1 files changed, 396 insertions, 0 deletions
diff --git a/lib/libm/common_source/erf.c b/lib/libm/common_source/erf.c
new file mode 100644
index 000000000000..308f1a952d73
--- /dev/null
+++ b/lib/libm/common_source/erf.c
@@ -0,0 +1,396 @@
+/*-
+ * Copyright (c) 1992, 1993
+ *	The Regents of the University of California.  All rights reserved.
+ *
+ * Redistribution and use in source and binary forms, with or without
+ * modification, are permitted provided that the following conditions
+ * are met:
+ * 1. Redistributions of source code must retain the above copyright
+ *    notice, this list of conditions and the following disclaimer.
+ * 2. Redistributions in binary form must reproduce the above copyright
+ *    notice, this list of conditions and the following disclaimer in the
+ *    documentation and/or other materials provided with the distribution.
+ * 3. All advertising materials mentioning features or use of this software
+ *    must display the following acknowledgement:
+ *	This product includes software developed by the University of
+ *	California, Berkeley and its contributors.
+ * 4. Neither the name of the University nor the names of its contributors
+ *    may be used to endorse or promote products derived from this software
+ *    without specific prior written permission.
+ *
+ * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
+ * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
+ * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
+ * ARE DISCLAIMED.  IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
+ * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
+ * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
+ * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
+ * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
+ * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
+ * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
+ * SUCH DAMAGE.
+ */
+
+#ifndef lint
+static char sccsid[] = "@(#)erf.c	8.1 (Berkeley) 6/4/93";
+#endif /* not lint */
+
+/* Modified Nov 30, 1992 P. McILROY:
+ *	Replaced expansions for x >= 1.25 (error 1.7ulp vs ~6ulp)
+ * Replaced even+odd with direct calculation for x < .84375,
+ * to avoid destructive cancellation.
+ *
+ * Performance of erfc(x):
+ * In 300000 trials in the range [.83, .84375] the
+ * maximum observed error was 3.6ulp.
+ *
+ * In [.84735,1.25] the maximum observed error was <2.5ulp in
+ * 100000 runs in the range [1.2, 1.25].
+ *
+ * In [1.25,26] (Not including subnormal results)
+ * the error is < 1.7ulp.
+ */
+
+/* double erf(double x)
+ * double erfc(double x)
+ *			     x
+ *		      2      |\
+ *     erf(x)  =  ---------  | exp(-t*t)dt
+ *		   sqrt(pi) \|
+ *			     0
+ *
+ *     erfc(x) =  1-erf(x)
+ *
+ * Method:
+ *      1. Reduce x to |x| by erf(-x) = -erf(x)
+ *	2. For x in [0, 0.84375]
+ *	    erf(x)  = x + x*P(x^2)
+ *          erfc(x) = 1 - erf(x)           if x<=0.25
+ *                  = 0.5 + ((0.5-x)-x*P)  if x in [0.25,0.84375]
+ *	   where
+ *			2		 2	  4		  20  
+ *              P =  P(x ) = (p0 + p1 * x + p2 * x + ... + p10 * x  )
+ * 	   is an approximation to (erf(x)-x)/x with precision
+ *
+ *						 -56.45
+ *			| P - (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 fixed
+ *	   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.
+ *
+ *      3. For x in [0.84375,1.25], let s = x - 1, and
+ *         c = 0.84506291151 rounded to single (24 bits)
+ *         	erf(x)  = c  + P1(s)/Q1(s)
+ *         	erfc(x) = (1-c)  - P1(s)/Q1(s)
+ *         	|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
+ *
+ *	4. For x in [1.25, 2]; [2, 4]
+ *         	erf(x)  = 1.0 - tiny
+ *		erfc(x)	= (1/x)exp(-x*x-(.5*log(pi) -.5z + R(z)/S(z))
+ *
+ *	Where z = 1/(x*x), R is degree 9, and S is degree 3;
+ *	
+ *      5. For x in [4,28]
+ *         	erf(x)  = 1.0 - tiny
+ *		erfc(x)	= (1/x)exp(-x*x-(.5*log(pi)+eps + zP(z))
+ *
+ *	Where P is degree 14 polynomial in 1/(x*x).
