lib/libm/src/e_j1.c

   1 /* @(#)e_j1.c 5.1 93/09/24 */
   2 /*
   3  * ====================================================
   4  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   5  *
   6  * Developed at SunPro, 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  * $NetBSD: e_j1.c,v 1.11 2002/05/26 22:01:50 wiz Exp $
  13  * $DragonFly: src/lib/libm/src/e_j1.c,v 1.1 2005/07/26 21:15:20 joerg Exp $
  14  */
  15
  16 /* j1(x), y1(x)
  17  * Bessel function of the first and second kinds of order zero.
  18  * Method -- j1(x):
  19  *      1. For tiny x, we use j1(x) = x/2 - x^3/16 + x^5/384 - ...
  20  *      2. Reduce x to |x| since j1(x)=-j1(-x),  and
  21  *         for x in (0,2)
  22  *              j1(x) = x/2 + x*z*R0/S0,  where z = x*x;
  23  *         (precision:  |j1/x - 1/2 - R0/S0 |<2**-61.51 )
  24  *         for x in (2,inf)
  25  *              j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x1)-q1(x)*sin(x1))
  26  *              y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1))
  27  *         where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1)
  28  *         as follow:
  29  *              cos(x1) =  cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
  30  *                      =  1/sqrt(2) * (sin(x) - cos(x))
  31  *              sin(x1) =  sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
  32  *                      = -1/sqrt(2) * (sin(x) + cos(x))
  33  *         (To avoid cancellation, use
  34  *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
  35  *          to compute the worse one.)
  36  *
  37  *      3 Special cases
  38  *              j1(nan)= nan
  39  *              j1(0) = 0
  40  *              j1(inf) = 0
  41  *
  42  * Method -- y1(x):
  43  *      1. screen out x<=0 cases: y1(0)=-inf, y1(x<0)=NaN
  44  *      2. For x<2.
  45  *         Since
  46  *              y1(x) = 2/pi*(j1(x)*(ln(x/2)+Euler)-1/x-x/2+5/64*x^3-...)
  47  *         therefore y1(x)-2/pi*j1(x)*ln(x)-1/x is an odd function.
  48  *         We use the following function to approximate y1,
  49  *              y1(x) = x*U(z)/V(z) + (2/pi)*(j1(x)*ln(x)-1/x), z= x^2
  50  *         where for x in [0,2] (abs err less than 2**-65.89)
  51  *              U(z) = U0[0] + U0[1]*z + ... + U0[4]*z^4
  52  *              V(z) = 1  + v0[0]*z + ... + v0[4]*z^5
  53  *         Note: For tiny x, 1/x dominate y1 and hence
  54  *              y1(tiny) = -2/pi/tiny, (choose tiny<2**-54)
  55  *      3. For x>=2.
  56  *              y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1))
  57  *         where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1)
  58  *         by method mentioned above.
  59  */
  60
  61 #include <math.h>
  62 #include "math_private.h"
  63
  64 static double pone(double), qone(double);
  65
  66 static const double
  67 huge    = 1e300,
  68 one     = 1.0,
  69 invsqrtpi=  5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
  70 tpi      =  6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
  71         /* R0/S0 on [0,2] */
  72 r00  = -6.25000000000000000000e-02, /* 0xBFB00000, 0x00000000 */
  73 r01  =  1.40705666955189706048e-03, /* 0x3F570D9F, 0x98472C61 */
  74 r02  = -1.59955631084035597520e-05, /* 0xBEF0C5C6, 0xBA169668 */
  75 r03  =  4.96727999609584448412e-08, /* 0x3E6AAAFA, 0x46CA0BD9 */
  76 s01  =  1.91537599538363460805e-02, /* 0x3F939D0B, 0x12637E53 */
  77 s02  =  1.85946785588630915560e-04, /* 0x3F285F56, 0xB9CDF664 */
  78 s03  =  1.17718464042623683263e-06, /* 0x3EB3BFF8, 0x333F8498 */
  79 s04  =  5.04636257076217042715e-09, /* 0x3E35AC88, 0xC97DFF2C */
