libpok-refman-current-html/html/e__j1_8c_source.html

00001 /*
00002  *                               POK header
00003  *
00004  * The following file is a part of the POK project. Any modification should
00005  * made according to the POK licence. You CANNOT use this file or a part of
00006  * this file is this part of a file for your own project
00007  *
00008  * For more information on the POK licence, please see our LICENCE FILE
00009  *
00010  * Please follow the coding guidelines described in doc/CODING_GUIDELINES
00011  *
00012  *                                      Copyright (c) 2007-2009 POK team
00013  *
00014  * Created by julien on Fri Jan 30 14:41:34 2009
00015  */
00016
00017 /* @(#)e_j1.c 5.1 93/09/24 */
00018 /*
00019  * ====================================================
00020  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
00021  *
00022  * Developed at SunPro, a Sun Microsystems, Inc. business.
00023  * Permission to use, copy, modify, and distribute this
00024  * software is freely granted, provided that this notice
00025  * is preserved.
00026  * ====================================================
00027  */
00028
00029 /* __ieee754_j1(x), __ieee754_y1(x)
00030  * Bessel function of the first and second kinds of order zero.
00031  * Method -- j1(x):
00032  *      1. For tiny x, we use j1(x) = x/2 - x^3/16 + x^5/384 - ...
00033  *      2. Reduce x to |x| since j1(x)=-j1(-x),  and
00034  *         for x in (0,2)
00035  *              j1(x) = x/2 + x*z*R0/S0,  where z = x*x;
00036  *         (precision:  |j1/x - 1/2 - R0/S0 |<2**-61.51 )
00037  *         for x in (2,inf)
00038  *              j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x1)-q1(x)*sin(x1))
00039  *              y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1))
00040  *         where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1)
00041  *         as follow:
00042  *              cos(x1) =  cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
00043  *                      =  1/sqrt(2) * (sin(x) - cos(x))
00044  *              sin(x1) =  sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
00045  *                      = -1/sqrt(2) * (sin(x) + cos(x))
00046  *         (To avoid cancellation, use
00047  *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
00048  *          to compute the worse one.)
00049  *
00050  *      3 Special cases
00051  *              j1(nan)= nan
00052  *              j1(0) = 0
00053  *              j1(inf) = 0
00054  *
00055  * Method -- y1(x):
00056  *      1. screen out x<=0 cases: y1(0)=-inf, y1(x<0)=NaN
00057  *      2. For x<2.
00058  *         Since
00059  *              y1(x) = 2/pi*(j1(x)*(ln(x/2)+Euler)-1/x-x/2+5/64*x^3-...)
00060  *         therefore y1(x)-2/pi*j1(x)*ln(x)-1/x is an odd function.
00061  *         We use the following function to approximate y1,
00062  *              y1(x) = x*U(z)/V(z) + (2/pi)*(j1(x)*ln(x)-1/x), z= x^2
00063  *         where for x in [0,2] (abs err less than 2**-65.89)
00064  *              U(z) = U0[0] + U0[1]*z + ... + U0[4]*z^4
00065  *              V(z) = 1  + v0[0]*z + ... + v0[4]*z^5
00066  *         Note: For tiny x, 1/x dominate y1 and hence
00067  *              y1(tiny) = -2/pi/tiny, (choose tiny<2**-54)
00068  *      3. For x>=2.
00069  *              y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1))
00070  *         where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1)
00071  *         by method mentioned above.
