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s_erf.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
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00037  *
00038  * ***** END LICENSE BLOCK ***** */
00039 
00040 /* @(#)s_erf.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 /* double erf(double x)
00053  * double erfc(double x)
00054  *                        x
00055  *                  2      |\
00056  *     erf(x)  =  ---------  | exp(-t*t)dt
00057  *               sqrt(pi) \| 
00058  *                        0
00059  *
00060  *     erfc(x) =  1-erf(x)
00061  *  Note that 
00062  *            erf(-x) = -erf(x)
00063  *            erfc(-x) = 2 - erfc(x)
00064  *
00065  * Method:
00066  *     1. For |x| in [0, 0.84375]
00067  *         erf(x)  = x + x*R(x^2)
00068  *          erfc(x) = 1 - erf(x)           if x in [-.84375,0.25]
00069  *                  = 0.5 + ((0.5-x)-x*R)  if x in [0.25,0.84375]
00070  *        where R = P/Q where P is an odd poly of degree 8 and
00071  *        Q is an odd poly of degree 10.
00072  *                                         -57.90
00073  *                   | R - (erf(x)-x)/x | <= 2
00074  *     
00075  *
00076  *        Remark. The formula is derived by noting
00077  *          erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....)
00078  *        and that
00079  *          2/sqrt(pi) = 1.128379167095512573896158903121545171688
00080  *        is close to one. The interval is chosen because the fix
00081  *        point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is
00082  *        near 0.6174), and by some experiment, 0.84375 is chosen to
00083  *        guarantee the error is less than one ulp for erf.
00084  *
00085  *      2. For |x| in [0.84375,1.25], let s = |x| - 1, and
00086  *         c = 0.84506291151 rounded to single (24 bits)
00087  *            erf(x)  = sign(x) * (c  + P1(s)/Q1(s))
00088  *            erfc(x) = (1-c)  - P1(s)/Q1(s) if x > 0
00089  *                     1+(c+P1(s)/Q1(s))    if x < 0
00090  *            |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06
00091  *        Remark: here we use the taylor series expansion at x=1.
00092  *            erf(1+s) = erf(1) + s*Poly(s)
00093  *                    = 0.845.. + P1(s)/Q1(s)
00094  *        That is, we use rational approximation to approximate
00095  *                   erf(1+s) - (c = (single)0.84506291151)
00096  *        Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
00097  *        where 
00098  *            P1(s) = degree 6 poly in s
00099  *            Q1(s) = degree 6 poly in s
00100  *
00101  *      3. For x in [1.25,1/0.35(~2.857143)], 
00102  *            erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1)
00103  *            erf(x)  = 1 - erfc(x)
00104  *        where 
00105  *            R1(z) = degree 7 poly in z, (z=1/x^2)
00106  *            S1(z) = degree 8 poly in z
00107  *
00108  *      4. For x in [1/0.35,28]
00109  *            erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0
00110  *                   = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0
00111  *                   = 2.0 - tiny         (if x <= -6)
00112  *            erf(x)  = sign(x)*(1.0 - erfc(x)) if x < 6, else
00113  *            erf(x)  = sign(x)*(1.0 - tiny)
00114  *        where
00115  *            R2(z) = degree 6 poly in z, (z=1/x^2)
00116  *            S2(z) = degree 7 poly in z
00117  *
00118  *      Note1:
00119  *        To compute exp(-x*x-0.5625+R/S), let s be a single
00120  *        precision number and s := x; then
00121  *            -x*x = -s*s + (s-x)*(s+x)
00122  *             exp(-x*x-0.5626+R/S) = 
00123  *                   exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S);
00124  *      Note2:
00125  *        Here 4 and 5 make use of the asymptotic series
00126  *                     exp(-x*x)
00127  *            erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) )
00128  *                     x*sqrt(pi)
00129  *        We use rational approximation to approximate
00130  *            g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625
00131  *        Here is the error bound for R1/S1 and R2/S2
00132  *            |R1/S1 - f(x)|  < 2**(-62.57)
00133  *            |R2/S2 - f(x)|  < 2**(-61.52)
00134  *
00135  *      5. For inf > x >= 28
00136  *            erf(x)  = sign(x) *(1 - tiny)  (raise inexact)
00137  *            erfc(x) = tiny*tiny (raise underflow) if x > 0
00138  *                   = 2 - tiny if x<0
00139  *
00140  *      7. Special case:
00141  *            erf(0)  = 0, erf(inf)  = 1, erf(-inf) = -1,
00142  *            erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2, 
00143  *            erfc/erf(NaN) is NaN
00144  */
00145 
00146 
00147 #include "fdlibm.h"
00148 
00149 #ifdef __STDC__
00150 static const double
00151 #else
00152 static double
00153 #endif
00154 tiny       = 1e-300,
00155 half=  5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