  80 s05  =  1.23542274426137913908e-11; /* 0x3DAB2ACF, 0xCFB97ED8 */
  81
  82 static const double zero    = 0.0;
  83
  84 double
  85 j1(double x)
  86 {
  87         double z, s,c,ss,cc,r,u,v,y;
  88         int32_t hx,ix;
  89
  90         GET_HIGH_WORD(hx,x);
  91         ix = hx&0x7fffffff;
  92         if(ix>=0x7ff00000) return one/x;
  93         y = fabs(x);
  94         if(ix >= 0x40000000) {  /* |x| >= 2.0 */
  95                 s = sin(y);
  96                 c = cos(y);
  97                 ss = -s-c;
  98                 cc = s-c;
  99                 if(ix<0x7fe00000) {  /* make sure y+y not overflow */
 100                     z = cos(y+y);
 101                     if ((s*c)>zero) cc = z/ss;
 102                     else            ss = z/cc;
 103                 }
 104         /*
 105          * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x)
 106          * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x)
 107          */
 108                 if(ix>0x48000000) z = (invsqrtpi*cc)/sqrt(y);
 109                 else {
 110                     u = pone(y); v = qone(y);
 111                     z = invsqrtpi*(u*cc-v*ss)/sqrt(y);
 112                 }
 113                 if(hx<0) return -z;
 114                 else     return  z;
 115         }
 116         if(ix<0x3e400000) {     /* |x|<2**-27 */
 117             if(huge+x>one) return 0.5*x;/* inexact if x!=0 necessary */
 118         }
 119         z = x*x;
 120         r =  z*(r00+z*(r01+z*(r02+z*r03)));
 121         s =  one+z*(s01+z*(s02+z*(s03+z*(s04+z*s05))));
 122         r *= x;
 123         return(x*0.5+r/s);
 124 }
 125
 126 static const double U0[5] = {
 127  -1.96057090646238940668e-01, /* 0xBFC91866, 0x143CBC8A */
 128   5.04438716639811282616e-02, /* 0x3FA9D3C7, 0x76292CD1 */
 129  -1.91256895875763547298e-03, /* 0xBF5F55E5, 0x4844F50F */
 130   2.35252600561610495928e-05, /* 0x3EF8AB03, 0x8FA6B88E */
 131  -9.19099158039878874504e-08, /* 0xBE78AC00, 0x569105B8 */
 132 };
 133 static const double V0[5] = {
 134   1.99167318236649903973e-02, /* 0x3F94650D, 0x3F4DA9F0 */
 135   2.02552581025135171496e-04, /* 0x3F2A8C89, 0x6C257764 */
 136   1.35608801097516229404e-06, /* 0x3EB6C05A, 0x894E8CA6 */
 137   6.22741452364621501295e-09, /* 0x3E3ABF1D, 0x5BA69A86 */
 138   1.66559246207992079114e-11, /* 0x3DB25039, 0xDACA772A */
 139 };
 140
 141 double
 142 y1(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     /* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */
 150         if(ix>=0x7ff00000) return  one/(x+x*x);
 151         if((ix|lx)==0) return -one/zero;
 152         if(hx<0) return zero/zero;
 153         if(ix >= 0x40000000) {  /* |x| >= 2.0 */
 154                 s = sin(x);
 155                 c = cos(x);
 156                 ss = -s-c;
 157                 cc = s-c;
 158                 if(ix<0x7fe00000) {  /* make sure x+x not overflow */
 159                     z = cos(x+x);
 160                     if ((s*c)>zero) cc = z/ss;
 161                     else            ss = z/cc;
 162                 }
 163         /* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0))
 164          * where x0 = x-3pi/4
 165          *      Better formula:
 166          *              cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
 167          *                      =  1/sqrt(2) * (sin(x) - cos(x))
 168          *              sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
 169          *                      = -1/sqrt(2) * (cos(x) + sin(x))
 170          * To avoid cancellation, use
 171          *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
 172          * to compute the worse one.
