sysdeps/ieee754/dbl-64/s_erf.c - nest-cam/v366/glibc - Git at Google

 /* @(#)s_erf.c 5.1 93/09/24 */
 /*
  * ====================================================
  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
  *
  * Developed at SunPro, a Sun Microsystems, Inc. business.
  * Permission to use, copy, modify, and distribute this
  * software is freely granted, provided that this notice
  * is preserved.
  * ====================================================
  */
 /* Modified by Naohiko Shimizu/Tokai University, Japan 1997/08/25,
    for performance improvement on pipelined processors.
 */

 #if defined(LIBM_SCCS) && !defined(lint)
 static char rcsid[] = "$NetBSD: s_erf.c,v 1.8 1995/05/10 20:47:05 jtc Exp $";
 #endif

 /* double erf(double x)
  * double erfc(double x)
  *			     x
  *		      2      |\
  *     erf(x)  =  ---------  | exp(-t*t)dt
  *	 	   sqrt(pi) \|
  *			     0
  *
  *     erfc(x) =  1-erf(x)
  *  Note that
  *		erf(-x) = -erf(x)
  *		erfc(-x) = 2 - erfc(x)
  *
  * Method:
  *	1. For |x| in [0, 0.84375]
  *	    erf(x)  = x + x*R(x^2)
  *          erfc(x) = 1 - erf(x)           if x in [-.84375,0.25]
  *                  = 0.5 + ((0.5-x)-x*R)  if x in [0.25,0.84375]
  *	   where R = P/Q where P is an odd poly of degree 8 and
  *	   Q is an odd poly of degree 10.
  *						 -57.90
  *			| R - (erf(x)-x)/x | <= 2
  *
  *
  *	   Remark. The formula is derived by noting
  *          erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....)
  *	   and that
  *          2/sqrt(pi) = 1.128379167095512573896158903121545171688
  *	   is close to one. The interval is chosen because the fix
  *	   point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is
  *	   near 0.6174), and by some experiment, 0.84375 is chosen to
  * 	   guarantee the error is less than one ulp for erf.
  *
  *      2. For |x| in [0.84375,1.25], let s = |x| - 1, and
  *         c = 0.84506291151 rounded to single (24 bits)
  *         	erf(x)  = sign(x) * (c  + P1(s)/Q1(s))
  *         	erfc(x) = (1-c)  - P1(s)/Q1(s) if x > 0
  *			  1+(c+P1(s)/Q1(s))    if x < 0
  *         	|P1/Q1 - (erf(|x|)-c)| <= 2**-59.06
  *	   Remark: here we use the taylor series expansion at x=1.
  *		erf(1+s) = erf(1) + s*Poly(s)
  *			 = 0.845.. + P1(s)/Q1(s)
  *	   That is, we use rational approximation to approximate
  *			erf(1+s) - (c = (single)0.84506291151)
  *	   Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
  *	   where
  *		P1(s) = degree 6 poly in s
  *		Q1(s) = degree 6 poly in s
  *
  *      3. For x in [1.25,1/0.35(~2.857143)],
  *         	erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1)
  *         	erf(x)  = 1 - erfc(x)
  *	   where
  *		R1(z) = degree 7 poly in z, (z=1/x^2)
  *		S1(z) = degree 8 poly in z
  *
  *      4. For x in [1/0.35,28]
  *         	erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0
  *			= 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0
  *			= 2.0 - tiny		(if x <= -6)
  *         	erf(x)  = sign(x)*(1.0 - erfc(x)) if x < 6, else
  *         	erf(x)  = sign(x)*(1.0 - tiny)
  *	   where
  *		R2(z) = degree 6 poly in z, (z=1/x^2)
  *		S2(z) = degree 7 poly in z
  *
  *      Note1:
  *	   To compute exp(-x*x-0.5625+R/S), let s be a single
  *	   precision number and s := x; then
  *		-x*x = -s*s + (s-x)*(s+x)
  *	        exp(-x*x-0.5626+R/S) =
  *			exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S);
  *      Note2:
  *	   Here 4 and 5 make use of the asymptotic series
  *			  exp(-x*x)
  *		erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) )
  *			  x*sqrt(pi)
  *	   We use rational approximation to approximate
  *      	g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625
  *	   Here is the error bound for R1/S1 and R2/S2
  *      	|R1/S1 - f(x)|  < 2**(-62.57)
  *      	|R2/S2 - f(x)|  < 2**(-61.52)
  *
  *      5. For inf > x >= 28
  *         	erf(x)  = sign(x) *(1 - tiny)  (raise inexact)
  *         	erfc(x) = tiny*tiny (raise underflow) if x > 0
  *			= 2 - tiny if x<0
  *
  *      7. Special case:
  *         	erf(0)  = 0, erf(inf)  = 1, erf(-inf) = -1,
  *         	erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2,
  *	   	erfc/erf(NaN) is NaN
  */


