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e_j0.c
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00001 /* -*- Mode: C; tab-width: 8; indent-tabs-mode: nil; c-basic-offset: 4 -*-
00002  *
00003  * ***** BEGIN LICENSE BLOCK *****
00004  * Version: MPL 1.1/GPL 2.0/LGPL 2.1
00005  *
00006  * The contents of this file are subject to the Mozilla Public License Version
00007  * 1.1 (the "License"); you may not use this file except in compliance with
00008  * the License. You may obtain a copy of the License at
00009  * http://www.mozilla.org/MPL/
00010  *
00011  * Software distributed under the License is distributed on an "AS IS" basis,
00012  * WITHOUT WARRANTY OF ANY KIND, either express or implied. See the License
00013  * for the specific language governing rights and limitations under the
00014  * License.
00015  *
00016  * The Original Code is Mozilla Communicator client code, released
00017  * March 31, 1998.
00018  *
00019  * The Initial Developer of the Original Code is
00020  * Sun Microsystems, Inc.
00021  * Portions created by the Initial Developer are Copyright (C) 1998
00022  * the Initial Developer. All Rights Reserved.
00023  *
00024  * Contributor(s):
00025  *
00026  * Alternatively, the contents of this file may be used under the terms of
00027  * either of the GNU General Public License Version 2 or later (the "GPL"),
00028  * or the GNU Lesser General Public License Version 2.1 or later (the "LGPL"),
00029  * in which case the provisions of the GPL or the LGPL are applicable instead
00030  * of those above. If you wish to allow use of your version of this file only
00031  * under the terms of either the GPL or the LGPL, and not to allow others to
00032  * use your version of this file under the terms of the MPL, indicate your
00033  * decision by deleting the provisions above and replace them with the notice
00034  * and other provisions required by the GPL or the LGPL. If you do not delete
00035  * the provisions above, a recipient may use your version of this file under
00036  * the terms of any one of the MPL, the GPL or the LGPL.
00037  *
00038  * ***** END LICENSE BLOCK ***** */
00039 
00040 /* @(#)e_j0.c 1.3 95/01/18 */
00041 /*
00042  * ====================================================
00043  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
00044  *
00045  * Developed at SunSoft, a Sun Microsystems, Inc. business.
00046  * Permission to use, copy, modify, and distribute this
00047  * software is freely granted, provided that this notice 
00048  * is preserved.
00049  * ====================================================
00050  */
00051 
00052 /* __ieee754_j0(x), __ieee754_y0(x)
00053  * Bessel function of the first and second kinds of order zero.
00054  * Method -- j0(x):
00055  *     1. For tiny x, we use j0(x) = 1 - x^2/4 + x^4/64 - ...
00056  *     2. Reduce x to |x| since j0(x)=j0(-x),  and
00057  *        for x in (0,2)
00058  *            j0(x) = 1-z/4+ z^2*R0/S0,  where z = x*x;
00059  *        (precision:  |j0-1+z/4-z^2R0/S0 |<2**-63.67 )
00060  *        for x in (2,inf)
00061  *            j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)-q0(x)*sin(x0))
00062  *        where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
00063  *        as follow:
00064  *            cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
00065  *                   = 1/sqrt(2) * (cos(x) + sin(x))
00066  *            sin(x0) = sin(x)cos(pi/4)-cos(x)sin(pi/4)
00067  *                   = 1/sqrt(2) * (sin(x) - cos(x))
00068  *        (To avoid cancellation, use
00069  *            sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
00070  *         to compute the worse one.)
00071  *        
00072  *     3 Special cases
00073  *            j0(nan)= nan
00074  *            j0(0) = 1
00075  *            j0(inf) = 0
00076  *            
00077  * Method -- y0(x):
00078  *     1. For x<2.
00079  *        Since 
00080  *            y0(x) = 2/pi*(j0(x)*(ln(x/2)+Euler) + x^2/4 - ...)
00081  *        therefore y0(x)-2/pi*j0(x)*ln(x) is an even function.
00082  *        We use the following function to approximate y0,
00083  *            y0(x) = U(z)/V(z) + (2/pi)*(j0(x)*ln(x)), z= x^2
00084  *        where 
00085  *            U(z) = u00 + u01*z + ... + u06*z^6
00086  *            V(z) = 1  + v01*z + ... + v04*z^4
00087  *        with absolute approximation error bounded by 2**-72.