 173          */
 174                 if(ix>0x48000000) z = (invsqrtpi*ss)/sqrt(x);
 175                 else {
 176                     u = pone(x); v = qone(x);
 177                     z = invsqrtpi*(u*ss+v*cc)/sqrt(x);
 178                 }
 179                 return z;
 180         }
 181         if(ix<=0x3c900000) {    /* x < 2**-54 */
 182             return(-tpi/x);
 183         }
 184         z = x*x;
 185         u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4])));
 186         v = one+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4]))));
 187         return(x*(u/v) + tpi*(j1(x)*log(x)-one/x));
 188 }
 189
 190 /* For x >= 8, the asymptotic expansions of pone is
 191  *      1 + 15/128 s^2 - 4725/2^15 s^4 - ...,   where s = 1/x.
 192  * We approximate pone by
 193  *      pone(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  *      | pone(x)-1-R/S | <= 2  ** ( -60.06)
 198  */
 199
 200 static const double pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
 201   0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
 202   1.17187499999988647970e-01, /* 0x3FBDFFFF, 0xFFFFFCCE */
 203   1.32394806593073575129e+01, /* 0x402A7A9D, 0x357F7FCE */
 204   4.12051854307378562225e+02, /* 0x4079C0D4, 0x652EA590 */
 205   3.87474538913960532227e+03, /* 0x40AE457D, 0xA3A532CC */
 206   7.91447954031891731574e+03, /* 0x40BEEA7A, 0xC32782DD */
 207 };
 208 static const double ps8[5] = {
 209   1.14207370375678408436e+02, /* 0x405C8D45, 0x8E656CAC */
 210   3.65093083420853463394e+03, /* 0x40AC85DC, 0x964D274F */
 211   3.69562060269033463555e+04, /* 0x40E20B86, 0x97C5BB7F */
 212   9.76027935934950801311e+04, /* 0x40F7D42C, 0xB28F17BB */
 213   3.08042720627888811578e+04, /* 0x40DE1511, 0x697A0B2D */
 214 };
 215
 216 static const double pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
 217   1.31990519556243522749e-11, /* 0x3DAD0667, 0xDAE1CA7D */
 218   1.17187493190614097638e-01, /* 0x3FBDFFFF, 0xE2C10043 */
 219   6.80275127868432871736e+00, /* 0x401B3604, 0x6E6315E3 */
 220   1.08308182990189109773e+02, /* 0x405B13B9, 0x452602ED */
 221   5.17636139533199752805e+02, /* 0x40802D16, 0xD052D649 */
 222   5.28715201363337541807e+02, /* 0x408085B8, 0xBB7E0CB7 */
 223 };
 224 static const double ps5[5] = {
 225   5.92805987221131331921e+01, /* 0x404DA3EA, 0xA8AF633D */
 226   9.91401418733614377743e+02, /* 0x408EFB36, 0x1B066701 */
 227   5.35326695291487976647e+03, /* 0x40B4E944, 0x5706B6FB */
 228   7.84469031749551231769e+03, /* 0x40BEA4B0, 0xB8A5BB15 */
 229   1.50404688810361062679e+03, /* 0x40978030, 0x036F5E51 */
 230 };
 231
 232 static const double pr3[6] = {
 233   3.02503916137373618024e-09, /* 0x3E29FC21, 0xA7AD9EDD */
 234   1.17186865567253592491e-01, /* 0x3FBDFFF5, 0x5B21D17B */
 235   3.93297750033315640650e+00, /* 0x400F76BC, 0xE85EAD8A */
 236   3.51194035591636932736e+01, /* 0x40418F48, 0x9DA6D129 */
 237   9.10550110750781271918e+01, /* 0x4056C385, 0x4D2C1837 */
 238   4.85590685197364919645e+01, /* 0x4048478F, 0x8EA83EE5 */
 239 };
 240 static const double ps3[5] = {
 241   3.47913095001251519989e+01, /* 0x40416549, 0xA134069C */
 242   3.36762458747825746741e+02, /* 0x40750C33, 0x07F1A75F */
