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glibc  2.9
e_j0.c
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00001 /* @(#)e_j0.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 /* Modified by Naohiko Shimizu/Tokai University, Japan 1997/08/26,
00013    for performance improvement on pipelined processors.
00014 */
00015 
00016 #if defined(LIBM_SCCS) && !defined(lint)
00017 static char rcsid[] = "$NetBSD: e_j0.c,v 1.8 1995/05/10 20:45:23 jtc Exp $";
00018 #endif
00019 
00020 /* __ieee754_j0(x), __ieee754_y0(x)
00021  * Bessel function of the first and second kinds of order zero.
00022  * Method -- j0(x):
00023  *     1. For tiny x, we use j0(x) = 1 - x^2/4 + x^4/64 - ...
00024  *     2. Reduce x to |x| since j0(x)=j0(-x),  and
00025  *        for x in (0,2)
00026  *            j0(x) = 1-z/4+ z^2*R0/S0,  where z = x*x;
00027  *        (precision:  |j0-1+z/4-z^2R0/S0 |<2**-63.67 )
00028  *        for x in (2,inf)
00029  *            j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)-q0(x)*sin(x0))
00030  *        where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
00031  *        as follow:
00032  *            cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
00033  *                   = 1/sqrt(2) * (cos(x) + sin(x))
00034  *            sin(x0) = sin(x)cos(pi/4)-cos(x)sin(pi/4)
00035  *                   = 1/sqrt(2) * (sin(x) - cos(x))
00036  *        (To avoid cancellation, use
00037  *            sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
00038  *         to compute the worse one.)
00039  *
00040  *     3 Special cases
00041  *            j0(nan)= nan
00042  *            j0(0) = 1
00043  *            j0(inf) = 0
00044  *
00045  * Method -- y0(x):
00046  *     1. For x<2.
00047  *        Since
00048  *            y0(x) = 2/pi*(j0(x)*(ln(x/2)+Euler) + x^2/4 - ...)
00049  *        therefore y0(x)-2/pi*j0(x)*ln(x) is an even function.
00050  *        We use the following function to approximate y0,
00051  *            y0(x) = U(z)/V(z) + (2/pi)*(j0(x)*ln(x)), z= x^2
00052  *        where
00053  *            U(z) = u00 + u01*z + ... + u06*z^6
00054  *            V(z) = 1  + v01*z + ... + v04*z^4
00055  *        with absolute approximation error bounded by 2**-72.
00056  *        Note: For tiny x, U/V = u0 and j0(x)~1, hence
00057  *            y0(tiny) = u0 + (2/pi)*ln(tiny), (choose tiny<2**-27)
00058  *     2. For x>=2.
00059  *            y0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)+q0(x)*sin(x0))
00060  *        where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
00061  *        by the method mentioned above.
00062  *     3. Special cases: y0(0)=-inf, y0(x<0)=NaN, y0(inf)=0.
00063  */
00064 
00065 #include "math.h"
00066 #include "math_private.h"
00067 
00068 #ifdef __STDC__
00069 static double pzero(double), qzero(double);
00070 #else
00071 static double pzero(), qzero();
00072 #endif
00073 
00074 #ifdef __STDC__
00075 static const double
00076 #else
00077 static double
00078 #endif
00079 huge   = 1e300,
00080 one    = 1.0,
00081 invsqrtpi=  5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
00082 tpi      =  6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
00083               /* R0/S0 on [0, 2.00] */
00084 R[]  =  {0.0, 0.0, 1.56249999999999947958e-02, /* 0x3F8FFFFF, 0xFFFFFFFD */
00085  -1.89979294238854721751e-04, /* 0xBF28E6A5, 0xB61AC6E9 */
00086   1.82954049532700665670e-06, /* 0x3EBEB1D1, 0x0C503919 */
00087  -4.61832688532103189199e-09}, /* 0xBE33D5E7, 0x73D63FCE */
00088 S[]  =  {0.0, 1.56191029464890010492e-02, /* 0x3F8FFCE8, 0x82C8C2A4 */
00089   1.16926784663337450260e-04, /* 0x3F1EA6D2, 0xDD57DBF4 */
00090   5.13546550207318111446e-07, /* 0x3EA13B54, 0xCE84D5A9 */
00091   1.16614003333790000205e-09}; /* 0x3E1408BC, 0xF4745D8F */
00092 
00093 #ifdef __STDC__
00094 static const double zero = 0.0;
00095 #else
00096 static double zero = 0.0;
00097 #endif
00098 
00099 #ifdef __STDC__
00100        double __ieee754_j0(double x)
00101 #else
00102        double __ieee754_j0(x)
00103        double x;
00104 #endif
00105 {
