1 /* 2 * ==================================================== 3 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 4 * 5 * Developed at SunSoft, a Sun Microsystems, Inc. business. 6 * Permission to use, copy, modify, and distribute this 7 * software is freely granted, provided that this notice 8 * is preserved. 9 * ==================================================== 10 */ 11 12 /* j0(x), y0(x) 13 * Bessel function of the first and second kinds of order zero. 14 * Method -- j0(x): 15 * 1. For tiny x, we use j0(x) = 1 - x^2/4 + x^4/64 - ... 16 * 2. Reduce x to |x| since j0(x)=j0(-x), and 17 * for x in (0,2) 18 * j0(x) = 1-z/4+ z^2*R0/S0, where z = x*x; 19 * (precision: |j0-1+z/4-z^2R0/S0 |<2**-63.67 ) 20 * for x in (2,inf) 21 * j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)-q0(x)*sin(x0)) 22 * where x0 = x-pi/4. It is better to compute sin(x0),cos(x0) 23 * as follow: 24 * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4) 25 * = 1/sqrt(2) * (cos(x) + sin(x)) 26 * sin(x0) = sin(x)cos(pi/4)-cos(x)sin(pi/4) 27 * = 1/sqrt(2) * (sin(x) - cos(x)) 28 * (To avoid cancellation, use 29 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) 30 * to compute the worse one.) 31 * 32 * 3 Special cases 33 * j0(nan)= nan 34 * j0(0) = 1 35 * j0(inf) = 0 36 * 37 * Method -- y0(x): 38 * 1. For x<2. 39 * Since 40 * y0(x) = 2/pi*(j0(x)*(ln(x/2)+Euler) + x^2/4 - ...) 41 * therefore y0(x)-2/pi*j0(x)*ln(x) is an even function. 42 * We use the following function to approximate y0, 43 * y0(x) = U(z)/V(z) + (2/pi)*(j0(x)*ln(x)), z= x^2 44 * where 45 * U(z) = u00 + u01*z + ... + u06*z^6 46 * V(z) = 1 + v01*z + ... + v04*z^4 47 * with absolute approximation error bounded by 2**-72. 48 * Note: For tiny x, U/V = u0 and j0(x)~1, hence 49 * y0(tiny) = u0 + (2/pi)*ln(tiny), (choose tiny<2**-27) 50 * 2. For x>=2. 51 * y0(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 * by the method mentioned above. 54 * 3. Special cases: y0(0)=-inf, y0(x<0)=NaN, y0(inf)=0. 55 */ 56 57 #include "math.h" 58 #include "math_private.h" 59 60 static __inline double pzero(double), qzero(double); 61 62 static const volatile double vone = 1, vzero = 0; 63 64 static const double 65 huge = 1e300, 66 one = 1.0, 67 invsqrtpi= 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */ 68 tpi = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */ 69 /* R0/S0 on [0, 2.00] */ 70 R02 = 1.56249999999999947958e-02, /* 0x3F8FFFFF, 0xFFFFFFFD */ 71 R03 = -1.89979294238854721751e-04, /* 0xBF28E6A5, 0xB61AC6E9 */ 72 R04 = 1.82954049532700665670e-06, /* 0x3EBEB1D1, 0x0C503919 */ 73 R05 = -4.61832688532103189199e-09, /* 0xBE33D5E7, 0x73D63FCE */ 74 S01 = 1.56191029464890010492e-02, /* 0x3F8FFCE8, 0x82C8C2A4 */ 75 S02 = 1.16926784663337450260e-04, /* 0x3F1EA6D2, 0xDD57DBF4 */ 76 S03 = 5.13546550207318111446e-07, /* 0x3EA13B54, 0xCE84D5A9 */ 77 S04 = 1.16614003333790000205e-09; /* 0x3E1408BC, 0xF4745D8F */ 78 79 static const double zero = 0, qrtr = 0.25; 80 81 double 82 j0(double x) 83 { 84 double