xref: /freebsd/lib/msun/bsdsrc/b_exp.c (revision ae83180158c4c937f170e31eff311b18c0286a93)
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 #ifndef lint
35 static char sccsid[] = "@(#)exp.c	8.1 (Berkeley) 6/4/93";
36 #endif /* not lint */
37 #include <sys/cdefs.h>
38 __FBSDID("$FreeBSD$");
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  * Constants:
77  * The hexadecimal values are the intended ones for the following constants.
78  * The decimal values may be used, provided that the compiler will convert
79  * from decimal to binary accurately enough to produce the hexadecimal values
80  * shown.
81  */
82 
83 #include "mathimpl.h"
84 
85 vc(ln2hi,  6.9314718055829871446E-1  ,7217,4031,0000,f7d0,   0, .B17217F7D00000)
86 vc(ln2lo,  1.6465949582897081279E-12 ,bcd5,2ce7,d9cc,e4f1, -39, .E7BCD5E4F1D9CC)
87 vc(lnhuge, 9.4961163736712506989E1   ,ec1d,43bd,9010,a73e,   7, .BDEC1DA73E9010)
88 vc(lntiny,-9.5654310917272452386E1   ,4f01,c3bf,33af,d72e,   7,-.BF4F01D72E33AF)
89 vc(invln2, 1.4426950408889634148E0   ,aa3b,40b8,17f1,295c,   1, .B8AA3B295C17F1)
90 vc(p1,     1.6666666666666602251E-1  ,aaaa,3f2a,a9f1,aaaa,  -2, .AAAAAAAAAAA9F1)
91 vc(p2,    -2.7777777777015591216E-3  ,0b60,bc36,ec94,b5f5,  -8,-.B60B60B5F5EC94)
92 vc(p3,     6.6137563214379341918E-5  ,b355,398a,f15f,792e, -13, .8AB355792EF15F)
93 vc(p4,    -1.6533902205465250480E-6  ,ea0e,b6dd,5f84,2e93, -19,-.DDEA0E2E935F84)
94 vc(p5,     4.1381367970572387085E-8  ,bb4b,3431,2683,95f5, -24, .B1BB4B95F52683)
95 
96 #ifdef vccast
97 #define    ln2hi    vccast(ln2hi)
98 #define    ln2lo    vccast(ln2lo)
99 #define   lnhuge    vccast(lnhuge)
100 #define   lntiny    vccast(lntiny)
101 #define   invln2    vccast(invln2)
102 #define       p1    vccast(p1)
103 #define       p2    vccast(p2)
104 #define       p3    vccast(p3)
105 #define       p4    vccast(p4)
106 #define       p5    vccast(p5)
107 #endif
108 
109 ic(p1,     1.6666666666666601904E-1,  -3,  1.555555555553E)
110 ic(p2,    -2.7777777777015593384E-3,  -9, -1.6C16C16BEBD93)
111 ic(p3,     6.6137563214379343612E-5, -14,  1.1566AAF25DE2C)
112 ic(p4,    -1.6533902205465251539E-6, -20, -1.BBD41C5D26BF1)
113 ic(p5,     4.1381367970572384604E-8, -25,  1.6376972BEA4D0)
114 ic(ln2hi,  6.9314718036912381649E-1,  -1,  1.62E42FEE00000)
115 ic(ln2lo,  1.9082149292705877000E-10,-33,  1.A39EF35793C76)
116 ic(lnhuge, 7.1602103751842355450E2,    9,  1.6602B15B7ECF2)
117 ic(lntiny,-7.5137154372698068983E2,    9, -1.77AF8EBEAE354)
118 ic(invln2, 1.4426950408889633870E0,    0,  1.71547652B82FE)
119 
120 #if 0
121 double exp(x)
122 double x;
123 {
124 	double  z,hi,lo,c;
125 	int k;
126 
127 #if !defined(vax)&&!defined(tahoe)
128 	if(x!=x) return(x);	/* x is NaN */
129 #endif	/* !defined(vax)&&!defined(tahoe) */
130 	if( x <= lnhuge ) {
131 		if( x >= lntiny ) {
132 
133 		    /* argument reduction : x --> x - k*ln2 */
134 
135 			k=invln2*x+copysign(0.5,x);	/* k=NINT(x/ln2) */
136 
137 		    /* express x-k*ln2 as hi-lo and let x=hi-lo rounded */
138 
139 			hi=x-k*ln2hi;
140 			x=hi-(lo=k*ln2lo);
141 
142 		    /* return 2^k*[1+x+x*c/(2+c)]  */
143 			z=x*x;
144 			c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5))));
145 			return  scalb(1.0+(hi-(lo-(x*c)/(2.0-c))),k);
146 
147 		}
148 		/* end of x > lntiny */
149 
150 		else
151 		     /* exp(-big#) underflows to zero */
152 		     if(finite(x))  return(scalb(1.0,-5000));
153 
154 		     /* exp(-INF) is zero */
155 		     else return(0.0);
156 	}
157 	/* end of x < lnhuge */
158 
159 	else
160 	/* exp(INF) is INF, exp(+big#) overflows to INF */
161 	    return( finite(x) ?  scalb(1.0,5000)  : x);
162 }
163 #endif
164 
165 /* returns exp(r = x + c) for |c| < |x| with no overlap.  */
166 
167 double __exp__D(x, c)
168 double x, c;
169 {
170 	double  z,hi,lo, t;
171 	int k;
172 
173 #if !defined(vax)&&!defined(tahoe)
174 	if (x!=x) return(x);	/* x is NaN */
175 #endif	/* !defined(vax)&&!defined(tahoe) */
176 	if ( x <= lnhuge ) {
177 		if ( x >= lntiny ) {
178 
179 		    /* argument reduction : x --> x - k*ln2 */
180 			z = invln2*x;
181 			k = z + copysign(.5, x);
182 
183 		    /* express (x+c)-k*ln2 as hi-lo and let x=hi-lo rounded */
184 
185 			hi=(x-k*ln2hi);			/* Exact. */
186 			x= hi - (lo = k*ln2lo-c);
187 		    /* return 2^k*[1+x+x*c/(2+c)]  */
188 			z=x*x;
189 			c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5))));
190 			c = (x*c)/(2.0-c);
191 
192 			return  scalb(1.+(hi-(lo - c)), k);
193 		}
194 		/* end of x > lntiny */
195 
196 		else
197 		     /* exp(-big#) underflows to zero */
198 		     if(finite(x))  return(scalb(1.0,-5000));
199 
200 		     /* exp(-INF) is zero */
201 		     else return(0.0);
202 	}
203 	/* end of x < lnhuge */
204 
205 	else
206 	/* exp(INF) is INF, exp(+big#) overflows to INF */
207 	    return( finite(x) ?  scalb(1.0,5000)  : x);
208 }
209