 #include "math.h"
 #include "math_private.h"

 #ifdef __STDC__
 static const double
 #else
 static double
 #endif
 tiny	    = 1e-300,
 half=  5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
 one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
 two =  2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
 	/* c = (float)0.84506291151 */
 erx =  8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */
 /*
  * Coefficients for approximation to  erf on [0,0.84375]
  */
 efx =  1.28379167095512586316e-01, /* 0x3FC06EBA, 0x8214DB69 */
 efx8=  1.02703333676410069053e+00, /* 0x3FF06EBA, 0x8214DB69 */
 pp[]  =  {1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */
  -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */
  -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */
  -5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */
  -2.37630166566501626084e-05}, /* 0xBEF8EAD6, 0x120016AC */
 qq[]  =  {0.0, 3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */
   6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */
   5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */
   1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */
  -3.96022827877536812320e-06}, /* 0xBED09C43, 0x42A26120 */
 /*
  * Coefficients for approximation to  erf  in [0.84375,1.25]
  */
 pa[]  = {-2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */
   4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */
  -3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */
   3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */
  -1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */
   3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */
  -2.16637559486879084300e-03}, /* 0xBF61BF38, 0x0A96073F */
 qa[]  =  {0.0, 1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */
   5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */
   7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */
   1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */
   1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */
   1.19844998467991074170e-02}, /* 0x3F888B54, 0x5735151D */
 /*
  * Coefficients for approximation to  erfc in [1.25,1/0.35]
  */
 ra[]  = {-9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */
  -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */
  -1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */
  -6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */
  -1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */
  -1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */
  -8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */
  -9.81432934416914548592e+00}, /* 0xC023A0EF, 0xC69AC25C */
 sa[]  =  {0.0,1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */
   1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */
   4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */
   6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */
   4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */
   1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */
   6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */
  -6.04244152148580987438e-02}, /* 0xBFAEEFF2, 0xEE749A62 */
 /*
  * Coefficients for approximation to  erfc in [1/.35,28]
  */
 rb[]  = {-9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */
  -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */
  -1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */
  -1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */
  -6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */
  -1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */
  -4.83519191608651397019e+02}, /* 0xC07E384E, 0x9BDC383F */
 sb[]  =  {0.0,3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */
   3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */
   1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */
   3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */
   2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */
   4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */
  -2.24409524465858183362e+01}; /* 0xC03670E2, 0x42712D62 */

 #ifdef __STDC__
 	double __erf(double x)
 #else
 	double __erf(x)
 	double x;
 #endif
 {
 	int32_t hx,ix,i;
 	double R,S,P,Q,s,y,z,r;
 	GET_HIGH_WORD(hx,x);
 	ix = hx&0x7fffffff;
 	if(ix>=0x7ff00000) {		/* erf(nan)=nan */
 	    i = ((u_int32_t)hx>>31)<<1;
 	    return (double)(1-i)+one/x;	/* erf(+-inf)=+-1 */
 	}

