1 2 /* @(#)e_j0.c 1.3 95/01/18 */ 3 /* 4 * ==================================================== 5 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 6 * 7 * Developed at SunSoft, a Sun Microsystems, Inc. business. 8 * Permission to use, copy, modify, and distribute this 9 * software is freely granted, provided that this notice 10 * is preserved. 11 * ==================================================== 12 */ 13 14 #include <assert.h> 15 16 #include "cdefs-compat.h" 17 //__FBSDID("$FreeBSD: src/lib/msun/src/e_j0.c,v 1.9 2008/02/22 02:30:35 das Exp $"); 18 19 /* __ieee754_j0(x), __ieee754_y0(x) 20 * Bessel function of the first and second kinds of order zero. 21 * Method -- j0(x): 22 * 1. For tiny x, we use j0(x) = 1 - x^2/4 + x^4/64 - ... 23 * 2. Reduce x to |x| since j0(x)=j0(-x), and 24 * for x in (0,2) 25 * j0(x) = 1-z/4+ z^2*R0/S0, where z = x*x; 26 * (precision: |j0-1+z/4-z^2R0/S0 |<2**-63.67 ) 27 * for x in (2,inf) 28 * j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)-q0(x)*sin(x0)) 29 * where x0 = x-pi/4. It is better to compute sin(x0),cos(x0) 30 * as follow: 31 * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4) 32 * = 1/sqrt(2) * (cos(x) + sin(x)) 33 * sin(x0) = sin(x)cos(pi/4)-cos(x)sin(pi/4) 34 * = 1/sqrt(2) * (sin(x) - cos(x)) 35 * (To avoid cancellation, use 36 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) 37 * to compute the worse one.) 38 * 39 * 3 Special cases 40 * j0(nan)= nan 41 * j0(0) = 1 42 * j0(inf) = 0 43 * 44 * Method -- y0(x): 45 * 1. For x<2. 46 * Since 47 * y0(x) = 2/pi*(j0(x)*(ln(x/2)+Euler) + x^2/4 - ...) 48 * therefore y0(x)-2/pi*j0(x)*ln(x) is an even function. 49 * We use the following function to approximate y0, 50 * y0(x) = U(z)/V(z) + (2/pi)*(j0(x)*ln(x)), z= x^2 51 * where 52 * U(z) = u00 + u01*z + ... + u06*z^6 53 * V(z) = 1 + v01*z + ... + v04*z^4 54 * with absolute approximation error bounded by 2**-72. 55 * Note: For tiny x, U/V = u0 and j0(x)~1, hence 56 * y0(tiny) = u0 + (2/pi)*ln(tiny), (choose tiny<2**-27) 57 * 2. For x>=2. 58 * y0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)+q0(x)*sin(x0)) 59 * where x0 = x-pi/4. It is better to compute sin(x0),cos(x0) 60 * by the method mentioned above. 61 * 3. Special cases: y0(0)=-inf, y0(x<0)=NaN, y0(inf)=0. 62 */ 63 64 #include <openlibm.h> 65 66 #include "math_private.h" 67 68 static double pzero(double), qzero(double); 69 70 static const double 71 huge = 1e300, 72 one = 1.0, 73 invsqrtpi= 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */ 74 tpi = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */ 75 /* R0/S0 on [0, 2.00] */ 76 R02 = 1.56249999999999947958e-02, /* 0x3F8FFFFF, 0xFFFFFFFD */ 77 R03 = -1.89979294238854721751e-04, /* 0xBF28E6A5, 0xB61AC6E9 */ 78 R04 = 1.82954049532700665670e-06, /* 0x3EBEB1D1, 0x0C503919 */ 79 R05 = -4.61832688532103189199e-09, /* 0xBE33D5E7, 0x73D63FCE */ 80 S01 = 1.56191029464890010492e-02, /* 0x3F8FFCE8, 0x82C8C2A4 */ 81 S02 = 1.16926784663337450260e-04, /* 0x3F1EA6D2, 0xDD57DBF4 */ 82 S03 = 5.13546550207318111446e-07, /* 0x3EA13B54, 0xCE84D5A9 */ 