00088  *        Note: For tiny x, U/V = u0 and j0(x)~1, hence
00089  *            y0(tiny) = u0 + (2/pi)*ln(tiny), (choose tiny<2**-27)
00090  *     2. For x>=2.
00091  *            y0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)+q0(x)*sin(x0))
00092  *        where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
00093  *        by the method mentioned above.
00094  *     3. Special cases: y0(0)=-inf, y0(x<0)=NaN, y0(inf)=0.
00095  */
00096 
00097 #include "fdlibm.h"
00098 
00099 #ifdef __STDC__
00100 static double pzero(double), qzero(double);
00101 #else
00102 static double pzero(), qzero();
00103 #endif
00104 
00105 #ifdef __STDC__
00106 static const double 
00107 #else
00108 static double 
00109 #endif
00110 really_big    = 1e300,
00111 one    = 1.0,
00112 invsqrtpi=  5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
00113 tpi      =  6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
00114               /* R0/S0 on [0, 2.00] */
00115 R02  =  1.56249999999999947958e-02, /* 0x3F8FFFFF, 0xFFFFFFFD */
00116 R03  = -1.89979294238854721751e-04, /* 0xBF28E6A5, 0xB61AC6E9 */
00117 R04  =  1.82954049532700665670e-06, /* 0x3EBEB1D1, 0x0C503919 */
00118 R05  = -4.61832688532103189199e-09, /* 0xBE33D5E7, 0x73D63FCE */
00119 S01  =  1.56191029464890010492e-02, /* 0x3F8FFCE8, 0x82C8C2A4 */
00120 S02  =  1.16926784663337450260e-04, /* 0x3F1EA6D2, 0xDD57DBF4 */
00121 S03  =  5.13546550207318111446e-07, /* 0x3EA13B54, 0xCE84D5A9 */
00122 S04  =  1.16614003333790000205e-09; /* 0x3E1408BC, 0xF4745D8F */
00123 
00124 static double zero = 0.0;
00125 
00126 #ifdef __STDC__
00127        double __ieee754_j0(double x) 
00128 #else
00129        double __ieee754_j0(x) 
00130        double x;
00131 #endif
00132 {
00133         fd_twoints un;
00134        double z, s,c,ss,cc,r,u,v;
00135        int hx,ix;
00136 
00137         un.d = x;
00138        hx = __HI(un);
00139        ix = hx&0x7fffffff;
00140        if(ix>=0x7ff00000) return one/(x*x);
00141        x = fd_fabs(x);
00142        if(ix >= 0x40000000) {      /* |x| >= 2.0 */
00143               s = fd_sin(x);
00144               c = fd_cos(x);
00145               ss = s-c;
00146               cc = s+c;
00147               if(ix<0x7fe00000) {  /* make sure x+x not overflow */
00148                   z = -fd_cos(x+x);
00149                   if ((s*c)<zero) cc = z/ss;
00150                   else          ss = z/cc;
00151               }
00152        /*
00153         * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
00154         * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
00155         */
00156               if(ix>0x48000000) z = (invsqrtpi*cc)/fd_sqrt(x);
00157               else {
00158                   u = pzero(x); v = qzero(x);
00159                   z = invsqrtpi*(u*cc-v*ss)/fd_sqrt(x);
00160               }
00161               return z;
00162        }
00163        if(ix<0x3f200000) {  /* |x| < 2**-13 */
00164            if(really_big+x>one) {  /* raise inexact if x != 0 */
00165                if(ix<0x3e400000) return one;     /* |x|<2**-27 */
00166                else        return one - 0.25*x*x;
00167            }
00168        }
00169        z = x*x;
00170        r =  z*(R02+z*(R03+z*(R04+z*R05)));
00171        s =  one+z*(S01+z*(S02+z*(S03+z*S04)));
00172        if(ix < 0x3FF00000) {       /* |x| < 1.00 */
00173            return one + z*(-0.25+(r/s));
00174        } else {
00175            u = 0.5*x;
00176            return((one+u)*(one-u)+z*(r/s));