+ *
+ *      Notes:
+ *	   Here 4 and 5 make use of the asymptotic series
+ *			  exp(-x*x)
+ *		erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) );
+ *			  x*sqrt(pi)
+ *
+ *		where for z = 1/(x*x)
+ *		P(z) ~ z/2*(-1 + z*3/2*(1 + z*5/2*(-1 + z*7/2*(1 +...))))
+ *
+ *	   Thus we use rational approximation to approximate
+ *              erfc*x*exp(x*x) ~ 1/sqrt(pi);
+ *
+ *		The error bound for the target function, G(z) for
+ *		the interval
+ *		[4, 28]:
+ * 		|eps + 1/(z)P(z) - G(z)| < 2**(-56.61)
+ *		for [2, 4]:
+ *      	|R(z)/S(z) - G(z)|	 < 2**(-58.24)
+ *		for [1.25, 2]:
+ *		|R(z)/S(z) - G(z)|	 < 2**(-58.12)
+ *
+ *      6. For inf > x >= 28
+ *         	erf(x)  = 1 - tiny  (raise inexact)
+ *         	erfc(x) = tiny*tiny (raise underflow)
+ *
+ *      7. Special cases:
+ *         	erf(0)  = 0, erf(inf)  = 1, erf(-inf) = -1,
+ *         	erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2, 
+ *	   	erfc/erf(NaN) is NaN
+ */
+
+#if defined(vax) || defined(tahoe)
+#define _IEEE	0
+#define TRUNC(x) (double) (float) (x)
+#else
+#define _IEEE	1
+#define TRUNC(x) *(((int *) &x) + 1) &= 0xf8000000
+#define infnan(x) 0.0
+#endif
+
+#ifdef _IEEE_LIBM
+/*
+ * redefining "___function" to "function" in _IEEE_LIBM mode
+ */
+#include "ieee_libm.h"
+#endif
+
+static double
+tiny	    = 1e-300,
+half	    = 0.5,
+one	    = 1.0,
+two	    = 2.0,
+c 	    = 8.45062911510467529297e-01, /* (float)0.84506291151 */
+/*
+ * Coefficients for approximation to erf in [0,0.84375]
+ */
+p0t8 = 1.02703333676410051049867154944018394163280,
+p0 =   1.283791670955125638123339436800229927041e-0001,
+p1 =  -3.761263890318340796574473028946097022260e-0001,
+p2 =   1.128379167093567004871858633779992337238e-0001,
+p3 =  -2.686617064084433642889526516177508374437e-0002,
+p4 =   5.223977576966219409445780927846432273191e-0003,
+p5 =  -8.548323822001639515038738961618255438422e-0004,
+p6 =   1.205520092530505090384383082516403772317e-0004,
+p7 =  -1.492214100762529635365672665955239554276e-0005,
+p8 =   1.640186161764254363152286358441771740838e-0006,
+p9 =  -1.571599331700515057841960987689515895479e-0007,
+p10=   1.073087585213621540635426191486561494058e-0008;
+/*
+ * Coefficients for approximation to erf in [0.84375,1.25] 
+ */
+static double
+pa0 =  -2.362118560752659485957248365514511540287e-0003,
+pa1 =   4.148561186837483359654781492060070469522e-0001,
+pa2 =  -3.722078760357013107593507594535478633044e-0001,
+pa3 =   3.183466199011617316853636418691420262160e-0001,
+pa4 =  -1.108946942823966771253985510891237782544e-0001,
+pa5 =   3.547830432561823343969797140537411825179e-0002,
+pa6 =  -2.166375594868790886906539848893221184820e-0003,
+qa1 =   1.064208804008442270765369280952419863524e-0001,
+qa2 =   5.403979177021710663441167681878575087235e-0001,
+qa3 =   7.182865441419627066207655332170665812023e-0002,
+qa4 =   1.261712198087616469108438860983447773726e-0001,
+qa5 =   1.363708391202905087876983523620537833157e-0002,
+qa6 =   1.198449984679910764099772682882189711364e-0002;
+/*
+ * log(sqrt(pi)) for large x expansions.
+ * The tail (lsqrtPI_lo) is included in the rational
+ * approximations.
