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      1 /*
      2  * Copyright (c) 1985, 1993
      3  *	The Regents of the University of California.  All rights reserved.
      4  *
      5  * Redistribution and use in source and binary forms, with or without
      6  * modification, are permitted provided that the following conditions
      7  * are met:
      8  * 1. Redistributions of source code must retain the above copyright
      9  *    notice, this list of conditions and the following disclaimer.
     10  * 2. Redistributions in binary form must reproduce the above copyright
     11  *    notice, this list of conditions and the following disclaimer in the
     12  *    documentation and/or other materials provided with the distribution.
     13  * 3. All advertising materials mentioning features or use of this software
     14  *    must display the following acknowledgement:
     15  *	This product includes software developed by the University of
     16  *	California, Berkeley and its contributors.
     17  * 4. Neither the name of the University nor the names of its contributors
     18  *    may be used to endorse or promote products derived from this software
     19  *    without specific prior written permission.
     20  *
     21  * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
     22  * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
     23  * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
     24  * ARE DISCLAIMED.  IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
     25  * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
     26  * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
     27  * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
     28  * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
     29  * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
     30  * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
     31  * SUCH DAMAGE.
     32  */
     33 
     34 #ifndef lint
     35 static char sccsid[] = "@(#)exp.c	8.1 (Berkeley) 6/4/93";
     36 #endif /* not lint */
     37 #include <sys/cdefs.h>
     38 /* __FBSDID("$FreeBSD: src/lib/msun/bsdsrc/b_exp.c,v 1.7 2004/12/16 20:40:37 das Exp $"); */
     39 
     40 
     41 /* EXP(X)
     42  * RETURN THE EXPONENTIAL OF X
     43  * DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS)
     44  * CODED IN C BY K.C. NG, 1/19/85;
     45  * REVISED BY K.C. NG on 2/6/85, 2/15/85, 3/7/85, 3/24/85, 4/16/85, 6/14/86.
     46  *
     47  * Required system supported functions:
     48  *	scalb(x,n)
     49  *	copysign(x,y)
     50  *	finite(x)
     51  *
     52  * Method:
     53  *	1. Argument Reduction: given the input x, find r and integer k such
     54  *	   that
     55  *	                   x = k*ln2 + r,  |r| <= 0.5*ln2 .
     56  *	   r will be represented as r := z+c for better accuracy.
     57  *
     58  *	2. Compute exp(r) by
     59  *
     60  *		exp(r) = 1 + r + r*R1/(2-R1),
     61  *	   where
     62  *		R1 = x - x^2*(p1+x^2*(p2+x^2*(p3+x^2*(p4+p5*x^2)))).
     63  *
     64  *	3. exp(x) = 2^k * exp(r) .
     65  *
     66  * Special cases:
     67  *	exp(INF) is INF, exp(NaN) is NaN;
     68  *	exp(-INF)=  0;
     69  *	for finite argument, only exp(0)=1 is exact.
     70  *
     71  * Accuracy:
     72  *	exp(x) returns the exponential of x nearly rounded. In a test run
     73  *	with 1,156,000 random arguments on a VAX, the maximum observed
     74  *	error was 0.869 ulps (units in the last place).
     75  */
     76 
     77 #include "mathimpl.h"
     78 
     79 const static double p1 = 0x1.555555555553ep-3;
     80 const static double p2 = -0x1.6c16c16bebd93p-9;
     81 const static double p3 = 0x1.1566aaf25de2cp-14;
     82 const static double p4 = -0x1.bbd41c5d26bf1p-20;
     83 const static double p5 = 0x1.6376972bea4d0p-25;
     84 const static double ln2hi = 0x1.62e42fee00000p-1;
     85 const static double ln2lo = 0x1.a39ef35793c76p-33;
     86 const static double lnhuge = 0x1.6602b15b7ecf2p9;
     87 const static double lntiny = -0x1.77af8ebeae354p9;
     88 const static double invln2 = 0x1.71547652b82fep0;
     89 
     90 #if 0
     91 double exp(x)
     92 double x;
     93 {
     94 	double  z,hi,lo,c;
     95 	int k;
     96 
     97 #if !defined(vax)&&!defined(tahoe)
     98 	if(x!=x) return(x);	/* x is NaN */
     99 #endif	/* !defined(vax)&&!defined(tahoe) */
    100 	if( x <= lnhuge ) {
    101 		if( x >= lntiny ) {
    102 
    103 		    /* argument reduction : x --> x - k*ln2 */
    104 
    105 			k=invln2*x+copysign(0.5,x);	/* k=NINT(x/ln2) */
    106 
    107 		    /* express x-k*ln2 as hi-lo and let x=hi-lo rounded */
    108 
    109 			hi=x-k*ln2hi;
    110 			x=hi-(lo=k*ln2lo);
    111 
    112 		    /* return 2^k*[1+x+x*c/(2+c)]  */
    113 			z=x*x;
    114 			c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5))));
    115 			return  scalb(1.0+(hi-(lo-(x*c)/(2.0-c))),k);
    116 
    117 		}
    118 		/* end of x > lntiny */
    119 
    120 		else
    121 		     /* exp(-big#) underflows to zero */
    122 		     if(finite(x))  return(scalb(1.0,-5000));
    123 
    124 		     /* exp(-INF) is zero */
    125 		     else return(0.0);
    126 	}
    127 	/* end of x < lnhuge */
    128 
    129 	else
    130 	/* exp(INF) is INF, exp(+big#) overflows to INF */
    131 	    return( finite(x) ?  scalb(1.0,5000)  : x);
    132 }
    133 #endif
    134 
    135 /* returns exp(r = x + c) for |c| < |x| with no overlap.  */
    136 
    137 double __exp__D(x, c)
    138 double x, c;
    139 {
    140 	double  z,hi,lo;
    141 	int k;
    142 
    143 	if (x != x)	/* x is NaN */
    144 		return(x);
    145 	if ( x <= lnhuge ) {
    146 		if ( x >= lntiny ) {
    147 
    148 		    /* argument reduction : x --> x - k*ln2 */
    149 			z = invln2*x;
    150 			k = z + copysign(.5, x);
    151 
    152 		    /* express (x+c)-k*ln2 as hi-lo and let x=hi-lo rounded */
    153 
    154 			hi=(x-k*ln2hi);			/* Exact. */
    155 			x= hi - (lo = k*ln2lo-c);
    156 		    /* return 2^k*[1+x+x*c/(2+c)]  */
    157 			z=x*x;
    158 			c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5))));
    159 			c = (x*c)/(2.0-c);
    160 
    161 			return  scalb(1.+(hi-(lo - c)), k);
    162 		}
    163 		/* end of x > lntiny */
    164 
    165 		else
    166 		     /* exp(-big#) underflows to zero */
    167 		     if(finite(x))  return(scalb(1.0,-5000));
    168 
    169 		     /* exp(-INF) is zero */
    170 		     else return(0.0);
    171 	}
    172 	/* end of x < lnhuge */
    173 
    174 	else
    175 	/* exp(INF) is INF, exp(+big#) overflows to INF */
    176 	    return( finite(x) ?  scalb(1.0,5000)  : x);
    177 }
    178