 	if(ix < 0x3feb0000) {		/* |x|<0.84375 */
 	    double r1,r2,s1,s2,s3,z2,z4;
 	    if(ix < 0x3e300000) { 	/* |x|<2**-28 */
 	        if (ix < 0x00800000)
 		    return 0.125*(8.0*x+efx8*x);  /*avoid underflow */
 		return x + efx*x;
 	    }
 	    z = x*x;
 #ifdef DO_NOT_USE_THIS
 	    r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
 	    s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
 #else
 	    r1 = pp[0]+z*pp[1]; z2=z*z;
 	    r2 = pp[2]+z*pp[3]; z4=z2*z2;
 	    s1 = one+z*qq[1];
 	    s2 = qq[2]+z*qq[3];
 	    s3 = qq[4]+z*qq[5];
             r = r1 + z2*r2 + z4*pp[4];
 	    s  = s1 + z2*s2 + z4*s3;
 #endif
 	    y = r/s;
 	    return x + x*y;
 	}
 	if(ix < 0x3ff40000) {		/* 0.84375 <= |x| < 1.25 */
 	    double s2,s4,s6,P1,P2,P3,P4,Q1,Q2,Q3,Q4;
 	    s = fabs(x)-one;
 #ifdef DO_NOT_USE_THIS
 	    P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
 	    Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
 #else
 	    P1 = pa[0]+s*pa[1]; s2=s*s;
 	    Q1 = one+s*qa[1];   s4=s2*s2;
 	    P2 = pa[2]+s*pa[3]; s6=s4*s2;
 	    Q2 = qa[2]+s*qa[3];
 	    P3 = pa[4]+s*pa[5];
 	    Q3 = qa[4]+s*qa[5];
 	    P4 = pa[6];
 	    Q4 = qa[6];
 	    P = P1 + s2*P2 + s4*P3 + s6*P4;
 	    Q = Q1 + s2*Q2 + s4*Q3 + s6*Q4;
 #endif
 	    if(hx>=0) return erx + P/Q; else return -erx - P/Q;
 	}
 	if (ix >= 0x40180000) {		/* inf>|x|>=6 */
 	    if(hx>=0) return one-tiny; else return tiny-one;
 	}
 	x = fabs(x);
  	s = one/(x*x);
 	if(ix< 0x4006DB6E) {	/* |x| < 1/0.35 */
 #ifdef DO_NOT_USE_THIS
 	    R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(
 				ra5+s*(ra6+s*ra7))))));
 	    S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(
 				sa5+s*(sa6+s*(sa7+s*sa8)))))));
 #else
 	    double R1,R2,R3,R4,S1,S2,S3,S4,s2,s4,s6,s8;
 	    R1 = ra[0]+s*ra[1];s2 = s*s;
 	    S1 = one+s*sa[1];  s4 = s2*s2;
 	    R2 = ra[2]+s*ra[3];s6 = s4*s2;
 	    S2 = sa[2]+s*sa[3];s8 = s4*s4;
 	    R3 = ra[4]+s*ra[5];
 	    S3 = sa[4]+s*sa[5];
 	    R4 = ra[6]+s*ra[7];
 	    S4 = sa[6]+s*sa[7];
 	    R = R1 + s2*R2 + s4*R3 + s6*R4;
 	    S = S1 + s2*S2 + s4*S3 + s6*S4 + s8*sa[8];
 #endif
 	} else {	/* |x| >= 1/0.35 */
 #ifdef DO_NOT_USE_THIS
 	    R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(
 				rb5+s*rb6)))));
 	    S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(
 				sb5+s*(sb6+s*sb7))))));
 #else
 	    double R1,R2,R3,S1,S2,S3,S4,s2,s4,s6;
 	    R1 = rb[0]+s*rb[1];s2 = s*s;
 	    S1 = one+s*sb[1];  s4 = s2*s2;
 	    R2 = rb[2]+s*rb[3];s6 = s4*s2;
 	    S2 = sb[2]+s*sb[3];
 	    R3 = rb[4]+s*rb[5];
 	    S3 = sb[4]+s*sb[5];
 	    S4 = sb[6]+s*sb[7];
 	    R = R1 + s2*R2 + s4*R3 + s6*rb[6];
 	    S = S1 + s2*S2 + s4*S3 + s6*S4;
 #endif
 	}
 	z  = x;
 	SET_LOW_WORD(z,0);
 	r  =  __ieee754_exp(-z*z-0.5625)*__ieee754_exp((z-x)*(z+x)+R/S);
 	if(hx>=0) return one-r/x; else return  r/x-one;
 }
 weak_alias (__erf, erf)
 #ifdef NO_LONG_DOUBLE
 strong_alias (__erf, __erfl)
 weak_alias (__erf, erfl)
 #endif

 #ifdef __STDC__
 	double __erfc(double x)
 #else
 	double __erfc(x)
 	double x;
 #endif
 {
 	int32_t hx,ix;
 	double R,S,P,Q,s,y,z,r;
 	GET_HIGH_WORD(hx,x);
 	ix = hx&0x7fffffff;
 	if(ix>=0x7ff00000) {			/* erfc(nan)=nan */
 						/* erfc(+-inf)=0,2 */
 	    return (double)(((u_int32_t)hx>>31)<<1)+one/x;
 	}