+*/
+static double
+   lsqrtPI_hi = .5723649429247000819387380943226;
+/*
+ * lsqrtPI_lo = .000000000000000005132975581353913;
+ *
+ * Coefficients for approximation to erfc in [2, 4]
+*/
+static double
+rb0  =	-1.5306508387410807582e-010,	/* includes lsqrtPI_lo */
+rb1  =	 2.15592846101742183841910806188e-008,
+rb2  =	 6.24998557732436510470108714799e-001,
+rb3  =	 8.24849222231141787631258921465e+000,
+rb4  =	 2.63974967372233173534823436057e+001,
+rb5  =	 9.86383092541570505318304640241e+000,
+rb6  =	-7.28024154841991322228977878694e+000,
+rb7  =	 5.96303287280680116566600190708e+000,
+rb8  =	-4.40070358507372993983608466806e+000,
+rb9  =	 2.39923700182518073731330332521e+000,
+rb10 =	-6.89257464785841156285073338950e-001,
+sb1  =	 1.56641558965626774835300238919e+001,
+sb2  =	 7.20522741000949622502957936376e+001,
+sb3  =	 9.60121069770492994166488642804e+001;
+/*
+ * Coefficients for approximation to erfc in [1.25, 2]
+*/
+static double
+rc0  =	-2.47925334685189288817e-007,	/* includes lsqrtPI_lo */
+rc1  =	 1.28735722546372485255126993930e-005,
+rc2  =	 6.24664954087883916855616917019e-001,
+rc3  =	 4.69798884785807402408863708843e+000,
+rc4  =	 7.61618295853929705430118701770e+000,
+rc5  =	 9.15640208659364240872946538730e-001,
+rc6  =	-3.59753040425048631334448145935e-001,
+rc7  =	 1.42862267989304403403849619281e-001,
+rc8  =	-4.74392758811439801958087514322e-002,
+rc9  =	 1.09964787987580810135757047874e-002,
+rc10 =	-1.28856240494889325194638463046e-003,
+sc1  =	 9.97395106984001955652274773456e+000,
+sc2  =	 2.80952153365721279953959310660e+001,
+sc3  =	 2.19826478142545234106819407316e+001;
+/*
+ * Coefficients for approximation to  erfc in [4,28]
+ */
+static double
+rd0  =	-2.1491361969012978677e-016,	/* includes lsqrtPI_lo */
+rd1  =	-4.99999999999640086151350330820e-001,
+rd2  =	 6.24999999772906433825880867516e-001,
+rd3  =	-1.54166659428052432723177389562e+000,
+rd4  =	 5.51561147405411844601985649206e+000,
+rd5  =	-2.55046307982949826964613748714e+001,
+rd6  =	 1.43631424382843846387913799845e+002,
+rd7  =	-9.45789244999420134263345971704e+002,
+rd8  =	 6.94834146607051206956384703517e+003,
+rd9  =	-5.27176414235983393155038356781e+004,
+rd10 =	 3.68530281128672766499221324921e+005,
+rd11 =	-2.06466642800404317677021026611e+006,
+rd12 =	 7.78293889471135381609201431274e+006,
+rd13 =	-1.42821001129434127360582351685e+007;
+
+double erf(x)
+	double x;
+{
+	double R,S,P,Q,ax,s,y,z,r,fabs(),exp();
+	if(!finite(x)) {		/* erf(nan)=nan */
+	    if (isnan(x))
+		return(x);
+	    return (x > 0 ? one : -one); /* erf(+/-inf)= +/-1 */
+	}
+	if ((ax = x) < 0)
+		ax = - ax;
+	if (ax < .84375) {
+	    if (ax < 3.7e-09) {
+		if (ax < 1.0e-308)
+		    return 0.125*(8.0*x+p0t8*x);  /*avoid underflow */
