RTAI API: base/math/e_j1.c Source File

00001 /* @(#)e_j1.c 5.1 93/09/24 */
00002 /*
00003  * ====================================================
00004  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
00005  *
00006  * Developed at SunPro, a Sun Microsystems, Inc. business.
00007  * Permission to use, copy, modify, and distribute this
00008  * software is freely granted, provided that this notice 
00009  * is preserved.
00010  * ====================================================
00011  */
00012 
00013 #if defined(LIBM_SCCS) && !defined(lint)
00014 static char rcsid[] = "$NetBSD: e_j1.c,v 1.8 1995/05/10 20:45:27 jtc Exp $";
00015 #endif
00016 
00017 /* __ieee754_j1(x), __ieee754_y1(x)
00018  * Bessel function of the first and second kinds of order zero.
00019  * Method -- j1(x):
00020  *  1. For tiny x, we use j1(x) = x/2 - x^3/16 + x^5/384 - ...
00021  *  2. Reduce x to |x| since j1(x)=-j1(-x),  and
00022  *     for x in (0,2)
00023  *      j1(x) = x/2 + x*z*R0/S0,  where z = x*x;
00024  *     (precision:  |j1/x - 1/2 - R0/S0 |<2**-61.51 )
00025  *     for x in (2,inf)
00026  *      j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x1)-q1(x)*sin(x1))
00027  *      y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1))
00028  *     where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1)
00029  *     as follow:
00030  *      cos(x1) =  cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
00031  *          =  1/sqrt(2) * (sin(x) - cos(x))
00032  *      sin(x1) =  sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
00033  *          = -1/sqrt(2) * (sin(x) + cos(x))
00034  *     (To avoid cancellation, use
00035  *      sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
00036  *      to compute the worse one.)
00037  *     
00038  *  3 Special cases
00039  *      j1(nan)= nan
00040  *      j1(0) = 0
00041  *      j1(inf) = 0
00042  *      
00043  * Method -- y1(x):
00044  *  1. screen out x<=0 cases: y1(0)=-inf, y1(x<0)=NaN 
00045  *  2. For x<2.
00046  *     Since 
00047  *      y1(x) = 2/pi*(j1(x)*(ln(x/2)+Euler)-1/x-x/2+5/64*x^3-...)
00048  *     therefore y1(x)-2/pi*j1(x)*ln(x)-1/x is an odd function.
00049  *     We use the following function to approximate y1,
00050  *      y1(x) = x*U(z)/V(z) + (2/pi)*(j1(x)*ln(x)-1/x), z= x^2
00051  *     where for x in [0,2] (abs err less than 2**-65.89)
00052  *      U(z) = U0[0] + U0[1]*z + ... + U0[4]*z^4
00053  *      V(z) = 1  + v0[0]*z + ... + v0[4]*z^5
00054  *     Note: For tiny x, 1/x dominate y1 and hence
00055  *      y1(tiny) = -2/pi/tiny, (choose tiny<2**-54)
00056  *  3. For x>=2.
00057  *      y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1))
00058  *     where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1)
00059  *     by method mentioned above.