 	if(ix < 0x3feb0000) {		/* |x|<0.84375 */
 	    double r1,r2,s1,s2,s3,z2,z4;
 	    if(ix < 0x3c700000)  	/* |x|<2**-56 */
 		return one-x;
 	    z = x*x;
 #ifdef DO_NOT_USE_THIS
 	    r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
 	    s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
 #else
 	    r1 = pp[0]+z*pp[1]; z2=z*z;
 	    r2 = pp[2]+z*pp[3]; z4=z2*z2;
 	    s1 = one+z*qq[1];
 	    s2 = qq[2]+z*qq[3];
 	    s3 = qq[4]+z*qq[5];
             r = r1 + z2*r2 + z4*pp[4];
 	    s  = s1 + z2*s2 + z4*s3;
 #endif
 	    y = r/s;
 	    if(hx < 0x3fd00000) {  	/* x<1/4 */
 		return one-(x+x*y);
 	    } else {
 		r = x*y;
 		r += (x-half);
 	        return half - r ;
 	    }
 	}
 	if(ix < 0x3ff40000) {		/* 0.84375 <= |x| < 1.25 */
 	    double s2,s4,s6,P1,P2,P3,P4,Q1,Q2,Q3,Q4;
 	    s = fabs(x)-one;
 #ifdef DO_NOT_USE_THIS
 	    P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
 	    Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
 #else
 	    P1 = pa[0]+s*pa[1]; s2=s*s;
 	    Q1 = one+s*qa[1];   s4=s2*s2;
 	    P2 = pa[2]+s*pa[3]; s6=s4*s2;
 	    Q2 = qa[2]+s*qa[3];
 	    P3 = pa[4]+s*pa[5];
 	    Q3 = qa[4]+s*qa[5];
 	    P4 = pa[6];
 	    Q4 = qa[6];
 	    P = P1 + s2*P2 + s4*P3 + s6*P4;
 	    Q = Q1 + s2*Q2 + s4*Q3 + s6*Q4;
 #endif
 	    if(hx>=0) {
 	        z  = one-erx; return z - P/Q;
 	    } else {
 		z = erx+P/Q; return one+z;
 	    }
 	}
 	if (ix < 0x403c0000) {		/* |x|<28 */
 	    x = fabs(x);
  	    s = one/(x*x);
 	    if(ix< 0x4006DB6D) {	/* |x| < 1/.35 ~ 2.857143*/
 #ifdef DO_NOT_USE_THIS
 	        R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(
 				ra5+s*(ra6+s*ra7))))));
 	        S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(
 				sa5+s*(sa6+s*(sa7+s*sa8)))))));
 #else
 		double R1,R2,R3,R4,S1,S2,S3,S4,s2,s4,s6,s8;
 	    R1 = ra[0]+s*ra[1];s2 = s*s;
 	    S1 = one+s*sa[1];  s4 = s2*s2;
 	    R2 = ra[2]+s*ra[3];s6 = s4*s2;
 	    S2 = sa[2]+s*sa[3];s8 = s4*s4;
 	    R3 = ra[4]+s*ra[5];
 	    S3 = sa[4]+s*sa[5];
 	    R4 = ra[6]+s*ra[7];
 	    S4 = sa[6]+s*sa[7];
 	    R = R1 + s2*R2 + s4*R3 + s6*R4;
 	    S = S1 + s2*S2 + s4*S3 + s6*S4 + s8*sa[8];
 #endif
 	    } else {			/* |x| >= 1/.35 ~ 2.857143 */
 		double R1,R2,R3,S1,S2,S3,S4,s2,s4,s6;
 		if(hx<0&&ix>=0x40180000) return two-tiny;/* x < -6 */
 #ifdef DO_NOT_USE_THIS
 	        R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(
 				rb5+s*rb6)))));
 	        S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(
 				sb5+s*(sb6+s*sb7))))));
 #else
 		R1 = rb[0]+s*rb[1];s2 = s*s;
 		S1 = one+s*sb[1];  s4 = s2*s2;
 		R2 = rb[2]+s*rb[3];s6 = s4*s2;
 		S2 = sb[2]+s*sb[3];
 		R3 = rb[4]+s*rb[5];
 		S3 = sb[4]+s*sb[5];
 		S4 = sb[6]+s*sb[7];
 		R = R1 + s2*R2 + s4*R3 + s6*rb[6];
 		S = S1 + s2*S2 + s4*S3 + s6*S4;
 #endif
 	    }
 	    z  = x;
 	    SET_LOW_WORD(z,0);
 	    r  =  __ieee754_exp(-z*z-0.5625)*
 			__ieee754_exp((z-x)*(z+x)+R/S);
 	    if(hx>0) return r/x; else return two-r/x;
 	} else {
 	    if(hx>0) return tiny*tiny; else return two-tiny;
 	}
 }
 weak_alias (__erfc, erfc)
 #ifdef NO_LONG_DOUBLE
 strong_alias (__erfc, __erfcl)
 weak_alias (__erfc, erfcl)
 #endif
	/* @(#)s_erf.c 5.1 93/09/24 */
	/*
	* ====================================================
	* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
	*
	* Developed at SunPro, a Sun Microsystems, Inc. business.
	* Permission to use, copy, modify, and distribute this
	* software is freely granted, provided that this notice
	* is preserved.
	* ====================================================
	*/
	/* Modified by Naohiko Shimizu/Tokai University, Japan 1997/08/25,
	for performance improvement on pipelined processors.
	*/

