b_exp.c revision 330897 
1/*-
2 * SPDX-License-Identifier: BSD-4-Clause
3 *
4 * Copyright (c) 1985, 1993
5 *	The Regents of the University of California.  All rights reserved.
6 *
7 * Redistribution and use in source and binary forms, with or without
8 * modification, are permitted provided that the following conditions
9 * are met:
10 * 1. Redistributions of source code must retain the above copyright
11 *    notice, this list of conditions and the following disclaimer.
12 * 2. Redistributions in binary form must reproduce the above copyright
13 *    notice, this list of conditions and the following disclaimer in the
14 *    documentation and/or other materials provided with the distribution.
15 * 3. All advertising materials mentioning features or use of this software
16 *    must display the following acknowledgement:
17 *	This product includes software developed by the University of
18 *	California, Berkeley and its contributors.
19 * 4. Neither the name of the University nor the names of its contributors
20 *    may be used to endorse or promote products derived from this software
21 *    without specific prior written permission.
22 *
23 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
24 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
25 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
26 * ARE DISCLAIMED.  IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
27 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
28 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
29 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
30 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
31 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
32 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
33 * SUCH DAMAGE.
34 */
35
36/* @(#)exp.c	8.1 (Berkeley) 6/4/93 */
37#include <sys/cdefs.h>
38__FBSDID("$FreeBSD: stable/11/lib/msun/bsdsrc/b_exp.c 330897 2018-03-14 03:19:51Z eadler $");
39
40
41/* EXP(X)
42 * RETURN THE EXPONENTIAL OF X
43 * DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS)
44 * CODED IN C BY K.C. NG, 1/19/85;
45 * REVISED BY K.C. NG on 2/6/85, 2/15/85, 3/7/85, 3/24/85, 4/16/85, 6/14/86.
46 *
47 * Required system supported functions:
48 *	scalb(x,n)
49 *	copysign(x,y)
50 *	finite(x)
51 *
52 * Method:
53 *	1. Argument Reduction: given the input x, find r and integer k such
54 *	   that
55 *	                   x = k*ln2 + r,  |r| <= 0.5*ln2 .
56 *	   r will be represented as r := z+c for better accuracy.
57 *
58 *	2. Compute exp(r) by
59 *
60 *		exp(r) = 1 + r + r*R1/(2-R1),
61 *	   where
62 *		R1 = x - x^2*(p1+x^2*(p2+x^2*(p3+x^2*(p4+p5*x^2)))).
63 *
64 *	3. exp(x) = 2^k * exp(r) .
65 *
66 * Special cases:
67 *	exp(INF) is INF, exp(NaN) is NaN;
68 *	exp(-INF)=  0;
69 *	for finite argument, only exp(0)=1 is exact.
70 *
71 * Accuracy:
72 *	exp(x) returns the exponential of x nearly rounded. In a test run
73 *	with 1,156,000 random arguments on a VAX, the maximum observed
74 *	error was 0.869 ulps (units in the last place).
75 */
76
77#include "mathimpl.h"
78
79static const double p1 = 0x1.555555555553ep-3;
80static const double p2 = -0x1.6c16c16bebd93p-9;
81static const double p3 = 0x1.1566aaf25de2cp-14;
82static const double p4 = -0x1.bbd41c5d26bf1p-20;
83static const double p5 = 0x1.6376972bea4d0p-25;
84static const double ln2hi = 0x1.62e42fee00000p-1;
85static const double ln2lo = 0x1.a39ef35793c76p-33;
86static const double lnhuge = 0x1.6602b15b7ecf2p9;
87static const double lntiny = -0x1.77af8ebeae354p9;
88static const double invln2 = 0x1.71547652b82fep0;
89
90#if 0
91double exp(x)
92double x;
93{
94	double  z,hi,lo,c;
95	int k;
96
97#if !defined(vax)&&!defined(tahoe)
98	if(x!=x) return(x);	/* x is NaN */
99#endif	/* !defined(vax)&&!defined(tahoe) */
100	if( x <= lnhuge ) {
101		if( x >= lntiny ) {
102
103		    /* argument reduction : x --> x - k*ln2 */
104
105			k=invln2*x+copysign(0.5,x);	/* k=NINT(x/ln2) */
106
107		    /* express x-k*ln2 as hi-lo and let x=hi-lo rounded */
108
109			hi=x-k*ln2hi;
110			x=hi-(lo=k*ln2lo);
111
112		    /* return 2^k*[1+x+x*c/(2+c)]  */
113			z=x*x;
114			c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5))));
115			return  scalb(1.0+(hi-(lo-(x*c)/(2.0-c))),k);
116
117		}
118		/* end of x > lntiny */
119
120		else
121		     /* exp(-big#) underflows to zero */
122		     if(finite(x))  return(scalb(1.0,-5000));
123
124		     /* exp(-INF) is zero */
125		     else return(0.0);
126	}
127	/* end of x < lnhuge */
128
129	else
130	/* exp(INF) is INF, exp(+big#) overflows to INF */
131	    return( finite(x) ?  scalb(1.0,5000)  : x);
132}
133#endif
134
135/* returns exp(r = x + c) for |c| < |x| with no overlap.  */
136
137double __exp__D(x, c)
138double x, c;
139{
140	double  z,hi,lo;
141	int k;
142
143	if (x != x)	/* x is NaN */
144		return(x);
145	if ( x <= lnhuge ) {
146		if ( x >= lntiny ) {
147
148		    /* argument reduction : x --> x - k*ln2 */
149			z = invln2*x;
150			k = z + copysign(.5, x);
151
152		    /* express (x+c)-k*ln2 as hi-lo and let x=hi-lo rounded */
153
154			hi=(x-k*ln2hi);			/* Exact. */
155			x= hi - (lo = k*ln2lo-c);
156		    /* return 2^k*[1+x+x*c/(2+c)]  */
157			z=x*x;
158			c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5))));
159			c = (x*c)/(2.0-c);
160
161			return  scalb(1.+(hi-(lo - c)), k);
162		}
163		/* end of x > lntiny */
164
165		else
166		     /* exp(-big#) underflows to zero */
167		     if(finite(x))  return(scalb(1.0,-5000));
168
169		     /* exp(-INF) is zero */
170		     else return(0.0);
171	}
172	/* end of x < lnhuge */
173
174	else
175	/* exp(INF) is INF, exp(+big#) overflows to INF */
176	    return( finite(x) ?  scalb(1.0,5000)  : x);
177}
178