* 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.
static char sccsid
[] = "@(#)expm1.c 1.1 (ELEFUNT) %G%";
* RETURN THE EXPONENTIAL OF X MINUS ONE
* 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, 3/7/85, 3/21/85, 4/16/85.
* Required system supported functions:
* 1. Argument Reduction: given the input x, find r and integer k such
* 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,c)
* 3. EXPM1(x) = 2^k * ( EXPM1(r) + 1-2^-k ).
* 1. When k=1 and z < -0.25, we use the following formula for
* EXPM1(x) = 2 * ( (z+0.5) + exp__E(z,c) )
* 2. To avoid rounding error in 1-2^-k where k is large, we use
* EXPM1(x) = 2^k * { [z+(exp__E(z,c)-2^-k )] + 1 }
* EXPM1(INF) is INF, EXPM1(NaN) is NaN;
* for finite argument, only EXPM1(0)=0 is exact.
* EXPM1(x) returns the exact (exp(x)-1) nearly rounded. In a test run with
* 1,166,000 random arguments on a VAX, the maximum observed error was
* .872 ulps (units of the last place).
* 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
#ifdef VAX /* VAX D format */
/* ln2hi = 6.9314718055829871446E-1 , Hex 2^ 0 * .B17217F7D00000 */
/* ln2lo = 1.6465949582897081279E-12 , Hex 2^-39 * .E7BCD5E4F1D9CC */
/* lnhuge = 9.4961163736712506989E1 , Hex 2^ 7 * .BDEC1DA73E9010 */
/* invln2 = 1.4426950408889634148E0 ; Hex 2^ 1 * .B8AA3B295C17F1 */
static long ln2hix
[] = { 0x72174031, 0x0000f7d0};
static long ln2lox
[] = { 0xbcd52ce7, 0xd9cce4f1};
static long lnhugex
[] = { 0xec1d43bd, 0x9010a73e};
static long invln2x
[] = { 0xaa3b40b8, 0x17f1295c};
#define ln2hi (*(double*)ln2hix)
#define ln2lo (*(double*)ln2lox)
#define lnhuge (*(double*)lnhugex)
#define invln2 (*(double*)invln2x)
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 */
invln2
= 1.4426950408889633870E0
; /*Hex 2^ 0 * 1.71547652B82FE */
double static one
=1.0, half
=1.0/2.0;
double scalb(), copysign(), exp__E(), z
,hi
,lo
,c
;
if(x
!=x
) return(x
); /* x is NaN */
/* argument reduction : x - k*ln2 */
k
= invln2
*x
+copysign(0.5,x
); /* k=NINT(x/ln2) */
if(k
==0) return(z
+exp__E(z
,c
));
{x
=z
+half
;x
+=exp__E(z
,c
); return(x
+x
);}
{z
+=exp__E(z
,c
); x
=half
+z
; return(x
+x
);}
{ x
=one
-scalb(one
,-k
); z
+= exp__E(z
,c
);}
{ x
= exp__E(z
,c
)-scalb(one
,-k
); x
+=z
; z
=one
;}
{ x
= exp__E(z
,c
)+z
; z
=one
;}
/* expm1(-big#) rounded to -1 (inexact) */
{ ln2hi
+ln2lo
; return(-one
);}
/* expm1(INF) is INF, expm1(+big#) overflows to INF */
return( finite(x
) ? scalb(one
,5000) : x
);