	#if defined(LIBM_SCCS) && !defined(lint)
	static char rcsid[] = "$NetBSD: s_erf.c,v 1.8 1995/05/10 20:47:05 jtc Exp $";
	#endif

	/* double erf(double x)
	* double erfc(double x)
	* x
	* 2 \|\
	* erf(x) = --------- \| exp(-t*t)dt
	* sqrt(pi) \\|
	* 0
	*
	* erfc(x) = 1-erf(x)
	* Note that
	* erf(-x) = -erf(x)
	* erfc(-x) = 2 - erfc(x)
	*
	* Method:
	* 1. For \|x\| in [0, 0.84375]
	* erf(x) = x + x*R(x^2)
	* erfc(x) = 1 - erf(x) if x in [-.84375,0.25]
	* = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375]
	* where R = P/Q where P is an odd poly of degree 8 and
	* Q is an odd poly of degree 10.
	* -57.90
	* \| R - (erf(x)-x)/x \| <= 2
	*
	*
	* Remark. The formula is derived by noting
	* erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....)
	* and that
	* 2/sqrt(pi) = 1.128379167095512573896158903121545171688
	* is close to one. The interval is chosen because the fix
	* point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is
	* near 0.6174), and by some experiment, 0.84375 is chosen to
	* guarantee the error is less than one ulp for erf.
	*
	* 2. For \|x\| in [0.84375,1.25], let s = \|x\| - 1, and
	* c = 0.84506291151 rounded to single (24 bits)
	* erf(x) = sign(x) * (c + P1(s)/Q1(s))
	* erfc(x) = (1-c) - P1(s)/Q1(s) if x > 0
	* 1+(c+P1(s)/Q1(s)) if x < 0
	* \|P1/Q1 - (erf(\|x\|)-c)\| <= 2**-59.06
	* Remark: here we use the taylor series expansion at x=1.
	* erf(1+s) = erf(1) + s*Poly(s)
	* = 0.845.. + P1(s)/Q1(s)
	* That is, we use rational approximation to approximate
	* erf(1+s) - (c = (single)0.84506291151)
	* Note that \|P1/Q1\|< 0.078 for x in [0.84375,1.25]
	* where
	* P1(s) = degree 6 poly in s
	* Q1(s) = degree 6 poly in s
	*
	* 3. For x in [1.25,1/0.35(~2.857143)],
	* erfc(x) = (1/x)exp(-xx-0.5625+R1/S1)
	* erf(x) = 1 - erfc(x)
	* where
	* R1(z) = degree 7 poly in z, (z=1/x^2)
	* S1(z) = degree 8 poly in z
	*
	* 4. For x in [1/0.35,28]
	* erfc(x) = (1/x)exp(-xx-0.5625+R2/S2) if x > 0
	* = 2.0 - (1/x)exp(-xx-0.5625+R2/S2) if -6<x<0
	* = 2.0 - tiny (if x <= -6)
	* erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6, else
	* erf(x) = sign(x)*(1.0 - tiny)
	* where
	* R2(z) = degree 6 poly in z, (z=1/x^2)
	* S2(z) = degree 7 poly in z
	*
	* Note1:
	* To compute exp(-x*x-0.5625+R/S), let s be a single
	* precision number and s := x; then
	* -xx = -ss + (s-x)*(s+x)
	* exp(-x*x-0.5626+R/S) =
	* exp(-ss-0.5625)exp((s-x)*(s+x)+R/S);
	* Note2:
	* Here 4 and 5 make use of the asymptotic series
	* exp(-x*x)
	* erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) )
	* x*sqrt(pi)
	* We use rational approximation to approximate
	* g(s)=f(1/x^2) = log(erfc(x)x) - xx + 0.5625
	* Here is the error bound for R1/S1 and R2/S2
	* \|R1/S1 - f(x)\| < 2**(-62.57)
	* \|R2/S2 - f(x)\| < 2**(-61.52)
	*
	* 5. For inf > x >= 28
	* erf(x) = sign(x) *(1 - tiny) (raise inexact)
	* erfc(x) = tiny*tiny (raise underflow) if x > 0
	* = 2 - tiny if x<0
	*
	* 7. Special case:
	* erf(0) = 0, erf(inf) = 1, erf(-inf) = -1,
	* erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2,
	* erfc/erf(NaN) is NaN
	*/