00106        double z, s,c,ss,cc,r,u,v,r1,r2,s1,s2,z2,z4;
00107        int32_t hx,ix;
00108 
00109        GET_HIGH_WORD(hx,x);
00110        ix = hx&0x7fffffff;
00111        if(ix>=0x7ff00000) return one/(x*x);
00112        x = fabs(x);
00113        if(ix >= 0x40000000) {      /* |x| >= 2.0 */
00114               __sincos (x, &s, &c);
00115               ss = s-c;
00116               cc = s+c;
00117               if(ix<0x7fe00000) {  /* make sure x+x not overflow */
00118                   z = -__cos(x+x);
00119                   if ((s*c)<zero) cc = z/ss;
00120                   else          ss = z/cc;
00121               }
00122        /*
00123         * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
00124         * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
00125         */
00126               if(ix>0x48000000) z = (invsqrtpi*cc)/__ieee754_sqrt(x);
00127               else {
00128                   u = pzero(x); v = qzero(x);
00129                   z = invsqrtpi*(u*cc-v*ss)/__ieee754_sqrt(x);
00130               }
00131               return z;
00132        }
00133        if(ix<0x3f200000) {  /* |x| < 2**-13 */
00134            if(huge+x>one) { /* raise inexact if x != 0 */
00135                if(ix<0x3e400000) return one;     /* |x|<2**-27 */
00136                else        return one - 0.25*x*x;
00137            }
00138        }
00139        z = x*x;
00140 #ifdef DO_NOT_USE_THIS
00141        r =  z*(R02+z*(R03+z*(R04+z*R05)));
00142        s =  one+z*(S01+z*(S02+z*(S03+z*S04)));
00143 #else
00144        r1 = z*R[2]; z2=z*z;
00145        r2 = R[3]+z*R[4]; z4=z2*z2;
00146        r  = r1 + z2*r2 + z4*R[5];
00147        s1 = one+z*S[1];
00148        s2 = S[2]+z*S[3];
00149        s = s1 + z2*s2 + z4*S[4];
00150 #endif
00151        if(ix < 0x3FF00000) {       /* |x| < 1.00 */
00152            return one + z*(-0.25+(r/s));
00153        } else {
00154            u = 0.5*x;
00155            return((one+u)*(one-u)+z*(r/s));
00156        }
00157 }
00158 
00159 #ifdef __STDC__
00160 static const double
00161 #else
00162 static double
00163 #endif
00164 U[]  = {-7.38042951086872317523e-02, /* 0xBFB2E4D6, 0x99CBD01F */
00165   1.76666452509181115538e-01, /* 0x3FC69D01, 0x9DE9E3FC */
00166  -1.38185671945596898896e-02, /* 0xBF8C4CE8, 0xB16CFA97 */
00167   3.47453432093683650238e-04, /* 0x3F36C54D, 0x20B29B6B */
00168  -3.81407053724364161125e-06, /* 0xBECFFEA7, 0x73D25CAD */
00169   1.95590137035022920206e-08, /* 0x3E550057, 0x3B4EABD4 */
00170  -3.98205194132103398453e-11}, /* 0xBDC5E43D, 0x693FB3C8 */
00171 V[]  =  {1.27304834834123699328e-02, /* 0x3F8A1270, 0x91C9C71A */
00172   7.60068627350353253702e-05, /* 0x3F13ECBB, 0xF578C6C1 */
00173   2.59150851840457805467e-07, /* 0x3E91642D, 0x7FF202FD */
00174   4.41110311332675467403e-10}; /* 0x3DFE5018, 0x3BD6D9EF */
00175 
00176 #ifdef __STDC__
00177        double __ieee754_y0(double x)
00178 #else
00179        double __ieee754_y0(x)
00180        double x;
00181 #endif
00182 {
00183        double z, s,c,ss,cc,u,v,z2,z4,z6,u1,u2,u3,v1,v2;
00184        int32_t hx,ix,lx;
00185 
00186        EXTRACT_WORDS(hx,lx,x);
00187         ix = 0x7fffffff&hx;
00188     /* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0, y0(0) is -inf.  */
00189        if(ix>=0x7ff00000) return  one/(x+x*x);
00190         if((ix|lx)==0) return -HUGE_VAL+x; /* -inf and overflow exception.  */
00191         if(hx<0) return zero/(zero*x);
00192         if(ix >= 0x40000000) {  /* |x| >= 2.0 */
00193         /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
00194          * where x0 = x-pi/4
00195          *      Better formula:
00196          *              cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
00197          *                      =  1/sqrt(2) * (sin(x) + cos(x))
00198          *              sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
00199          *                      =  1/sqrt(2) * (sin(x) - cos(x))
00200          * To avoid cancellation, use
00201          *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
00202          * to compute the worse one.