+		return x + p0*x;
+	    }
+	    y = x*x;
+	    r = y*(p1+y*(p2+y*(p3+y*(p4+y*(p5+
+			y*(p6+y*(p7+y*(p8+y*(p9+y*p10)))))))));
+	    return x + x*(p0+r);
+	}
+	if (ax < 1.25) {		/* 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 (x>=0)
+		return (c + P/Q);
+	    else
+		return (-c - P/Q);
+	}
+	if (ax >= 6.0) {		/* inf>|x|>=6 */
+	    if (x >= 0.0)
+		return (one-tiny);
+	    else
+		return (tiny-one);
+	}
+    /* 1.25 <= |x| < 6 */
+	z = -ax*ax;
+	s = -one/z;
+	if (ax < 2.0) {
+		R = rc0+s*(rc1+s*(rc2+s*(rc3+s*(rc4+s*(rc5+
+			s*(rc6+s*(rc7+s*(rc8+s*(rc9+s*rc10)))))))));
+		S = one+s*(sc1+s*(sc2+s*sc3));
+	} else {
+		R = rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+
+			s*(rb6+s*(rb7+s*(rb8+s*(rb9+s*rb10)))))))));
+		S = one+s*(sb1+s*(sb2+s*sb3));
+	}
+	y = (R/S -.5*s) - lsqrtPI_hi;
+	z += y;
+	z = exp(z)/ax;
+	if (x >= 0)
+		return (one-z);
+	else
+		return (z-one);
+}
+
+double erfc(x) 
+	double x;
+{
+	double R,S,P,Q,s,ax,y,z,r,fabs(),__exp__D();
+	if (!finite(x)) {
+		if (isnan(x))		/* erfc(NaN) = NaN */
+			return(x);
+		else if (x > 0)		/* erfc(+-inf)=0,2 */
+			return 0.0;
+		else
+			return 2.0;
+	}
+	if ((ax = x) < 0)
+		ax = -ax;
+	if (ax < .84375) {			/* |x|<0.84375 */
+	    if (ax < 1.38777878078144568e-17)  	/* |x|<2**-56 */
+		return one-x;
+	    y = x*x;
+	    r = y*(p1+y*(p2+y*(p3+y*(p4+y*(p5+
+			y*(p6+y*(p7+y*(p8+y*(p9+y*p10)))))))));
+	    if (ax < .0625) {  	/* |x|<2**-4 */
+		return (one-(x+x*(p0+r)));
+	    } else {
+		r = x*(p0+r);
+		r += (x-half);
+	        return (half - r);
+	    }
+	}
+	if (ax < 1.25) {		/* 0.84375 <= |x| < 1.25 */
+	    s = ax-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 (x>=0) {
+	        z  = one-c; return z - P/Q; 
+	    } else {
+		z = c+P/Q; return one+z;
+	    }
+	}
+	if (ax >= 28)	/* Out of range */
+ 		if (x>0)
+			return (tiny*tiny);
+		else
+			return (two-tiny);
+	z = ax;
+	TRUNC(z);
+	y = z - ax; y *= (ax+z);
+	z *= -z;			/* Here z + y = -x^2 */
+		s = one/(-z-y);		/* 1/(x*x) */
+	if (ax >= 4) {			/* 6 <= ax */
+		R = s*(rd1+s*(rd2+s*(rd3+s*(rd4+s*(rd5+
+			s*(rd6+s*(rd7+s*(rd8+s*(rd9+s*(rd10
+			+s*(rd11+s*(rd12+s*rd13))))))))))));
+		y += rd0;
+	} else if (ax >= 2) {
+		R = rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+
+			s*(rb6+s*(rb7+s*(rb8+s*(rb9+s*rb10)))))))));
+		S = one+s*(sb1+s*(sb2+s*sb3));
+		y += R/S;
+		R = -.5*s;
+	} else {
+		R = rc0+s*(rc1+s*(rc2+s*(rc3+s*(rc4+s*(rc5+
+			s*(rc6+s*(rc7+s*(rc8+s*(rc9+s*rc10)))))))));
+		S = one+s*(sc1+s*(sc2+s*sc3));
+		y += R/S;
+		R = -.5*s;
+	}
+	/* return exp(-x^2 - lsqrtPI_hi + R + y)/x;	*/
+	s = ((R + y) - lsqrtPI_hi) + z;
+	y = (((z-s) - lsqrtPI_hi) + R) + y;
+	r = __exp__D(s, y)/x;
+	if (x>0)
+		return r;
+	else
+		return two-r;
+}