00060  */
00061 
00062 #include "math.h"
00063 #include "mathP.h"
00064 
00065 #ifdef __STDC__
00066 static double pone(double), qone(double);
00067 #else
00068 static double pone(), qone();
00069 #endif
00070 
00071 #ifdef __STDC__
00072 static const double 
00073 #else
00074 static double 
00075 #endif
00076 huge    = 1e300,
00077 one = 1.0,
00078 invsqrtpi=  5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
00079 tpi      =  6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
00080     /* R0/S0 on [0,2] */
00081 r00  = -6.25000000000000000000e-02, /* 0xBFB00000, 0x00000000 */
00082 r01  =  1.40705666955189706048e-03, /* 0x3F570D9F, 0x98472C61 */
00083 r02  = -1.59955631084035597520e-05, /* 0xBEF0C5C6, 0xBA169668 */
00084 r03  =  4.96727999609584448412e-08, /* 0x3E6AAAFA, 0x46CA0BD9 */
00085 s01  =  1.91537599538363460805e-02, /* 0x3F939D0B, 0x12637E53 */
00086 s02  =  1.85946785588630915560e-04, /* 0x3F285F56, 0xB9CDF664 */
00087 s03  =  1.17718464042623683263e-06, /* 0x3EB3BFF8, 0x333F8498 */
00088 s04  =  5.04636257076217042715e-09, /* 0x3E35AC88, 0xC97DFF2C */
00089 s05  =  1.23542274426137913908e-11; /* 0x3DAB2ACF, 0xCFB97ED8 */
00090 
00091 #ifdef __STDC__
00092 static const double zero    = 0.0;
00093 #else
00094 static double zero    = 0.0;
00095 #endif
00096 
00097 #ifdef __STDC__
00098     double __ieee754_j1(double x) 
00099 #else
00100     double __ieee754_j1(x) 
00101     double x;
00102 #endif
00103 {
00104     double z, s,c,ss,cc,r,u,v,y;
00105     int32_t hx,ix;
00106 
00107     GET_HIGH_WORD(hx,x);
00108     ix = hx&0x7fffffff;
00109     if(ix>=0x7ff00000) return one/x;
00110     y = fabs(x);
00111     if(ix >= 0x40000000) {  /* |x| >= 2.0 */
00112         s = sin(y);
00113         c = cos(y);
00114         ss = -s-c;
00115         cc = s-c;
00116         if(ix<0x7fe00000) {  /* make sure y+y not overflow */
00117             z = cos(y+y);
00118             if ((s*c)>zero) cc = z/ss;
00119             else        ss = z/cc;
00120         }
00121     /*
00122      * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x)
00123      * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x)
00124      */
00125         if(ix>0x48000000) z = (invsqrtpi*cc)/sqrt(y);
00126         else {
00127             u = pone(y); v = qone(y);
00128             z = invsqrtpi*(u*cc-v*ss)/sqrt(y);
00129         }
00130         if(hx<0) return -z;
00131         else     return  z;
00132     }
00133     if(ix<0x3e400000) { /* |x|<2**-27 */
00134         if(huge+x>one) return 0.5*x;/* inexact if x!=0 necessary */
00135     }
00136     z = x*x;
00137     r =  z*(r00+z*(r01+z*(r02+z*r03)));
00138     s =  one+z*(s01+z*(s02+z*(s03+z*(s04+z*s05))));
00139     r *= x;
00140     return(x*0.5+r/s);
00141 }
00142 
00143 #ifdef __STDC__
00144 static const double U0[5] = {
00145 #else
00146 static double U0[5] = {
00147 #endif
00148  -1.96057090646238940668e-01, /* 0xBFC91866, 0x143CBC8A */
00149   5.04438716639811282616e-02, /* 0x3FA9D3C7, 0x76292CD1 */
00150  -1.91256895875763547298e-03, /* 0xBF5F55E5, 0x4844F50F */
00151   2.35252600561610495928e-05, /* 0x3EF8AB03, 0x8FA6B88E */
00152  -9.19099158039878874504e-08, /* 0xBE78AC00, 0x569105B8 */
00153 };
00154 #ifdef __STDC__
00155 static const double V0[5] = {
00156 #else
00157 static double V0[5] = {
00158 #endif
00159   1.99167318236649903973e-02, /* 0x3F94650D, 0x3F4DA9F0 */
00160   2.02552581025135171496e-04, /* 0x3F2A8C89, 0x6C257764 */
00161   1.35608801097516229404e-06, /* 0x3EB6C05A, 0x894E8CA6 */
00162   6.22741452364621501295e-09, /* 0x3E3ABF1D, 0x5BA69A86 */
00163   1.66559246207992079114e-11, /* 0x3DB25039, 0xDACA772A */
00164 };
00165 
00166 #ifdef __STDC__
00167     double __ieee754_y1(double x) 
00168 #else
00169     double __ieee754_y1(x) 
00170     double x;
00171 #endif