 243   1.04687139975775130551e+03, /* 0x40905B7C, 0x5037D523 */
 244   8.90811346398256432622e+02, /* 0x408BD67D, 0xA32E31E9 */
 245   1.03787932439639277504e+02, /* 0x4059F26D, 0x7C2EED53 */
 246 };
 247
 248 static const double pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
 249   1.07710830106873743082e-07, /* 0x3E7CE9D4, 0xF65544F4 */
 250   1.17176219462683348094e-01, /* 0x3FBDFF42, 0xBE760D83 */
 251   2.36851496667608785174e+00, /* 0x4002F2B7, 0xF98FAEC0 */
 252   1.22426109148261232917e+01, /* 0x40287C37, 0x7F71A964 */
 253   1.76939711271687727390e+01, /* 0x4031B1A8, 0x177F8EE2 */
 254   5.07352312588818499250e+00, /* 0x40144B49, 0xA574C1FE */
 255 };
 256 static const double ps2[5] = {
 257   2.14364859363821409488e+01, /* 0x40356FBD, 0x8AD5ECDC */
 258   1.25290227168402751090e+02, /* 0x405F5293, 0x14F92CD5 */
 259   2.32276469057162813669e+02, /* 0x406D08D8, 0xD5A2DBD9 */
 260   1.17679373287147100768e+02, /* 0x405D6B7A, 0xDA1884A9 */
 261   8.36463893371618283368e+00, /* 0x4020BAB1, 0xF44E5192 */
 262 };
 263
 264 static double
 265 pone(double x)
 266 {
 267         const double *p,*q;
 268         double z,r,s;
 269         int32_t ix;
 270
 271         p = q = 0;
 272         GET_HIGH_WORD(ix,x);
 273         ix &= 0x7fffffff;
 274         if(ix>=0x40200000)     {p = pr8; q= ps8;}
 275         else if(ix>=0x40122E8B){p = pr5; q= ps5;}
 276         else if(ix>=0x4006DB6D){p = pr3; q= ps3;}
 277         else if(ix>=0x40000000){p = pr2; q= ps2;}
 278         z = one/(x*x);
 279         r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
 280         s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
 281         return one+ r/s;
 282 }
 283
 284
 285 /* For x >= 8, the asymptotic expansions of qone is
 286  *      3/8 s - 105/1024 s^3 - ..., where s = 1/x.
 287  * We approximate pone by
 288  *      qone(x) = s*(0.375 + (R/S))
 289  * where  R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10
 290  *        S = 1 + qs1*s^2 + ... + qs6*s^12
 291  * and
 292  *      | qone(x)/s -0.375-R/S | <= 2  ** ( -61.13)
 293  */
 294
 295 static const double qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
 296   0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
 297  -1.02539062499992714161e-01, /* 0xBFBA3FFF, 0xFFFFFDF3 */
 298  -1.62717534544589987888e+01, /* 0xC0304591, 0xA26779F7 */
 299  -7.59601722513950107896e+02, /* 0xC087BCD0, 0x53E4B576 */
 300  -1.18498066702429587167e+04, /* 0xC0C724E7, 0x40F87415 */
 301  -4.84385124285750353010e+04, /* 0xC0E7A6D0, 0x65D09C6A */
 302 };
 303 static const double qs8[6] = {
 304   1.61395369700722909556e+02, /* 0x40642CA6, 0xDE5BCDE5 */
 305   7.82538599923348465381e+03, /* 0x40BE9162, 0xD0D88419 */
 306   1.33875336287249578163e+05, /* 0x4100579A, 0xB0B75E98 */
 307   7.19657723683240939863e+05, /* 0x4125F653, 0x72869C19 */
 308   6.66601232617776375264e+05, /* 0x412457D2, 0x7719AD5C */
 309  -2.94490264303834643215e+05, /* 0xC111F969, 0x0EA5AA18 */
 310 };
 311
 312 static const double qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
 313  -2.08979931141764104297e-11, /* 0xBDB6FA43, 0x1AA1A098 */
