4.4BSD/usr/src/old/libm/libm/exp.c
/*
* Copyright (c) 1985 Regents of the University of California.
*
* Use and reproduction of this software are granted in accordance with
* the terms and conditions specified in the Berkeley Software License
* Agreement (in particular, this entails acknowledgement of the programs'
* source, and inclusion of this notice) with the additional understanding
* that 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 "sendbug 4bsd-bugs@BERKELEY", to the authors.
*/
#ifndef lint
static char sccsid[] = "@(#)exp.c 4.3 (Berkeley) 8/21/85";
#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.
*
* Required system supported functions:
* scalb(x,n)
* copysign(x,y)
* finite(x)
*
* Kernel function:
* exp__E(x,c)
*
* 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 expm1(r)=exp(r)-1 by
*
* expm1(r=z+c) := z + exp__E(z,r)
*
* 3. exp(x) = 2^k * ( expm1(r) + 1 ).
*
* 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 .768 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.
*/
#ifdef VAX /* VAX D format */
/* double static */
/* 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 */
static long ln2hix[] = { 0x72174031, 0x0000f7d0};
static long ln2lox[] = { 0xbcd52ce7, 0xd9cce4f1};
static long lnhugex[] = { 0xec1d43bd, 0x9010a73e};
static long lntinyx[] = { 0x4f01c3bf, 0x33afd72e};
static long invln2x[] = { 0xaa3b40b8, 0x17f1295c};
#define ln2hi (*(double*)ln2hix)
#define ln2lo (*(double*)ln2lox)
#define lnhuge (*(double*)lnhugex)
#define lntiny (*(double*)lntinyx)
#define invln2 (*(double*)invln2x)
#else /* IEEE double */
double static
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
double exp(x)
double x;
{
double scalb(), copysign(), exp__E(), z,hi,lo,c;
int k,finite();
#ifndef VAX
if(x!=x) return(x); /* x is NaN */
#endif
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 z+c */
hi=x-k*ln2hi;
z=hi-(lo=k*ln2lo);
c=(hi-z)-lo;
/* return 2^k*[expm1(x) + 1] */
z += exp__E(z,c);
return (scalb(z+1.0,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);
}