178 }; 179 static const GENERIC v0[5] = { 180 1.0, 181 4.678678931512549002587702477349214886475e-0003, 182 9.486828955529948534822800829497565178985e-0006, 183 1.001495929158861646659010844136682454906e-0008, 184 4.725338116256021660204443235685358593611e-0012, 185 }; 186 187 GENERIC 188 y0(GENERIC x) { 189 GENERIC z, /* d, */ s, c, ss, cc, u, v; 190 int i; 191 192 if (isnan(x)) 193 return (x*x); /* + -> * for Cheetah */ 194 if (x <= zero) { 195 if (x == zero) 196 /* d= -one/(x-x); */ 197 return (_SVID_libm_err(x, x, 8)); 198 else 199 /* d = zero/(x-x); */ 200 return (_SVID_libm_err(x, x, 9)); 201 } 202 if (x > 8.0) { 203 if (!finite(x)) 204 return (zero); 205 s = sin(x); 206 c = cos(x); 207 /* 208 * j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)-q0(x)*sin(x0)) 209 * where x0 = x-pi/4 210 * Better formula: 211 * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4) 212 * = 1/sqrt(2) * (cos(x) + sin(x)) 213 * sin(x0) = sin(x)cos(pi/4)-cos(x)sin(pi/4) 214 * = 1/sqrt(2) * (sin(x) - cos(x)) 215 * To avoid cancellation, use 216 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) 217 * to compute the worse one. 218 */ 219 if (x > 8.9e307) { /* x+x may overflow */ 220 ss = s-c; 221 cc = s+c; 222 } else if (signbit(s) != signbit(c)) { 223 ss = s - c; 224 cc = -cos(x+x)/ss; 225 } else { 226 cc = s + c; 227 ss = -cos(x+x)/cc; 228 } 229 /* 230 * j0(x) = 1/sqrt(pi*x) * (P(0,x)*cc - Q(0,x)*ss) 231 * y0(x) = 1/sqrt(pi*x) * (P(0,x)*ss + Q(0,x)*cc) 232 */ 233 if (x > 1.0e40) 234 z = (invsqrtpi*ss)/sqrt(x); 235 else 236 z = invsqrtpi*(pzero(x)*ss+qzero(x)*cc)/sqrt(x); 237 if (x > X_TLOSS) 238 return (_SVID_libm_err(x, z, 35)); 239 else 240 return (z); 241 242 } 243 if (x <= tiny) { 244 return (u0[0] + tpi*log(x)); 245 } 246 z = x*x; 247 for (u = u0[12], i = 11; i >= 0; i--) u = u*z + u0[i]; 248 v = v0[0]+z*(v0[1]+z*(v0[2]+z*(v0[3]+z*v0[4]))); 249 return (u/v + tpi*(j0(x)*log(x))); 250 } 251 252 static const GENERIC pr[7] = { /* [8 -- inf] pzero 6550 */ 253 .4861344183386052721391238447e5, 254 .1377662549407112278133438945e6, 255 .1222466364088289731869114004e6, 256 .4107070084315176135583353374e5, 257 .5026073801860637125889039915e4, 258 .1783193659125479654541542419e3, 259 .88010344055383421691677564e0, 260 }; 261 static const GENERIC ps[7] = { /* [8 -- inf] pzero 6550 */ 262 .4861344183386052721414037058e5, 263 .1378196632630384670477582699e6, 264 .1223967185341006542748936787e6, 265 .4120150243795353639995862617e5, 266 .5068271181053546392490184353e4, 267 .1829817905472769960535671664e3, 268 1.0, 269 }; 270 static const GENERIC huge = 1.0e10; 271 272 static GENERIC 273 pzero(GENERIC x) { 274 GENERIC s, r, t, z; 275 int i; 276 if (x > huge) 277 return (one); 278 t = eight/x; z = t*t; 279 r = pr[5]+z*pr[6]; 280 s = ps[5]+z; 281 for (i = 4; i >= 0; i--) { 282 r = r*z + pr[i]; 283 s = s*z + ps[i]; 284 } 285 return (r/s); 286 } 287 288 static const GENERIC qr[7] = { /* [8 -- inf] qzero 6950 */ 289 -.1731210995701068539185611951e3, 290 -.5522559165936166961235240613e3, 291 -.5604935606637346590614529613e3, 292 -.2200430300226009379477365011e3, 293 -.323869355375648849771296746e2, 294 -.14294979207907956223499258e1, 295 -.834690374102384988158918e-2, 296 }; 297 static const GENERIC qs[7] = { /* [8 -- inf] qzero 6950 */ 298 .1107975037248683865326709645e5, 299 .3544581680627082674651471873e5, 300 .3619118937918394132179019059e5, 301 .1439895563565398007471485822e5, 302 .2190277023344363955930226234e4, 303 .106695157020407986137501682e3, 304 1.0, 305 }; 306 307 static GENERIC 308 qzero(GENERIC x) { 309 GENERIC s, r, t, z; 310 int i; 311 if (x > huge) 312 return (-0.125/x); 313 t = eight/x; z = t*t; 314 r = qr[5]+z*qr[6]; 315 s = qs[5]+z; 316 for (i = 4; i >= 0; i--) { 317 r = r*z + qr[i]; 318 s = s*z + qs[i]; 319 } 320 return (t*(r/s)); 321 } 322