00177        }
00178 }
00179 
00180 #ifdef __STDC__
00181 static const double
00182 #else
00183 static double
00184 #endif
00185 u00  = -7.38042951086872317523e-02, /* 0xBFB2E4D6, 0x99CBD01F */
00186 u01  =  1.76666452509181115538e-01, /* 0x3FC69D01, 0x9DE9E3FC */
00187 u02  = -1.38185671945596898896e-02, /* 0xBF8C4CE8, 0xB16CFA97 */
00188 u03  =  3.47453432093683650238e-04, /* 0x3F36C54D, 0x20B29B6B */
00189 u04  = -3.81407053724364161125e-06, /* 0xBECFFEA7, 0x73D25CAD */
00190 u05  =  1.95590137035022920206e-08, /* 0x3E550057, 0x3B4EABD4 */
00191 u06  = -3.98205194132103398453e-11, /* 0xBDC5E43D, 0x693FB3C8 */
00192 v01  =  1.27304834834123699328e-02, /* 0x3F8A1270, 0x91C9C71A */
00193 v02  =  7.60068627350353253702e-05, /* 0x3F13ECBB, 0xF578C6C1 */
00194 v03  =  2.59150851840457805467e-07, /* 0x3E91642D, 0x7FF202FD */
00195 v04  =  4.41110311332675467403e-10; /* 0x3DFE5018, 0x3BD6D9EF */
00196 
00197 #ifdef __STDC__
00198        double __ieee754_y0(double x) 
00199 #else
00200        double __ieee754_y0(x) 
00201        double x;
00202 #endif
00203 {
00204         fd_twoints un;
00205        double z, s,c,ss,cc,u,v;
00206        int hx,ix,lx;
00207 
00208         un.d = x;
00209         hx = __HI(un);
00210         ix = 0x7fffffff&hx;
00211         lx = __LO(un);
00212     /* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0  */
00213        if(ix>=0x7ff00000) return  one/(x+x*x); 
00214         if((ix|lx)==0) return -one/zero;
00215         if(hx<0) return zero/zero;
00216         if(ix >= 0x40000000) {  /* |x| >= 2.0 */
00217         /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
00218          * where x0 = x-pi/4
00219          *      Better formula:
00220          *              cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
00221          *                      =  1/sqrt(2) * (sin(x) + cos(x))
00222          *              sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
00223          *                      =  1/sqrt(2) * (sin(x) - cos(x))
00224          * To avoid cancellation, use
00225          *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
00226          * to compute the worse one.
00227          */
00228                 s = fd_sin(x);
00229                 c = fd_cos(x);
00230                 ss = s-c;
00231                 cc = s+c;
00232        /*
00233         * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
00234         * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
00235         */
00236                 if(ix<0x7fe00000) {  /* make sure x+x not overflow */
00237                     z = -fd_cos(x+x);
00238                     if ((s*c)<zero) cc = z/ss;
00239                     else            ss = z/cc;
00240                 }
00241                 if(ix>0x48000000) z = (invsqrtpi*ss)/fd_sqrt(x);
00242                 else {
00243                     u = pzero(x); v = qzero(x);
00244                     z = invsqrtpi*(u*ss+v*cc)/fd_sqrt(x);
00245                 }
00246                 return z;
00247        }
00248        if(ix<=0x3e400000) { /* x < 2**-27 */
00249            return(u00 + tpi*__ieee754_log(x));
00250        }
00251        z = x*x;
00252        u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06)))));
00253        v = one+z*(v01+z*(v02+z*(v03+z*v04)));
00254        return(u/v + tpi*(__ieee754_j0(x)*__ieee754_log(x)));
00255 }
00256 
00257 /* The asymptotic expansions of pzero is
00258  *     1 - 9/128 s^2 + 11025/98304 s^4 - ...,    where s = 1/x.