00172 {
00173     double z, s,c,ss,cc,u,v;
00174     int32_t hx,ix,lx;
00175 
00176     EXTRACT_WORDS(hx,lx,x);
00177         ix = 0x7fffffff&hx;
00178     /* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */
00179     if(ix>=0x7ff00000) return  one/(x+x*x); 
00180         if((ix|lx)==0) return -one/zero;
00181         if(hx<0) return zero/zero;
00182         if(ix >= 0x40000000) {  /* |x| >= 2.0 */
00183                 s = sin(x);
00184                 c = cos(x);
00185                 ss = -s-c;
00186                 cc = s-c;
00187                 if(ix<0x7fe00000) {  /* make sure x+x not overflow */
00188                     z = cos(x+x);
00189                     if ((s*c)>zero) cc = z/ss;
00190                     else            ss = z/cc;
00191                 }
00192         /* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0))
00193          * where x0 = x-3pi/4
00194          *      Better formula:
00195          *              cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
00196          *                      =  1/sqrt(2) * (sin(x) - cos(x))
00197          *              sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
00198          *                      = -1/sqrt(2) * (cos(x) + sin(x))
00199          * To avoid cancellation, use
00200          *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
00201          * to compute the worse one.
00202          */
00203                 if(ix>0x48000000) z = (invsqrtpi*ss)/sqrt(x);
00204                 else {
00205                     u = pone(x); v = qone(x);
00206                     z = invsqrtpi*(u*ss+v*cc)/sqrt(x);
00207                 }
00208                 return z;
00209         } 
00210         if(ix<=0x3c900000) {    /* x < 2**-54 */
00211             return(-tpi/x);
00212         } 
00213         z = x*x;
00214         u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4])));
00215         v = one+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4]))));
00216         return(x*(u/v) + tpi*(__ieee754_j1(x)*__ieee754_log(x)-one/x));
00217 }
00218 
00219 /* For x >= 8, the asymptotic expansions of pone is
00220  *  1 + 15/128 s^2 - 4725/2^15 s^4 - ...,   where s = 1/x.
00221  * We approximate pone by
00222  *  pone(x) = 1 + (R/S)
00223  * where  R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10
00224  *    S = 1 + ps0*s^2 + ... + ps4*s^10
00225  * and
00226  *  | pone(x)-1-R/S | <= 2  ** ( -60.06)
00227  */
00228 
00229 #ifdef __STDC__
00230 static const double pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
00231 #else
00232 static double pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
00233 #endif
00234   0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
00235   1.17187499999988647970e-01, /* 0x3FBDFFFF, 0xFFFFFCCE */
00236   1.32394806593073575129e+01, /* 0x402A7A9D, 0x357F7FCE */
00237   4.12051854307378562225e+02, /* 0x4079C0D4, 0x652EA590 */
00238   3.87474538913960532227e+03, /* 0x40AE457D, 0xA3A532CC */
00239   7.91447954031891731574e+03, /* 0x40BEEA7A, 0xC32782DD */
00240 };
00241 #ifdef __STDC__
00242 static const double ps8[5] = {
00243 #else
00244 static double ps8[5] = {
00245 #endif
00246   1.14207370375678408436e+02, /* 0x405C8D45, 0x8E656CAC */
00247   3.65093083420853463394e+03, /* 0x40AC85DC, 0x964D274F */
00248   3.69562060269033463555e+04, /* 0x40E20B86, 0x97C5BB7F */
00249   9.76027935934950801311e+04, /* 0x40F7D42C, 0xB28F17BB */
00250   3.08042720627888811578e+04, /* 0x40DE1511, 0x697A0B2D */
00251 };
00252 
00253 #ifdef __STDC__
00254 static const double pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
00255 #else
00256 static double pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
00257 #endif
00258   1.31990519556243522749e-11, /* 0x3DAD0667, 0xDAE1CA7D */