00072  */
00073
00074 #ifdef POK_NEEDS_LIBMATH
00075
00076 #include <libm.h>
00077 #include "math_private.h"
00078
00079 static double pone(double), qone(double);
00080
00081 static const double
00082 huge    = 1e300,
00083 one     = 1.0,
00084 invsqrtpi=  5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
00085 tpi      =  6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
00086         /* R0/S0 on [0,2] */
00087 r00  = -6.25000000000000000000e-02, /* 0xBFB00000, 0x00000000 */
00088 r01  =  1.40705666955189706048e-03, /* 0x3F570D9F, 0x98472C61 */
00089 r02  = -1.59955631084035597520e-05, /* 0xBEF0C5C6, 0xBA169668 */
00090 r03  =  4.96727999609584448412e-08, /* 0x3E6AAAFA, 0x46CA0BD9 */
00091 s01  =  1.91537599538363460805e-02, /* 0x3F939D0B, 0x12637E53 */
00092 s02  =  1.85946785588630915560e-04, /* 0x3F285F56, 0xB9CDF664 */
00093 s03  =  1.17718464042623683263e-06, /* 0x3EB3BFF8, 0x333F8498 */
00094 s04  =  5.04636257076217042715e-09, /* 0x3E35AC88, 0xC97DFF2C */
00095 s05  =  1.23542274426137913908e-11; /* 0x3DAB2ACF, 0xCFB97ED8 */
00096
00097 static const double zero    = 0.0;
00098
00099 double
00100 __ieee754_j1(double x)
00101 {
00102         double z, s,c,ss,cc,r,u,v,y;
00103         int32_t hx,ix;
00104
00105         GET_HIGH_WORD(hx,x);
00106         ix = hx&0x7fffffff;
00107         if(ix>=0x7ff00000) return one/x;
00108         y = fabs(x);
00109         if(ix >= 0x40000000) {  /* |x| >= 2.0 */
00110                 s = sin(y);
00111                 c = cos(y);
00112                 ss = -s-c;
00113                 cc = s-c;
00114                 if(ix<0x7fe00000) {  /* make sure y+y not overflow */
00115                     z = cos(y+y);
00116                     if ((s*c)>zero) cc = z/ss;
00117                     else            ss = z/cc;
00118                 }
00119         /*
00120          * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x)
00121          * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x)
00122          */
00123                 if(ix>0x48000000) z = (invsqrtpi*cc)/sqrt(y);
00124                 else {
00125                     u = pone(y); v = qone(y);
00126                     z = invsqrtpi*(u*cc-v*ss)/sqrt(y);
00127                 }
00128                 if(hx<0) return -z;
00129                 else     return  z;
00130         }
00131         if(ix<0x3e400000) {     /* |x|<2**-27 */
00132             if(huge+x>one) return 0.5*x;/* inexact if x!=0 necessary */
00133         }
00134         z = x*x;
00135         r =  z*(r00+z*(r01+z*(r02+z*r03)));
00136         s =  one+z*(s01+z*(s02+z*(s03+z*(s04+z*s05))));
00137         r *= x;
00138         return(x*0.5+r/s);
00139 }
00140
00141 static const double U0[5] = {
00142  -1.96057090646238940668e-01, /* 0xBFC91866, 0x143CBC8A */
00143   5.04438716639811282616e-02, /* 0x3FA9D3C7, 0x76292CD1 */
00144  -1.91256895875763547298e-03, /* 0xBF5F55E5, 0x4844F50F */
00145   2.35252600561610495928e-05, /* 0x3EF8AB03, 0x8FA6B88E */
00146  -9.19099158039878874504e-08, /* 0xBE78AC00, 0x569105B8 */
00147 };
00148 static const double V0[5] = {
00149   1.99167318236649903973e-02, /* 0x3F94650D, 0x3F4DA9F0 */
00150   2.02552581025135171496e-04, /* 0x3F2A8C89, 0x6C257764 */
00151   1.35608801097516229404e-06, /* 0x3EB6C05A, 0x894E8CA6 */
00152   6.22741452364621501295e-09, /* 0x3E3ABF1D, 0x5BA69A86 */
00153   1.66559246207992079114e-11, /* 0x3DB25039, 0xDACA772A */
00154 };
00155
00156 double
00157 __ieee754_y1(double x)
00158 {
00159         double z, s,c,ss,cc,u,v;
00160         int32_t hx,ix,lx;
00161
00162         EXTRACT_WORDS(hx,lx,x);
00163         ix = 0x7fffffff&hx;
00164     /* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */
00165         if(ix>=0x7ff00000) return  one/(x+x*x);
00166         if((ix|lx)==0) return -one/zero;
00167         if(hx<0) return zero/zero;
00168         if(ix >= 0x40000000) {  /* |x| >= 2.0 */
00169                 s = sin(x);
00170                 c = cos(x);
00171                 ss = -s-c;
00172                 cc = s-c;
00173                 if(ix<0x7fe00000) {  /* make sure x+x not overflow */
00174                     z = cos(x+x);
00175                     if ((s*c)>zero) cc = z/ss;
00176                     else            ss = z/cc;
00177                 }
00178         /* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0))
00179          * where x0 = x-3pi/4
00180          *      Better formula:
00181          *              cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
00182          *                      =  1/sqrt(2) * (sin(x) - cos(x))
00183          *              sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
00184          *                      = -1/sqrt(2) * (cos(x) + sin(x))
00185          * To avoid cancellation, use
00186          *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
00187          * to compute the worse one.