00203          */
00204               __sincos (x, &s, &c);
00205                 ss = s-c;
00206                 cc = s+c;
00207        /*
00208         * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
00209         * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
00210         */
00211                 if(ix<0x7fe00000) {  /* make sure x+x not overflow */
00212                     z = -__cos(x+x);
00213                     if ((s*c)<zero) cc = z/ss;
00214                     else            ss = z/cc;
00215                 }
00216                 if(ix>0x48000000) z = (invsqrtpi*ss)/__ieee754_sqrt(x);
00217                 else {
00218                     u = pzero(x); v = qzero(x);
00219                     z = invsqrtpi*(u*ss+v*cc)/__ieee754_sqrt(x);
00220                 }
00221                 return z;
00222        }
00223        if(ix<=0x3e400000) { /* x < 2**-27 */
00224            return(U[0] + tpi*__ieee754_log(x));
00225        }
00226        z = x*x;
00227 #ifdef DO_NOT_USE_THIS
00228        u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06)))));
00229        v = one+z*(v01+z*(v02+z*(v03+z*v04)));
00230 #else
00231        u1 = U[0]+z*U[1]; z2=z*z;
00232        u2 = U[2]+z*U[3]; z4=z2*z2;
00233        u3 = U[4]+z*U[5]; z6=z4*z2;
00234        u = u1 + z2*u2 + z4*u3 + z6*U[6];
00235        v1 = one+z*V[0];
00236        v2 = V[1]+z*V[2];
00237        v = v1 + z2*v2 + z4*V[3];
00238 #endif
00239        return(u/v + tpi*(__ieee754_j0(x)*__ieee754_log(x)));
00240 }
00241 
00242 /* The asymptotic expansions of pzero is
00243  *     1 - 9/128 s^2 + 11025/98304 s^4 - ...,    where s = 1/x.
00244  * For x >= 2, We approximate pzero by
00245  *     pzero(x) = 1 + (R/S)
00246  * where  R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10
00247  *       S = 1 + pS0*s^2 + ... + pS4*s^10
00248  * and
00249  *     | pzero(x)-1-R/S | <= 2  ** ( -60.26)
00250  */
00251 #ifdef __STDC__
00252 static const double pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
00253 #else
00254 static double pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
00255 #endif
00256   0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
00257  -7.03124999999900357484e-02, /* 0xBFB1FFFF, 0xFFFFFD32 */
00258  -8.08167041275349795626e+00, /* 0xC02029D0, 0xB44FA779 */
00259  -2.57063105679704847262e+02, /* 0xC0701102, 0x7B19E863 */
00260  -2.48521641009428822144e+03, /* 0xC0A36A6E, 0xCD4DCAFC */
00261  -5.25304380490729545272e+03, /* 0xC0B4850B, 0x36CC643D */
00262 };
00263 #ifdef __STDC__
00264 static const double pS8[5] = {
00265 #else
00266 static double pS8[5] = {
00267 #endif
00268   1.16534364619668181717e+02, /* 0x405D2233, 0x07A96751 */
00269   3.83374475364121826715e+03, /* 0x40ADF37D, 0x50596938 */
00270   4.05978572648472545552e+04, /* 0x40E3D2BB, 0x6EB6B05F */
00271   1.16752972564375915681e+05, /* 0x40FC810F, 0x8F9FA9BD */
00272   4.76277284146730962675e+04, /* 0x40E74177, 0x4F2C49DC */
00273 };
00274 
00275 #ifdef __STDC__
00276 static const double pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
00277 #else
00278 static double pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
00279 #endif
00280  -1.14125464691894502584e-11, /* 0xBDA918B1, 0x47E495CC */
00281  -7.03124940873599280078e-02, /* 0xBFB1FFFF, 0xE69AFBC6 */
00282  -4.15961064470587782438e+00, /* 0xC010A370, 0xF90C6BBF */
00283  -6.76747652265167261021e+01, /* 0xC050EB2F, 0x5A7D1783 */
00284  -3.31231299649172967747e+02, /* 0xC074B3B3, 0x6742CC63 */
00285  -3.46433388365604912451e+02, /* 0xC075A6EF, 0x28A38BD7 */
00286 };
00287 #ifdef __STDC__
00288 static const double pS5[5] = {
00289 #else
00290 static double pS5[5] = {
00291 #endif
