/* @(#)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.
 * ====================================================
 */

/*
FUNCTION
        <<erf>>, <<erff>>, <<erfc>>, <<erfcf>>---error function
INDEX
	erf
INDEX
	erff
INDEX
	erfc
INDEX
	erfcf

SYNOPSIS
	#include <math.h>
	double erf(double <[x]>);
	float erff(float <[x]>);
	double erfc(double <[x]>);
	float erfcf(float <[x]>);

DESCRIPTION
	<<erf>> calculates an approximation to the ``error function'',
	which estimates the probability that an observation will fall within
	<[x]> standard deviations of the mean (assuming a normal
	distribution).
	@tex
	The error function is defined as
	$${2\over\sqrt\pi}\times\int_0^x e^{-t^2}dt$$
	 @end tex

	<<erfc>> calculates the complementary probability; that is,
	<<erfc(<[x]>)>> is <<1 - erf(<[x]>)>>.  <<erfc>> is computed directly,
	so that you can use it to avoid the loss of precision that would
	result from subtracting large probabilities (on large <[x]>) from 1.

	<<erff>> and <<erfcf>> differ from <<erf>> and <<erfc>> only in the
	argument and result types.

RETURNS
	For positive arguments, <<erf>> and all its variants return a
	probability---a number between 0 and 1.

PORTABILITY
	None of the variants of <<erf>> are ANSI C.
*/

/* 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 _NEED_FLOAT64

static const __float64
    tiny = _F_64(1e-300),
    half = _F_64(5.00000000000000000000e-01), /* 0x3FE00000, 0x00000000 */
    one = _F_64(1.00000000000000000000e+00), /* 0x3FF00000, 0x00000000 */
    two = _F_64(2.00000000000000000000e+00), /* 0x40000000, 0x00000000 */
    /* c = (float)0.84506291151 */
    erx = _F_64(8.45062911510467529297e-01), /* 0x3FEB0AC1, 0x60000000 */
    /*
 * Coefficients for approximation to  erf on [0,0.84375]
 */
    efx = _F_64(1.28379167095512586316e-01), /* 0x3FC06EBA, 0x8214DB69 */
    efx8 = _F_64(1.02703333676410069053e+00), /* 0x3FF06EBA, 0x8214DB69 */
    pp0 = _F_64(1.28379167095512558561e-01), /* 0x3FC06EBA, 0x8214DB68 */
    pp1 = _F_64(-3.25042107247001499370e-01), /* 0xBFD4CD7D, 0x691CB913 */
    pp2 = _F_64(-2.84817495755985104766e-02), /* 0xBF9D2A51, 0xDBD7194F */
    pp3 = _F_64(-5.77027029648944159157e-03), /* 0xBF77A291, 0x236668E4 */
    pp4 = _F_64(-2.37630166566501626084e-05), /* 0xBEF8EAD6, 0x120016AC */
    qq1 = _F_64(3.97917223959155352819e-01), /* 0x3FD97779, 0xCDDADC09 */
    qq2 = _F_64(6.50222499887672944485e-02), /* 0x3FB0A54C, 0x5536CEBA */
    qq3 = _F_64(5.08130628187576562776e-03), /* 0x3F74D022, 0xC4D36B0F */
    qq4 = _F_64(1.32494738004321644526e-04), /* 0x3F215DC9, 0x221C1A10 */
    qq5 = _F_64(-3.96022827877536812320e-06), /* 0xBED09C43, 0x42A26120 */
    /*
 * Coefficients for approximation to  erf  in [0.84375,1.25]
 */
    pa0 = _F_64(-2.36211856075265944077e-03), /* 0xBF6359B8, 0xBEF77538 */
    pa1 = _F_64(4.14856118683748331666e-01), /* 0x3FDA8D00, 0xAD92B34D */
    pa2 = _F_64(-3.72207876035701323847e-01), /* 0xBFD7D240, 0xFBB8C3F1 */
    pa3 = _F_64(3.18346619901161753674e-01), /* 0x3FD45FCA, 0x805120E4 */
    pa4 = _F_64(-1.10894694282396677476e-01), /* 0xBFBC6398, 0x3D3E28EC */
    pa5 = _F_64(3.54783043256182359371e-02), /* 0x3FA22A36, 0x599795EB */
    pa6 = _F_64(-2.16637559486879084300e-03), /* 0xBF61BF38, 0x0A96073F */
    qa1 = _F_64(1.06420880400844228286e-01), /* 0x3FBB3E66, 0x18EEE323 */
    qa2 = _F_64(5.40397917702171048937e-01), /* 0x3FE14AF0, 0x92EB6F33 */
    qa3 = _F_64(7.18286544141962662868e-02), /* 0x3FB2635C, 0xD99FE9A7 */
    qa4 = _F_64(1.26171219808761642112e-01), /* 0x3FC02660, 0xE763351F */
    qa5 = _F_64(1.36370839120290507362e-02), /* 0x3F8BEDC2, 0x6B51DD1C */
    qa6 = _F_64(1.19844998467991074170e-02), /* 0x3F888B54, 0x5735151D */
    /*
 * Coefficients for approximation to  erfc in [1.25,1/0.35]
 */
    ra0 = _F_64(-9.86494403484714822705e-03), /* 0xBF843412, 0x600D6435 */
    ra1 = _F_64(-6.93858572707181764372e-01), /* 0xBFE63416, 0xE4BA7360 */
    ra2 = _F_64(-1.05586262253232909814e+01), /* 0xC0251E04, 0x41B0E726 */
    ra3 = _F_64(-6.23753324503260060396e+01), /* 0xC04F300A, 0xE4CBA38D */
    ra4 = _F_64(-1.62396669462573470355e+02), /* 0xC0644CB1, 0x84282266 */
    ra5 = _F_64(-1.84605092906711035994e+02), /* 0xC067135C, 0xEBCCABB2 */
    ra6 = _F_64(-8.12874355063065934246e+01), /* 0xC0545265, 0x57E4D2F2 */
    ra7 = _F_64(-9.81432934416914548592e+00), /* 0xC023A0EF, 0xC69AC25C */
    sa1 = _F_64(1.96512716674392571292e+01), /* 0x4033A6B9, 0xBD707687 */
    sa2 = _F_64(1.37657754143519042600e+02), /* 0x4061350C, 0x526AE721 */
    sa3 = _F_64(4.34565877475229228821e+02), /* 0x407B290D, 0xD58A1A71 */
    sa4 = _F_64(6.45387271733267880336e+02), /* 0x40842B19, 0x21EC2868 */
    sa5 = _F_64(4.29008140027567833386e+02), /* 0x407AD021, 0x57700314 */
    sa6 = _F_64(1.08635005541779435134e+02), /* 0x405B28A3, 0xEE48AE2C */
    sa7 = _F_64(6.57024977031928170135e+00), /* 0x401A47EF, 0x8E484A93 */
    sa8 = _F_64(-6.04244152148580987438e-02), /* 0xBFAEEFF2, 0xEE749A62 */
    /*
 * Coefficients for approximation to  erfc in [1/.35,28]
 */
    rb0 = _F_64(-9.86494292470009928597e-03), /* 0xBF843412, 0x39E86F4A */
    rb1 = _F_64(-7.99283237680523006574e-01), /* 0xBFE993BA, 0x70C285DE */
    rb2 = _F_64(-1.77579549177547519889e+01), /* 0xC031C209, 0x555F995A */
    rb3 = _F_64(-1.60636384855821916062e+02), /* 0xC064145D, 0x43C5ED98 */
    rb4 = _F_64(-6.37566443368389627722e+02), /* 0xC083EC88, 0x1375F228 */
    rb5 = _F_64(-1.02509513161107724954e+03), /* 0xC0900461, 0x6A2E5992 */
    rb6 = _F_64(-4.83519191608651397019e+02), /* 0xC07E384E, 0x9BDC383F */
    sb1 = _F_64(3.03380607434824582924e+01), /* 0x403E568B, 0x261D5190 */
    sb2 = _F_64(3.25792512996573918826e+02), /* 0x40745CAE, 0x221B9F0A */
    sb3 = _F_64(1.53672958608443695994e+03), /* 0x409802EB, 0x189D5118 */
    sb4 = _F_64(3.19985821950859553908e+03), /* 0x40A8FFB7, 0x688C246A */
    sb5 = _F_64(2.55305040643316442583e+03), /* 0x40A3F219, 0xCEDF3BE6 */
    sb6 = _F_64(4.74528541206955367215e+02), /* 0x407DA874, 0xE79FE763 */
    sb7 = _F_64(-2.24409524465858183362e+01); /* 0xC03670E2, 0x42712D62 */

