1*84d9c625SLionel Sambuc /* $NetBSD: b_exp.c,v 1.1 2012/05/05 17:54:14 christos Exp $ */
2*84d9c625SLionel Sambuc
3*84d9c625SLionel Sambuc /*
4*84d9c625SLionel Sambuc * Copyright (c) 1985, 1993
5*84d9c625SLionel Sambuc * The Regents of the University of California. All rights reserved.
6*84d9c625SLionel Sambuc *
7*84d9c625SLionel Sambuc * Redistribution and use in source and binary forms, with or without
8*84d9c625SLionel Sambuc * modification, are permitted provided that the following conditions
9*84d9c625SLionel Sambuc * are met:
10*84d9c625SLionel Sambuc * 1. Redistributions of source code must retain the above copyright
11*84d9c625SLionel Sambuc * notice, this list of conditions and the following disclaimer.
12*84d9c625SLionel Sambuc * 2. Redistributions in binary form must reproduce the above copyright
13*84d9c625SLionel Sambuc * notice, this list of conditions and the following disclaimer in the
14*84d9c625SLionel Sambuc * documentation and/or other materials provided with the distribution.
15*84d9c625SLionel Sambuc * 3. All advertising materials mentioning features or use of this software
16*84d9c625SLionel Sambuc * must display the following acknowledgement:
17*84d9c625SLionel Sambuc * This product includes software developed by the University of
18*84d9c625SLionel Sambuc * California, Berkeley and its contributors.
19*84d9c625SLionel Sambuc * 4. Neither the name of the University nor the names of its contributors
20*84d9c625SLionel Sambuc * may be used to endorse or promote products derived from this software
21*84d9c625SLionel Sambuc * without specific prior written permission.
22*84d9c625SLionel Sambuc *
23*84d9c625SLionel Sambuc * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
24*84d9c625SLionel Sambuc * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
25*84d9c625SLionel Sambuc * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
26*84d9c625SLionel Sambuc * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
27*84d9c625SLionel Sambuc * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
28*84d9c625SLionel Sambuc * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
29*84d9c625SLionel Sambuc * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
30*84d9c625SLionel Sambuc * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
31*84d9c625SLionel Sambuc * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
32*84d9c625SLionel Sambuc * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
33*84d9c625SLionel Sambuc * SUCH DAMAGE.
34*84d9c625SLionel Sambuc */
35*84d9c625SLionel Sambuc
36*84d9c625SLionel Sambuc /* @(#)exp.c 8.1 (Berkeley) 6/4/93 */
37*84d9c625SLionel Sambuc #include <sys/cdefs.h>
38*84d9c625SLionel Sambuc #if 0
39*84d9c625SLionel Sambuc __FBSDID("$FreeBSD: release/9.0.0/lib/msun/bsdsrc/b_exp.c 176449 2008-02-22 02:26:51Z das $");
40*84d9c625SLionel Sambuc #else
41*84d9c625SLionel Sambuc __RCSID("$NetBSD: b_exp.c,v 1.1 2012/05/05 17:54:14 christos Exp $");
42*84d9c625SLionel Sambuc #endif
43*84d9c625SLionel Sambuc
44*84d9c625SLionel Sambuc
45*84d9c625SLionel Sambuc /* EXP(X)
46*84d9c625SLionel Sambuc * RETURN THE EXPONENTIAL OF X
47*84d9c625SLionel Sambuc * DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS)
48*84d9c625SLionel Sambuc * CODED IN C BY K.C. NG, 1/19/85;
49*84d9c625SLionel Sambuc * REVISED BY K.C. NG on 2/6/85, 2/15/85, 3/7/85, 3/24/85, 4/16/85, 6/14/86.
50*84d9c625SLionel Sambuc *
51*84d9c625SLionel Sambuc * Required system supported functions:
52*84d9c625SLionel Sambuc * scalb(x,n)
53*84d9c625SLionel Sambuc * copysign(x,y)
54*84d9c625SLionel Sambuc * finite(x)
55*84d9c625SLionel Sambuc *
56*84d9c625SLionel Sambuc * Method:
57*84d9c625SLionel Sambuc * 1. Argument Reduction: given the input x, find r and integer k such
58*84d9c625SLionel Sambuc * that
59*84d9c625SLionel Sambuc * x = k*ln2 + r, |r| <= 0.5*ln2 .
60*84d9c625SLionel Sambuc * r will be represented as r := z+c for better accuracy.
61*84d9c625SLionel Sambuc *
62*84d9c625SLionel Sambuc * 2. Compute exp(r) by
63*84d9c625SLionel Sambuc *
64*84d9c625SLionel Sambuc * exp(r) = 1 + r + r*R1/(2-R1),
65*84d9c625SLionel Sambuc * where
66*84d9c625SLionel Sambuc * R1 = x - x^2*(p1+x^2*(p2+x^2*(p3+x^2*(p4+p5*x^2)))).
67*84d9c625SLionel Sambuc *
68*84d9c625SLionel Sambuc * 3. exp(x) = 2^k * exp(r) .