00292   6.07539382692300335975e+01, /* 0x404E6081, 0x0C98C5DE */
00293   1.05125230595704579173e+03, /* 0x40906D02, 0x5C7E2864 */
00294   5.97897094333855784498e+03, /* 0x40B75AF8, 0x8FBE1D60 */
00295   9.62544514357774460223e+03, /* 0x40C2CCB8, 0xFA76FA38 */
00296   2.40605815922939109441e+03, /* 0x40A2CC1D, 0xC70BE864 */
00297 };
00298 
00299 #ifdef __STDC__
00300 static const double pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
00301 #else
00302 static double pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
00303 #endif
00304  -2.54704601771951915620e-09, /* 0xBE25E103, 0x6FE1AA86 */
00305  -7.03119616381481654654e-02, /* 0xBFB1FFF6, 0xF7C0E24B */
00306  -2.40903221549529611423e+00, /* 0xC00345B2, 0xAEA48074 */
00307  -2.19659774734883086467e+01, /* 0xC035F74A, 0x4CB94E14 */
00308  -5.80791704701737572236e+01, /* 0xC04D0A22, 0x420A1A45 */
00309  -3.14479470594888503854e+01, /* 0xC03F72AC, 0xA892D80F */
00310 };
00311 #ifdef __STDC__
00312 static const double pS3[5] = {
00313 #else
00314 static double pS3[5] = {
00315 #endif
00316   3.58560338055209726349e+01, /* 0x4041ED92, 0x84077DD3 */
00317   3.61513983050303863820e+02, /* 0x40769839, 0x464A7C0E */
00318   1.19360783792111533330e+03, /* 0x4092A66E, 0x6D1061D6 */
00319   1.12799679856907414432e+03, /* 0x40919FFC, 0xB8C39B7E */
00320   1.73580930813335754692e+02, /* 0x4065B296, 0xFC379081 */
00321 };
00322 
00323 #ifdef __STDC__
00324 static const double pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
00325 #else
00326 static double pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
00327 #endif
00328  -8.87534333032526411254e-08, /* 0xBE77D316, 0xE927026D */
00329  -7.03030995483624743247e-02, /* 0xBFB1FF62, 0x495E1E42 */
00330  -1.45073846780952986357e+00, /* 0xBFF73639, 0x8A24A843 */
00331  -7.63569613823527770791e+00, /* 0xC01E8AF3, 0xEDAFA7F3 */
00332  -1.11931668860356747786e+01, /* 0xC02662E6, 0xC5246303 */
00333  -3.23364579351335335033e+00, /* 0xC009DE81, 0xAF8FE70F */
00334 };
00335 #ifdef __STDC__
00336 static const double pS2[5] = {
00337 #else
00338 static double pS2[5] = {
00339 #endif
00340   2.22202997532088808441e+01, /* 0x40363865, 0x908B5959 */
00341   1.36206794218215208048e+02, /* 0x4061069E, 0x0EE8878F */
00342   2.70470278658083486789e+02, /* 0x4070E786, 0x42EA079B */
00343   1.53875394208320329881e+02, /* 0x40633C03, 0x3AB6FAFF */
00344   1.46576176948256193810e+01, /* 0x402D50B3, 0x44391809 */
00345 };
00346 
00347 #ifdef __STDC__
00348        static double pzero(double x)
00349 #else
00350        static double pzero(x)
00351        double x;
00352 #endif
00353 {
00354 #ifdef __STDC__
00355        const double *p,*q;
00356 #else
00357        double *p,*q;
00358 #endif
00359        double z,r,s,z2,z4,r1,r2,r3,s1,s2,s3;
00360        int32_t ix;
00361        GET_HIGH_WORD(ix,x);
00362        ix &= 0x7fffffff;
00363        if(ix>=0x40200000)     {p = pR8; q= pS8;}
00364        else if(ix>=0x40122E8B){p = pR5; q= pS5;}
00365        else if(ix>=0x4006DB6D){p = pR3; q= pS3;}
00366        else if(ix>=0x40000000){p = pR2; q= pS2;}
00367        z = one/(x*x);
00368 #ifdef DO_NOT_USE_THIS
00369        r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
00370        s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
00371 #else
00372        r1 = p[0]+z*p[1]; z2=z*z;
00373        r2 = p[2]+z*p[3]; z4=z2*z2;
00374        r3 = p[4]+z*p[5];
00375        r = r1 + z2*r2 + z4*r3;
00376        s1 = one+z*q[0];
00377        s2 = q[1]+z*q[2];
00378        s3 = q[3]+z*q[4];
00379        s = s1 + z2*s2 + z4*s3;
00380 #endif
00381        return one+ r/s;
00382 }
00383 
00384 
00385 /* For x >= 8, the asymptotic expansions of qzero is
00386  *     -1/8 s + 75/1024 s^3 - ..., where s = 1/x.