__float64
erf64(__float64 x)
{
    __int32_t hx, ix, i;
    __float64 R, S, P, Q, s, y, z, r;
    GET_HIGH_WORD(hx, x);
    ix = hx & 0x7fffffff;
    if (ix >= 0x7ff00000) { /* erf(nan)=nan */
        i = ((__uint32_t)hx >> 31) << 1;
        return (__float64)(1 - i) + one / x; /* erf(+-inf)=+-1 */
    }

    if (ix < 0x3feb0000) { /* |x|<0.84375 */
        if (ix < 0x3e300000) { /* |x|<2**-28 */
            if (ix < 0x00800000)
                return _F_64(0.125) * (_F_64(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 = fabs64(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 = fabs64(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;
    SET_LOW_WORD(z, 0);
    r = exp(-z * z - _F_64(0.5625)) * exp((z - x) * (z + x) + R / S);
    if (hx >= 0)
        return one - r / x;
    else
        return r / x - one;
}

_MATH_ALIAS_d_d(erf)

__float64
erfc64(__float64 x)
{
    __int32_t hx, ix;
    __float64 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 (__float64)(((__uint32_t)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 = fabs64(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 = fabs64(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;
        SET_LOW_WORD(z, 0);
        r = exp(-z * z - _F_64(0.5625)) * exp((z - x) * (z + x) + R / S);
        if (hx > 0)
            return r / x;
        else
            return two - r / x;
    } else {
        if (hx > 0)
            return __math_uflow(0);
        else
            return two - tiny;
    }
}

_MATH_ALIAS_d_d(erfc)

#endif /* _NEED_FLOAT64 */