00259  * For x >= 2, We approximate pzero by
00260  *     pzero(x) = 1 + (R/S)
00261  * where  R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10
00262  *       S = 1 + pS0*s^2 + ... + pS4*s^10
00263  * and
00264  *     | pzero(x)-1-R/S | <= 2  ** ( -60.26)
00265  */
00266 #ifdef __STDC__
00267 static const double pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
00268 #else
00269 static double pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
00270 #endif
00271   0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
00272  -7.03124999999900357484e-02, /* 0xBFB1FFFF, 0xFFFFFD32 */
00273  -8.08167041275349795626e+00, /* 0xC02029D0, 0xB44FA779 */
00274  -2.57063105679704847262e+02, /* 0xC0701102, 0x7B19E863 */
00275  -2.48521641009428822144e+03, /* 0xC0A36A6E, 0xCD4DCAFC */
00276  -5.25304380490729545272e+03, /* 0xC0B4850B, 0x36CC643D */
00277 };
00278 #ifdef __STDC__
00279 static const double pS8[5] = {
00280 #else
00281 static double pS8[5] = {
00282 #endif
00283   1.16534364619668181717e+02, /* 0x405D2233, 0x07A96751 */
00284   3.83374475364121826715e+03, /* 0x40ADF37D, 0x50596938 */
00285   4.05978572648472545552e+04, /* 0x40E3D2BB, 0x6EB6B05F */
00286   1.16752972564375915681e+05, /* 0x40FC810F, 0x8F9FA9BD */
00287   4.76277284146730962675e+04, /* 0x40E74177, 0x4F2C49DC */
00288 };
00289 
00290 #ifdef __STDC__
00291 static const double pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
00292 #else
00293 static double pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
00294 #endif
00295  -1.14125464691894502584e-11, /* 0xBDA918B1, 0x47E495CC */
00296  -7.03124940873599280078e-02, /* 0xBFB1FFFF, 0xE69AFBC6 */
00297  -4.15961064470587782438e+00, /* 0xC010A370, 0xF90C6BBF */
00298  -6.76747652265167261021e+01, /* 0xC050EB2F, 0x5A7D1783 */
00299  -3.31231299649172967747e+02, /* 0xC074B3B3, 0x6742CC63 */
00300  -3.46433388365604912451e+02, /* 0xC075A6EF, 0x28A38BD7 */
00301 };
00302 #ifdef __STDC__
00303 static const double pS5[5] = {
00304 #else
00305 static double pS5[5] = {
00306 #endif
00307   6.07539382692300335975e+01, /* 0x404E6081, 0x0C98C5DE */
00308   1.05125230595704579173e+03, /* 0x40906D02, 0x5C7E2864 */
00309   5.97897094333855784498e+03, /* 0x40B75AF8, 0x8FBE1D60 */
00310   9.62544514357774460223e+03, /* 0x40C2CCB8, 0xFA76FA38 */
00311   2.40605815922939109441e+03, /* 0x40A2CC1D, 0xC70BE864 */
00312 };
00313 
00314 #ifdef __STDC__
00315 static const double pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
00316 #else
00317 static double pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
00318 #endif
00319  -2.54704601771951915620e-09, /* 0xBE25E103, 0x6FE1AA86 */
00320  -7.03119616381481654654e-02, /* 0xBFB1FFF6, 0xF7C0E24B */
00321  -2.40903221549529611423e+00, /* 0xC00345B2, 0xAEA48074 */
00322  -2.19659774734883086467e+01, /* 0xC035F74A, 0x4CB94E14 */
00323  -5.80791704701737572236e+01, /* 0xC04D0A22, 0x420A1A45 */
00324  -3.14479470594888503854e+01, /* 0xC03F72AC, 0xA892D80F */
00325 };
00326 #ifdef __STDC__
00327 static const double pS3[5] = {
00328 #else
00329 static double pS3[5] = {
00330 #endif
00331   3.58560338055209726349e+01, /* 0x4041ED92, 0x84077DD3 */
00332   3.61513983050303863820e+02, /* 0x40769839, 0x464A7C0E */
00333   1.19360783792111533330e+03, /* 0x4092A66E, 0x6D1061D6 */
00334   1.12799679856907414432e+03, /* 0x40919FFC, 0xB8C39B7E */
00335   1.73580930813335754692e+02, /* 0x4065B296, 0xFC379081 */
00336 };
00337 
00338 #ifdef __STDC__
00339 static const double pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
00340 #else
00341 static double pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
00342 #endif
00343  -8.87534333032526411254e-08, /* 0xBE77D316, 0xE927026D */