83 S04 = 1.16614003333790000205e-09; /* 0x3E1408BC, 0xF4745D8F */ 84 85 static const double zero = 0.0; 86 87 DLLEXPORT double 88 __ieee754_j0(double x) 89 { 90 double z, s,c,ss,cc,r,u,v; 91 int32_t hx,ix; 92 93 GET_HIGH_WORD(hx,x); 94 ix = hx&0x7fffffff; 95 if(ix>=0x7ff00000) return one/(x*x); 96 x = fabs(x); 97 if(ix >= 0x40000000) { /* |x| >= 2.0 */ 98 s = sin(x); 99 c = cos(x); 100 ss = s-c; 101 cc = s+c; 102 if(ix<0x7fe00000) { /* make sure x+x not overflow */ 103 z = -cos(x+x); 104 if ((s*c)<zero) cc = z/ss; 105 else ss = z/cc; 106 } 107 /* 108 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x) 109 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x) 110 */ 111 if(ix>0x48000000) z = (invsqrtpi*cc)/sqrt(x); 112 else { 113 u = pzero(x); v = qzero(x); 114 z = invsqrtpi*(u*cc-v*ss)/sqrt(x); 115 } 116 return z; 117 } 118 if(ix<0x3f200000) { /* |x| < 2**-13 */ 119 if(huge+x>one) { /* raise inexact if x != 0 */ 120 if(ix<0x3e400000) return one; /* |x|<2**-27 */ 121 else return one - 0.25*x*x; 122 } 123 } 124 z = x*x; 125 r = z*(R02+z*(R03+z*(R04+z*R05))); 126 s = one+z*(S01+z*(S02+z*(S03+z*S04))); 127 if(ix < 0x3FF00000) { /* |x| < 1.00 */ 128 return one + z*(-0.25+(r/s)); 129 } else { 130 u = 0.5*x; 131 return((one+u)*(one-u)+z*(r/s)); 132 } 133 } 134 135 static const double 136 u00 = -7.38042951086872317523e-02, /* 0xBFB2E4D6, 0x99CBD01F */ 137 u01 = 1.76666452509181115538e-01, /* 0x3FC69D01, 0x9DE9E3FC */ 138 u02 = -1.38185671945596898896e-02, /* 0xBF8C4CE8, 0xB16CFA97 */ 139 u03 = 3.47453432093683650238e-04, /* 0x3F36C54D, 0x20B29B6B */ 140 u04 = -3.81407053724364161125e-06, /* 0xBECFFEA7, 0x73D25CAD */ 141 u05 = 1.95590137035022920206e-08, /* 0x3E550057, 0x3B4EABD4 */ 142 u06 = -3.98205194132103398453e-11, /* 0xBDC5E43D, 0x693FB3C8 */ 143 v01 = 1.27304834834123699328e-02, /* 0x3F8A1270, 0x91C9C71A */ 144 v02 = 7.60068627350353253702e-05, /* 0x3F13ECBB, 0xF578C6C1 */ 145 v03 = 2.59150851840457805467e-07, /* 0x3E91642D, 0x7FF202FD */ 146 v04 = 4.41110311332675467403e-10; /* 0x3DFE5018, 0x3BD6D9EF */ 147 148 DLLEXPORT double 149 __ieee754_y0(double x) 150 { 151 double z, s,c,ss,cc,u,v; 152 int32_t hx,ix,lx; 153 154 EXTRACT_WORDS(hx,lx,x); 155 ix = 0x7fffffff&hx; 156 /* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0 */ 157 if(ix>=0x7ff00000) return one/(x+x*x); 158 if((ix|lx)==0) return -one/zero; 159 if(hx<0) return zero/zero; 160 if(ix >= 0x40000000) { /* |x| >= 2.0 */ 161 /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0)) 162 * where x0 = x-pi/4 163 * Better formula: 164 * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4) 165 * = 1/sqrt(2) * (sin(x) + cos(x)) 166 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4) 167 * = 1/sqrt(2) * (sin(x) - cos(x)) 168 * To avoid cancellation, use 169 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) 170 * to compute the worse one. 171 */ 172 s = sin(x); 173 c = cos(x); 174 ss = s-c; 175 cc = s+c; 176 /* 177 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x) 178 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x) 179 */ 180 