z, s,c,ss,cc,r,u,v; 85 int32_t hx,ix; 86 87 GET_HIGH_WORD(hx,x); 88 ix = hx&0x7fffffff; 89 if(ix>=0x7ff00000) return one/(x*x); 90 x = fabs(x); 91 if(ix >= 0x40000000) { /* |x| >= 2.0 */ 92 sincos(x, &s, &c); 93 ss = s-c; 94 cc = s+c; 95 if(ix<0x7fe00000) { /* Make sure x+x does not overflow. */ 96 z = -cos(x+x); 97 if ((s*c)<zero) cc = z/ss; 98 else ss = z/cc; 99 } 100 /* 101 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x) 102 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x) 103 */ 104 if(ix>0x48000000) z = (invsqrtpi*cc)/sqrt(x); 105 else { 106 u = pzero(x); v = qzero(x); 107 z = invsqrtpi*(u*cc-v*ss)/sqrt(x); 108 } 109 return z; 110 } 111 if(ix<0x3f200000) { /* |x| < 2**-13 */ 112 if(huge+x>one) { /* raise inexact if x != 0 */ 113 if(ix<0x3e400000) return one; /* |x|<2**-27 */ 114 else return one - x*x/4; 115 } 116 } 117 z = x*x; 118 r = z*(R02+z*(R03+z*(R04+z*R05))); 119 s = one+z*(S01+z*(S02+z*(S03+z*S04))); 120 if(ix < 0x3FF00000) { /* |x| < 1.00 */ 121 return one + z*((r/s)-qrtr); 122 } else { 123 u = x/2; 124 return((one+u)*(one-u)+z*(r/s)); 125 } 126 } 127 128 static const double 129 u00 = -7.38042951086872317523e-02, /* 0xBFB2E4D6, 0x99CBD01F */ 130 u01 = 1.76666452509181115538e-01, /* 0x3FC69D01, 0x9DE9E3FC */ 131 u02 = -1.38185671945596898896e-02, /* 0xBF8C4CE8, 0xB16CFA97 */ 132 u03 = 3.47453432093683650238e-04, /* 0x3F36C54D, 0x20B29B6B */ 133 u04 = -3.81407053724364161125e-06, /* 0xBECFFEA7, 0x73D25CAD */ 134 u05 = 1.95590137035022920206e-08, /* 0x3E550057, 0x3B4EABD4 */ 135 u06 = -3.98205194132103398453e-11, /* 0xBDC5E43D, 0x693FB3C8 */ 136 v01 = 1.27304834834123699328e-02, /* 0x3F8A1270, 0x91C9C71A */ 137 v02 = 7.60068627350353253702e-05, /* 0x3F13ECBB, 0xF578C6C1 */ 138 v03 = 2.59150851840457805467e-07, /* 0x3E91642D, 0x7FF202FD */ 139 v04 = 4.41110311332675467403e-10; /* 0x3DFE5018, 0x3BD6D9EF */ 140 141 double 142 y0(double x) 143 { 144 double z, s,c,ss,cc,u,v; 145 int32_t hx,ix,lx; 146 147 EXTRACT_WORDS(hx,lx,x); 148 ix = 0x7fffffff&hx; 149 /* 150 * y0(NaN) = NaN. 151 * y0(Inf) = 0. 152 * y0(-Inf) = NaN and raise invalid exception. 153 */ 154 if(ix>=0x7ff00000) return vone/(x+x*x); 155 /* y0(+-0) = -inf and raise divide-by-zero exception. */ 156 if((ix|lx)==0) return -one/vzero; 157 /* y0(x<0) = NaN and raise invalid exception. */ 158 if(hx<0) return vzero/vzero; 159 if(ix >= 0x40000000) { /* |x| >= 2.0 */ 160 /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0)) 161 * where x0 = x-pi/4 162 * Better formula: 163 * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4) 164 * = 1/sqrt(2) * (sin(x) + cos(x)) 165 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4) 166 * = 1/sqrt(2) * (sin(x) - cos(x)) 167 * To avoid cancellation, use 168 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) 169 * to compute the worse one. 170 */ 171 sincos(x, &s, &c); 172 ss = s-c; 173 cc = s+c; 174 /* 175 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x) 176 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x) 177 */ 178 