171 lines
5.5 KiB
C
171 lines
5.5 KiB
C
/*
|
|
* Copyright (c) 1985, 1993
|
|
* The Regents of the University of California. All rights reserved.
|
|
*
|
|
* Redistribution and use in source and binary forms, with or without
|
|
* modification, are permitted provided that the following conditions
|
|
* are met:
|
|
* 1. Redistributions of source code must retain the above copyright
|
|
* notice, this list of conditions and the following disclaimer.
|
|
* 2. Redistributions in binary form must reproduce the above copyright
|
|
* notice, this list of conditions and the following disclaimer in the
|
|
* documentation and/or other materials provided with the distribution.
|
|
* 3. All advertising materials mentioning features or use of this software
|
|
* must display the following acknowledgement:
|
|
* This product includes software developed by the University of
|
|
* California, Berkeley and its contributors.
|
|
* 4. Neither the name of the University nor the names of its contributors
|
|
* may be used to endorse or promote products derived from this software
|
|
* without specific prior written permission.
|
|
*
|
|
* THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
|
|
* ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
|
|
* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
|
|
* ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
|
|
* FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
|
|
* DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
|
|
* OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
|
|
* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
|
|
* LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
|
|
* OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
|
|
* SUCH DAMAGE.
|
|
*/
|
|
|
|
#include <sys/cdefs.h>
|
|
__FBSDID("$FreeBSD$");
|
|
|
|
#ifndef lint
|
|
static char sccsid[] = "@(#)expm1.c 8.1 (Berkeley) 6/4/93";
|
|
#endif /* not lint */
|
|
|
|
/* EXPM1(X)
|
|
* 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:
|
|
* 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,c)
|
|
*
|
|
* 3. EXPM1(x) = 2^k * ( EXPM1(r) + 1-2^-k ).
|
|
*
|
|
* Remarks:
|
|
* 1. When k=1 and z < -0.25, we use the following formula for
|
|
* better accuracy:
|
|
* 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 }
|
|
* when k>56.
|
|
*
|
|
* Special cases:
|
|
* EXPM1(INF) is INF, EXPM1(NaN) is NaN;
|
|
* EXPM1(-INF)= -1;
|
|
* for finite argument, only EXPM1(0)=0 is exact.
|
|
*
|
|
* Accuracy:
|
|
* 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).
|
|
*
|
|
* 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.
|
|
*/
|
|
|
|
#include "mathimpl.h"
|
|
|
|
vc(ln2hi, 6.9314718055829871446E-1 ,7217,4031,0000,f7d0, 0, .B17217F7D00000)
|
|
vc(ln2lo, 1.6465949582897081279E-12 ,bcd5,2ce7,d9cc,e4f1, -39, .E7BCD5E4F1D9CC)
|
|
vc(lnhuge, 9.4961163736712506989E1 ,ec1d,43bd,9010,a73e, 7, .BDEC1DA73E9010)
|
|
vc(invln2, 1.4426950408889634148E0 ,aa3b,40b8,17f1,295c, 1, .B8AA3B295C17F1)
|
|
|
|
ic(ln2hi, 6.9314718036912381649E-1, -1, 1.62E42FEE00000)
|
|
ic(ln2lo, 1.9082149292705877000E-10, -33, 1.A39EF35793C76)
|
|
ic(lnhuge, 7.1602103751842355450E2, 9, 1.6602B15B7ECF2)
|
|
ic(invln2, 1.4426950408889633870E0, 0, 1.71547652B82FE)
|
|
|
|
#ifdef vccast
|
|
#define ln2hi vccast(ln2hi)
|
|
#define ln2lo vccast(ln2lo)
|
|
#define lnhuge vccast(lnhuge)
|
|
#define invln2 vccast(invln2)
|
|
#endif
|
|
|
|
double expm1(x)
|
|
double x;
|
|
{
|
|
const static double one=1.0, half=1.0/2.0;
|
|
double z,hi,lo,c;
|
|
int k;
|
|
#if defined(vax)||defined(tahoe)
|
|
static prec=56;
|
|
#else /* defined(vax)||defined(tahoe) */
|
|
static prec=53;
|
|
#endif /* defined(vax)||defined(tahoe) */
|
|
|
|
#if !defined(vax)&&!defined(tahoe)
|
|
if(x!=x) return(x); /* x is NaN */
|
|
#endif /* !defined(vax)&&!defined(tahoe) */
|
|
|
|
if( x <= lnhuge ) {
|
|
if( x >= -40.0 ) {
|
|
|
|
/* argument reduction : x - k*ln2 */
|
|
k= invln2 *x+copysign(0.5,x); /* k=NINT(x/ln2) */
|
|
hi=x-k*ln2hi ;
|
|
z=hi-(lo=k*ln2lo);
|
|
c=(hi-z)-lo;
|
|
|
|
if(k==0) return(z+__exp__E(z,c));
|
|
if(k==1)
|
|
if(z< -0.25)
|
|
{x=z+half;x +=__exp__E(z,c); return(x+x);}
|
|
else
|
|
{z+=__exp__E(z,c); x=half+z; return(x+x);}
|
|
/* end of k=1 */
|
|
|
|
else {
|
|
if(k<=prec)
|
|
{ x=one-scalb(one,-k); z += __exp__E(z,c);}
|
|
else if(k<100)
|
|
{ x = __exp__E(z,c)-scalb(one,-k); x+=z; z=one;}
|
|
else
|
|
{ x = __exp__E(z,c)+z; z=one;}
|
|
|
|
return (scalb(x+z,k));
|
|
}
|
|
}
|
|
/* end of x > lnunfl */
|
|
|
|
else
|
|
/* expm1(-big#) rounded to -1 (inexact) */
|
|
if(finite(x))
|
|
{ ln2hi+ln2lo; return(-one);}
|
|
|
|
/* expm1(-INF) is -1 */
|
|
else return(-one);
|
|
}
|
|
/* end of x < lnhuge */
|
|
|
|
else
|
|
/* expm1(INF) is INF, expm1(+big#) overflows to INF */
|
|
return( finite(x) ? scalb(one,5000) : x);
|
|
}
|