00344  -7.03030995483624743247e-02, /* 0xBFB1FF62, 0x495E1E42 */
00345  -1.45073846780952986357e+00, /* 0xBFF73639, 0x8A24A843 */
00346  -7.63569613823527770791e+00, /* 0xC01E8AF3, 0xEDAFA7F3 */
00347  -1.11931668860356747786e+01, /* 0xC02662E6, 0xC5246303 */
00348  -3.23364579351335335033e+00, /* 0xC009DE81, 0xAF8FE70F */
00349 };
00350 #ifdef __STDC__
00351 static const double pS2[5] = {
00352 #else
00353 static double pS2[5] = {
00354 #endif
00355   2.22202997532088808441e+01, /* 0x40363865, 0x908B5959 */
00356   1.36206794218215208048e+02, /* 0x4061069E, 0x0EE8878F */
00357   2.70470278658083486789e+02, /* 0x4070E786, 0x42EA079B */
00358   1.53875394208320329881e+02, /* 0x40633C03, 0x3AB6FAFF */
00359   1.46576176948256193810e+01, /* 0x402D50B3, 0x44391809 */
00360 };
00361 
00362 #ifdef __STDC__
00363        static double pzero(double x)
00364 #else
00365        static double pzero(x)
00366        double x;
00367 #endif
00368 {
00369 #ifdef __STDC__
00370        const double *p,*q;
00371 #else
00372        double *p,*q;
00373 #endif
00374         fd_twoints u;
00375        double z,r,s;
00376        int ix;
00377         u.d = x;
00378        ix = 0x7fffffff&__HI(u);
00379        if(ix>=0x40200000)     {p = pR8; q= pS8;}
00380        else if(ix>=0x40122E8B){p = pR5; q= pS5;}
00381        else if(ix>=0x4006DB6D){p = pR3; q= pS3;}
00382        else if(ix>=0x40000000){p = pR2; q= pS2;}
00383        z = one/(x*x);
00384        r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
00385        s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
00386        return one+ r/s;
00387 }
00388               
00389 
00390 /* For x >= 8, the asymptotic expansions of qzero is
00391  *     -1/8 s + 75/1024 s^3 - ..., where s = 1/x.
00392  * We approximate pzero by
00393  *     qzero(x) = s*(-1.25 + (R/S))
00394  * where  R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10
00395  *       S = 1 + qS0*s^2 + ... + qS5*s^12
00396  * and
00397  *     | qzero(x)/s +1.25-R/S | <= 2  ** ( -61.22)
00398  */
00399 #ifdef __STDC__
00400 static const double qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
00401 #else
00402 static double qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
00403 #endif
00404   0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
00405   7.32421874999935051953e-02, /* 0x3FB2BFFF, 0xFFFFFE2C */
00406   1.17682064682252693899e+01, /* 0x40278952, 0x5BB334D6 */
00407   5.57673380256401856059e+02, /* 0x40816D63, 0x15301825 */
00408   8.85919720756468632317e+03, /* 0x40C14D99, 0x3E18F46D */
00409   3.70146267776887834771e+04, /* 0x40E212D4, 0x0E901566 */
00410 };
00411 #ifdef __STDC__
00412 static const double qS8[6] = {
00413 #else
00414 static double qS8[6] = {
00415 #endif
00416   1.63776026895689824414e+02, /* 0x406478D5, 0x365B39BC */
00417   8.09834494656449805916e+03, /* 0x40BFA258, 0x4E6B0563 */
00418   1.42538291419120476348e+05, /* 0x41016652, 0x54D38C3F */
00419   8.03309257119514397345e+05, /* 0x412883DA, 0x83A52B43 */
00420   8.40501579819060512818e+05, /* 0x4129A66B, 0x28DE0B3D */
00421  -3.43899293537866615225e+05, /* 0xC114FD6D, 0x2C9530C5 */
00422 };
00423 
00424 #ifdef __STDC__
00425 static const double qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
00426 #else
00427 static double qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
00428 #endif
00429   1.84085963594515531381e-11, /* 0x3DB43D8F, 0x29CC8CD9 */
00430   7.32421766612684765896e-02, /* 0x3FB2BFFF, 0xD172B04C */
00431   5.83563508962056953777e+00, /* 0x401757B0, 0xB9953DD3 */
00432   1.35111577286449829671e+02, /* 0x4060E392, 0x0A8788E9 */
00433   1.02724376596164097464e+03, /* 0x40900CF9, 0x9DC8C481 */