if(ix<0x7fe00000) { /* make sure x+x not overflow */ 179 z = -cos(x+x); 180 if ((s*c)<zero) cc = z/ss; 181 else ss = z/cc; 182 } 183 if(ix>0x48000000) z = (invsqrtpi*ss)/sqrt(x); 184 else { 185 u = pzero(x); v = qzero(x); 186 z = invsqrtpi*(u*ss+v*cc)/sqrt(x); 187 } 188 return z; 189 } 190 if(ix<=0x3e400000) { /* x < 2**-27 */ 191 return(u00 + tpi*log(x)); 192 } 193 z = x*x; 194 u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06))))); 195 v = one+z*(v01+z*(v02+z*(v03+z*v04))); 196 return(u/v + tpi*(j0(x)*log(x))); 197 } 198 199 /* The asymptotic expansions of pzero is 200 * 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x. 201 * For x >= 2, We approximate pzero by 202 * pzero(x) = 1 + (R/S) 203 * where R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10 204 * S = 1 + pS0*s^2 + ... + pS4*s^10 205 * and 206 * | pzero(x)-1-R/S | <= 2 ** ( -60.26) 207 */ 208 static const double pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 209 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */ 210 -7.03124999999900357484e-02, /* 0xBFB1FFFF, 0xFFFFFD32 */ 211 -8.08167041275349795626e+00, /* 0xC02029D0, 0xB44FA779 */ 212 -2.57063105679704847262e+02, /* 0xC0701102, 0x7B19E863 */ 213 -2.48521641009428822144e+03, /* 0xC0A36A6E, 0xCD4DCAFC */ 214 -5.25304380490729545272e+03, /* 0xC0B4850B, 0x36CC643D */ 215 }; 216 static const double pS8[5] = { 217 1.16534364619668181717e+02, /* 0x405D2233, 0x07A96751 */ 218 3.83374475364121826715e+03, /* 0x40ADF37D, 0x50596938 */ 219 4.05978572648472545552e+04, /* 0x40E3D2BB, 0x6EB6B05F */ 220 1.16752972564375915681e+05, /* 0x40FC810F, 0x8F9FA9BD */ 221 4.76277284146730962675e+04, /* 0x40E74177, 0x4F2C49DC */ 222 }; 223 224 static const double pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 225 -1.14125464691894502584e-11, /* 0xBDA918B1, 0x47E495CC */ 226 -7.03124940873599280078e-02, /* 0xBFB1FFFF, 0xE69AFBC6 */ 227 -4.15961064470587782438e+00, /* 0xC010A370, 0xF90C6BBF */ 228 -6.76747652265167261021e+01, /* 0xC050EB2F, 0x5A7D1783 */ 229 -3.31231299649172967747e+02, /* 0xC074B3B3, 0x6742CC63 */ 230 -3.46433388365604912451e+02, /* 0xC075A6EF, 0x28A38BD7 */ 231 }; 232 static const double pS5[5] = { 233 6.07539382692300335975e+01, /* 0x404E6081, 0x0C98C5DE */ 234 1.05125230595704579173e+03, /* 0x40906D02, 0x5C7E2864 */ 235 5.97897094333855784498e+03, /* 0x40B75AF8, 0x8FBE1D60 */ 236 9.62544514357774460223e+03, /* 0x40C2CCB8, 0xFA76FA38 */ 237 2.40605815922939109441e+03, /* 0x40A2CC1D, 0xC70BE864 */ 238 }; 239 240 static const double pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ 241 -2.54704601771951915620e-09, /* 0xBE25E103, 0x6FE1AA86 */ 242 -7.03119616381481654654e-02, /* 0xBFB1FFF6, 0xF7C0E24B */ 243 -2.40903221549529611423e+00, /* 0xC00345B2, 0xAEA48074 */ 244 -2.19659774734883086467e+01, /* 0xC035F74A, 0x4CB94E14 */ 245 -5.80791704701737572236e+01, /* 0xC04D0A22, 0x420A1A45 */ 246 -3.14479470594888503854e+01, /* 0xC03F72AC, 0xA892D80F */ 247 }; 248 static const double pS3[5] = { 249 