src/math/j0.c

   1 /* origin: FreeBSD /usr/src/lib/msun/src/e_j0.c */
   2 /*
   3  * ====================================================
   4  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   5  *
   6  * Developed at SunSoft, a Sun Microsystems, Inc. business.
   7  * Permission to use, copy, modify, and distribute this
   8  * software is freely granted, provided that this notice
   9  * is preserved.
  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 "libm.h"
  58
  59 static double pzero(double), qzero(double);
  60
  61 static const double
  62 invsqrtpi = 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
  63 tpi       = 6.36619772367581382433e-01; /* 0x3FE45F30, 0x6DC9C883 */
  64
  65 /* common method when |x|>=2 */
  66 static double common(uint32_t ix, double x, int y0)
  67 {
  68         double s,c,ss,cc,z;
  69
  70         /*
  71          * j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x-pi/4)-q0(x)*sin(x-pi/4))
  72          * y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x-pi/4)+q0(x)*cos(x-pi/4))
  73          *
  74          * sin(x-pi/4) = (sin(x) - cos(x))/sqrt(2)
  75          * cos(x-pi/4) = (sin(x) + cos(x))/sqrt(2)
  76          * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
  77          */
  78         s = sin(x);
  79         c = cos(x);
  80         if (y0)
  81                 c = -c;
  82         cc = s+c;
  83         /* avoid overflow in 2*x, big ulp error when x>=0x1p1023 */
  84         if (ix < 0x7fe00000) {
  85                 ss = s-c;
  86                 z = -cos(2*x);
  87                 if (s*c < 0)
  88                         cc = z/ss;
  89                 else
  90                         ss = z/cc;
  91                 if (ix < 0x48000000) {
  92                         if (y0)
  93                                 ss = -ss;
  94                         cc = pzero(x)*cc-qzero(x)*ss;
  95                 }
  96         }
  97         return invsqrtpi*cc/sqrt(x);
  98 }
  99
 100 /* R0/S0 on [0, 2.00] */
 101 static const double
 102 R02 =  1.56249999999999947958e-02, /* 0x3F8FFFFF, 0xFFFFFFFD */
 103 R03 = -1.89979294238854721751e-04, /* 0xBF28E6A5, 0xB61AC6E9 */
 104 R04 =  1.82954049532700665670e-06, /* 0x3EBEB1D1, 0x0C503919 */
 105 R05 = -4.61832688532103189199e-09, /* 0xBE33D5E7, 0x73D63FCE */
 106 S01 =  1.56191029464890010492e-02, /* 0x3F8FFCE8, 0x82C8C2A4 */
 107 S02 =  1.16926784663337450260e-04, /* 0x3F1EA6D2, 0xDD57DBF4 */
 108 S03 =  5.13546550207318111446e-07, /* 0x3EA13B54, 0xCE84D5A9 */
 109 S04 =  1.16614003333790000205e-09; /* 0x3E1408BC, 0xF4745D8F */
 110
 111 double j0(double x)
 112 {
 113         double z,r,s;
 114         uint32_t ix;
 115
 116         GET_HIGH_WORD(ix, x);
 117         ix &= 0x7fffffff;
 118
 119         /* j0(+-inf)=0, j0(nan)=nan */
 120         if (ix >= 0x7ff00000)
 121                 return 1/(x*x);
 122         x = fabs(x);
 123
 124         if (ix >= 0x40000000) {  /* |x| >= 2 */
 125                 /* large ulp error near zeros: 2.4, 5.52, 8.6537,.. */
 126                 return common(ix,x,0);
 127         }
 128
 129         /* 1 - x*x/4 + x*x*R(x^2)/S(x^2) */
 130         if (ix >= 0x3f200000) {  /* |x| >= 2**-13 */
 131                 /* up to 4ulp error close to 2 */
 132                 z = x*x;
 133                 r = z*(R02+z*(R03+z*(R04+z*R05)));
