* 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
[] = "@(#)exp__E.c 1.2 (Berkeley) 8/21/85";
* ASSUMPTION: c << x SO THAT fl(x+c)=x.
* (c is the correction term for x)
* / exp(x+c) - 1 - x , 1E-19 < |x| < .3465736
* 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:
* 1. Rational approximation. Let r=x+c.
* exp(r) - 1 = ---------------------- ,
* exp__E(r) is computed using
* x*x (x/2)*W - ( Q - ( 2*P + x*P ) )
* --- + (c + x*[---------------------------------- + c ])
* where P := p1*x^2 + p2*x^4,
* Q := q1*x^2 + q2*x^4 (for 56 bits precision, add q3*x^6)
* (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
* sinh(r/2) = r/2 + r * P , cosh(r/2) = 1 + Q . )
* The coefficients were obtained by a special Remez algorithm.
* | exp(x) - 1 | 2**(-57), (IEEE double)
* | ------------ - (exp__E(x,0)+x)/x | <=
* | x | 2**(-69). (VAX D)
* 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 */
/* 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)
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 */
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
) {
q
= z
*( q1
+z
*( q2
+z
* q3
));
c
+= x
*((xh
*w
-(q
-(p
+xp
)))/(one
-w
)+c
);
if(x
!=zero
) one
+small
; /* raise the inexact flag */
return(copysign(zero
,x
));