 314  -1.02539050241375426231e-01, /* 0xBFBA3FFF, 0xCB597FEF */
 315  -8.05644828123936029840e+00, /* 0xC0201CE6, 0xCA03AD4B */
 316  -1.83669607474888380239e+02, /* 0xC066F56D, 0x6CA7B9B0 */
 317  -1.37319376065508163265e+03, /* 0xC09574C6, 0x6931734F */
 318  -2.61244440453215656817e+03, /* 0xC0A468E3, 0x88FDA79D */
 319 };
 320 static const double qs5[6] = {
 321   8.12765501384335777857e+01, /* 0x405451B2, 0xFF5A11B2 */
 322   1.99179873460485964642e+03, /* 0x409F1F31, 0xE77BF839 */
 323   1.74684851924908907677e+04, /* 0x40D10F1F, 0x0D64CE29 */
 324   4.98514270910352279316e+04, /* 0x40E8576D, 0xAABAD197 */
 325   2.79480751638918118260e+04, /* 0x40DB4B04, 0xCF7C364B */
 326  -4.71918354795128470869e+03, /* 0xC0B26F2E, 0xFCFFA004 */
 327 };
 328
 329 static const double qr3[6] = {
 330  -5.07831226461766561369e-09, /* 0xBE35CFA9, 0xD38FC84F */
 331  -1.02537829820837089745e-01, /* 0xBFBA3FEB, 0x51AEED54 */
 332  -4.61011581139473403113e+00, /* 0xC01270C2, 0x3302D9FF */
 333  -5.78472216562783643212e+01, /* 0xC04CEC71, 0xC25D16DA */
 334  -2.28244540737631695038e+02, /* 0xC06C87D3, 0x4718D55F */
 335  -2.19210128478909325622e+02, /* 0xC06B66B9, 0x5F5C1BF6 */
 336 };
 337 static const double qs3[6] = {
 338   4.76651550323729509273e+01, /* 0x4047D523, 0xCCD367E4 */
 339   6.73865112676699709482e+02, /* 0x40850EEB, 0xC031EE3E */
 340   3.38015286679526343505e+03, /* 0x40AA684E, 0x448E7C9A */
 341   5.54772909720722782367e+03, /* 0x40B5ABBA, 0xA61D54A6 */
 342   1.90311919338810798763e+03, /* 0x409DBC7A, 0x0DD4DF4B */
 343  -1.35201191444307340817e+02, /* 0xC060E670, 0x290A311F */
 344 };
 345
 346 static const double qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
 347  -1.78381727510958865572e-07, /* 0xBE87F126, 0x44C626D2 */
 348  -1.02517042607985553460e-01, /* 0xBFBA3E8E, 0x9148B010 */
 349  -2.75220568278187460720e+00, /* 0xC0060484, 0x69BB4EDA */
 350  -1.96636162643703720221e+01, /* 0xC033A9E2, 0xC168907F */
 351  -4.23253133372830490089e+01, /* 0xC04529A3, 0xDE104AAA */
 352  -2.13719211703704061733e+01, /* 0xC0355F36, 0x39CF6E52 */
 353 };
 354 static const double qs2[6] = {
 355   2.95333629060523854548e+01, /* 0x403D888A, 0x78AE64FF */
 356   2.52981549982190529136e+02, /* 0x406F9F68, 0xDB821CBA */
 357   7.57502834868645436472e+02, /* 0x4087AC05, 0xCE49A0F7 */
 358   7.39393205320467245656e+02, /* 0x40871B25, 0x48D4C029 */
 359   1.55949003336666123687e+02, /* 0x40637E5E, 0x3C3ED8D4 */
 360  -4.95949898822628210127e+00, /* 0xC013D686, 0xE71BE86B */
 361 };
 362
 363 static double
 364 qone(double x)
 365 {
 366         const double *p,*q;
 367         double  s,r,z;
 368         int32_t ix;
 369
 370         p = q = 0;
 371         GET_HIGH_WORD(ix,x);
 372         ix &= 0x7fffffff;
 373         if(ix>=0x40200000)     {p = qr8; q= qs8;}
 374         else if(ix>=0x40122E8B){p = qr5; q= qs5;}
 375         else if(ix>=0x4006DB6D){p = qr3; q= qs3;}
 376         else if(ix>=0x40000000){p = qr2; q= qs2;}
 377         z = one/(x*x);
 378         r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
 379         s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
 380         return (.375 + r/s)/x;
 381 }