compiler/stdc/math/e_j0.c

   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 #ifndef lint
  15 static char rcsid[] = "$FreeBSD: src/lib/msun/src/e_j0.c,v 1.9 2008/02/22 02:30:35 das Exp $";
  16 #endif
  17
  18 #include <assert.h>
  19
  20 /* __ieee754_j0(x), __ieee754_y0(x)
  21  * Bessel function of the first and second kinds of order zero.
  22  * Method -- j0(x):
  23  *      1. For tiny x, we use j0(x) = 1 - x^2/4 + x^4/64 - ...
  24  *      2. Reduce x to |x| since j0(x)=j0(-x),  and
  25  *         for x in (0,2)
  26  *              j0(x) = 1-z/4+ z^2*R0/S0,  where z = x*x;
  27  *         (precision:  |j0-1+z/4-z^2R0/S0 |<2**-63.67 )
  28  *         for x in (2,inf)
  29  *              j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)-q0(x)*sin(x0))
  30  *         where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
  31  *         as follow:
  32  *              cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
  33  *                      = 1/sqrt(2) * (cos(x) + sin(x))
  34  *              sin(x0) = sin(x)cos(pi/4)-cos(x)sin(pi/4)
  35  *                      = 1/sqrt(2) * (sin(x) - cos(x))
  36  *         (To avoid cancellation, use
  37  *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
  38  *          to compute the worse one.)
  39  *
  40  *      3 Special cases
  41  *              j0(nan)= nan
  42  *              j0(0) = 1
  43  *              j0(inf) = 0
  44  *
  45  * Method -- y0(x):
  46  *      1. For x<2.
  47  *         Since
  48  *              y0(x) = 2/pi*(j0(x)*(ln(x/2)+Euler) + x^2/4 - ...)
  49  *         therefore y0(x)-2/pi*j0(x)*ln(x) is an even function.
  50  *         We use the following function to approximate y0,
  51  *              y0(x) = U(z)/V(z) + (2/pi)*(j0(x)*ln(x)), z= x^2
  52  *         where
  53  *              U(z) = u00 + u01*z + ... + u06*z^6
  54  *              V(z) = 1  + v01*z + ... + v04*z^4
  55  *         with absolute approximation error bounded by 2**-72.
  56  *         Note: For tiny x, U/V = u0 and j0(x)~1, hence
  57  *              y0(tiny) = u0 + (2/pi)*ln(tiny), (choose tiny<2**-27)
  58  *      2. For x>=2.
  59  *              y0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)+q0(x)*sin(x0))
  60  *         where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
  61  *         by the method mentioned above.
  62  *      3. Special cases: y0(0)=-inf, y0(x<0)=NaN, y0(inf)=0.
  63  */
  64
  65 #include "math.h"
  66 #include "math_private.h"
  67
  68 static __inline double pzero(double), qzero(double);
  69
  70 static const volatile double vone __attribute__ ((__section__(".rodata,\"a\" " SECTIONCOMMENT))) = 1.0, vzero __attribute__ ((__section__(".rodata,\"a\" " SECTIONCOMMENT))) = 0.0;
  71
  72 static const double
  73 huge    = 1e300,
  74 one     = 1.0,
  75 invsqrtpi=  5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
  76 tpi      =  6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
  77                 /* R0/S0 on [0, 2.00] */
  78 R02  =  1.56249999999999947958e-02, /* 0x3F8FFFFF, 0xFFFFFFFD */
  79 R03  = -1.89979294238854721751e-04, /* 0xBF28E6A5, 0xB61AC6E9 */
  80 R04  =  1.82954049532700665670e-06, /* 0x3EBEB1D1, 0x0C503919 */
  81 R05  = -4.61832688532103189199e-09, /* 0xBE33D5E7, 0x73D63FCE */
  82 S01  =  1.56191029464890010492e-02, /* 0x3F8FFCE8, 0x82C8C2A4 */
  83 S02  =  1.16926784663337450260e-04, /* 0x3F1EA6D2, 0xDD57DBF4 */
  84 S03  =  5.13546550207318111446e-07, /* 0x3EA13B54, 0xCE84D5A9 */
  85 S04  =  1.16614003333790000205e-09; /* 0x3E1408BC, 0xF4745D8F */
  86
  87 static const double zero = 0.0;
  88
  89 double
  90 __ieee754_j0(double x)
  91 {
  92         double z, s,c,ss,cc,r,u,v;
  93         int32_t hx,ix;