 134                 s = 1+z*(S01+z*(S02+z*(S03+z*S04)));
 135                 return (1+x/2)*(1-x/2) + z*(r/s);
 136         }
 137
 138         /* 1 - x*x/4 */
 139         /* prevent underflow */
 140         /* inexact should be raised when x!=0, this is not done correctly */
 141         if (ix >= 0x38000000)  /* |x| >= 2**-127 */
 142                 x = 0.25*x*x;
 143         return 1 - x;
 144 }
 145
 146 static const double
 147 u00  = -7.38042951086872317523e-02, /* 0xBFB2E4D6, 0x99CBD01F */
 148 u01  =  1.76666452509181115538e-01, /* 0x3FC69D01, 0x9DE9E3FC */
 149 u02  = -1.38185671945596898896e-02, /* 0xBF8C4CE8, 0xB16CFA97 */
 150 u03  =  3.47453432093683650238e-04, /* 0x3F36C54D, 0x20B29B6B */
 151 u04  = -3.81407053724364161125e-06, /* 0xBECFFEA7, 0x73D25CAD */
 152 u05  =  1.95590137035022920206e-08, /* 0x3E550057, 0x3B4EABD4 */
 153 u06  = -3.98205194132103398453e-11, /* 0xBDC5E43D, 0x693FB3C8 */
 154 v01  =  1.27304834834123699328e-02, /* 0x3F8A1270, 0x91C9C71A */
 155 v02  =  7.60068627350353253702e-05, /* 0x3F13ECBB, 0xF578C6C1 */
 156 v03  =  2.59150851840457805467e-07, /* 0x3E91642D, 0x7FF202FD */
 157 v04  =  4.41110311332675467403e-10; /* 0x3DFE5018, 0x3BD6D9EF */
 158
 159 double y0(double x)
 160 {
 161         double z,u,v;
 162         uint32_t ix,lx;
 163
 164         EXTRACT_WORDS(ix, lx, x);
 165
 166         /* y0(nan)=nan, y0(<0)=nan, y0(0)=-inf, y0(inf)=0 */
 167         if ((ix<<1 | lx) == 0)
 168                 return -1/0.0;
 169         if (ix>>31)
 170                 return 0/0.0;
 171         if (ix >= 0x7ff00000)
 172                 return 1/x;
 173
 174         if (ix >= 0x40000000) {  /* x >= 2 */
 175                 /* large ulp errors near zeros: 3.958, 7.086,.. */
 176                 return common(ix,x,1);
 177         }
 178
 179         /* U(x^2)/V(x^2) + (2/pi)*j0(x)*log(x) */
 180         if (ix >= 0x3e400000) {  /* x >= 2**-27 */
 181                 /* large ulp error near the first zero, x ~= 0.89 */
 182                 z = x*x;
 183                 u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06)))));
 184                 v = 1.0+z*(v01+z*(v02+z*(v03+z*v04)));
 185                 return u/v + tpi*(j0(x)*log(x));
 186         }
 187         return u00 + tpi*log(x);
 188 }
 189
 190 /* The asymptotic expansions of pzero is
 191  *      1 - 9/128 s^2 + 11025/98304 s^4 - ...,  where s = 1/x.
 192  * For x >= 2, We approximate pzero by
 193  *      pzero(x) = 1 + (R/S)
 194  * where  R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10
 195  *        S = 1 + pS0*s^2 + ... + pS4*s^10
 196  * and
 197  *      | pzero(x)-1-R/S | <= 2  ** ( -60.26)
 198  */
 199 static const double pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
 200   0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
 201  -7.03124999999900357484e-02, /* 0xBFB1FFFF, 0xFFFFFD32 */
 202  -8.08167041275349795626e+00, /* 0xC02029D0, 0xB44FA779 */
 203  -2.57063105679704847262e+02, /* 0xC0701102, 0x7B19E863 */
 204  -2.48521641009428822144e+03, /* 0xC0A36A6E, 0xCD4DCAFC */
 205  -5.25304380490729545272e+03, /* 0xC0B4850B, 0x36CC643D */
 206 };
 207 static const double pS8[5] = {
 208   1.16534364619668181717e+02, /* 0x405D2233, 0x07A96751 */
 209   3.83374475364121826715e+03, /* 0x40ADF37D, 0x50596938 */
 210   4.05978572648472545552e+04, /* 0x40E3D2BB, 0x6EB6B05F */
 211   1.16752972564375915681e+05, /* 0x40FC810F, 0x8F9FA9BD */
