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BSD 4.3tahoe
/*
* Copyright (c) 1985 Regents of the University of California.
* All rights reserved.
*
* Redistribution and use in source and binary forms are permitted
* provided that the above copyright notice and this paragraph are
* duplicated in all such forms and that any documentation,
* advertising materials, and other materials related to such
* distribution and use acknowledge that the software was developed
* by the University of California, Berkeley. The name of the
* University may not be used to endorse or promote products derived
* from this software without specific prior written permission.
* THIS SOFTWARE IS PROVIDED ``AS IS'' AND WITHOUT ANY EXPRESS OR
* IMPLIED WARRANTIES, INCLUDING, WITHOUT LIMITATION, THE IMPLIED
* WARRANTIES OF MERCHANTIBILITY AND FITNESS FOR A PARTICULAR PURPOSE.
*
* All recipients should regard themselves as participants in an ongoing
* research project and hence should feel obligated to report their
* experiences (good or bad) with these elementary function codes, using
* the sendbug(8) program, to the authors.
*/
#ifndef lint
static char sccsid[] = "@(#)exp.c 5.3 (Berkeley) 6/30/88";
#endif /* not lint */
/* EXP(X)
* RETURN THE EXPONENTIAL OF X
* DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS)
* CODED IN C BY K.C. NG, 1/19/85;
* REVISED BY K.C. NG on 2/6/85, 2/15/85, 3/7/85, 3/24/85, 4/16/85, 6/14/86.
*
* Required system supported functions:
* scalb(x,n)
* copysign(x,y)
* finite(x)
*
* Method:
* 1. Argument Reduction: given the input x, find r and integer k such
* that
* x = k*ln2 + r, |r| <= 0.5*ln2 .
* r will be represented as r := z+c for better accuracy.
*
* 2. Compute exp(r) by
*
* exp(r) = 1 + r + r*R1/(2-R1),
* where
* R1 = x - x^2*(p1+x^2*(p2+x^2*(p3+x^2*(p4+p5*x^2)))).
*
* 3. exp(x) = 2^k * exp(r) .
*
* Special cases:
* exp(INF) is INF, exp(NaN) is NaN;
* exp(-INF)= 0;
* for finite argument, only exp(0)=1 is exact.
*
* Accuracy:
* exp(x) returns the exponential of x nearly rounded. In a test run
* with 1,156,000 random arguments on a VAX, the maximum observed
* error was 0.869 ulps (units in the last place).
*
* 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.
*/
#if defined(vax)||defined(tahoe) /* VAX D format */
#ifdef vax
#define _0x(A,B) 0x/**/A/**/B
#else /* vax */
#define _0x(A,B) 0x/**/B/**/A
#endif /* vax */
/* static double */
/* ln2hi = 6.9314718055829871446E-1 , Hex 2^ 0 * .B17217F7D00000 */
/* ln2lo = 1.6465949582897081279E-12 , Hex 2^-39 * .E7BCD5E4F1D9CC */
/* lnhuge = 9.4961163736712506989E1 , Hex 2^ 7 * .BDEC1DA73E9010 */
/* lntiny = -9.5654310917272452386E1 , Hex 2^ 7 * -.BF4F01D72E33AF */
/* invln2 = 1.4426950408889634148E0 ; Hex 2^ 1 * .B8AA3B295C17F1 */
/* p1 = 1.6666666666666602251E-1 , Hex 2^-2 * .AAAAAAAAAAA9F1 */