  94
  95         GET_HIGH_WORD(hx,x);
  96         ix = hx&0x7fffffff;
  97         if(ix>=0x7ff00000) return one/(x*x);
  98         x = fabs(x);
  99         if(ix >= 0x40000000) {  /* |x| >= 2.0 */
 100                 s = sin(x);
 101                 c = cos(x);
 102                 ss = s-c;
 103                 cc = s+c;
 104                 if(ix<0x7fe00000) {  /* make sure x+x not overflow */
 105                     z = -cos(x+x);
 106                     if ((s*c)<zero) cc = z/ss;
 107                     else            ss = z/cc;
 108                 }
 109         /*
 110          * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
 111          * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
 112          */
 113                 if(ix>0x48000000) z = (invsqrtpi*cc)/sqrt(x);
 114                 else {
 115                     u = pzero(x); v = qzero(x);
 116                     z = invsqrtpi*(u*cc-v*ss)/sqrt(x);
 117                 }
 118                 return z;
 119         }
 120         if(ix<0x3f200000) {     /* |x| < 2**-13 */
 121             if(huge+x>one) {    /* raise inexact if x != 0 */
 122                 if(ix<0x3e400000) return one;   /* |x|<2**-27 */
 123                 else          return one - x*x/4;
 124             }
 125         }
 126         z = x*x;
 127         r =  z*(R02+z*(R03+z*(R04+z*R05)));
 128         s =  one+z*(S01+z*(S02+z*(S03+z*S04)));
 129         if(ix < 0x3FF00000) {   /* |x| < 1.00 */
 130             return one + z*(-0.25+(r/s));
 131         } else {
 132             u = 0.5*x;
 133             return((one+u)*(one-u)+z*(r/s));
 134         }
 135 }
 136
 137 static const double
 138 u00  = -7.38042951086872317523e-02, /* 0xBFB2E4D6, 0x99CBD01F */
 139 u01  =  1.76666452509181115538e-01, /* 0x3FC69D01, 0x9DE9E3FC */
 140 u02  = -1.38185671945596898896e-02, /* 0xBF8C4CE8, 0xB16CFA97 */
 141 u03  =  3.47453432093683650238e-04, /* 0x3F36C54D, 0x20B29B6B */
 142 u04  = -3.81407053724364161125e-06, /* 0xBECFFEA7, 0x73D25CAD */
 143 u05  =  1.95590137035022920206e-08, /* 0x3E550057, 0x3B4EABD4 */
 144 u06  = -3.98205194132103398453e-11, /* 0xBDC5E43D, 0x693FB3C8 */
 145 v01  =  1.27304834834123699328e-02, /* 0x3F8A1270, 0x91C9C71A */
 146 v02  =  7.60068627350353253702e-05, /* 0x3F13ECBB, 0xF578C6C1 */
 147 v03  =  2.59150851840457805467e-07, /* 0x3E91642D, 0x7FF202FD */
 148 v04  =  4.41110311332675467403e-10; /* 0x3DFE5018, 0x3BD6D9EF */
 149
 150 double
 151 __ieee754_y0(double x)
 152 {
 153         double z, s,c,ss,cc,u,v;
 154         int32_t hx,ix,lx;
 155
 156         EXTRACT_WORDS(hx,lx,x);
 157         ix = 0x7fffffff&hx;
 158         /*
 159          * y0(NaN) = NaN.
 160          * y0(Inf) = 0.
 161          * y0(-Inf) = NaN and raise invalid exception.
 162          */
 163         if(ix>=0x7ff00000) return vone/(x+x*x);
 164         /* y0(+-0) = -inf and raise divide-by-zero exception. */
 165         if((ix|lx)==0) return -one/vzero;
 166         /* y0(x<0) = NaN and raise invalid exception. */
 167         if(hx<0) return vzero/vzero;
 168         if(ix >= 0x40000000) {  /* |x| >= 2.0 */
 169         /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
 170          * where x0 = x-pi/4
 171          *      Better formula:
 172          *              cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
 173          *                      =  1/sqrt(2) * (sin(x) + cos(x))
 174          *              sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
 175          *                      =  1/sqrt(2) * (sin(x) - cos(x))
 176          * To avoid cancellation, use
 177          *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
 178          * to compute the worse one.