3.58560338055209726349e+01, /* 0x4041ED92, 0x84077DD3 */ 250 3.61513983050303863820e+02, /* 0x40769839, 0x464A7C0E */ 251 1.19360783792111533330e+03, /* 0x4092A66E, 0x6D1061D6 */ 252 1.12799679856907414432e+03, /* 0x40919FFC, 0xB8C39B7E */ 253 1.73580930813335754692e+02, /* 0x4065B296, 0xFC379081 */ 254 }; 255 256 static const double pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 257 -8.87534333032526411254e-08, /* 0xBE77D316, 0xE927026D */ 258 -7.03030995483624743247e-02, /* 0xBFB1FF62, 0x495E1E42 */ 259 -1.45073846780952986357e+00, /* 0xBFF73639, 0x8A24A843 */ 260 -7.63569613823527770791e+00, /* 0xC01E8AF3, 0xEDAFA7F3 */ 261 -1.11931668860356747786e+01, /* 0xC02662E6, 0xC5246303 */ 262 -3.23364579351335335033e+00, /* 0xC009DE81, 0xAF8FE70F */ 263 }; 264 static const double pS2[5] = { 265 2.22202997532088808441e+01, /* 0x40363865, 0x908B5959 */ 266 1.36206794218215208048e+02, /* 0x4061069E, 0x0EE8878F */ 267 2.70470278658083486789e+02, /* 0x4070E786, 0x42EA079B */ 268 1.53875394208320329881e+02, /* 0x40633C03, 0x3AB6FAFF */ 269 1.46576176948256193810e+01, /* 0x402D50B3, 0x44391809 */ 270 }; 271 272 static __inline double 273 pzero(double x) 274 { 275 const double *p,*q; 276 double z,r,s; 277 int32_t ix; 278 GET_HIGH_WORD(ix,x); 279 ix &= 0x7fffffff; 280 if(ix>=0x40200000) {p = pR8; q= pS8;} 281 else if(ix>=0x40122E8B){p = pR5; q= pS5;} 282 else if(ix>=0x4006DB6D){p = pR3; q= pS3;} 283 else {p = pR2; q= pS2;} /* ix>=0x40000000 */ 284 z = one/(x*x); 285 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); 286 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4])))); 287 return one+ r/s; 288 } 289 290 291 /* For x >= 8, the asymptotic expansions of qzero is 292 * -1/8 s + 75/1024 s^3 - ..., where s = 1/x. 293 * We approximate pzero by 294 * qzero(x) = s*(-1.25 + (R/S)) 295 * where R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10 296 * S = 1 + qS0*s^2 + ... + qS5*s^12 297 * and 298 * | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22) 299 */ 300 static const double qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 301 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */ 302 7.32421874999935051953e-02, /* 0x3FB2BFFF, 0xFFFFFE2C */ 303 1.17682064682252693899e+01, /* 0x40278952, 0x5BB334D6 */ 304 5.57673380256401856059e+02, /* 0x40816D63, 0x15301825 */ 305 8.85919720756468632317e+03, /* 0x40C14D99, 0x3E18F46D */ 306 3.70146267776887834771e+04, /* 0x40E212D4, 0x0E901566 */ 307 }; 308 static const double qS8[6] = { 309 1.63776026895689824414e+02, /* 0x406478D5, 0x365B39BC */ 310 8.09834494656449805916e+03, /* 0x40BFA258, 0x4E6B0563 */ 311 1.42538291419120476348e+05, /* 0x41016652, 0x54D38C3F */ 312 8.03309257119514397345e+05, /* 0x412883DA, 0x83A52B43 */ 313 8.40501579819060512818e+05, /* 0x4129A66B, 0x28DE0B3D */ 314 -3.43899293537866615225e+05, /* 0xC114FD6D, 0x2C9530C5 */ 315 }; 316 317 static const double qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 318 1.84085963594515531381e-11, /* 0x3DB43D8F, 0x29CC8CD9 */ 319 