* 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.
"@(#)exp.c 4.3 (Berkeley) 8/21/85; 1.8 (ucb.elefunt) %G%";
* 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, 6/14/86.
* 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.
* exp(r) = 1 + r + r*R1/(2-R1),
* R1 = x - x^2*(p1+x^2*(p2+x^2*(p3+x^2*(p4+p5*x^2)))).
* 3. exp(x) = 2^k * exp(r) .
* exp(INF) is INF, exp(NaN) is NaN;
* for finite argument, only exp(0)=1 is exact.
* 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 0.869 ulps (units in 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
#if defined(vax)||defined(tahoe) /* VAX D format */
#define _0x(A,B) 0x/**/A/**/B
#define _0x(A,B) 0x/**/B/**/A
/* 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 */
/* p1 = 1.6666666666666602251E-1 , Hex 2^-2 * .AAAAAAAAAAA9F1 */
/* p2 = -2.7777777777015591216E-3 , Hex 2^-8 * -.B60B60B5F5EC94 */
/* p3 = 6.6137563214379341918E-5 , Hex 2^-13 * .8AB355792EF15F */
/* p4 = -1.6533902205465250480E-6 , Hex 2^-19 * -.DDEA0E2E935F84 */
/* p5 = 4.1381367970572387085E-8 , Hex 2^-24 * .B1BB4B95F52683 */
static long ln2hix
[] = { _0x(7217,4031), _0x(0000,f7d0
)};
static long ln2lox
[] = { _0x(bcd5
,2ce7
), _0x(d9cc
,e4f1
)};
static long lnhugex
[] = { _0x(ec1d
,43bd
), _0x(9010,a73e
)};
static long lntinyx
[] = { _0x(4f01
,c3bf
), _0x(33af
,d72e
)};
static long invln2x
[] = { _0x(aa3b
,40b8
), _0x(17f1
,295c
)};
static long p1x
[] = { _0x(aaaa
,3f2a
), _0x(a9f1
,aaaa
)};
static long p2x
[] = { _0x(0b60,bc36
), _0x(ec94
,b5f5
)};
static long p3x
[] = { _0x(b355
,398a
), _0x(f15f
,792e
)};
static long p4x
[] = { _0x(ea0e
,b6dd
), _0x(5f84
,2e93
)};
static long p5x
[] = { _0x(bb4b
,3431), _0x(2683,95f5
)};
#define ln2hi (*(double*)ln2hix)
#define ln2lo (*(double*)ln2lox)
#define lnhuge (*(double*)lnhugex)
#define lntiny (*(double*)lntinyx)
#define invln2 (*(double*)invln2x)
#define p1 (*(double*)p1x)
#define p2 (*(double*)p2x)
#define p3 (*(double*)p3x)
#define p4 (*(double*)p4x)
#define p5 (*(double*)p5x)
#else /* defined(vax)||defined(tahoe) */
p1
= 1.6666666666666601904E-1 , /*Hex 2^-3 * 1.555555555553E */
p2
= -2.7777777777015593384E-3 , /*Hex 2^-9 * -1.6C16C16BEBD93 */
p3
= 6.6137563214379343612E-5 , /*Hex 2^-14 * 1.1566AAF25DE2C */
p4
= -1.6533902205465251539E-6 , /*Hex 2^-20 * -1.BBD41C5D26BF1 */
p5
= 4.1381367970572384604E-8 , /*Hex 2^-25 * 1.6376972BEA4D0 */
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 /* defined(vax)||defined(tahoe) */
double scalb(), copysign(), z
,hi
,lo
,c
;
#if !defined(vax)&&!defined(tahoe)
if(x
!=x
) return(x
); /* x is NaN */
#endif /* !defined(vax)&&!defined(tahoe) */
/* argument reduction : x --> x - k*ln2 */
k
=invln2
*x
+copysign(0.5,x
); /* k=NINT(x/ln2) */
/* express x-k*ln2 as hi-lo and let x=hi-lo rounded */
/* return 2^k*[1+x+x*c/(2+c)] */
c
= x
- z
*(p1
+z
*(p2
+z
*(p3
+z
*(p4
+z
*p5
))));
return scalb(1.0+(hi
-(lo
-(x
*c
)/(2.0-c
))),k
);
/* exp(-big#) underflows to zero */
if(finite(x
)) return(scalb(1.0,-5000));
/* exp(INF) is INF, exp(+big#) overflows to INF */
return( finite(x
) ? scalb(1.0,5000) : x
);