 179          */
 180                 s = sin(x);
 181                 c = cos(x);
 182                 ss = s-c;
 183                 cc = s+c;
 184         /*
 185          * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
 186          * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
 187          */
 188                 if(ix<0x7fe00000) {  /* make sure x+x not overflow */
 189                     z = -cos(x+x);
 190                     if ((s*c)<zero) cc = z/ss;
 191                     else            ss = z/cc;
 192                 }
 193                 if(ix>0x48000000) z = (invsqrtpi*ss)/sqrt(x);
 194                 else {
 195                     u = pzero(x); v = qzero(x);
 196                     z = invsqrtpi*(u*ss+v*cc)/sqrt(x);
 197                 }
 198                 return z;
 199         }
 200         if(ix<=0x3e400000) {    /* x < 2**-27 */
 201             return(u00 + tpi*__ieee754_log(x));
 202         }
 203         z = x*x;
 204         u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06)))));
 205         v = one+z*(v01+z*(v02+z*(v03+z*v04)));
 206         return(u/v + tpi*(__ieee754_j0(x)*__ieee754_log(x)));
 207 }
 208
 209 /* The asymptotic expansions of pzero is
 210  *      1 - 9/128 s^2 + 11025/98304 s^4 - ...,  where s = 1/x.
 211  * For x >= 2, We approximate pzero by
 212  *      pzero(x) = 1 + (R/S)
 213  * where  R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10
 214  *        S = 1 + pS0*s^2 + ... + pS4*s^10
 215  * and
 216  *      | pzero(x)-1-R/S | <= 2  ** ( -60.26)
 217  */
 218 static const double pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
 219   0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
 220  -7.03124999999900357484e-02, /* 0xBFB1FFFF, 0xFFFFFD32 */
 221  -8.08167041275349795626e+00, /* 0xC02029D0, 0xB44FA779 */
 222  -2.57063105679704847262e+02, /* 0xC0701102, 0x7B19E863 */
 223  -2.48521641009428822144e+03, /* 0xC0A36A6E, 0xCD4DCAFC */
 224  -5.25304380490729545272e+03, /* 0xC0B4850B, 0x36CC643D */
 225 };
 226 static const double pS8[5] = {
 227   1.16534364619668181717e+02, /* 0x405D2233, 0x07A96751 */
 228   3.83374475364121826715e+03, /* 0x40ADF37D, 0x50596938 */
 229   4.05978572648472545552e+04, /* 0x40E3D2BB, 0x6EB6B05F */
 230   1.16752972564375915681e+05, /* 0x40FC810F, 0x8F9FA9BD */
 231   4.76277284146730962675e+04, /* 0x40E74177, 0x4F2C49DC */
 232 };
 233
 234 static const double pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
 235  -1.14125464691894502584e-11, /* 0xBDA918B1, 0x47E495CC */
 236  -7.03124940873599280078e-02, /* 0xBFB1FFFF, 0xE69AFBC6 */
 237  -4.15961064470587782438e+00, /* 0xC010A370, 0xF90C6BBF */
 238  -6.76747652265167261021e+01, /* 0xC050EB2F, 0x5A7D1783 */
 239  -3.31231299649172967747e+02, /* 0xC074B3B3, 0x6742CC63 */
 240  -3.46433388365604912451e+02, /* 0xC075A6EF, 0x28A38BD7 */
 241 };
 242 static const double pS5[5] = {
 243   6.07539382692300335975e+01, /* 0x404E6081, 0x0C98C5DE */
 244   1.05125230595704579173e+03, /* 0x40906D02, 0x5C7E2864 */
 245   5.97897094333855784498e+03, /* 0x40B75AF8, 0x8FBE1D60 */
 246   9.62544514357774460223e+03, /* 0x40C2CCB8, 0xFA76FA38 */
 247   2.40605815922939109441e+03, /* 0x40A2CC1D, 0xC70BE864 */
 248 };
 249
 250 static const double pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
 251  -2.54704601771951915620e-09, /* 0xBE25E103, 0x6FE1AA86 */