	#include "math.h"
	#include "math_private.h"

	#ifdef __STDC__
	static const double
	#else
	static double
	#endif
	tiny = 1e-300,
	half= 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
	one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
	two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
	/* c = (float)0.84506291151 */
	erx = 8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */
	/*
	* Coefficients for approximation to erf on [0,0.84375]
	*/
	efx = 1.28379167095512586316e-01, /* 0x3FC06EBA, 0x8214DB69 */
	efx8= 1.02703333676410069053e+00, /* 0x3FF06EBA, 0x8214DB69 */
	pp[] = {1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */
	-3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */
	-2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */
	-5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */
	-2.37630166566501626084e-05}, /* 0xBEF8EAD6, 0x120016AC */
	qq[] = {0.0, 3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */
	6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */
	5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */
	1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */
	-3.96022827877536812320e-06}, /* 0xBED09C43, 0x42A26120 */
	/*
	* Coefficients for approximation to erf in [0.84375,1.25]
	*/
	pa[] = {-2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */
	4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */
	-3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */
	3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */
	-1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */
	3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */
	-2.16637559486879084300e-03}, /* 0xBF61BF38, 0x0A96073F */
	qa[] = {0.0, 1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */
	5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */
	7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */
	1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */
	1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */
	1.19844998467991074170e-02}, /* 0x3F888B54, 0x5735151D */
	/*
	* Coefficients for approximation to erfc in [1.25,1/0.35]
	*/
	ra[] = {-9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */
	-6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */
	-1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */
	-6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */
	-1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */
	-1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */
	-8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */
	-9.81432934416914548592e+00}, /* 0xC023A0EF, 0xC69AC25C */
	sa[] = {0.0,1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */
	1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */
	4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */
	6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */
	4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */
	1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */
	6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */
	-6.04244152148580987438e-02}, /* 0xBFAEEFF2, 0xEE749A62 */
	/*
	* Coefficients for approximation to erfc in [1/.35,28]
	*/
	rb[] = {-9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */
	-7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */
	-1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */
	-1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */
	-6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */
	-1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */
	-4.83519191608651397019e+02}, /* 0xC07E384E, 0x9BDC383F */
	sb[] = {0.0,3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */
	3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */
	1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */
	3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */
	2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */
	4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */
	-2.24409524465858183362e+01}; /* 0xC03670E2, 0x42712D62 */

	#ifdef __STDC__
	double __erf(double x)
	#else
	double __erf(x)
	double x;
	#endif
	{
	int32_t hx,ix,i;
	double R,S,P,Q,s,y,z,r;
	GET_HIGH_WORD(hx,x);
	ix = hx&0x7fffffff;
	if(ix>=0x7ff00000) { /* erf(nan)=nan */
	i = ((u_int32_t)hx>>31)<<1;
	return (double)(1-i)+one/x; /* erf(+-inf)=+-1 */
	}

