Totally rework the math library, this time based on the MacOs X

math library (which is itself based on the math lib from FreeBSD).
 -Erik

1 files changed, 314 insertions, 0 deletions

diff --git a/libm/s_erf.c b/libm/s_erf.c
new file mode 100644
index 000000000..e0bf2a115
--- /dev/null
+++ b/libm/s_erf.c
@@ -0,0 +1,314 @@
+/* @(#)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.
+ * ====================================================
+ */
+
+#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 */
+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
+{
+	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;
+}
+
+#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 */
+	    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;
+	}
+}