 252  -7.03119616381481654654e-02, /* 0xBFB1FFF6, 0xF7C0E24B */
 253  -2.40903221549529611423e+00, /* 0xC00345B2, 0xAEA48074 */
 254  -2.19659774734883086467e+01, /* 0xC035F74A, 0x4CB94E14 */
 255  -5.80791704701737572236e+01, /* 0xC04D0A22, 0x420A1A45 */
 256  -3.14479470594888503854e+01, /* 0xC03F72AC, 0xA892D80F */
 257 };
 258 static const double pS3[5] = {
 259   3.58560338055209726349e+01, /* 0x4041ED92, 0x84077DD3 */
 260   3.61513983050303863820e+02, /* 0x40769839, 0x464A7C0E */
 261   1.19360783792111533330e+03, /* 0x4092A66E, 0x6D1061D6 */
 262   1.12799679856907414432e+03, /* 0x40919FFC, 0xB8C39B7E */
 263   1.73580930813335754692e+02, /* 0x4065B296, 0xFC379081 */
 264 };
 265
 266 static const double pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
 267  -8.87534333032526411254e-08, /* 0xBE77D316, 0xE927026D */
 268  -7.03030995483624743247e-02, /* 0xBFB1FF62, 0x495E1E42 */
 269  -1.45073846780952986357e+00, /* 0xBFF73639, 0x8A24A843 */
 270  -7.63569613823527770791e+00, /* 0xC01E8AF3, 0xEDAFA7F3 */
 271  -1.11931668860356747786e+01, /* 0xC02662E6, 0xC5246303 */
 272  -3.23364579351335335033e+00, /* 0xC009DE81, 0xAF8FE70F */
 273 };
 274 static const double pS2[5] = {
 275   2.22202997532088808441e+01, /* 0x40363865, 0x908B5959 */
 276   1.36206794218215208048e+02, /* 0x4061069E, 0x0EE8878F */
 277   2.70470278658083486789e+02, /* 0x4070E786, 0x42EA079B */
 278   1.53875394208320329881e+02, /* 0x40633C03, 0x3AB6FAFF */
 279   1.46576176948256193810e+01, /* 0x402D50B3, 0x44391809 */
 280 };
 281
 282         static double pzero(double x)
 283 {
 284         const double *p,*q;
 285         double z,r,s;
 286         int32_t ix;
 287         GET_HIGH_WORD(ix,x);
 288         ix &= 0x7fffffff;
 289         if(ix>=0x40200000)     {p = pR8; q= pS8;}
 290         else if(ix>=0x40122E8B){p = pR5; q= pS5;}
 291         else if(ix>=0x4006DB6D){p = pR3; q= pS3;}
 292         else                   {p = pR2; q= pS2;}       /* ix>=0x40000000 */
 293         z = one/(x*x);
 294         r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
 295         s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
 296         return one+ r/s;
 297 }
 298
 299
 300 /* For x >= 8, the asymptotic expansions of qzero is
 301  *      -1/8 s + 75/1024 s^3 - ..., where s = 1/x.
 302  * We approximate pzero by
 303  *      qzero(x) = s*(-1.25 + (R/S))
 304  * where  R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10
 305  *        S = 1 + qS0*s^2 + ... + qS5*s^12
 306  * and
 307  *      | qzero(x)/s +1.25-R/S | <= 2  ** ( -61.22)
 308  */
 309 static const double qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
 310   0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
 311   7.32421874999935051953e-02, /* 0x3FB2BFFF, 0xFFFFFE2C */
 312   1.17682064682252693899e+01, /* 0x40278952, 0x5BB334D6 */
 313   5.57673380256401856059e+02, /* 0x40816D63, 0x15301825 */
 314   8.85919720756468632317e+03, /* 0x40C14D99, 0x3E18F46D */
 315   3.70146267776887834771e+04, /* 0x40E212D4, 0x0E901566 */
 316 };
 317 static const double qS8[6] = {
 318   1.63776026895689824414e+02, /* 0x406478D5, 0x365B39BC */
 319   8.09834494656449805916e+03, /* 0x40BFA258, 0x4E6B0563 */
 320   1.42538291419120476348e+05, /* 0x41016652, 0x54D38C3F */
 321   8.03309257119514397345e+05, /* 0x412883DA, 0x83A52B43 */
 322   8.40501579819060512818e+05, /* 0x4129A66B, 0x28DE0B3D */