if(ix<0x7fe00000) { /* make sure x+x not overflow */ 181 z = -cos(x+x); 182 if ((s*c)<zero) cc = z/ss; 183 else ss = z/cc; 184 } 185 if(ix>0x48000000) z = (invsqrtpi*ss)/sqrt(x); 186 else { 187 u = pzero(x); v = qzero(x); 188 z = invsqrtpi*(u*ss+v*cc)/sqrt(x); 189 } 190 return z; 191 } 192 if(ix<=0x3e400000) { /* x < 2**-27 */ 193 return(u00 + tpi*__ieee754_log(x)); 194 } 195 z = x*x; 196 u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06))))); 197 v = one+z*(v01+z*(v02+z*(v03+z*v04))); 198 return(u/v + tpi*(__ieee754_j0(x)*__ieee754_log(x))); 199 } 200 201 /* The asymptotic expansions of pzero is 202 * 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x. 203 * For x >= 2, We approximate pzero by 204 * pzero(x) = 1 + (R/S) 205 * where R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10 206 * S = 1 + pS0*s^2 + ... + pS4*s^10 207 * and 208 * | pzero(x)-1-R/S | <= 2 ** ( -60.26) 209 */ 210 static const double pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 211 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */ 212 -7.03124999999900357484e-02, /* 0xBFB1FFFF, 0xFFFFFD32 */ 213 -8.08167041275349795626e+00, /* 0xC02029D0, 0xB44FA779 */ 214 -2.57063105679704847262e+02, /* 0xC0701102, 0x7B19E863 */ 215 -2.48521641009428822144e+03, /* 0xC0A36A6E, 0xCD4DCAFC */ 216 -5.25304380490729545272e+03, /* 0xC0B4850B, 0x36CC643D */ 217 }; 218 static const double pS8[5] = { 219 1.16534364619668181717e+02, /* 0x405D2233, 0x07A96751 */ 220 3.83374475364121826715e+03, /* 0x40ADF37D, 0x50596938 */ 221 4.05978572648472545552e+04, /* 0x40E3D2BB, 0x6EB6B05F */ 222 1.16752972564375915681e+05, /* 0x40FC810F, 0x8F9FA9BD */ 223 4.76277284146730962675e+04, /* 0x40E74177, 0x4F2C49DC */ 224 }; 225 226 static const double pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 227 -1.14125464691894502584e-11, /* 0xBDA918B1, 0x47E495CC */ 228 -7.03124940873599280078e-02, /* 0xBFB1FFFF, 0xE69AFBC6 */ 229 -4.15961064470587782438e+00, /* 0xC010A370, 0xF90C6BBF */ 230 -6.76747652265167261021e+01, /* 0xC050EB2F, 0x5A7D1783 */ 231 -3.31231299649172967747e+02, /* 0xC074B3B3, 0x6742CC63 */ 232 -3.46433388365604912451e+02, /* 0xC075A6EF, 0x28A38BD7 */ 233 }; 234 static const double pS5[5] = { 235 6.07539382692300335975e+01, /* 0x404E6081, 0x0C98C5DE */ 236 1.05125230595704579173e+03, /* 0x40906D02, 0x5C7E2864 */ 237 5.97897094333855784498e+03, /* 0x40B75AF8, 0x8FBE1D60 */ 238 9.62544514357774460223e+03, /* 0x40C2CCB8, 0xFA76FA38 */ 239 2.40605815922939109441e+03, /* 0x40A2CC1D, 0xC70BE864 */ 240 }; 241 242 static const double pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ 243 -2.54704601771951915620e-09, /* 0xBE25E103, 0x6FE1AA86 */ 244 -7.03119616381481654654e-02, /* 0xBFB1FFF6, 0xF7C0E24B */ 245 -2.40903221549529611423e+00, /* 0xC00345B2, 0xAEA48074 */ 246 -2.19659774734883086467e+01, /* 0xC035F74A, 0x4CB94E14 */ 247 -5.80791704701737572236e+01, /* 0xC04D0A22, 0x420A1A45 */ 248 -3.14479470594888503854e+01, /* 0xC03F72AC, 0xA892D80F */ 249 }; 250 static const double pS3[5] = { 251 3.58560338055209726349e+01, /* 