00188          */
00189                 if(ix>0x48000000) z = (invsqrtpi*ss)/sqrt(x);
00190                 else {
00191                     u = pone(x); v = qone(x);
00192                     z = invsqrtpi*(u*ss+v*cc)/sqrt(x);
00193                 }
00194                 return z;
00195         }
00196         if(ix<=0x3c900000) {    /* x < 2**-54 */
00197             return(-tpi/x);
00198         }
00199         z = x*x;
00200         u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4])));
00201         v = one+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4]))));
00202         return(x*(u/v) + tpi*(__ieee754_j1(x)*__ieee754_log(x)-one/x));
00203 }
00204
00205 /* For x >= 8, the asymptotic expansions of pone is
00206  *      1 + 15/128 s^2 - 4725/2^15 s^4 - ...,   where s = 1/x.
00207  * We approximate pone by
00208  *      pone(x) = 1 + (R/S)
00209  * where  R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10
00210  *        S = 1 + ps0*s^2 + ... + ps4*s^10
00211  * and
00212  *      | pone(x)-1-R/S | <= 2  ** ( -60.06)
00213  */
00214
00215 static const double pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
00216   0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
00217   1.17187499999988647970e-01, /* 0x3FBDFFFF, 0xFFFFFCCE */
00218   1.32394806593073575129e+01, /* 0x402A7A9D, 0x357F7FCE */
00219   4.12051854307378562225e+02, /* 0x4079C0D4, 0x652EA590 */
00220   3.87474538913960532227e+03, /* 0x40AE457D, 0xA3A532CC */
00221   7.91447954031891731574e+03, /* 0x40BEEA7A, 0xC32782DD */
00222 };
00223 static const double ps8[5] = {
00224   1.14207370375678408436e+02, /* 0x405C8D45, 0x8E656CAC */
00225   3.65093083420853463394e+03, /* 0x40AC85DC, 0x964D274F */
00226   3.69562060269033463555e+04, /* 0x40E20B86, 0x97C5BB7F */
00227   9.76027935934950801311e+04, /* 0x40F7D42C, 0xB28F17BB */
00228   3.08042720627888811578e+04, /* 0x40DE1511, 0x697A0B2D */
00229 };
00230
00231 static const double pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
00232   1.31990519556243522749e-11, /* 0x3DAD0667, 0xDAE1CA7D */
00233   1.17187493190614097638e-01, /* 0x3FBDFFFF, 0xE2C10043 */
00234   6.80275127868432871736e+00, /* 0x401B3604, 0x6E6315E3 */
00235   1.08308182990189109773e+02, /* 0x405B13B9, 0x452602ED */
00236   5.17636139533199752805e+02, /* 0x40802D16, 0xD052D649 */
00237   5.28715201363337541807e+02, /* 0x408085B8, 0xBB7E0CB7 */
00238 };
00239 static const double ps5[5] = {
00240   5.92805987221131331921e+01, /* 0x404DA3EA, 0xA8AF633D */
00241   9.91401418733614377743e+02, /* 0x408EFB36, 0x1B066701 */
00242   5.35326695291487976647e+03, /* 0x40B4E944, 0x5706B6FB */
00243   7.84469031749551231769e+03, /* 0x40BEA4B0, 0xB8A5BB15 */
00244   1.50404688810361062679e+03, /* 0x40978030, 0x036F5E51 */
00245 };
00246
00247 static const double pr3[6] = {
00248   3.02503916137373618024e-09, /* 0x3E29FC21, 0xA7AD9EDD */
00249   1.17186865567253592491e-01, /* 0x3FBDFFF5, 0x5B21D17B */
00250   3.93297750033315640650e+00, /* 0x400F76BC, 0xE85EAD8A */
00251   3.51194035591636932736e+01, /* 0x40418F48, 0x9DA6D129 */
00252   9.10550110750781271918e+01, /* 0x4056C385, 0x4D2C1837 */
00253   4.85590685197364919645e+01, /* 0x4048478F, 0x8EA83EE5 */
00254 };
00255 static const double ps3[5] = {