00259   1.17187493190614097638e-01, /* 0x3FBDFFFF, 0xE2C10043 */
00260   6.80275127868432871736e+00, /* 0x401B3604, 0x6E6315E3 */
00261   1.08308182990189109773e+02, /* 0x405B13B9, 0x452602ED */
00262   5.17636139533199752805e+02, /* 0x40802D16, 0xD052D649 */
00263   5.28715201363337541807e+02, /* 0x408085B8, 0xBB7E0CB7 */
00264 };
00265 #ifdef __STDC__
00266 static const double ps5[5] = {
00267 #else
00268 static double ps5[5] = {
00269 #endif
00270   5.92805987221131331921e+01, /* 0x404DA3EA, 0xA8AF633D */
00271   9.91401418733614377743e+02, /* 0x408EFB36, 0x1B066701 */
00272   5.35326695291487976647e+03, /* 0x40B4E944, 0x5706B6FB */
00273   7.84469031749551231769e+03, /* 0x40BEA4B0, 0xB8A5BB15 */
00274   1.50404688810361062679e+03, /* 0x40978030, 0x036F5E51 */
00275 };
00276 
00277 #ifdef __STDC__
00278 static const double pr3[6] = {
00279 #else
00280 static double pr3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
00281 #endif
00282   3.02503916137373618024e-09, /* 0x3E29FC21, 0xA7AD9EDD */
00283   1.17186865567253592491e-01, /* 0x3FBDFFF5, 0x5B21D17B */
00284   3.93297750033315640650e+00, /* 0x400F76BC, 0xE85EAD8A */
00285   3.51194035591636932736e+01, /* 0x40418F48, 0x9DA6D129 */
00286   9.10550110750781271918e+01, /* 0x4056C385, 0x4D2C1837 */
00287   4.85590685197364919645e+01, /* 0x4048478F, 0x8EA83EE5 */
00288 };
00289 #ifdef __STDC__
00290 static const double ps3[5] = {
00291 #else
00292 static double ps3[5] = {
00293 #endif
00294   3.47913095001251519989e+01, /* 0x40416549, 0xA134069C */
00295   3.36762458747825746741e+02, /* 0x40750C33, 0x07F1A75F */
00296   1.04687139975775130551e+03, /* 0x40905B7C, 0x5037D523 */
00297   8.90811346398256432622e+02, /* 0x408BD67D, 0xA32E31E9 */
00298   1.03787932439639277504e+02, /* 0x4059F26D, 0x7C2EED53 */
00299 };
00300 
00301 #ifdef __STDC__
00302 static const double pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
00303 #else
00304 static double pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
00305 #endif
00306   1.07710830106873743082e-07, /* 0x3E7CE9D4, 0xF65544F4 */
00307   1.17176219462683348094e-01, /* 0x3FBDFF42, 0xBE760D83 */
00308   2.36851496667608785174e+00, /* 0x4002F2B7, 0xF98FAEC0 */
00309   1.22426109148261232917e+01, /* 0x40287C37, 0x7F71A964 */
00310   1.76939711271687727390e+01, /* 0x4031B1A8, 0x177F8EE2 */
00311   5.07352312588818499250e+00, /* 0x40144B49, 0xA574C1FE */
00312 };
00313 #ifdef __STDC__
00314 static const double ps2[5] = {
00315 #else
00316 static double ps2[5] = {
00317 #endif
00318   2.14364859363821409488e+01, /* 0x40356FBD, 0x8AD5ECDC */
00319   1.25290227168402751090e+02, /* 0x405F5293, 0x14F92CD5 */
00320   2.32276469057162813669e+02, /* 0x406D08D8, 0xD5A2DBD9 */
00321   1.17679373287147100768e+02, /* 0x405D6B7A, 0xDA1884A9 */
00322   8.36463893371618283368e+00, /* 0x4020BAB1, 0xF44E5192 */
00323 };
00324 
00325 #ifdef __STDC__
00326     static double pone(double x)
00327 #else
00328     static double pone(x)
00329     double x;
00330 #endif
00331 {
00332 #ifdef __STDC__
00333     const double *p = 0,*q = 0;
00334 #else
00335     double *p = 0,*q = 0;
00336 #endif
00337     double z,r,s;
00338         int32_t ix;
00339     GET_HIGH_WORD(ix,x);
00340     ix &= 0x7fffffff;
00341         if(ix>=0x40200000)     {p = pr8; q= ps8;}
00342         else if(ix>=0x40122E8B){p = pr5; q= ps5;}
00343         else if(ix>=0x4006DB6D){p = pr3; q= ps3;}
00344         else if(ix>=0x40000000){p = pr2; q= ps2;}
00345         z = one/(x*x);
00346         r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
00347         s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
00348         return one+ r/s;
00349 }
00350         
00351 
00352 /* For x >= 8, the asymptotic expansions of qone is
00353  *  3/8 s - 105/1024 s^3 - ..., where s = 1/x.