0x4041ED92, 0x84077DD3 */ 252 3.61513983050303863820e+02, /* 0x40769839, 0x464A7C0E */ 253 1.19360783792111533330e+03, /* 0x4092A66E, 0x6D1061D6 */ 254 1.12799679856907414432e+03, /* 0x40919FFC, 0xB8C39B7E */ 255 1.73580930813335754692e+02, /* 0x4065B296, 0xFC379081 */ 256 }; 257 258 static const double pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 259 -8.87534333032526411254e-08, /* 0xBE77D316, 0xE927026D */ 260 -7.03030995483624743247e-02, /* 0xBFB1FF62, 0x495E1E42 */ 261 -1.45073846780952986357e+00, /* 0xBFF73639, 0x8A24A843 */ 262 -7.63569613823527770791e+00, /* 0xC01E8AF3, 0xEDAFA7F3 */ 263 -1.11931668860356747786e+01, /* 0xC02662E6, 0xC5246303 */ 264 -3.23364579351335335033e+00, /* 0xC009DE81, 0xAF8FE70F */ 265 }; 266 static const double pS2[5] = { 267 2.22202997532088808441e+01, /* 0x40363865, 0x908B5959 */ 268 1.36206794218215208048e+02, /* 0x4061069E, 0x0EE8878F */ 269 2.70470278658083486789e+02, /* 0x4070E786, 0x42EA079B */ 270 1.53875394208320329881e+02, /* 0x40633C03, 0x3AB6FAFF */ 271 1.46576176948256193810e+01, /* 0x402D50B3, 0x44391809 */ 272 }; 273 274 /* Note: This function is only called for ix>=0x40000000 (see above) */ 275 static double pzero(double x) 276 { 277 const double *p,*q; 278 double z,r,s; 279 int32_t ix; 280 GET_HIGH_WORD(ix,x); 281 ix &= 0x7fffffff; 282 assert(ix>=0x40000000 && ix<=0x48000000); 283 if(ix>=0x40200000) {p = pR8; q= pS8;} 284 else if(ix>=0x40122E8B){p = pR5; q= pS5;} 285 else if(ix>=0x4006DB6D){p = pR3; q= pS3;} 286 else {p = pR2; q= pS2;} 287 z = one/(x*x); 288 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); 289 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4])))); 290 return one+ r/s; 291 } 292 293 294 /* For x >= 8, the asymptotic expansions of qzero is 295 * -1/8 s + 75/1024 s^3 - ..., where s = 1/x. 296 * We approximate pzero by 297 * qzero(x) = s*(-1.25 + (R/S)) 298 * where R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10 299 * S = 1 + qS0*s^2 + ... + qS5*s^12 300 * and 301 * | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22) 302 */ 303 static const double qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 304 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */ 305 7.32421874999935051953e-02, /* 0x3FB2BFFF, 0xFFFFFE2C */ 306 1.17682064682252693899e+01, /* 0x40278952, 0x5BB334D6 */ 307 5.57673380256401856059e+02, /* 0x40816D63, 0x15301825 */ 308 8.85919720756468632317e+03, /* 0x40C14D99, 0x3E18F46D */ 309 3.70146267776887834771e+04, /* 0x40E212D4, 0x0E901566 */ 310 }; 311 static const double qS8[6] = { 312 1.63776026895689824414e+02, /* 0x406478D5, 0x365B39BC */ 313 8.09834494656449805916e+03, /* 0x40BFA258, 0x4E6B0563 */ 314 1.42538291419120476348e+05, /* 0x41016652, 0x54D38C3F */ 315 8.03309257119514397345e+05, /* 0x412883DA, 0x83A52B43 */ 316 8.40501579819060512818e+05, /* 0x4129A66B, 0x28DE0B3D */ 317 -3.43899293537866615225e+05, /* 0xC114FD6D, 0x2C9530C5 */ 318 }; 319 320 static const double qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 321 1.84085963594515531381e-11, /* 0x3DB43D8F, 0x29CC8CD9 */ 322 