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