00387  * We approximate pzero by
00388  *     qzero(x) = s*(-1.25 + (R/S))
00389  * where  R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10
00390  *       S = 1 + qS0*s^2 + ... + qS5*s^12
00391  * and
00392  *     | qzero(x)/s +1.25-R/S | <= 2  ** ( -61.22)
00393  */
00394 #ifdef __STDC__
00395 static const double qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
00396 #else
00397 static double qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
00398 #endif
00399   0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
00400   7.32421874999935051953e-02, /* 0x3FB2BFFF, 0xFFFFFE2C */
00401   1.17682064682252693899e+01, /* 0x40278952, 0x5BB334D6 */
00402   5.57673380256401856059e+02, /* 0x40816D63, 0x15301825 */
00403   8.85919720756468632317e+03, /* 0x40C14D99, 0x3E18F46D */
00404   3.70146267776887834771e+04, /* 0x40E212D4, 0x0E901566 */
00405 };
00406 #ifdef __STDC__
00407 static const double qS8[6] = {
00408 #else
00409 static double qS8[6] = {
00410 #endif
00411   1.63776026895689824414e+02, /* 0x406478D5, 0x365B39BC */
00412   8.09834494656449805916e+03, /* 0x40BFA258, 0x4E6B0563 */
00413   1.42538291419120476348e+05, /* 0x41016652, 0x54D38C3F */
00414   8.03309257119514397345e+05, /* 0x412883DA, 0x83A52B43 */
00415   8.40501579819060512818e+05, /* 0x4129A66B, 0x28DE0B3D */
00416  -3.43899293537866615225e+05, /* 0xC114FD6D, 0x2C9530C5 */
00417 };
00418 
00419 #ifdef __STDC__
00420 static const double qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
00421 #else
00422 static double qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
00423 #endif
00424   1.84085963594515531381e-11, /* 0x3DB43D8F, 0x29CC8CD9 */
00425   7.32421766612684765896e-02, /* 0x3FB2BFFF, 0xD172B04C */
00426   5.83563508962056953777e+00, /* 0x401757B0, 0xB9953DD3 */
00427   1.35111577286449829671e+02, /* 0x4060E392, 0x0A8788E9 */
00428   1.02724376596164097464e+03, /* 0x40900CF9, 0x9DC8C481 */
00429   1.98997785864605384631e+03, /* 0x409F17E9, 0x53C6E3A6 */
00430 };
00431 #ifdef __STDC__
00432 static const double qS5[6] = {
00433 #else
00434 static double qS5[6] = {
00435 #endif
00436   8.27766102236537761883e+01, /* 0x4054B1B3, 0xFB5E1543 */
00437   2.07781416421392987104e+03, /* 0x40A03BA0, 0xDA21C0CE */
00438   1.88472887785718085070e+04, /* 0x40D267D2, 0x7B591E6D */
00439   5.67511122894947329769e+04, /* 0x40EBB5E3, 0x97E02372 */
00440   3.59767538425114471465e+04, /* 0x40E19118, 0x1F7A54A0 */
00441  -5.35434275601944773371e+03, /* 0xC0B4EA57, 0xBEDBC609 */
00442 };
00443 
00444 #ifdef __STDC__
00445 static const double qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
00446 #else
00447 static double qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
00448 #endif
00449   4.37741014089738620906e-09, /* 0x3E32CD03, 0x6ADECB82 */
00450   7.32411180042911447163e-02, /* 0x3FB2BFEE, 0x0E8D0842 */
00451   3.34423137516170720929e+00, /* 0x400AC0FC, 0x61149CF5 */
00452   4.26218440745412650017e+01, /* 0x40454F98, 0x962DAEDD */
