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glibc  2.9
s_erf.c
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00001 /* @(#)s_erf.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 1997/08/25,
00013    for performance improvement on pipelined processors.
00014 */
00015 
00016 #if defined(LIBM_SCCS) && !defined(lint)
00017 static char rcsid[] = "$NetBSD: s_erf.c,v 1.8 1995/05/10 20:47:05 jtc Exp $";
00018 #endif
00019 
00020 /* double erf(double x)
00021  * double erfc(double x)
00022  *                        x
00023  *                  2      |\
00024  *     erf(x)  =  ---------  | exp(-t*t)dt
00025  *               sqrt(pi) \|
00026  *                        0
00027  *
00028  *     erfc(x) =  1-erf(x)
00029  *  Note that
00030  *            erf(-x) = -erf(x)
00031  *            erfc(-x) = 2 - erfc(x)
00032  *
00033  * Method:
00034  *     1. For |x| in [0, 0.84375]
00035  *         erf(x)  = x + x*R(x^2)
00036  *          erfc(x) = 1 - erf(x)           if x in [-.84375,0.25]
00037  *                  = 0.5 + ((0.5-x)-x*R)  if x in [0.25,0.84375]
00038  *        where R = P/Q where P is an odd poly of degree 8 and
00039  *        Q is an odd poly of degree 10.
00040  *                                         -57.90
00041  *                   | R - (erf(x)-x)/x | <= 2
00042  *
00043  *
00044  *        Remark. The formula is derived by noting
00045  *          erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....)
00046  *        and that
00047  *          2/sqrt(pi) = 1.128379167095512573896158903121545171688
00048  *        is close to one. The interval is chosen because the fix
00049  *        point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is
00050  *        near 0.6174), and by some experiment, 0.84375 is chosen to
00051  *        guarantee the error is less than one ulp for erf.
00052  *
00053  *      2. For |x| in [0.84375,1.25], let s = |x| - 1, and
00054  *         c = 0.84506291151 rounded to single (24 bits)
00055  *            erf(x)  = sign(x) * (c  + P1(s)/Q1(s))
00056  *            erfc(x) = (1-c)  - P1(s)/Q1(s) if x > 0
00057  *                     1+(c+P1(s)/Q1(s))    if x < 0
00058  *            |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06
00059  *        Remark: here we use the taylor series expansion at x=1.
00060  *            erf(1+s) = erf(1) + s*Poly(s)
00061  *                    = 0.845.. + P1(s)/Q1(s)
00062  *        That is, we use rational approximation to approximate
00063  *                   erf(1+s) - (c = (single)0.84506291151)
00064  *        Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
00065  *        where
00066  *            P1(s) = degree 6 poly in s
00067  *            Q1(s) = degree 6 poly in s
00068  *
00069  *      3. For x in [1.25,1/0.35(~2.857143)],
00070  *            erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1)
00071  *            erf(x)  = 1 - erfc(x)
00072  *        where
00073  *            R1(z) = degree 7 poly in z, (z=1/x^2)
00074  *            S1(z) = degree 8 poly in z
00075  *
00076  *      4. For x in [1/0.35,28]
00077  *            erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0
00078  *                   = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0
00079  *                   = 2.0 - tiny         (if x <= -6)
00080  *            erf(x)  = sign(x)*(1.0 - erfc(x)) if x < 6, else
00081  *            erf(x)  = sign(x)*(1.0 - tiny)
00082  *        where
00083  *            R2(z) = degree 6 poly in z, (z=1/x^2)
00084  *            S2(z) = degree 7 poly in z
00085  *
00086  *      Note1:
00087  *        To compute exp(-x*x-0.5625+R/S), let s be a single
00088  *        precision number and s := x; then
00089  *            -x*x = -s*s + (s-x)*(s+x)
00090  *             exp(-x*x-0.5626+R/S) =
00091  *                   exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S);
00092  *      Note2:
00093  *        Here 4 and 5 make use of the asymptotic series
00094  *                     exp(-x*x)
00095  *            erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) )
00096  *                     x*sqrt(pi)
00097  *        We use rational approximation to approximate
00098  *            g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625
00099  *        Here is the error bound for R1/S1 and R2/S2
00100  *            |R1/S1 - f(x)|  < 2**(-62.57)
00101  *            |R2/S2 - f(x)|  < 2**(-61.52)
00102  *
00103  *      5. For inf > x >= 28
00104  *            erf(x)  = sign(x) *(1 - tiny)  (raise inexact)
00105  *            erfc(x) = tiny*tiny (raise underflow) if x > 0
00106  *                   = 2 - tiny if x<0
00107  *
00108  *      7. Special case:
00109  *            erf(0)  = 0, erf(inf)  = 1, erf(-inf) = -1,
00110  *            erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2,
00111  *            erfc/erf(NaN) is NaN
00112  */
00113 
00114 
00115 #include "math.h"
00116 #include "math_private.h"
00117 
00118 #ifdef __STDC__
00119 static const double
00120 #else
00121 static double
00122 #endif
00123 tiny       = 1e-300,
00124 half=  5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
00125 one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
00126 two =  2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
00127        /* c = (float)0.84506291151 */
00128 erx =  8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */
00129 /*
00130  * Coefficients for approximation to  erf on [0,0.84375]
00131  */
00132 efx =  1.28379167095512586316e-01, /* 0x3FC06EBA, 0x8214DB69 */
00133 efx8=  1.02703333676410069053e+00, /* 0x3FF06EBA, 0x8214DB69 */
00134 pp[]  =  {1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */
00135  -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */
00136  -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */
00137  -5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */
00138  -2.37630166566501626084e-05}, /* 0xBEF8EAD6, 0x120016AC */
00139 qq[]  =  {0.0, 3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */
00140   6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */
00141   5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */
00142   1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */
00143  -3.96022827877536812320e-06}, /* 0xBED09C43, 0x42A26120 */
00144 /*
00145  * Coefficients for approximation to  erf  in [0.84375,1.25]
00146  */
00147 pa[]  = {-2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */
00148   4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */
00149  -3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */
00150   3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */
00151  -1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */
00152   3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */
00153  -2.16637559486879084300e-03}, /* 0xBF61BF38, 0x0A96073F */
00154 qa[]  =  {0.0, 1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */
00155   5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */
00156   7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */
00157   1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */
00158   1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */
00159   1.19844998467991074170e-02}, /* 0x3F888B54, 0x5735151D */
00160 /*
00161  * Coefficients for approximation to  erfc in [1.25,1/0.35]
00162  */
00163 ra[]  = {-9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */
00164  -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */
00165  -1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */
00166  -6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */
00167  -1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */
00168  -1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */
00169  -8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */
00170  -9.81432934416914548592e+00}, /* 0xC023A0EF, 0xC69AC25C */
00171 sa[]  =  {0.0,1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */
00172   1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */
00173   4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */
00174   6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */
00175   4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */
00176   1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */
00177   6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */
00178  -6.04244152148580987438e-02}, /* 0xBFAEEFF2, 0xEE749A62 */
00179 /*
00180  * Coefficients for approximation to  erfc in [1/.35,28]
00181  */
00182 rb[]  = {-9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */
00183  -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */
00184  -1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */
00185  -1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */
00186  -6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */
00187  -1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */
00188  -4.83519191608651397019e+02}, /* 0xC07E384E, 0x9BDC383F */
00189 sb[]  =  {0.0,3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */
00190   3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */
00191   1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */
00192   3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */
00193   2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */
00194   4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */
00195  -2.24409524465858183362e+01}; /* 0xC03670E2, 0x42712D62 */
00196 
00197 #ifdef __STDC__