00256   3.47913095001251519989e+01, /* 0x40416549, 0xA134069C */
00257   3.36762458747825746741e+02, /* 0x40750C33, 0x07F1A75F */
00258   1.04687139975775130551e+03, /* 0x40905B7C, 0x5037D523 */
00259   8.90811346398256432622e+02, /* 0x408BD67D, 0xA32E31E9 */
00260   1.03787932439639277504e+02, /* 0x4059F26D, 0x7C2EED53 */
00261 };
00262
00263 static const double pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
00264   1.07710830106873743082e-07, /* 0x3E7CE9D4, 0xF65544F4 */
00265   1.17176219462683348094e-01, /* 0x3FBDFF42, 0xBE760D83 */
00266   2.36851496667608785174e+00, /* 0x4002F2B7, 0xF98FAEC0 */
00267   1.22426109148261232917e+01, /* 0x40287C37, 0x7F71A964 */
00268   1.76939711271687727390e+01, /* 0x4031B1A8, 0x177F8EE2 */
00269   5.07352312588818499250e+00, /* 0x40144B49, 0xA574C1FE */
00270 };
00271 static const double ps2[5] = {
00272   2.14364859363821409488e+01, /* 0x40356FBD, 0x8AD5ECDC */
00273   1.25290227168402751090e+02, /* 0x405F5293, 0x14F92CD5 */
00274   2.32276469057162813669e+02, /* 0x406D08D8, 0xD5A2DBD9 */
00275   1.17679373287147100768e+02, /* 0x405D6B7A, 0xDA1884A9 */
00276   8.36463893371618283368e+00, /* 0x4020BAB1, 0xF44E5192 */
00277 };
00278
00279 static double
00280 pone(double x)
00281 {
00282         const double *p,*q;
00283         double z,r,s;
00284         int32_t ix;
00285
00286         p = q = 0;
00287         GET_HIGH_WORD(ix,x);
00288         ix &= 0x7fffffff;
00289         if(ix>=0x40200000)     {p = pr8; q= ps8;}
00290         else if(ix>=0x40122E8B){p = pr5; q= ps5;}
00291         else if(ix>=0x4006DB6D){p = pr3; q= ps3;}
00292         else if(ix>=0x40000000){p = pr2; q= ps2;}
00293         z = one/(x*x);
00294         r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
00295         s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
00296         return one+ r/s;
00297 }
00298
00299
00300 /* For x >= 8, the asymptotic expansions of qone is
00301  *      3/8 s - 105/1024 s^3 - ..., where s = 1/x.
00302  * We approximate pone by
00303  *      qone(x) = s*(0.375 + (R/S))
00304  * where  R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10
00305  *        S = 1 + qs1*s^2 + ... + qs6*s^12
00306  * and
00307  *      | qone(x)/s -0.375-R/S | <= 2  ** ( -61.13)
00308  */
00309
00310 static const double qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
00311   0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
00312  -1.02539062499992714161e-01, /* 0xBFBA3FFF, 0xFFFFFDF3 */
00313  -1.62717534544589987888e+01, /* 0xC0304591, 0xA26779F7 */
00314  -7.59601722513950107896e+02, /* 0xC087BCD0, 0x53E4B576 */
00315  -1.18498066702429587167e+04, /* 0xC0C724E7, 0x40F87415 */
00316  -4.84385124285750353010e+04, /* 0xC0E7A6D0, 0x65D09C6A */
00317 };
00318 static const double qs8[6] = {
00319   1.61395369700722909556e+02, /* 0x40642CA6, 0xDE5BCDE5 */
00320   7.82538599923348465381e+03, /* 0x40BE9162, 0xD0D88419 */
00321   1.33875336287249578163e+05, /* 0x4100579A, 0xB0B75E98 */
00322   7.19657723683240939863e+05, /* 0x4125F653, 0x72869C19 */
00323   6.66601232617776375264e+05, /* 0x412457D2, 0x7719AD5C */
00324  -2.94490264303834643215e+05, /* 0xC111F969, 0x0EA5AA18 */
00325 };
00326
00327 static const double qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
00328  -2.08979931141764104297e-11, /* 0xBDB6FA43, 0x1AA1A098 */
