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      1 /*	$NetBSD: b_exp.c,v 1.1 2012/05/05 17:54:14 christos Exp $	*/
      2 
      3 /*
      4  * Copyright (c) 1985, 1993
      5  *	The Regents of the University of California.  All rights reserved.
      6  *
      7  * Redistribution and use in source and binary forms, with or without
      8  * modification, are permitted provided that the following conditions
      9  * are met:
     10  * 1. Redistributions of source code must retain the above copyright
     11  *    notice, this list of conditions and the following disclaimer.
     12  * 2. Redistributions in binary form must reproduce the above copyright
     13  *    notice, this list of conditions and the following disclaimer in the
     14  *    documentation and/or other materials provided with the distribution.
     15  * 3. All advertising materials mentioning features or use of this software
     16  *    must display the following acknowledgement:
     17  *	This product includes software developed by the University of
     18  *	California, Berkeley and its contributors.
     19  * 4. Neither the name of the University nor the names of its contributors
     20  *    may be used to endorse or promote products derived from this software
     21  *    without specific prior written permission.
     22  *
     23  * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
     24  * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
     25  * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
     26  * ARE DISCLAIMED.  IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
     27  * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
     28  * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
     29  * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
     30  * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
     31  * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
     32  * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
     33  * SUCH DAMAGE.
     34  */
     35 
     36 /* @(#)exp.c	8.1 (Berkeley) 6/4/93 */
     37 #include <sys/cdefs.h>
     38 #if 0
     39 __FBSDID("$FreeBSD: release/9.0.0/lib/msun/bsdsrc/b_exp.c 176449 2008-02-22 02:26:51Z das $");
     40 #else
     41 __RCSID("$NetBSD: b_exp.c,v 1.1 2012/05/05 17:54:14 christos Exp $");
     42 #endif
     43 
     44 
     45 /* EXP(X)
     46  * RETURN THE EXPONENTIAL OF X
     47  * DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS)
     48  * CODED IN C BY K.C. NG, 1/19/85;
     49  * REVISED BY K.C. NG on 2/6/85, 2/15/85, 3/7/85, 3/24/85, 4/16/85, 6/14/86.
     50  *
     51  * Required system supported functions:
     52  *	scalb(x,n)
     53  *	copysign(x,y)
     54  *	finite(x)
     55  *
     56  * Method:
     57  *	1. Argument Reduction: given the input x, find r and integer k such
     58  *	   that
     59  *	                   x = k*ln2 + r,  |r| <= 0.5*ln2 .
     60  *	   r will be represented as r := z+c for better accuracy.
     61  *
     62  *	2. Compute exp(r) by
     63  *
     64  *		exp(r) = 1 + r + r*R1/(2-R1),
     65  *	   where
     66  *		R1 = x - x^2*(p1+x^2*(p2+x^2*(p3+x^2*(p4+p5*x^2)))).
     67  *
     68  *	3. exp(x) = 2^k * exp(r) .
     69  *
     70  * Special cases:
     71  *	exp(INF) is INF, exp(NaN) is NaN;
     72  *	exp(-INF)=  0;
     73  *	for finite argument, only exp(0)=1 is exact.
     74  *
     75  * Accuracy:
     76  *	exp(x) returns the exponential of x nearly rounded. In a test run
     77  *	with 1,156,000 random arguments on a VAX, the maximum observed
     78  *	error was 0.869 ulps (units in the last place).
     79  */
     80 
     81 #include "math.h"
     82 #include "math_private.h"
     83 
     84 static const double p1 = 0x1.555555555553ep-3;
     85 static const double p2 = -0x1.6c16c16bebd93p-9;
     86 static const double p3 = 0x1.1566aaf25de2cp-14;
     87 static const double p4 = -0x1.bbd41c5d26bf1p-20;
     88 static const double p5 = 0x1.6376972bea4d0p-25;
     89 static const double ln2hi = 0x1.62e42fee00000p-1;
     90 static const double ln2lo = 0x1.a39ef35793c76p-33;
     91 static const double lnhuge = 0x1.6602b15b7ecf2p9;
     92 static const double lntiny = -0x1.77af8ebeae354p9;
     93 static const double invln2 = 0x1.71547652b82fep0;
     94 
     95 /* returns exp(r = x + c) for |c| < |x| with no overlap.  */
     96 
     97 double
     98 __exp__D(double x, double c)
     99 {
    100 	double  z,hi,lo;
    101 	int k;
    102 
    103 	if (x != x)	/* x is NaN */
    104 		return(x);
    105 	if ( x <= lnhuge ) {
    106 		if ( x >= lntiny ) {
    107 
    108 		    /* argument reduction : x --> x - k*ln2 */
    109 			z = invln2*x;
    110 			k = z + copysign(.5, x);
    111 
    112 		    /* express (x+c)-k*ln2 as hi-lo and let x=hi-lo rounded */
    113 
    114 			hi=(x-k*ln2hi);			/* Exact. */
    115 			x= hi - (lo = k*ln2lo-c);
    116 		    /* return 2^k*[1+x+x*c/(2+c)]  */
    117 			z=x*x;
    118 			c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5))));
    119 			c = (x*c)/(2.0-c);
    120 
    121 			return  scalb(1.+(hi-(lo - c)), k);
    122 		}
    123 		/* end of x > lntiny */
    124 
    125 		else
    126 		     /* exp(-big#) underflows to zero */
    127 		     if(finite(x))  return(scalb(1.0,-5000));
    128 
    129 		     /* exp(-INF) is zero */
    130 		     else return(0.0);
    131 	}
    132 	/* end of x < lnhuge */
    133 
    134 	else
    135 	/* exp(INF) is INF, exp(+big#) overflows to INF */
    136 	    return( finite(x) ?  scalb(1.0,5000)  : x);
    137 }
    138