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

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