4.4BSD/usr/src/old/libm/libm/exp__E.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__E.c 1.2 (Berkeley) 8/21/85";
#endif not lint
/* exp__E(x,c)
* ASSUMPTION: c << x SO THAT fl(x+c)=x.
* (c is the correction term for x)
* exp__E RETURNS
*
* / exp(x+c) - 1 - x , 1E-19 < |x| < .3465736
* exp__E(x,c) = |
* \ 0 , |x| < 1E-19.
*
* DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS)
* KERNEL FUNCTION OF EXP, EXPM1, POW FUNCTIONS
* CODED IN C BY K.C. NG, 1/31/85;
* REVISED BY K.C. NG on 3/16/85, 4/16/85.
*
* Required system supported function:
* copysign(x,y)
*
* Method:
* 1. Rational approximation. Let r=x+c.
* Based on
* 2 * sinh(r/2)
* exp(r) - 1 = ---------------------- ,
* cosh(r/2) - sinh(r/2)
* exp__E(r) is computed using
* x*x (x/2)*W - ( Q - ( 2*P + x*P ) )
* --- + (c + x*[---------------------------------- + c ])
* 2 1 - W
* where P := p1*x^2 + p2*x^4,
* Q := q1*x^2 + q2*x^4 (for 56 bits precision, add q3*x^6)
* W := x/2-(Q-x*P),
*
* (See the listing below for the values of p1,p2,q1,q2,q3. The poly-
* nomials P and Q may be regarded as the approximations to sinh
* and cosh :
* sinh(r/2) = r/2 + r * P , cosh(r/2) = 1 + Q . )
*
* The coefficients were obtained by a special Remez algorithm.
*
* Approximation error:
*
* | exp(x) - 1 | 2**(-57), (IEEE double)
* | ------------ - (exp__E(x,0)+x)/x | <=
* | x | 2**(-69). (VAX D)
*
* 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 */
/* static double */
/* p1 = 1.5150724356786683059E-2 , Hex 2^ -6 * .F83ABE67E1066A */
/* p2 = 6.3112487873718332688E-5 , Hex 2^-13 * .845B4248CD0173 */
/* q1 = 1.1363478204690669916E-1 , Hex 2^ -3 * .E8B95A44A2EC45 */
/* q2 = 1.2624568129896839182E-3 , Hex 2^ -9 * .A5790572E4F5E7 */
/* q3 = 1.5021856115869022674E-6 ; Hex 2^-19 * .C99EB4604AC395 */
static long p1x[] = { 0x3abe3d78, 0x066a67e1};
static long p2x[] = { 0x5b423984, 0x017348cd};
static long q1x[] = { 0xb95a3ee8, 0xec4544a2};
static long q2x[] = { 0x79053ba5, 0xf5e772e4};
static long q3x[] = { 0x9eb436c9, 0xc395604a};
#define p1 (*(double*)p1x)
#define p2 (*(double*)p2x)
#define q1 (*(double*)q1x)
#define q2 (*(double*)q2x)
#define q3 (*(double*)q3x)
#else /* IEEE double */
static double
p1 = 1.3887401997267371720E-2 , /*Hex 2^ -7 * 1.C70FF8B3CC2CF */
p2 = 3.3044019718331897649E-5 , /*Hex 2^-15 * 1.15317DF4526C4 */
q1 = 1.1110813732786649355E-1 , /*Hex 2^ -4 * 1.C719538248597 */
q2 = 9.9176615021572857300E-4 ; /*Hex 2^-10 * 1.03FC4CB8C98E8 */
#endif
double exp__E(x,c)
double x,c;
{
double static zero=0.0, one=1.0, half=1.0/2.0, small=1.0E-19;
double copysign(),z,p,q,xp,xh,w;
if(copysign(x,one)>small) {
z = x*x ;
p = z*( p1 +z* p2 );
#ifdef VAX
q = z*( q1 +z*( q2 +z* q3 ));
#else /* IEEE double */
q = z*( q1 +z* q2 );
#endif
xp= x*p ;
xh= x*half ;
w = xh-(q-xp) ;
p = p+p;
c += x*((xh*w-(q-(p+xp)))/(one-w)+c);
return(z*half+c);
}
/* end of |x| > small */
else {
if(x!=zero) one+small; /* raise the inexact flag */
return(copysign(zero,x));
}
}