/* @(#)s_erf.c 1.3 95/01/18 */

/*

 * ====================================================

 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.

 *

 * Developed at SunSoft, a Sun Microsystems, Inc. business.

 * Permission to use, copy, modify, and distribute this

 * software is freely granted, provided that this notice 

 * is preserved.

 * ====================================================

 */

/* 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 "fdlibm.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 */

pp0  =  1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */

pp1  = -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */

pp2  = -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */

pp3  = -5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */

pp4  = -2.37630166566501626084e-05, /* 0xBEF8EAD6, 0x120016AC */

qq1  =  3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */

qq2  =  6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */

qq3  =  5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */

qq4  =  1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */

qq5  = -3.96022827877536812320e-06, /* 0xBED09C43, 0x42A26120 */

/*

 * Coefficients for approximation to  erf  in [0.84375,1.25] 

 */

pa0  = -2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */

pa1  =  4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */

pa2  = -3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */

pa3  =  3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */

pa4  = -1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */

pa5  =  3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */

pa6  = -2.16637559486879084300e-03, /* 0xBF61BF38, 0x0A96073F */

qa1  =  1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */

qa2  =  5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */

qa3  =  7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */

qa4  =  1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */

qa5  =  1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */

qa6  =  1.19844998467991074170e-02, /* 0x3F888B54, 0x5735151D */

/*

 * Coefficients for approximation to  erfc in [1.25,1/0.35]

 */

ra0  = -9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */

ra1  = -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */

ra2  = -1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */

ra3  = -6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */

ra4  = -1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */

ra5  = -1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */

ra6  = -8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */

ra7  = -9.81432934416914548592e+00, /* 0xC023A0EF, 0xC69AC25C */

sa1  =  1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */

sa2  =  1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */

sa3  =  4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */

sa4  =  6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */

sa5  =  4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */

sa6  =  1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */

sa7  =  6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */

sa8  = -6.04244152148580987438e-02, /* 0xBFAEEFF2, 0xEE749A62 */

/*

 * Coefficients for approximation to  erfc in [1/.35,28]

 */

rb0  = -9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */

rb1  = -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */

rb2  = -1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */

rb3  = -1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */

rb4  = -6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */

rb5  = -1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */

rb6  = -4.83519191608651397019e+02, /* 0xC07E384E, 0x9BDC383F */

sb1  =  3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */

sb2  =  3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */

sb3  =  1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */

sb4  =  3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */

sb5  =  2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */

sb6  =  4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */

sb7  = -2.24409524465858183362e+01; /* 0xC03670E2, 0x42712D62 */

#ifdef __STDC__

	double erf(double x) 

#else

	double erf(x) 

	double x;

#endif

{

	int hx,ix,i;

	double R,S,P,Q,s,y,z,r;

	hx = __HI(x);

	ix = hx&0x7fffffff;

	if(ix>=0x7ff00000) {		/* erf(nan)=nan */

	    i = ((unsigned)hx>>31)<<1;

	    return (double)(1-i)+one/x;	/* erf(+-inf)=+-1 */

	}

	if(ix < 0x3feb0000) {		/* |x|<0.84375 */

	    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;

	    r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));

	    s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));

	    y = r/s;

	    return x + x*y;

	}

	if(ix < 0x3ff40000) {		/* 0.84375 <= |x| < 1.25 */

	    s = fabs(x)-one;

	    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)))));

	    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 */

	    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 {	/* |x| >= 1/0.35 */

	    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))))));

	}

	z  = x;  

	__LO(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;

}

#ifdef __STDC__

	double erfc(double x) 

#else

	double erfc(x) 

	double x;

#endif

{

	int hx,ix;

	double R,S,P,Q,s,y,z,r;

	hx = __HI(x);

	ix = hx&0x7fffffff;

	if(ix>=0x7ff00000) {			/* erfc(nan)=nan */

						/* erfc(+-inf)=0,2 */

	    return (double)(((unsigned)hx>>31)<<1)+one/x;

	}

	if(ix < 0x3feb0000) {		/* |x|<0.84375 */

	    if(ix < 0x3c700000)  	/* |x|<2**-56 */

		return one-x;

	    z = x*x;

	    r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));

	    s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));

	    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 */

	    s = fabs(x)-one;

	    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)))));

	    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*/

	        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 {			/* |x| >= 1/.35 ~ 2.857143 */

		if(hx<0&&ix>=0x40180000) return two-tiny;/* x < -6 */

	        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))))));

	    }

	    z  = x;

	    __LO(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;

	}

}