7.32421766612684765896e-02, /* 0x3FB2BFFF, 0xD172B04C */ 320 5.83563508962056953777e+00, /* 0x401757B0, 0xB9953DD3 */ 321 1.35111577286449829671e+02, /* 0x4060E392, 0x0A8788E9 */ 322 1.02724376596164097464e+03, /* 0x40900CF9, 0x9DC8C481 */ 323 1.98997785864605384631e+03, /* 0x409F17E9, 0x53C6E3A6 */ 324 }; 325 static const double qS5[6] = { 326 8.27766102236537761883e+01, /* 0x4054B1B3, 0xFB5E1543 */ 327 2.07781416421392987104e+03, /* 0x40A03BA0, 0xDA21C0CE */ 328 1.88472887785718085070e+04, /* 0x40D267D2, 0x7B591E6D */ 329 5.67511122894947329769e+04, /* 0x40EBB5E3, 0x97E02372 */ 330 3.59767538425114471465e+04, /* 0x40E19118, 0x1F7A54A0 */ 331 -5.35434275601944773371e+03, /* 0xC0B4EA57, 0xBEDBC609 */ 332 }; 333 334 static const double qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ 335 4.37741014089738620906e-09, /* 0x3E32CD03, 0x6ADECB82 */ 336 7.32411180042911447163e-02, /* 0x3FB2BFEE, 0x0E8D0842 */ 337 3.34423137516170720929e+00, /* 0x400AC0FC, 0x61149CF5 */ 338 4.26218440745412650017e+01, /* 0x40454F98, 0x962DAEDD */ 339 1.70808091340565596283e+02, /* 0x406559DB, 0xE25EFD1F */ 340 1.66733948696651168575e+02, /* 0x4064D77C, 0x81FA21E0 */ 341 }; 342 static const double qS3[6] = { 343 4.87588729724587182091e+01, /* 0x40486122, 0xBFE343A6 */ 344 7.09689221056606015736e+02, /* 0x40862D83, 0x86544EB3 */ 345 3.70414822620111362994e+03, /* 0x40ACF04B, 0xE44DFC63 */ 346 6.46042516752568917582e+03, /* 0x40B93C6C, 0xD7C76A28 */ 347 2.51633368920368957333e+03, /* 0x40A3A8AA, 0xD94FB1C0 */ 348 -1.49247451836156386662e+02, /* 0xC062A7EB, 0x201CF40F */ 349 }; 350 351 static const double qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 352 1.50444444886983272379e-07, /* 0x3E84313B, 0x54F76BDB */ 353 7.32234265963079278272e-02, /* 0x3FB2BEC5, 0x3E883E34 */ 354 1.99819174093815998816e+00, /* 0x3FFFF897, 0xE727779C */ 355 1.44956029347885735348e+01, /* 0x402CFDBF, 0xAAF96FE5 */ 356 3.16662317504781540833e+01, /* 0x403FAA8E, 0x29FBDC4A */ 357 1.62527075710929267416e+01, /* 0x403040B1, 0x71814BB4 */ 358 }; 359 static const double qS2[6] = { 360 3.03655848355219184498e+01, /* 0x403E5D96, 0xF7C07AED */ 361 2.69348118608049844624e+02, /* 0x4070D591, 0xE4D14B40 */ 362 8.44783757595320139444e+02, /* 0x408A6645, 0x22B3BF22 */ 363 8.82935845112488550512e+02, /* 0x408B977C, 0x9C5CC214 */ 364 2.12666388511798828631e+02, /* 0x406A9553, 0x0E001365 */ 365 -5.31095493882666946917e+00, /* 0xC0153E6A, 0xF8B32931 */ 366 }; 367 368 static __inline double 369 qzero(double x) 370 { 371 static const double eighth = 0.125; 372 const double *p,*q; 373 double s,r,z; 374 int32_t ix; 375 GET_HIGH_WORD(ix,x); 376 ix &= 0x7fffffff; 377 if(ix>=0x40200000) {p = qR8; q= qS8;} 378 else if(ix>=0x40122E8B){p = qR5; q= qS5;} 379 else if(ix>=0x4006DB6D){p = qR3; q= qS3;} 380 else {p = qR2; q= qS2;} /* ix>=0x40000000 */ 381 z = one/(x*x); 382 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); 383 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5]))))); 384 return (r/s-eighth)/x; 385 } 386