00156 one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
00157 two =  2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
00158        /* c = (float)0.84506291151 */
00159 erx =  8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */
00160 /*
00161  * Coefficients for approximation to  erf on [0,0.84375]
00162  */
00163 efx =  1.28379167095512586316e-01, /* 0x3FC06EBA, 0x8214DB69 */
00164 efx8=  1.02703333676410069053e+00, /* 0x3FF06EBA, 0x8214DB69 */
00165 pp0  =  1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */
00166 pp1  = -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */
00167 pp2  = -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */
00168 pp3  = -5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */
00169 pp4  = -2.37630166566501626084e-05, /* 0xBEF8EAD6, 0x120016AC */
00170 qq1  =  3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */
00171 qq2  =  6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */
00172 qq3  =  5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */
00173 qq4  =  1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */
00174 qq5  = -3.96022827877536812320e-06, /* 0xBED09C43, 0x42A26120 */
00175 /*
00176  * Coefficients for approximation to  erf  in [0.84375,1.25] 
00177  */
00178 pa0  = -2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */
00179 pa1  =  4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */
00180 pa2  = -3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */
00181 pa3  =  3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */
00182 pa4  = -1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */
00183 pa5  =  3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */
00184 pa6  = -2.16637559486879084300e-03, /* 0xBF61BF38, 0x0A96073F */
00185 qa1  =  1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */
00186 qa2  =  5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */
00187 qa3  =  7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */
00188 qa4  =  1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */
00189 qa5  =  1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */
00190 qa6  =  1.19844998467991074170e-02, /* 0x3F888B54, 0x5735151D */
00191 /*
00192  * Coefficients for approximation to  erfc in [1.25,1/0.35]
00193  */
00194 ra0  = -9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */
00195 ra1  = -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */
00196 ra2  = -1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */
00197 ra3  = -6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */
00198 ra4  = -1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */
00199 ra5  = -1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */
00200 ra6  = -8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */
00201 ra7  = -9.81432934416914548592e+00, /* 0xC023A0EF, 0xC69AC25C */
00202 sa1  =  1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */
00203 sa2  =  1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */
00204 sa3  =  4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */
00205 sa4  =  6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */
00206 sa5  =  4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */
00207 sa6  =  1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */
00208 sa7  =  6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */
00209 sa8  = -6.04244152148580987438e-02, /* 0xBFAEEFF2, 0xEE749A62 */
00210 /*
00211  * Coefficients for approximation to  erfc in [1/.35,28]
00212  */
00213 rb0  = -9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */
00214 rb1  = -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */
00215 rb2  = -1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */
00216 rb3  = -1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */
00217 rb4  = -6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */
00218 rb5  = -1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */
00219 rb6  = -4.83519191608651397019e+02, /* 0xC07E384E, 0x9BDC383F */
00220 sb1  =  3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */
00221 sb2  =  3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */
00222 sb3  =  1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */
00223 sb4  =  3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */
00224 sb5  =  2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */
00225 sb6  =  4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */
00226 sb7  = -2.24409524465858183362e+01; /* 0xC03670E2, 0x42712D62 */
00227 
00228 #ifdef __STDC__
00229        double fd_erf(double x) 
00230 #else
00231        double fd_erf(x) 
00232        double x;
00233 #endif
00234 {
00235         fd_twoints u;
00236        int hx,ix,i;
00237        double R,S,P,Q,s,y,z,r;
00238         u.d = x;
00239        hx = __HI(u);
00240        ix = hx&0x7fffffff;
00241        if(ix>=0x7ff00000) {        /* erf(nan)=nan */
00242            i = ((unsigned)hx>>31)<<1;