	if(ix < 0x3feb0000) { /* \|x\|<0.84375 */
	double r1,r2,s1,s2,s3,z2,z4;
	if(ix < 0x3e300000) { /* \|x\|<2*-28 /
	if (ix < 0x00800000)
	return 0.125(8.0x+efx8x); /avoid underflow */
	return x + efx*x;
	}
	z = x*x;
	#ifdef DO_NOT_USE_THIS
	r = pp0+z(pp1+z(pp2+z(pp3+zpp4)));
	s = one+z(qq1+z(qq2+z(qq3+z(qq4+z*qq5))));
	#else
	r1 = pp[0]+zpp[1]; z2=zz;
	r2 = pp[2]+zpp[3]; z4=z2z2;
	s1 = one+z*qq[1];
	s2 = qq[2]+z*qq[3];
	s3 = qq[4]+z*qq[5];
	r = r1 + z2r2 + z4pp[4];
	s = s1 + z2s2 + z4s3;
	#endif
	y = r/s;
	return x + x*y;
	}
	if(ix < 0x3ff40000) { /* 0.84375 <= \|x\| < 1.25 */
	double s2,s4,s6,P1,P2,P3,P4,Q1,Q2,Q3,Q4;
	s = fabs(x)-one;
	#ifdef DO_NOT_USE_THIS
	P = pa0+s(pa1+s(pa2+s(pa3+s(pa4+s(pa5+spa6)))));
	Q = one+s(qa1+s(qa2+s(qa3+s(qa4+s(qa5+sqa6)))));
	#else
	P1 = pa[0]+spa[1]; s2=ss;
	Q1 = one+sqa[1]; s4=s2s2;
	P2 = pa[2]+spa[3]; s6=s4s2;
	Q2 = qa[2]+s*qa[3];
	P3 = pa[4]+s*pa[5];
	Q3 = qa[4]+s*qa[5];
	P4 = pa[6];
	Q4 = qa[6];
	P = P1 + s2P2 + s4P3 + s6*P4;
	Q = Q1 + s2Q2 + s4Q3 + s6*Q4;
	#endif
	if(hx>=0) return erx + P/Q; else return -erx - P/Q;
	}
	if (ix >= 0x40180000) { /* inf>\|x\|>=6 */
	if(hx>=0) return one-tiny; else return tiny-one;
	}
	x = fabs(x);
	s = one/(x*x);
	if(ix< 0x4006DB6E) { /* \|x\| < 1/0.35 */
	#ifdef DO_NOT_USE_THIS
	R=ra0+s(ra1+s(ra2+s(ra3+s(ra4+s*(
	ra5+s(ra6+sra7))))));
	S=one+s(sa1+s(sa2+s(sa3+s(sa4+s*(
	sa5+s(sa6+s(sa7+s*sa8)))))));
	#else
	double R1,R2,R3,R4,S1,S2,S3,S4,s2,s4,s6,s8;
	R1 = ra[0]+sra[1];s2 = ss;
	S1 = one+ssa[1]; s4 = s2s2;
	R2 = ra[2]+sra[3];s6 = s4s2;
	S2 = sa[2]+ssa[3];s8 = s4s4;
	R3 = ra[4]+s*ra[5];
	S3 = sa[4]+s*sa[5];
	R4 = ra[6]+s*ra[7];
	S4 = sa[6]+s*sa[7];
	R = R1 + s2R2 + s4R3 + s6*R4;
	S = S1 + s2S2 + s4S3 + s6S4 + s8sa[8];
	#endif
	} else { /* \|x\| >= 1/0.35 */
	#ifdef DO_NOT_USE_THIS
	R=rb0+s(rb1+s(rb2+s(rb3+s(rb4+s*(
	rb5+s*rb6)))));
	S=one+s(sb1+s(sb2+s(sb3+s(sb4+s*(
	sb5+s(sb6+ssb7))))));
	#else
	double R1,R2,R3,S1,S2,S3,S4,s2,s4,s6;
	R1 = rb[0]+srb[1];s2 = ss;
	S1 = one+ssb[1]; s4 = s2s2;
	R2 = rb[2]+srb[3];s6 = s4s2;
	S2 = sb[2]+s*sb[3];
	R3 = rb[4]+s*rb[5];
	S3 = sb[4]+s*sb[5];
	S4 = sb[6]+s*sb[7];
	R = R1 + s2R2 + s4R3 + s6*rb[6];
	S = S1 + s2S2 + s4S3 + s6*S4;
	#endif
	}
	z = x;
	SET_LOW_WORD(z,0);
	r = __ieee754_exp(-zz-0.5625)__ieee754_exp((z-x)*(z+x)+R/S);
	if(hx>=0) return one-r/x; else return r/x-one;
	}
	weak_alias (__erf, erf)
	#ifdef NO_LONG_DOUBLE
	strong_alias (__erf, __erfl)
	weak_alias (__erf, erfl)
	#endif

	#ifdef __STDC__
	double __erfc(double x)
	#else
	double __erfc(x)
	double x;
	#endif
	{
	int32_t hx,ix;
	double R,S,P,Q,s,y,z,r;
	GET_HIGH_WORD(hx,x);
	ix = hx&0x7fffffff;
	if(ix>=0x7ff00000) { /* erfc(nan)=nan */
	/* erfc(+-inf)=0,2 */
	return (double)(((u_int32_t)hx>>31)<<1)+one/x;
	}