00453   1.70808091340565596283e+02, /* 0x406559DB, 0xE25EFD1F */
00454   1.66733948696651168575e+02, /* 0x4064D77C, 0x81FA21E0 */
00455 };
00456 #ifdef __STDC__
00457 static const double qS3[6] = {
00458 #else
00459 static double qS3[6] = {
00460 #endif
00461   4.87588729724587182091e+01, /* 0x40486122, 0xBFE343A6 */
00462   7.09689221056606015736e+02, /* 0x40862D83, 0x86544EB3 */
00463   3.70414822620111362994e+03, /* 0x40ACF04B, 0xE44DFC63 */
00464   6.46042516752568917582e+03, /* 0x40B93C6C, 0xD7C76A28 */
00465   2.51633368920368957333e+03, /* 0x40A3A8AA, 0xD94FB1C0 */
00466  -1.49247451836156386662e+02, /* 0xC062A7EB, 0x201CF40F */
00467 };
00468 
00469 #ifdef __STDC__
00470 static const double qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
00471 #else
00472 static double qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
00473 #endif
00474   1.50444444886983272379e-07, /* 0x3E84313B, 0x54F76BDB */
00475   7.32234265963079278272e-02, /* 0x3FB2BEC5, 0x3E883E34 */
00476   1.99819174093815998816e+00, /* 0x3FFFF897, 0xE727779C */
00477   1.44956029347885735348e+01, /* 0x402CFDBF, 0xAAF96FE5 */
00478   3.16662317504781540833e+01, /* 0x403FAA8E, 0x29FBDC4A */
00479   1.62527075710929267416e+01, /* 0x403040B1, 0x71814BB4 */
00480 };
00481 #ifdef __STDC__
00482 static const double qS2[6] = {
00483 #else
00484 static double qS2[6] = {
00485 #endif
00486   3.03655848355219184498e+01, /* 0x403E5D96, 0xF7C07AED */
00487   2.69348118608049844624e+02, /* 0x4070D591, 0xE4D14B40 */
00488   8.44783757595320139444e+02, /* 0x408A6645, 0x22B3BF22 */
00489   8.82935845112488550512e+02, /* 0x408B977C, 0x9C5CC214 */
00490   2.12666388511798828631e+02, /* 0x406A9553, 0x0E001365 */
00491  -5.31095493882666946917e+00, /* 0xC0153E6A, 0xF8B32931 */
00492 };
00493 
00494 #ifdef __STDC__
00495        static double qzero(double x)
00496 #else
00497        static double qzero(x)
00498        double x;
00499 #endif
00500 {
00501 #ifdef __STDC__
00502        const double *p,*q;
00503 #else
00504        double *p,*q;
00505 #endif
00506        double s,r,z,z2,z4,z6,r1,r2,r3,s1,s2,s3;
00507        int32_t ix;
00508        GET_HIGH_WORD(ix,x);
00509        ix &= 0x7fffffff;
00510        if(ix>=0x40200000)     {p = qR8; q= qS8;}
00511        else if(ix>=0x40122E8B){p = qR5; q= qS5;}
00512        else if(ix>=0x4006DB6D){p = qR3; q= qS3;}
00513        else if(ix>=0x40000000){p = qR2; q= qS2;}
00514        z = one/(x*x);
00515 #ifdef DO_NOT_USE_THIS
00516        r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
00517        s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
00518 #else
00519        r1 = p[0]+z*p[1]; z2=z*z;
00520        r2 = p[2]+z*p[3]; z4=z2*z2;
00521        r3 = p[4]+z*p[5]; z6=z4*z2;
00522        r= r1 + z2*r2 + z4*r3;
00523        s1 = one+z*q[0];
00524        s2 = q[1]+z*q[2];
00525        s3 = q[3]+z*q[4];
00526        s = s1 + z2*s2 + z4*s3 +z6*q[5];
00527 #endif
00528        return (-.125 + r/s)/x;
00529 }