00434   1.98997785864605384631e+03, /* 0x409F17E9, 0x53C6E3A6 */
00435 };
00436 #ifdef __STDC__
00437 static const double qS5[6] = {
00438 #else
00439 static double qS5[6] = {
00440 #endif
00441   8.27766102236537761883e+01, /* 0x4054B1B3, 0xFB5E1543 */
00442   2.07781416421392987104e+03, /* 0x40A03BA0, 0xDA21C0CE */
00443   1.88472887785718085070e+04, /* 0x40D267D2, 0x7B591E6D */
00444   5.67511122894947329769e+04, /* 0x40EBB5E3, 0x97E02372 */
00445   3.59767538425114471465e+04, /* 0x40E19118, 0x1F7A54A0 */
00446  -5.35434275601944773371e+03, /* 0xC0B4EA57, 0xBEDBC609 */
00447 };
00448 
00449 #ifdef __STDC__
00450 static const double qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
00451 #else
00452 static double qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
00453 #endif
00454   4.37741014089738620906e-09, /* 0x3E32CD03, 0x6ADECB82 */
00455   7.32411180042911447163e-02, /* 0x3FB2BFEE, 0x0E8D0842 */
00456   3.34423137516170720929e+00, /* 0x400AC0FC, 0x61149CF5 */
00457   4.26218440745412650017e+01, /* 0x40454F98, 0x962DAEDD */
00458   1.70808091340565596283e+02, /* 0x406559DB, 0xE25EFD1F */
00459   1.66733948696651168575e+02, /* 0x4064D77C, 0x81FA21E0 */
00460 };
00461 #ifdef __STDC__
00462 static const double qS3[6] = {
00463 #else
00464 static double qS3[6] = {
00465 #endif
00466   4.87588729724587182091e+01, /* 0x40486122, 0xBFE343A6 */
00467   7.09689221056606015736e+02, /* 0x40862D83, 0x86544EB3 */
00468   3.70414822620111362994e+03, /* 0x40ACF04B, 0xE44DFC63 */
00469   6.46042516752568917582e+03, /* 0x40B93C6C, 0xD7C76A28 */
00470   2.51633368920368957333e+03, /* 0x40A3A8AA, 0xD94FB1C0 */
00471  -1.49247451836156386662e+02, /* 0xC062A7EB, 0x201CF40F */
00472 };
00473 
00474 #ifdef __STDC__
00475 static const double qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
00476 #else
00477 static double qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
00478 #endif
00479   1.50444444886983272379e-07, /* 0x3E84313B, 0x54F76BDB */
00480   7.32234265963079278272e-02, /* 0x3FB2BEC5, 0x3E883E34 */
00481   1.99819174093815998816e+00, /* 0x3FFFF897, 0xE727779C */
00482   1.44956029347885735348e+01, /* 0x402CFDBF, 0xAAF96FE5 */
00483   3.16662317504781540833e+01, /* 0x403FAA8E, 0x29FBDC4A */
00484   1.62527075710929267416e+01, /* 0x403040B1, 0x71814BB4 */
00485 };
00486 #ifdef __STDC__
00487 static const double qS2[6] = {
00488 #else
00489 static double qS2[6] = {
00490 #endif
00491   3.03655848355219184498e+01, /* 0x403E5D96, 0xF7C07AED */
00492   2.69348118608049844624e+02, /* 0x4070D591, 0xE4D14B40 */
00493   8.44783757595320139444e+02, /* 0x408A6645, 0x22B3BF22 */
00494   8.82935845112488550512e+02, /* 0x408B977C, 0x9C5CC214 */
00495   2.12666388511798828631e+02, /* 0x406A9553, 0x0E001365 */
00496  -5.31095493882666946917e+00, /* 0xC0153E6A, 0xF8B32931 */
00497 };
00498 
00499 #ifdef __STDC__
00500        static double qzero(double x)
00501 #else
00502        static double qzero(x)
00503        double x;
00504 #endif
00505 {
00506 #ifdef __STDC__
00507        const double *p,*q;
00508 #else
00509        double *p,*q;
00510 #endif
00511         fd_twoints u;
00512        double s,r,z;
00513        int ix;
00514         u.d = x;
00515        ix = 0x7fffffff&__HI(u);
00516        if(ix>=0x40200000)     {p = qR8; q= qS8;}
00517        else if(ix>=0x40122E8B){p = qR5; q= qS5;}
00518        else if(ix>=0x4006DB6D){p = qR3; q= qS3;}
00519        else if(ix>=0x40000000){p = qR2; q= qS2;}
00520        z = one/(x*x);
00521        r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
00522        s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
00523        return (-.125 + r/s)/x;
00524 }