	if(ix < 0x3feb0000) { /* \|x\|<0.84375 */
	double r1,r2,s1,s2,s3,z2,z4;
	if(ix < 0x3c700000) /* \|x\|<2*-56 /
	return one-x;
	z = x*x;
	#ifdef DO_NOT_USE_THIS
	r = pp0+z(pp1+z(pp2+z(pp3+zpp4)));
	s = one+z(qq1+z(qq2+z(qq3+z(qq4+z*qq5))));
	#else
	r1 = pp[0]+zpp[1]; z2=zz;
	r2 = pp[2]+zpp[3]; z4=z2z2;
	s1 = one+z*qq[1];
	s2 = qq[2]+z*qq[3];
	s3 = qq[4]+z*qq[5];
	r = r1 + z2r2 + z4pp[4];
	s = s1 + z2s2 + z4s3;
	#endif
	y = r/s;
	if(hx < 0x3fd00000) { /* x<1/4 */
	return one-(x+x*y);
	} else {
	r = x*y;
	r += (x-half);
	return half - r ;
	}
	}
	if(ix < 0x3ff40000) { /* 0.84375 <= \|x\| < 1.25 */
	double s2,s4,s6,P1,P2,P3,P4,Q1,Q2,Q3,Q4;
	s = fabs(x)-one;
	#ifdef DO_NOT_USE_THIS
	P = pa0+s(pa1+s(pa2+s(pa3+s(pa4+s(pa5+spa6)))));
	Q = one+s(qa1+s(qa2+s(qa3+s(qa4+s(qa5+sqa6)))));
	#else
	P1 = pa[0]+spa[1]; s2=ss;
	Q1 = one+sqa[1]; s4=s2s2;
	P2 = pa[2]+spa[3]; s6=s4s2;
	Q2 = qa[2]+s*qa[3];
	P3 = pa[4]+s*pa[5];
	Q3 = qa[4]+s*qa[5];
	P4 = pa[6];
	Q4 = qa[6];
	P = P1 + s2P2 + s4P3 + s6*P4;
	Q = Q1 + s2Q2 + s4Q3 + s6*Q4;
	#endif
	if(hx>=0) {
	z = one-erx; return z - P/Q;
	} else {
	z = erx+P/Q; return one+z;
	}
	}
	if (ix < 0x403c0000) { /* \|x\|<28 */
	x = fabs(x);
	s = one/(x*x);
	if(ix< 0x4006DB6D) { /* \|x\| < 1/.35 ~ 2.857143*/
	#ifdef DO_NOT_USE_THIS
	R=ra0+s(ra1+s(ra2+s(ra3+s(ra4+s*(
	ra5+s(ra6+sra7))))));
	S=one+s(sa1+s(sa2+s(sa3+s(sa4+s*(
	sa5+s(sa6+s(sa7+s*sa8)))))));
	#else
	double R1,R2,R3,R4,S1,S2,S3,S4,s2,s4,s6,s8;
	R1 = ra[0]+sra[1];s2 = ss;
	S1 = one+ssa[1]; s4 = s2s2;
	R2 = ra[2]+sra[3];s6 = s4s2;
	S2 = sa[2]+ssa[3];s8 = s4s4;
	R3 = ra[4]+s*ra[5];
	S3 = sa[4]+s*sa[5];
	R4 = ra[6]+s*ra[7];
	S4 = sa[6]+s*sa[7];
	R = R1 + s2R2 + s4R3 + s6*R4;
	S = S1 + s2S2 + s4S3 + s6S4 + s8sa[8];
	#endif
	} else { /* \|x\| >= 1/.35 ~ 2.857143 */
	double R1,R2,R3,S1,S2,S3,S4,s2,s4,s6;
	if(hx<0&&ix>=0x40180000) return two-tiny;/* x < -6 */
	#ifdef DO_NOT_USE_THIS
	R=rb0+s(rb1+s(rb2+s(rb3+s(rb4+s*(
	rb5+s*rb6)))));
	S=one+s(sb1+s(sb2+s(sb3+s(sb4+s*(
	sb5+s(sb6+ssb7))))));
	#else
	R1 = rb[0]+srb[1];s2 = ss;
	S1 = one+ssb[1]; s4 = s2s2;
	R2 = rb[2]+srb[3];s6 = s4s2;
	S2 = sb[2]+s*sb[3];
	R3 = rb[4]+s*rb[5];
	S3 = sb[4]+s*sb[5];
	S4 = sb[6]+s*sb[7];
	R = R1 + s2R2 + s4R3 + s6*rb[6];
	S = S1 + s2S2 + s4S3 + s6*S4;
	#endif
	}
	z = x;
	SET_LOW_WORD(z,0);
	r = __ieee754_exp(-zz-0.5625)
	__ieee754_exp((z-x)*(z+x)+R/S);
	if(hx>0) return r/x; else return two-r/x;
	} else {
	if(hx>0) return tiny*tiny; else return two-tiny;
	}
	}
	weak_alias (__erfc, erfc)
	#ifdef NO_LONG_DOUBLE
	strong_alias (__erfc, __erfcl)
	weak_alias (__erfc, erfcl)
	#endif