/* p2 = -2.7777777777015591216E-3 , Hex 2^-8 * -.B60B60B5F5EC94 */
/* p3 = 6.6137563214379341918E-5 , Hex 2^-13 * .8AB355792EF15F */
/* p4 = -1.6533902205465250480E-6 , Hex 2^-19 * -.DDEA0E2E935F84 */
/* p5 = 4.1381367970572387085E-8 , Hex 2^-24 * .B1BB4B95F52683 */
static long ln2hix[] = { _0x(7217,4031), _0x(0000,f7d0)};
static long ln2lox[] = { _0x(bcd5,2ce7), _0x(d9cc,e4f1)};
static long lnhugex[] = { _0x(ec1d,43bd), _0x(9010,a73e)};
static long lntinyx[] = { _0x(4f01,c3bf), _0x(33af,d72e)};
static long invln2x[] = { _0x(aa3b,40b8), _0x(17f1,295c)};
static long p1x[] = { _0x(aaaa,3f2a), _0x(a9f1,aaaa)};
static long p2x[] = { _0x(0b60,bc36), _0x(ec94,b5f5)};
static long p3x[] = { _0x(b355,398a), _0x(f15f,792e)};
static long p4x[] = { _0x(ea0e,b6dd), _0x(5f84,2e93)};
static long p5x[] = { _0x(bb4b,3431), _0x(2683,95f5)};
#define ln2hi (*(double*)ln2hix)
#define ln2lo (*(double*)ln2lox)
#define lnhuge (*(double*)lnhugex)
#define lntiny (*(double*)lntinyx)
#define invln2 (*(double*)invln2x)
#define p1 (*(double*)p1x)
#define p2 (*(double*)p2x)
#define p3 (*(double*)p3x)
#define p4 (*(double*)p4x)
#define p5 (*(double*)p5x)
#else /* defined(vax)||defined(tahoe) */
static double
p1 = 1.6666666666666601904E-1 , /*Hex 2^-3 * 1.555555555553E */
p2 = -2.7777777777015593384E-3 , /*Hex 2^-9 * -1.6C16C16BEBD93 */
p3 = 6.6137563214379343612E-5 , /*Hex 2^-14 * 1.1566AAF25DE2C */
p4 = -1.6533902205465251539E-6 , /*Hex 2^-20 * -1.BBD41C5D26BF1 */
p5 = 4.1381367970572384604E-8 , /*Hex 2^-25 * 1.6376972BEA4D0 */
ln2hi = 6.9314718036912381649E-1 , /*Hex 2^ -1 * 1.62E42FEE00000 */
ln2lo = 1.9082149292705877000E-10 , /*Hex 2^-33 * 1.A39EF35793C76 */
lnhuge = 7.1602103751842355450E2 , /*Hex 2^ 9 * 1.6602B15B7ECF2 */
lntiny = -7.5137154372698068983E2 , /*Hex 2^ 9 * -1.77AF8EBEAE354 */
invln2 = 1.4426950408889633870E0 ; /*Hex 2^ 0 * 1.71547652B82FE */
#endif /* defined(vax)||defined(tahoe) */
double exp(x)
double x;
{
double scalb(), copysign(), z,hi,lo,c;
int k,finite();
#if !defined(vax)&&!defined(tahoe)
if(x!=x) return(x); /* x is NaN */
#endif /* !defined(vax)&&!defined(tahoe) */
if( x <= lnhuge ) {
if( x >= lntiny ) {
/* argument reduction : x --> x - k*ln2 */
k=invln2*x+copysign(0.5,x); /* k=NINT(x/ln2) */
/* express x-k*ln2 as hi-lo and let x=hi-lo rounded */
hi=x-k*ln2hi;
x=hi-(lo=k*ln2lo);
/* return 2^k*[1+x+x*c/(2+c)] */
z=x*x;
c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5))));
return scalb(1.0+(hi-(lo-(x*c)/(2.0-c))),k);
}
/* end of x > lntiny */
else
/* exp(-big#) underflows to zero */
if(finite(x)) return(scalb(1.0,-5000));
/* exp(-INF) is zero */
else return(0.0);
}
/* end of x < lnhuge */
else
/* exp(INF) is INF, exp(+big#) overflows to INF */
return( finite(x) ? scalb(1.0,5000) : x);
}
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