00243            return (double)(1-i)+one/x;    /* erf(+-inf)=+-1 */
00244        }
00245 
00246        if(ix < 0x3feb0000) {              /* |x|<0.84375 */
00247            if(ix < 0x3e300000) {   /* |x|<2**-28 */
00248                if (ix < 0x00800000) 
00249                   return 0.125*(8.0*x+efx8*x);  /*avoid underflow */
00250               return x + efx*x;
00251            }
00252            z = x*x;
00253            r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
00254            s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
00255            y = r/s;
00256            return x + x*y;
00257        }
00258        if(ix < 0x3ff40000) {              /* 0.84375 <= |x| < 1.25 */
00259            s = fd_fabs(x)-one;
00260            P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
00261            Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
00262            if(hx>=0) return erx + P/Q; else return -erx - P/Q;
00263        }
00264        if (ix >= 0x40180000) {            /* inf>|x|>=6 */
00265            if(hx>=0) return one-tiny; else return tiny-one;
00266        }
00267        x = fd_fabs(x);
00268        s = one/(x*x);
00269        if(ix< 0x4006DB6E) { /* |x| < 1/0.35 */
00270            R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(
00271                             ra5+s*(ra6+s*ra7))))));
00272            S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(
00273                             sa5+s*(sa6+s*(sa7+s*sa8)))))));
00274        } else {      /* |x| >= 1/0.35 */
00275            R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(
00276                             rb5+s*rb6)))));
00277            S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(
00278                             sb5+s*(sb6+s*sb7))))));
00279        }
00280        z  = x;  
00281         u.d = z;
00282        __LO(u) = 0;
00283         z = u.d;
00284        r  =  __ieee754_exp(-z*z-0.5625)*__ieee754_exp((z-x)*(z+x)+R/S);
00285        if(hx>=0) return one-r/x; else return  r/x-one;
00286 }
00287 
00288 #ifdef __STDC__
00289        double erfc(double x) 
00290 #else
00291        double erfc(x) 
00292        double x;
00293 #endif
00294 {
00295         fd_twoints u;
00296        int hx,ix;
00297        double R,S,P,Q,s,y,z,r;
00298         u.d = x;
00299        hx = __HI(u);
00300        ix = hx&0x7fffffff;
00301        if(ix>=0x7ff00000) {               /* erfc(nan)=nan */
00302                                           /* erfc(+-inf)=0,2 */
00303            return (double)(((unsigned)hx>>31)<<1)+one/x;
00304        }
00305 
00306        if(ix < 0x3feb0000) {              /* |x|<0.84375 */
00307            if(ix < 0x3c700000)     /* |x|<2**-56 */
00308               return one-x;
00309            z = x*x;
00310            r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
00311            s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
00312            y = r/s;
00313            if(hx < 0x3fd00000) {   /* x<1/4 */
00314               return one-(x+x*y);
00315            } else {
00316               r = x*y;
00317               r += (x-half);
00318                return half - r ;
00319            }
00320        }
00321        if(ix < 0x3ff40000) {              /* 0.84375 <= |x| < 1.25 */
00322            s = fd_fabs(x)-one;
00323            P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
00324            Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
00325            if(hx>=0) {
00326                z  = one-erx; return z - P/Q; 
00327            } else {
00328               z = erx+P/Q; return one+z;
00329            }
00330        }
00331        if (ix < 0x403c0000) {             /* |x|<28 */
00332            x = fd_fabs(x);
00333            s = one/(x*x);
00334            if(ix< 0x4006DB6D) {    /* |x| < 1/.35 ~ 2.857143*/
00335                R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(
00336                             ra5+s*(ra6+s*ra7))))));
00337                S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(
00338                             sa5+s*(sa6+s*(sa7+s*sa8)))))));
00339            } else {                /* |x| >= 1/.35 ~ 2.857143 */
00340               if(hx<0&&ix>=0x40180000) return two-tiny;/* x < -6 */
00341                R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(
00342                             rb5+s*rb6)))));
00343                S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(
00344                             sb5+s*(sb6+s*sb7))))));
00345            }
00346            z  = x;
00347             u.d = z;
00348            __LO(u)  = 0;
00349             z = u.d;
00350            r  =  __ieee754_exp(-z*z-0.5625)*
00351                      __ieee754_exp((z-x)*(z+x)+R/S);
00352            if(hx>0) return r/x; else return two-r/x;
00353        } else {
00354            if(hx>0) return tiny*tiny; else return two-tiny;
00355        }
00356 }