 212   4.76277284146730962675e+04, /* 0x40E74177, 0x4F2C49DC */
 213 };
 214
 215 static const double pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
 216  -1.14125464691894502584e-11, /* 0xBDA918B1, 0x47E495CC */
 217  -7.03124940873599280078e-02, /* 0xBFB1FFFF, 0xE69AFBC6 */
 218  -4.15961064470587782438e+00, /* 0xC010A370, 0xF90C6BBF */
 219  -6.76747652265167261021e+01, /* 0xC050EB2F, 0x5A7D1783 */
 220  -3.31231299649172967747e+02, /* 0xC074B3B3, 0x6742CC63 */
 221  -3.46433388365604912451e+02, /* 0xC075A6EF, 0x28A38BD7 */
 222 };
 223 static const double pS5[5] = {
 224   6.07539382692300335975e+01, /* 0x404E6081, 0x0C98C5DE */
 225   1.05125230595704579173e+03, /* 0x40906D02, 0x5C7E2864 */
 226   5.97897094333855784498e+03, /* 0x40B75AF8, 0x8FBE1D60 */
 227   9.62544514357774460223e+03, /* 0x40C2CCB8, 0xFA76FA38 */
 228   2.40605815922939109441e+03, /* 0x40A2CC1D, 0xC70BE864 */
 229 };
 230
 231 static const double pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
 232  -2.54704601771951915620e-09, /* 0xBE25E103, 0x6FE1AA86 */
 233  -7.03119616381481654654e-02, /* 0xBFB1FFF6, 0xF7C0E24B */
 234  -2.40903221549529611423e+00, /* 0xC00345B2, 0xAEA48074 */
 235  -2.19659774734883086467e+01, /* 0xC035F74A, 0x4CB94E14 */
 236  -5.80791704701737572236e+01, /* 0xC04D0A22, 0x420A1A45 */
 237  -3.14479470594888503854e+01, /* 0xC03F72AC, 0xA892D80F */
 238 };
 239 static const double pS3[5] = {
 240   3.58560338055209726349e+01, /* 0x4041ED92, 0x84077DD3 */
 241   3.61513983050303863820e+02, /* 0x40769839, 0x464A7C0E */
 242   1.19360783792111533330e+03, /* 0x4092A66E, 0x6D1061D6 */
 243   1.12799679856907414432e+03, /* 0x40919FFC, 0xB8C39B7E */
 244   1.73580930813335754692e+02, /* 0x4065B296, 0xFC379081 */
 245 };
 246
 247 static const double pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
 248  -8.87534333032526411254e-08, /* 0xBE77D316, 0xE927026D */
 249  -7.03030995483624743247e-02, /* 0xBFB1FF62, 0x495E1E42 */
 250  -1.45073846780952986357e+00, /* 0xBFF73639, 0x8A24A843 */
 251  -7.63569613823527770791e+00, /* 0xC01E8AF3, 0xEDAFA7F3 */
 252  -1.11931668860356747786e+01, /* 0xC02662E6, 0xC5246303 */
 253  -3.23364579351335335033e+00, /* 0xC009DE81, 0xAF8FE70F */
 254 };
 255 static const double pS2[5] = {
 256   2.22202997532088808441e+01, /* 0x40363865, 0x908B5959 */
 257   1.36206794218215208048e+02, /* 0x4061069E, 0x0EE8878F */
 258   2.70470278658083486789e+02, /* 0x4070E786, 0x42EA079B */
 259   1.53875394208320329881e+02, /* 0x40633C03, 0x3AB6FAFF */
 260   1.46576176948256193810e+01, /* 0x402D50B3, 0x44391809 */
 261 };
 262
 263 static double pzero(double x)
 264 {
 265         const double *p,*q;
 266         double_t z,r,s;
 267         uint32_t ix;
 268
 269         GET_HIGH_WORD(ix, x);
 270         ix &= 0x7fffffff;
 271         if      (ix >= 0x40200000){p = pR8; q = pS8;}
 272         else if (ix >= 0x40122E8B){p = pR5; q = pS5;}
 273         else if (ix >= 0x4006DB6D){p = pR3; q = pS3;}
 274         else /*ix >= 0x40000000*/ {p = pR2; q = pS2;}
 275         z = 1.0/(x*x);
 276         r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
 277         s = 1.0+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
 278         return 1.0 + r/s;
 279 }
 280
 281
 282 /* For x >= 8, the asymptotic expansions of qzero is
 283  *      -1/8 s + 75/1024 s^3 - ..., where s = 1/x.