00198        double __erf(double x)
00199 #else
00200        double __erf(x)
00201        double x;
00202 #endif
00203 {
00204        int32_t hx,ix,i;
00205        double R,S,P,Q,s,y,z,r;
00206        GET_HIGH_WORD(hx,x);
00207        ix = hx&0x7fffffff;
00208        if(ix>=0x7ff00000) {        /* erf(nan)=nan */
00209            i = ((u_int32_t)hx>>31)<<1;
00210            return (double)(1-i)+one/x;    /* erf(+-inf)=+-1 */
00211        }
00212 
00213        if(ix < 0x3feb0000) {              /* |x|<0.84375 */
00214            double r1,r2,s1,s2,s3,z2,z4;
00215            if(ix < 0x3e300000) {   /* |x|<2**-28 */
00216                if (ix < 0x00800000)
00217                   return 0.125*(8.0*x+efx8*x);  /*avoid underflow */
00218               return x + efx*x;
00219            }
00220            z = x*x;
00221 #ifdef DO_NOT_USE_THIS
00222            r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
00223            s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
00224 #else
00225            r1 = pp[0]+z*pp[1]; z2=z*z;
00226            r2 = pp[2]+z*pp[3]; z4=z2*z2;
00227            s1 = one+z*qq[1];
00228            s2 = qq[2]+z*qq[3];
00229            s3 = qq[4]+z*qq[5];
00230             r = r1 + z2*r2 + z4*pp[4];
00231            s  = s1 + z2*s2 + z4*s3;
00232 #endif
00233            y = r/s;
00234            return x + x*y;
00235        }
00236        if(ix < 0x3ff40000) {              /* 0.84375 <= |x| < 1.25 */
00237            double s2,s4,s6,P1,P2,P3,P4,Q1,Q2,Q3,Q4;
00238            s = fabs(x)-one;
00239 #ifdef DO_NOT_USE_THIS
00240            P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
00241            Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
00242 #else
00243            P1 = pa[0]+s*pa[1]; s2=s*s;
00244            Q1 = one+s*qa[1];   s4=s2*s2;
00245            P2 = pa[2]+s*pa[3]; s6=s4*s2;
00246            Q2 = qa[2]+s*qa[3];
00247            P3 = pa[4]+s*pa[5];
00248            Q3 = qa[4]+s*qa[5];
00249            P4 = pa[6];
00250            Q4 = qa[6];
00251            P = P1 + s2*P2 + s4*P3 + s6*P4;
00252            Q = Q1 + s2*Q2 + s4*Q3 + s6*Q4;
00253 #endif
00254            if(hx>=0) return erx + P/Q; else return -erx - P/Q;
00255        }
00256        if (ix >= 0x40180000) {            /* inf>|x|>=6 */
00257            if(hx>=0) return one-tiny; else return tiny-one;
00258        }
00259        x = fabs(x);
00260        s = one/(x*x);
00261        if(ix< 0x4006DB6E) { /* |x| < 1/0.35 */
00262 #ifdef DO_NOT_USE_THIS
00263            R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(
00264                             ra5+s*(ra6+s*ra7))))));
00265            S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(
00266                             sa5+s*(sa6+s*(sa7+s*sa8)))))));
00267 #else
00268            double R1,R2,R3,R4,S1,S2,S3,S4,s2,s4,s6,s8;
00269            R1 = ra[0]+s*ra[1];s2 = s*s;
00270            S1 = one+s*sa[1];  s4 = s2*s2;
00271            R2 = ra[2]+s*ra[3];s6 = s4*s2;
00272            S2 = sa[2]+s*sa[3];s8 = s4*s4;
00273            R3 = ra[4]+s*ra[5];
00274            S3 = sa[4]+s*sa[5];
00275            R4 = ra[6]+s*ra[7];
00276            S4 = sa[6]+s*sa[7];
00277            R = R1 + s2*R2 + s4*R3 + s6*R4;
00278            S = S1 + s2*S2 + s4*S3 + s6*S4 + s8*sa[8];
00279 #endif
00280        } else {      /* |x| >= 1/0.35 */
00281 #ifdef DO_NOT_USE_THIS
00282            R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(
00283                             rb5+s*rb6)))));
00284            S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(
00285                             sb5+s*(sb6+s*sb7))))));
00286 #else
00287            double R1,R2,R3,S1,S2,S3,S4,s2,s4,s6;
00288            R1 = rb[0]+s*rb[1];s2 = s*s;
00289            S1 = one+s*sb[1];  s4 = s2*s2;
00290            R2 = rb[2]+s*rb[3];s6 = s4*s2;
00291            S2 = sb[2]+s*sb[3];
00292            R3 = rb[4]+s*rb[5];
00293            S3 = sb[4]+s*sb[5];
00294            S4 = sb[6]+s*sb[7];
00295            R = R1 + s2*R2 + s4*R3 + s6*rb[6];
00296            S = S1 + s2*S2 + s4*S3 + s6*S4;
00297 #endif
00298        }
00299        z  = x;
00300        SET_LOW_WORD(z,0);
00301        r  =  __ieee754_exp(-z*z-0.5625)*__ieee754_exp((z-x)*(z+x)+R/S);
00302        if(hx>=0) return one-r/x; else return  r/x-one;
00303 }
00304 weak_alias (__erf, erf)
00305 #ifdef NO_LONG_DOUBLE
00306 strong_alias (__erf, __erfl)
00307 weak_alias (__erf, erfl)
00308 #endif
00309 
00310 #ifdef __STDC__
00311        double __erfc(double x)
00312 #else
00313        double __erfc(x)
00314        double x;
00315 #endif
00316 {
00317        int32_t hx,ix;
00318        double R,S,P,Q,s,y,z,r;