7.32421766612684765896e-02, /* 0x3FB2BFFF, 0xD172B04C */ 323 5.83563508962056953777e+00, /* 0x401757B0, 0xB9953DD3 */ 324 1.35111577286449829671e+02, /* 0x4060E392, 0x0A8788E9 */ 325 1.02724376596164097464e+03, /* 0x40900CF9, 0x9DC8C481 */ 326 1.98997785864605384631e+03, /* 0x409F17E9, 0x53C6E3A6 */ 327 }; 328 static const double qS5[6] = { 329 8.27766102236537761883e+01, /* 0x4054B1B3, 0xFB5E1543 */ 330 2.07781416421392987104e+03, /* 0x40A03BA0, 0xDA21C0CE */ 331 1.88472887785718085070e+04, /* 0x40D267D2, 0x7B591E6D */ 332 5.67511122894947329769e+04, /* 0x40EBB5E3, 0x97E02372 */ 333 3.59767538425114471465e+04, /* 0x40E19118, 0x1F7A54A0 */ 334 -5.35434275601944773371e+03, /* 0xC0B4EA57, 0xBEDBC609 */ 335 }; 336 337 static const double qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ 338 4.37741014089738620906e-09, /* 0x3E32CD03, 0x6ADECB82 */ 339 7.32411180042911447163e-02, /* 0x3FB2BFEE, 0x0E8D0842 */ 340 3.34423137516170720929e+00, /* 0x400AC0FC, 0x61149CF5 */ 341 4.26218440745412650017e+01, /* 0x40454F98, 0x962DAEDD */ 342 1.70808091340565596283e+02, /* 0x406559DB, 0xE25EFD1F */ 343 1.66733948696651168575e+02, /* 0x4064D77C, 0x81FA21E0 */ 344 }; 345 static const double qS3[6] = { 346 4.87588729724587182091e+01, /* 0x40486122, 0xBFE343A6 */ 347 7.09689221056606015736e+02, /* 0x40862D83, 0x86544EB3 */ 348 3.70414822620111362994e+03, /* 0x40ACF04B, 0xE44DFC63 */ 349 6.46042516752568917582e+03, /* 0x40B93C6C, 0xD7C76A28 */ 350 2.51633368920368957333e+03, /* 0x40A3A8AA, 0xD94FB1C0 */ 351 -1.49247451836156386662e+02, /* 0xC062A7EB, 0x201CF40F */ 352 }; 353 354 static const double qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 355 1.50444444886983272379e-07, /* 0x3E84313B, 0x54F76BDB */ 356 7.32234265963079278272e-02, /* 0x3FB2BEC5, 0x3E883E34 */ 357 1.99819174093815998816e+00, /* 0x3FFFF897, 0xE727779C */ 358 1.44956029347885735348e+01, /* 0x402CFDBF, 0xAAF96FE5 */ 359 3.16662317504781540833e+01, /* 0x403FAA8E, 0x29FBDC4A */ 360 1.62527075710929267416e+01, /* 0x403040B1, 0x71814BB4 */ 361 }; 362 static const double qS2[6] = { 363 3.03655848355219184498e+01, /* 0x403E5D96, 0xF7C07AED */ 364 2.69348118608049844624e+02, /* 0x4070D591, 0xE4D14B40 */ 365 8.44783757595320139444e+02, /* 0x408A6645, 0x22B3BF22 */ 366 8.82935845112488550512e+02, /* 0x408B977C, 0x9C5CC214 */ 367 2.12666388511798828631e+02, /* 0x406A9553, 0x0E001365 */ 368 -5.31095493882666946917e+00, /* 0xC0153E6A, 0xF8B32931 */ 369 }; 370 371 /* Note: This function is only called for ix>=0x40000000 (see above) */ 372 static double qzero(double x) 373 { 374 const double *p,*q; 375 double s,r,z; 376 int32_t ix; 377 GET_HIGH_WORD(ix,x); 378 ix &= 0x7fffffff; 379 assert(ix>=0x40000000 && ix<=0x48000000); 380 if(ix>=0x40200000) {p = qR8; q= qS8;} 381 else if(ix>=0x40122E8B){p = qR5; q= qS5;} 382 else if(ix>=0x4006DB6D){p = qR3; q= qS3;} 383 else {p = qR2; q= qS2;} 384 z = one/(x*x); 385 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); 386 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5]))))); 387 return (-.125 + r/s)/x; 388 } 389