text: /* @(#)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.
   * ====================================================
   */
  
  #include <sys/cdefs.h>
  __FBSDID("$FreeBSD: soc2013/dpl/head/lib/msun/src/s_erf.c 176494 2008-02-22 02:30:36Z das $");
  
  /* 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"
  
  static const double
  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 */
  
  double
  erf(double x)
  {
  	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 */
  	    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;
  	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;
  }
  
  double
  erfc(double x)
  {
  	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 */
  	    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;
  	    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;
  	}
  }
  