69*84d9c625SLionel Sambuc *
70*84d9c625SLionel Sambuc * Special cases:
71*84d9c625SLionel Sambuc * exp(INF) is INF, exp(NaN) is NaN;
72*84d9c625SLionel Sambuc * exp(-INF)= 0;
73*84d9c625SLionel Sambuc * for finite argument, only exp(0)=1 is exact.
74*84d9c625SLionel Sambuc *
75*84d9c625SLionel Sambuc * Accuracy:
76*84d9c625SLionel Sambuc * exp(x) returns the exponential of x nearly rounded. In a test run
77*84d9c625SLionel Sambuc * with 1,156,000 random arguments on a VAX, the maximum observed
78*84d9c625SLionel Sambuc * error was 0.869 ulps (units in the last place).
79*84d9c625SLionel Sambuc */
80*84d9c625SLionel Sambuc
81*84d9c625SLionel Sambuc #include "math.h"
82*84d9c625SLionel Sambuc #include "math_private.h"
83*84d9c625SLionel Sambuc
84*84d9c625SLionel Sambuc static const double p1 = 0x1.555555555553ep-3;
85*84d9c625SLionel Sambuc static const double p2 = -0x1.6c16c16bebd93p-9;
86*84d9c625SLionel Sambuc static const double p3 = 0x1.1566aaf25de2cp-14;
87*84d9c625SLionel Sambuc static const double p4 = -0x1.bbd41c5d26bf1p-20;
88*84d9c625SLionel Sambuc static const double p5 = 0x1.6376972bea4d0p-25;
89*84d9c625SLionel Sambuc static const double ln2hi = 0x1.62e42fee00000p-1;
90*84d9c625SLionel Sambuc static const double ln2lo = 0x1.a39ef35793c76p-33;
91*84d9c625SLionel Sambuc static const double lnhuge = 0x1.6602b15b7ecf2p9;
92*84d9c625SLionel Sambuc static const double lntiny = -0x1.77af8ebeae354p9;
93*84d9c625SLionel Sambuc static const double invln2 = 0x1.71547652b82fep0;
94*84d9c625SLionel Sambuc
95*84d9c625SLionel Sambuc /* returns exp(r = x + c) for |c| < |x| with no overlap. */
96*84d9c625SLionel Sambuc
97*84d9c625SLionel Sambuc double
__exp__D(double x,double c)98*84d9c625SLionel Sambuc __exp__D(double x, double c)
99*84d9c625SLionel Sambuc {
100*84d9c625SLionel Sambuc double z,hi,lo;
101*84d9c625SLionel Sambuc int k;
102*84d9c625SLionel Sambuc
103*84d9c625SLionel Sambuc if (x != x) /* x is NaN */
104*84d9c625SLionel Sambuc return(x);
105*84d9c625SLionel Sambuc if ( x <= lnhuge ) {
106*84d9c625SLionel Sambuc if ( x >= lntiny ) {
107*84d9c625SLionel Sambuc
108*84d9c625SLionel Sambuc /* argument reduction : x --> x - k*ln2 */
109*84d9c625SLionel Sambuc z = invln2*x;
110*84d9c625SLionel Sambuc k = z + copysign(.5, x);
111*84d9c625SLionel Sambuc
112*84d9c625SLionel Sambuc /* express (x+c)-k*ln2 as hi-lo and let x=hi-lo rounded */
113*84d9c625SLionel Sambuc
114*84d9c625SLionel Sambuc hi=(x-k*ln2hi); /* Exact. */
115*84d9c625SLionel Sambuc x= hi - (lo = k*ln2lo-c);
116*84d9c625SLionel Sambuc /* return 2^k*[1+x+x*c/(2+c)] */
117*84d9c625SLionel Sambuc z=x*x;
118*84d9c625SLionel Sambuc c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5))));
119*84d9c625SLionel Sambuc c = (x*c)/(2.0-c);
120*84d9c625SLionel Sambuc
121*84d9c625SLionel Sambuc return scalb(1.+(hi-(lo - c)), k);
122*84d9c625SLionel Sambuc }
123*84d9c625SLionel Sambuc /* end of x > lntiny */
124*84d9c625SLionel Sambuc
125*84d9c625SLionel Sambuc else
126*84d9c625SLionel Sambuc /* exp(-big#) underflows to zero */
127*84d9c625SLionel Sambuc if(finite(x)) return(scalb(1.0,-5000));
128*84d9c625SLionel Sambuc
129*84d9c625SLionel Sambuc /* exp(-INF) is zero */
130*84d9c625SLionel Sambuc else return(0.0);
131*84d9c625SLionel Sambuc }
132*84d9c625SLionel Sambuc /* end of x < lnhuge */
133*84d9c625SLionel Sambuc
134*84d9c625SLionel Sambuc else
135*84d9c625SLionel Sambuc /* exp(INF) is INF, exp(+big#) overflows to INF */
136*84d9c625SLionel Sambuc return( finite(x) ? scalb(1.0,5000) : x);
137*84d9c625SLionel Sambuc }
138