 284  * We approximate pzero by
 285  *      qzero(x) = s*(-1.25 + (R/S))
 286  * where  R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10
 287  *        S = 1 + qS0*s^2 + ... + qS5*s^12
 288  * and
 289  *      | qzero(x)/s +1.25-R/S | <= 2  ** ( -61.22)
 290  */
 291 static const double qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
 292   0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
 293   7.32421874999935051953e-02, /* 0x3FB2BFFF, 0xFFFFFE2C */
 294   1.17682064682252693899e+01, /* 0x40278952, 0x5BB334D6 */
 295   5.57673380256401856059e+02, /* 0x40816D63, 0x15301825 */
 296   8.85919720756468632317e+03, /* 0x40C14D99, 0x3E18F46D */
 297   3.70146267776887834771e+04, /* 0x40E212D4, 0x0E901566 */
 298 };
 299 static const double qS8[6] = {
 300   1.63776026895689824414e+02, /* 0x406478D5, 0x365B39BC */
 301   8.09834494656449805916e+03, /* 0x40BFA258, 0x4E6B0563 */
 302   1.42538291419120476348e+05, /* 0x41016652, 0x54D38C3F */
 303   8.03309257119514397345e+05, /* 0x412883DA, 0x83A52B43 */
 304   8.40501579819060512818e+05, /* 0x4129A66B, 0x28DE0B3D */
 305  -3.43899293537866615225e+05, /* 0xC114FD6D, 0x2C9530C5 */
 306 };
 307
 308 static const double qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
 309   1.84085963594515531381e-11, /* 0x3DB43D8F, 0x29CC8CD9 */
 310   7.32421766612684765896e-02, /* 0x3FB2BFFF, 0xD172B04C */
 311   5.83563508962056953777e+00, /* 0x401757B0, 0xB9953DD3 */
 312   1.35111577286449829671e+02, /* 0x4060E392, 0x0A8788E9 */
 313   1.02724376596164097464e+03, /* 0x40900CF9, 0x9DC8C481 */
 314   1.98997785864605384631e+03, /* 0x409F17E9, 0x53C6E3A6 */
 315 };
 316 static const double qS5[6] = {
 317   8.27766102236537761883e+01, /* 0x4054B1B3, 0xFB5E1543 */
 318   2.07781416421392987104e+03, /* 0x40A03BA0, 0xDA21C0CE */
 319   1.88472887785718085070e+04, /* 0x40D267D2, 0x7B591E6D */
 320   5.67511122894947329769e+04, /* 0x40EBB5E3, 0x97E02372 */
 321   3.59767538425114471465e+04, /* 0x40E19118, 0x1F7A54A0 */
 322  -5.35434275601944773371e+03, /* 0xC0B4EA57, 0xBEDBC609 */
 323 };
 324
 325 static const double qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
 326   4.37741014089738620906e-09, /* 0x3E32CD03, 0x6ADECB82 */
 327   7.32411180042911447163e-02, /* 0x3FB2BFEE, 0x0E8D0842 */
 328   3.34423137516170720929e+00, /* 0x400AC0FC, 0x61149CF5 */
 329   4.26218440745412650017e+01, /* 0x40454F98, 0x962DAEDD */
 330   1.70808091340565596283e+02, /* 0x406559DB, 0xE25EFD1F */
 331   1.66733948696651168575e+02, /* 0x4064D77C, 0x81FA21E0 */
 332 };
 333 static const double qS3[6] = {
 334   4.87588729724587182091e+01, /* 0x40486122, 0xBFE343A6 */
 335   7.09689221056606015736e+02, /* 0x40862D83, 0x86544EB3 */
 336   3.70414822620111362994e+03, /* 0x40ACF04B, 0xE44DFC63 */
 337   6.46042516752568917582e+03, /* 0x40B93C6C, 0xD7C76A28 */
 338   2.51633368920368957333e+03, /* 0x40A3A8AA, 0xD94FB1C0 */
 339  -1.49247451836156386662e+02, /* 0xC062A7EB, 0x201CF40F */
 340 };
 341
 342 static const double qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
 343   1.50444444886983272379e-07, /* 0x3E84313B, 0x54F76BDB */
 344   7.32234265963079278272e-02, /* 0x3FB2BEC5, 0x3E883E34 */
 345   1.99819174093815998816e+00, /* 0x3FFFF897, 0xE727779C */
 346   1.44956029347885735348e+01, /* 0x402CFDBF, 0xAAF96FE5 */
 347   3.16662317504781540833e+01, /* 0x403FAA8E, 0x29FBDC4A */
 348   1.62527075710929267416e+01, /* 0x403040B1, 0x71814BB4 */
 349 };
 350 static const double qS2[6] = {
 351   3.03655848355219184498e+01, /* 0x403E5D96, 0xF7C07AED */
 352   2.69348118608049844624e+02, /* 0x4070D591, 0xE4D14B40 */
 353   8.44783757595320139444e+02, /* 0x408A6645, 0x22B3BF22 */
 354   8.82935845112488550512e+02, /* 0x408B977C, 0x9C5CC214 */
 355   2.12666388511798828631e+02, /* 0x406A9553, 0x0E001365 */
 356  -5.31095493882666946917e+00, /* 0xC0153E6A, 0xF8B32931 */
 357 };
 358
 359 static double qzero(double x)
 360 {
 361         const double *p,*q;
 362         double_t s,r,z;
 363         uint32_t ix;
 364
 365         GET_HIGH_WORD(ix, x);
 366         ix &= 0x7fffffff;
 367         if      (ix >= 0x40200000){p = qR8; q = qS8;}
 368         else if (ix >= 0x40122E8B){p = qR5; q = qS5;}
 369         else if (ix >= 0x4006DB6D){p = qR3; q = qS3;}
 370         else /*ix >= 0x40000000*/ {p = qR2; q = qS2;}
 371         z = 1.0/(x*x);
 372         r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
 373         s = 1.0+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
 374         return (-.125 + r/s)/x;
 375 }