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authorRoland McGrath <roland@gnu.org>1995-02-18 01:27:10 +0000
committerRoland McGrath <roland@gnu.org>1995-02-18 01:27:10 +0000
commit28f540f45bbacd939bfd07f213bcad2bf730b1bf (patch)
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+/*
+ * Copyright (c) 1985, 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[] = "@(#)exp__E.c	8.1 (Berkeley) 6/4/93";
+#endif /* not lint */
+
+/* exp__E(x,c)
+ * ASSUMPTION: c << x  SO THAT  fl(x+c)=x.
+ * (c is the correction term for x)
+ * exp__E RETURNS
+ *
+ *			 /  exp(x+c) - 1 - x ,  1E-19 < |x| < .3465736
+ *       exp__E(x,c) = 	| 		     
+ *			 \  0 ,  |x| < 1E-19.
+ *
+ * DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS)
+ * KERNEL FUNCTION OF EXP, EXPM1, POW FUNCTIONS
+ * CODED IN C BY K.C. NG, 1/31/85;
+ * REVISED BY K.C. NG on 3/16/85, 4/16/85.
+ *
+ * Required system supported function:
+ *	copysign(x,y)	
+ *
+ * Method:
+ *	1. Rational approximation. Let r=x+c.
+ *	   Based on
+ *                                   2 * sinh(r/2)     
+ *                exp(r) - 1 =   ----------------------   ,
+ *                               cosh(r/2) - sinh(r/2)
+ *	   exp__E(r) is computed using
+ *                   x*x            (x/2)*W - ( Q - ( 2*P  + x*P ) )
+ *                   --- + (c + x*[---------------------------------- + c ])
+ *                    2                          1 - W
+ * 	   where  P := p1*x^2 + p2*x^4,
+ *	          Q := q1*x^2 + q2*x^4 (for 56 bits precision, add q3*x^6)
+ *	          W := x/2-(Q-x*P),
+ *
+ *	   (See the listing below for the values of p1,p2,q1,q2,q3. The poly-
+ *	    nomials P and Q may be regarded as the approximations to sinh
+ *	    and cosh :
+ *		sinh(r/2) =  r/2 + r * P  ,  cosh(r/2) =  1 + Q . )
+ *
+ *         The coefficients were obtained by a special Remez algorithm.
+ *
+ * Approximation error:
+ *
+ *   |	exp(x) - 1			   |        2**(-57),  (IEEE double)
+ *   | ------------  -  (exp__E(x,0)+x)/x  |  <= 
+ *   |	     x			           |	    2**(-69).  (VAX D)
+ *
+ * Constants:
+ * The hexadecimal values are the intended ones for the following constants.
+ * The decimal values may be used, provided that the compiler will convert
+ * from decimal to binary accurately enough to produce the hexadecimal values
+ * shown.
+ */
+
+#include "mathimpl.h"
+
+vc(p1, 1.5150724356786683059E-2 ,3abe,3d78,066a,67e1,  -6, .F83ABE67E1066A)
+vc(p2, 6.3112487873718332688E-5 ,5b42,3984,0173,48cd, -13, .845B4248CD0173)
+vc(q1, 1.1363478204690669916E-1 ,b95a,3ee8,ec45,44a2,  -3, .E8B95A44A2EC45)
+vc(q2, 1.2624568129896839182E-3 ,7905,3ba5,f5e7,72e4,  -9, .A5790572E4F5E7)
+vc(q3, 1.5021856115869022674E-6 ,9eb4,36c9,c395,604a, -19, .C99EB4604AC395)
+
+ic(p1, 1.3887401997267371720E-2,  -7, 1.C70FF8B3CC2CF)
+ic(p2, 3.3044019718331897649E-5, -15, 1.15317DF4526C4)
+ic(q1, 1.1110813732786649355E-1,  -4, 1.C719538248597)
+ic(q2, 9.9176615021572857300E-4, -10, 1.03FC4CB8C98E8)
+
+#ifdef vccast
+#define       p1    vccast(p1)
+#define       p2    vccast(p2)
+#define       q1    vccast(q1)
+#define       q2    vccast(q2)
+#define       q3    vccast(q3)
+#endif
+
+double __exp__E(x,c)
+double x,c;
+{
+	const static double zero=0.0, one=1.0, half=1.0/2.0, small=1.0E-19;
+	double z,p,q,xp,xh,w;
+	if(copysign(x,one)>small) {
+           z = x*x  ;
+	   p = z*( p1 +z* p2 );
+#if defined(vax)||defined(tahoe)
+           q = z*( q1 +z*( q2 +z* q3 ));
+#else	/* defined(vax)||defined(tahoe) */
+           q = z*( q1 +z*  q2 );
+#endif	/* defined(vax)||defined(tahoe) */
+           xp= x*p     ; 
+	   xh= x*half  ;
+           w = xh-(q-xp)  ;
+	   p = p+p;
+	   c += x*((xh*w-(q-(p+xp)))/(one-w)+c);
+	   return(z*half+c);
+	}
+	/* end of |x| > small */
+
+	else {
+	    if(x!=zero) one+small;	/* raise the inexact flag */
+	    return(copysign(zero,x));
+	}
+}