00319        GET_HIGH_WORD(hx,x);
00320        ix = hx&0x7fffffff;
00321        if(ix>=0x7ff00000) {               /* erfc(nan)=nan */
00322                                           /* erfc(+-inf)=0,2 */
00323            return (double)(((u_int32_t)hx>>31)<<1)+one/x;
00324        }
00325 
00326        if(ix < 0x3feb0000) {              /* |x|<0.84375 */
00327            double r1,r2,s1,s2,s3,z2,z4;
00328            if(ix < 0x3c700000)     /* |x|<2**-56 */
00329               return one-x;
00330            z = x*x;
00331 #ifdef DO_NOT_USE_THIS
00332            r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
00333            s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
00334 #else
00335            r1 = pp[0]+z*pp[1]; z2=z*z;
00336            r2 = pp[2]+z*pp[3]; z4=z2*z2;
00337            s1 = one+z*qq[1];
00338            s2 = qq[2]+z*qq[3];
00339            s3 = qq[4]+z*qq[5];
00340             r = r1 + z2*r2 + z4*pp[4];
00341            s  = s1 + z2*s2 + z4*s3;
00342 #endif
00343            y = r/s;
00344            if(hx < 0x3fd00000) {   /* x<1/4 */
00345               return one-(x+x*y);
00346            } else {
00347               r = x*y;
00348               r += (x-half);
00349                return half - r ;
00350            }
00351        }
00352        if(ix < 0x3ff40000) {              /* 0.84375 <= |x| < 1.25 */
00353            double s2,s4,s6,P1,P2,P3,P4,Q1,Q2,Q3,Q4;
00354            s = fabs(x)-one;
00355 #ifdef DO_NOT_USE_THIS
00356            P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
00357            Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
00358 #else
00359            P1 = pa[0]+s*pa[1]; s2=s*s;
00360            Q1 = one+s*qa[1];   s4=s2*s2;
00361            P2 = pa[2]+s*pa[3]; s6=s4*s2;
00362            Q2 = qa[2]+s*qa[3];
00363            P3 = pa[4]+s*pa[5];
00364            Q3 = qa[4]+s*qa[5];
00365            P4 = pa[6];
00366            Q4 = qa[6];
00367            P = P1 + s2*P2 + s4*P3 + s6*P4;
00368            Q = Q1 + s2*Q2 + s4*Q3 + s6*Q4;
00369 #endif
00370            if(hx>=0) {
00371                z  = one-erx; return z - P/Q;
00372            } else {
00373               z = erx+P/Q; return one+z;
00374            }
00375        }
00376        if (ix < 0x403c0000) {             /* |x|<28 */
00377            x = fabs(x);
00378            s = one/(x*x);
00379            if(ix< 0x4006DB6D) {    /* |x| < 1/.35 ~ 2.857143*/
00380 #ifdef DO_NOT_USE_THIS
00381                R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(
00382                             ra5+s*(ra6+s*ra7))))));
00383                S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(
00384                             sa5+s*(sa6+s*(sa7+s*sa8)))))));
00385 #else
00386               double R1,R2,R3,R4,S1,S2,S3,S4,s2,s4,s6,s8;
00387            R1 = ra[0]+s*ra[1];s2 = s*s;
00388            S1 = one+s*sa[1];  s4 = s2*s2;
00389            R2 = ra[2]+s*ra[3];s6 = s4*s2;
00390            S2 = sa[2]+s*sa[3];s8 = s4*s4;
00391            R3 = ra[4]+s*ra[5];
00392            S3 = sa[4]+s*sa[5];
00393            R4 = ra[6]+s*ra[7];
00394            S4 = sa[6]+s*sa[7];
00395            R = R1 + s2*R2 + s4*R3 + s6*R4;
00396            S = S1 + s2*S2 + s4*S3 + s6*S4 + s8*sa[8];
00397 #endif
00398            } else {                /* |x| >= 1/.35 ~ 2.857143 */
00399               double R1,R2,R3,S1,S2,S3,S4,s2,s4,s6;
00400               if(hx<0&&ix>=0x40180000) return two-tiny;/* x < -6 */
00401 #ifdef DO_NOT_USE_THIS
00402                R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(
00403                             rb5+s*rb6)))));
00404                S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(
00405                             sb5+s*(sb6+s*sb7))))));
00406 #else
00407               R1 = rb[0]+s*rb[1];s2 = s*s;
00408               S1 = one+s*sb[1];  s4 = s2*s2;
00409               R2 = rb[2]+s*rb[3];s6 = s4*s2;
00410               S2 = sb[2]+s*sb[3];
00411               R3 = rb[4]+s*rb[5];
00412               S3 = sb[4]+s*sb[5];
00413               S4 = sb[6]+s*sb[7];
00414               R = R1 + s2*R2 + s4*R3 + s6*rb[6];
00415               S = S1 + s2*S2 + s4*S3 + s6*S4;
00416 #endif
00417            }
00418            z  = x;
00419            SET_LOW_WORD(z,0);
00420            r  =  __ieee754_exp(-z*z-0.5625)*
00421                      __ieee754_exp((z-x)*(z+x)+R/S);
00422            if(hx>0) return r/x; else return two-r/x;
00423        } else {
00424            if(hx>0) return tiny*tiny; else return two-tiny;
00425        }
00426 }
00427 weak_alias (__erfc, erfc)
00428 #ifdef NO_LONG_DOUBLE
00429 strong_alias (__erfc, __erfcl)
00430 weak_alias (__erfc, erfcl)
00431 #endif