00354  * We approximate pone by
00355  *  qone(x) = s*(0.375 + (R/S))
00356  * where  R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10
00357  *    S = 1 + qs1*s^2 + ... + qs6*s^12
00358  * and
00359  *  | qone(x)/s -0.375-R/S | <= 2  ** ( -61.13)
00360  */
00361 
00362 #ifdef __STDC__
00363 static const double qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
00364 #else
00365 static double qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
00366 #endif
00367   0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
00368  -1.02539062499992714161e-01, /* 0xBFBA3FFF, 0xFFFFFDF3 */
00369  -1.62717534544589987888e+01, /* 0xC0304591, 0xA26779F7 */
00370  -7.59601722513950107896e+02, /* 0xC087BCD0, 0x53E4B576 */
00371  -1.18498066702429587167e+04, /* 0xC0C724E7, 0x40F87415 */
00372  -4.84385124285750353010e+04, /* 0xC0E7A6D0, 0x65D09C6A */
00373 };
00374 #ifdef __STDC__
00375 static const double qs8[6] = {
00376 #else
00377 static double qs8[6] = {
00378 #endif
00379   1.61395369700722909556e+02, /* 0x40642CA6, 0xDE5BCDE5 */
00380   7.82538599923348465381e+03, /* 0x40BE9162, 0xD0D88419 */
00381   1.33875336287249578163e+05, /* 0x4100579A, 0xB0B75E98 */
00382   7.19657723683240939863e+05, /* 0x4125F653, 0x72869C19 */
00383   6.66601232617776375264e+05, /* 0x412457D2, 0x7719AD5C */
00384  -2.94490264303834643215e+05, /* 0xC111F969, 0x0EA5AA18 */
00385 };
00386 
00387 #ifdef __STDC__
00388 static const double qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
00389 #else
00390 static double qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
00391 #endif
00392  -2.08979931141764104297e-11, /* 0xBDB6FA43, 0x1AA1A098 */
00393  -1.02539050241375426231e-01, /* 0xBFBA3FFF, 0xCB597FEF */
00394  -8.05644828123936029840e+00, /* 0xC0201CE6, 0xCA03AD4B */
00395  -1.83669607474888380239e+02, /* 0xC066F56D, 0x6CA7B9B0 */
00396  -1.37319376065508163265e+03, /* 0xC09574C6, 0x6931734F */
00397  -2.61244440453215656817e+03, /* 0xC0A468E3, 0x88FDA79D */
00398 };
00399 #ifdef __STDC__
00400 static const double qs5[6] = {
00401 #else
00402 static double qs5[6] = {
00403 #endif
00404   8.12765501384335777857e+01, /* 0x405451B2, 0xFF5A11B2 */
00405   1.99179873460485964642e+03, /* 0x409F1F31, 0xE77BF839 */
00406   1.74684851924908907677e+04, /* 0x40D10F1F, 0x0D64CE29 */
00407   4.98514270910352279316e+04, /* 0x40E8576D, 0xAABAD197 */
00408   2.79480751638918118260e+04, /* 0x40DB4B04, 0xCF7C364B */
00409  -4.71918354795128470869e+03, /* 0xC0B26F2E, 0xFCFFA004 */
00410 };
00411 
00412 #ifdef __STDC__
00413 static const double qr3[6] = {
00414 #else
00415 static double qr3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
00416 #endif
00417  -5.07831226461766561369e-09, /* 0xBE35CFA9, 0xD38FC84F */