00329  -1.02539050241375426231e-01, /* 0xBFBA3FFF, 0xCB597FEF */
00330  -8.05644828123936029840e+00, /* 0xC0201CE6, 0xCA03AD4B */
00331  -1.83669607474888380239e+02, /* 0xC066F56D, 0x6CA7B9B0 */
00332  -1.37319376065508163265e+03, /* 0xC09574C6, 0x6931734F */
00333  -2.61244440453215656817e+03, /* 0xC0A468E3, 0x88FDA79D */
00334 };
00335 static const double qs5[6] = {
00336   8.12765501384335777857e+01, /* 0x405451B2, 0xFF5A11B2 */
00337   1.99179873460485964642e+03, /* 0x409F1F31, 0xE77BF839 */
00338   1.74684851924908907677e+04, /* 0x40D10F1F, 0x0D64CE29 */
00339   4.98514270910352279316e+04, /* 0x40E8576D, 0xAABAD197 */
00340   2.79480751638918118260e+04, /* 0x40DB4B04, 0xCF7C364B */
00341  -4.71918354795128470869e+03, /* 0xC0B26F2E, 0xFCFFA004 */
00342 };
00343
00344 static const double qr3[6] = {
00345  -5.07831226461766561369e-09, /* 0xBE35CFA9, 0xD38FC84F */
00346  -1.02537829820837089745e-01, /* 0xBFBA3FEB, 0x51AEED54 */
00347  -4.61011581139473403113e+00, /* 0xC01270C2, 0x3302D9FF */
00348  -5.78472216562783643212e+01, /* 0xC04CEC71, 0xC25D16DA */
00349  -2.28244540737631695038e+02, /* 0xC06C87D3, 0x4718D55F */
00350  -2.19210128478909325622e+02, /* 0xC06B66B9, 0x5F5C1BF6 */
00351 };
00352 static const double qs3[6] = {
00353   4.76651550323729509273e+01, /* 0x4047D523, 0xCCD367E4 */
00354   6.73865112676699709482e+02, /* 0x40850EEB, 0xC031EE3E */
00355   3.38015286679526343505e+03, /* 0x40AA684E, 0x448E7C9A */
00356   5.54772909720722782367e+03, /* 0x40B5ABBA, 0xA61D54A6 */
00357   1.90311919338810798763e+03, /* 0x409DBC7A, 0x0DD4DF4B */
00358  -1.35201191444307340817e+02, /* 0xC060E670, 0x290A311F */
00359 };
00360
00361 static const double qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
00362  -1.78381727510958865572e-07, /* 0xBE87F126, 0x44C626D2 */
00363  -1.02517042607985553460e-01, /* 0xBFBA3E8E, 0x9148B010 */
00364  -2.75220568278187460720e+00, /* 0xC0060484, 0x69BB4EDA */
00365  -1.96636162643703720221e+01, /* 0xC033A9E2, 0xC168907F */
00366  -4.23253133372830490089e+01, /* 0xC04529A3, 0xDE104AAA */
00367  -2.13719211703704061733e+01, /* 0xC0355F36, 0x39CF6E52 */
00368 };
00369 static const double qs2[6] = {
00370   2.95333629060523854548e+01, /* 0x403D888A, 0x78AE64FF */
00371   2.52981549982190529136e+02, /* 0x406F9F68, 0xDB821CBA */
00372   7.57502834868645436472e+02, /* 0x4087AC05, 0xCE49A0F7 */
00373   7.39393205320467245656e+02, /* 0x40871B25, 0x48D4C029 */
00374   1.55949003336666123687e+02, /* 0x40637E5E, 0x3C3ED8D4 */
00375  -4.95949898822628210127e+00, /* 0xC013D686, 0xE71BE86B */
00376 };
00377
00378 static double
00379 qone(double x)
00380 {
00381         const double *p,*q;
00382         double  s,r,z;
00383         int32_t ix;
00384
00385         p = q = 0;
00386         GET_HIGH_WORD(ix,x);
00387         ix &= 0x7fffffff;
00388         if(ix>=0x40200000)     {p = qr8; q= qs8;}
00389         else if(ix>=0x40122E8B){p = qr5; q= qs5;}
00390         else if(ix>=0x4006DB6D){p = qr3; q= qs3;}
00391         else if(ix>=0x40000000){p = qr2; q= qs2;}
00392         z = one/(x*x);
00393         r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
00394         s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
00395         return (.375 + r/s)/x;
00396 }
00397 #endif
00398