 323  -3.43899293537866615225e+05, /* 0xC114FD6D, 0x2C9530C5 */
 324 };
 325
 326 static const double qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
 327   1.84085963594515531381e-11, /* 0x3DB43D8F, 0x29CC8CD9 */
 328   7.32421766612684765896e-02, /* 0x3FB2BFFF, 0xD172B04C */
 329   5.83563508962056953777e+00, /* 0x401757B0, 0xB9953DD3 */
 330   1.35111577286449829671e+02, /* 0x4060E392, 0x0A8788E9 */
 331   1.02724376596164097464e+03, /* 0x40900CF9, 0x9DC8C481 */
 332   1.98997785864605384631e+03, /* 0x409F17E9, 0x53C6E3A6 */
 333 };
 334 static const double qS5[6] = {
 335   8.27766102236537761883e+01, /* 0x4054B1B3, 0xFB5E1543 */
 336   2.07781416421392987104e+03, /* 0x40A03BA0, 0xDA21C0CE */
 337   1.88472887785718085070e+04, /* 0x40D267D2, 0x7B591E6D */
 338   5.67511122894947329769e+04, /* 0x40EBB5E3, 0x97E02372 */
 339   3.59767538425114471465e+04, /* 0x40E19118, 0x1F7A54A0 */
 340  -5.35434275601944773371e+03, /* 0xC0B4EA57, 0xBEDBC609 */
 341 };
 342
 343 static const double qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
 344   4.37741014089738620906e-09, /* 0x3E32CD03, 0x6ADECB82 */
 345   7.32411180042911447163e-02, /* 0x3FB2BFEE, 0x0E8D0842 */
 346   3.34423137516170720929e+00, /* 0x400AC0FC, 0x61149CF5 */
 347   4.26218440745412650017e+01, /* 0x40454F98, 0x962DAEDD */
 348   1.70808091340565596283e+02, /* 0x406559DB, 0xE25EFD1F */
 349   1.66733948696651168575e+02, /* 0x4064D77C, 0x81FA21E0 */
 350 };
 351 static const double qS3[6] = {
 352   4.87588729724587182091e+01, /* 0x40486122, 0xBFE343A6 */
 353   7.09689221056606015736e+02, /* 0x40862D83, 0x86544EB3 */
 354   3.70414822620111362994e+03, /* 0x40ACF04B, 0xE44DFC63 */
 355   6.46042516752568917582e+03, /* 0x40B93C6C, 0xD7C76A28 */
 356   2.51633368920368957333e+03, /* 0x40A3A8AA, 0xD94FB1C0 */
 357  -1.49247451836156386662e+02, /* 0xC062A7EB, 0x201CF40F */
 358 };
 359
 360 static const double qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
 361   1.50444444886983272379e-07, /* 0x3E84313B, 0x54F76BDB */
 362   7.32234265963079278272e-02, /* 0x3FB2BEC5, 0x3E883E34 */
 363   1.99819174093815998816e+00, /* 0x3FFFF897, 0xE727779C */
 364   1.44956029347885735348e+01, /* 0x402CFDBF, 0xAAF96FE5 */
 365   3.16662317504781540833e+01, /* 0x403FAA8E, 0x29FBDC4A */
 366   1.62527075710929267416e+01, /* 0x403040B1, 0x71814BB4 */
 367 };
 368 static const double qS2[6] = {
 369   3.03655848355219184498e+01, /* 0x403E5D96, 0xF7C07AED */
 370   2.69348118608049844624e+02, /* 0x4070D591, 0xE4D14B40 */
 371   8.44783757595320139444e+02, /* 0x408A6645, 0x22B3BF22 */
 372   8.82935845112488550512e+02, /* 0x408B977C, 0x9C5CC214 */
 373   2.12666388511798828631e+02, /* 0x406A9553, 0x0E001365 */
 374  -5.31095493882666946917e+00, /* 0xC0153E6A, 0xF8B32931 */
 375 };
 376
 377         static double qzero(double x)
 378 {
 379         const double *p,*q;
 380         double s,r,z;
 381         int32_t ix;
 382         GET_HIGH_WORD(ix,x);
 383         ix &= 0x7fffffff;
 384         if(ix>=0x40200000)     {p = qR8; q= qS8;}
 385         else if(ix>=0x40122E8B){p = qR5; q= qS5;}
 386         else if(ix>=0x4006DB6D){p = qR3; q= qS3;}
 387         else                   {p = qR2; q= qS2;}       /* ix>=0x40000000 */
 388         z = one/(x*x);
 389         r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
 390         s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
 391         return (-.125 + r/s)/x;
 392 }