00418  -1.02537829820837089745e-01, /* 0xBFBA3FEB, 0x51AEED54 */
00419  -4.61011581139473403113e+00, /* 0xC01270C2, 0x3302D9FF */
00420  -5.78472216562783643212e+01, /* 0xC04CEC71, 0xC25D16DA */
00421  -2.28244540737631695038e+02, /* 0xC06C87D3, 0x4718D55F */
00422  -2.19210128478909325622e+02, /* 0xC06B66B9, 0x5F5C1BF6 */
00423 };
00424 #ifdef __STDC__
00425 static const double qs3[6] = {
00426 #else
00427 static double qs3[6] = {
00428 #endif
00429   4.76651550323729509273e+01, /* 0x4047D523, 0xCCD367E4 */
00430   6.73865112676699709482e+02, /* 0x40850EEB, 0xC031EE3E */
00431   3.38015286679526343505e+03, /* 0x40AA684E, 0x448E7C9A */
00432   5.54772909720722782367e+03, /* 0x40B5ABBA, 0xA61D54A6 */
00433   1.90311919338810798763e+03, /* 0x409DBC7A, 0x0DD4DF4B */
00434  -1.35201191444307340817e+02, /* 0xC060E670, 0x290A311F */
00435 };
00436 
00437 #ifdef __STDC__
00438 static const double qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
00439 #else
00440 static double qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
00441 #endif
00442  -1.78381727510958865572e-07, /* 0xBE87F126, 0x44C626D2 */
00443  -1.02517042607985553460e-01, /* 0xBFBA3E8E, 0x9148B010 */
00444  -2.75220568278187460720e+00, /* 0xC0060484, 0x69BB4EDA */
00445  -1.96636162643703720221e+01, /* 0xC033A9E2, 0xC168907F */
00446  -4.23253133372830490089e+01, /* 0xC04529A3, 0xDE104AAA */
00447  -2.13719211703704061733e+01, /* 0xC0355F36, 0x39CF6E52 */
00448 };
00449 #ifdef __STDC__
00450 static const double qs2[6] = {
00451 #else
00452 static double qs2[6] = {
00453 #endif
00454   2.95333629060523854548e+01, /* 0x403D888A, 0x78AE64FF */
00455   2.52981549982190529136e+02, /* 0x406F9F68, 0xDB821CBA */
00456   7.57502834868645436472e+02, /* 0x4087AC05, 0xCE49A0F7 */
00457   7.39393205320467245656e+02, /* 0x40871B25, 0x48D4C029 */
00458   1.55949003336666123687e+02, /* 0x40637E5E, 0x3C3ED8D4 */
00459  -4.95949898822628210127e+00, /* 0xC013D686, 0xE71BE86B */
00460 };
00461 
00462 #ifdef __STDC__
00463     static double qone(double x)
00464 #else
00465     static double qone(x)
00466     double x;
00467 #endif
00468 {
00469 #ifdef __STDC__
00470     const double *p = 0,*q = 0;
00471 #else
00472     double *p = 0,*q = 0;
00473 #endif
00474     double  s,r,z;
00475     int32_t ix;
00476     GET_HIGH_WORD(ix,x);
00477     ix &= 0x7fffffff;
00478     if(ix>=0x40200000)     {p = qr8; q= qs8;}
00479     else if(ix>=0x40122E8B){p = qr5; q= qs5;}
00480     else if(ix>=0x4006DB6D){p = qr3; q= qs3;}
00481     else if(ix>=0x40000000){p = qr2; q= qs2;}
00482     z = one/(x*x);
00483     r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
00484     s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
00485     return (.375 + r/s)/x;
00486 }