1 /* $NetBSD: n_exp.c,v 1.9 2014/10/10 20:58:09 martin Exp $ */
2 /*
3 * Copyright (c) 1985, 1993
4 * The Regents of the University of California. All rights reserved.
5 *
6 * Redistribution and use in source and binary forms, with or without
7 * modification, are permitted provided that the following conditions
8 * are met:
9 * 1. Redistributions of source code must retain the above copyright
10 * notice, this list of conditions and the following disclaimer.
11 * 2. Redistributions in binary form must reproduce the above copyright
12 * notice, this list of conditions and the following disclaimer in the
13 * documentation and/or other materials provided with the distribution.
14 * 3. Neither the name of the University nor the names of its contributors
15 * may be used to endorse or promote products derived from this software
16 * without specific prior written permission.
17 *
18 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
19 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
20 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
21 * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
22 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
23 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
24 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
25 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
26 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
27 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
28 * SUCH DAMAGE.
29 */
30
31 #ifndef lint
32 #if 0
33 static char sccsid[] = "@(#)exp.c 8.1 (Berkeley) 6/4/93";
34 #endif
35 #endif /* not lint */
36
37 /* EXP(X)
38 * RETURN THE EXPONENTIAL OF X
39 * DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS)
40 * CODED IN C BY K.C. NG, 1/19/85;
41 * REVISED BY K.C. NG on 2/6/85, 2/15/85, 3/7/85, 3/24/85, 4/16/85, 6/14/86.
42 *
43 * Required system supported functions:
44 * scalb(x,n)
45 * copysign(x,y)
46 * finite(x)
47 *
48 * Method:
49 * 1. Argument Reduction: given the input x, find r and integer k such
50 * that
51 * x = k*ln2 + r, |r| <= 0.5*ln2 .
52 * r will be represented as r := z+c for better accuracy.
53 *
54 * 2. Compute exp(r) by
55 *
56 * exp(r) = 1 + r + r*R1/(2-R1),
57 * where
58 * R1 = x - x^2*(p1+x^2*(p2+x^2*(p3+x^2*(p4+p5*x^2)))).
59 *
60 * 3. exp(x) = 2^k * exp(r) .
61 *
62 * Special cases:
63 * exp(INF) is INF, exp(NaN) is NaN;
64 * exp(-INF)= 0;
65 * for finite argument, only exp(0)=1 is exact.
66 *
67 * Accuracy:
68 * exp(x) returns the exponential of x nearly rounded. In a test run
69 * with 1,156,000 random arguments on a VAX, the maximum observed
70 * error was 0.869 ulps (units in the last place).
71 *
72 * Constants:
73 * The hexadecimal values are the intended ones for the following constants.
74 * The decimal values may be used, provided that the compiler will convert
75 * from decimal to binary accurately enough to produce the hexadecimal values
76 * shown.
77 */
78
79 #define _LIBM_STATIC
80 #include "../src/namespace.h"
81 #include "mathimpl.h"
82
83 #ifdef __weak_alias
84 __weak_alias(exp, _exp);
85 __weak_alias(_expl, _exp);
86 __weak_alias(expf, _expf);
87 #endif
88
89 vc(ln2hi, 6.9314718055829871446E-1 ,7217,4031,0000,f7d0, 0, .B17217F7D00000)
90 vc(ln2lo, 1.6465949582897081279E-12 ,bcd5,2ce7,d9cc,e4f1, -39, .E7BCD5E4F1D9CC)
91 vc(lnhuge, 9.4961163736712506989E1 ,ec1d,43bd,9010,a73e, 7, .BDEC1DA73E9010)
92 vc(lntiny,-9.5654310917272452386E1 ,4f01,c3bf,33af,d72e, 7,-.BF4F01D72E33AF)
93 vc(invln2, 1.4426950408889634148E0 ,aa3b,40b8,17f1,295c, 1, .B8AA3B295C17F1)
94 vc(p1, 1.6666666666666602251E-1 ,aaaa,3f2a,a9f1,aaaa, -2, .AAAAAAAAAAA9F1)
95 vc(p2, -2.7777777777015591216E-3 ,0b60,bc36,ec94,b5f5, -8,-.B60B60B5F5EC94)
96 vc(p3, 6.6137563214379341918E-5 ,b355,398a,f15f,792e, -13, .8AB355792EF15F)
97 vc(p4, -1.6533902205465250480E-6 ,ea0e,b6dd,5f84,2e93, -19,-.DDEA0E2E935F84)
98 vc(p5, 4.1381367970572387085E-8 ,bb4b,3431,2683,95f5, -24, .B1BB4B95F52683)
99
100 #ifdef vccast
101 #define ln2hi vccast(ln2hi)
102 #define ln2lo vccast(ln2lo)
103 #define lnhuge vccast(lnhuge)
104 #define lntiny vccast(lntiny)
105 #define invln2 vccast(invln2)
106 #define p1 vccast(p1)
107 #define p2 vccast(p2)
108 #define p3 vccast(p3)
109 #define p4 vccast(p4)
110 #define p5 vccast(p5)
111 #endif
112
113 ic(p1, 1.6666666666666601904E-1, -3, 1.555555555553E)
114 ic(p2, -2.7777777777015593384E-3, -9, -1.6C16C16BEBD93)
115 ic(p3, 6.6137563214379343612E-5, -14, 1.1566AAF25DE2C)
116 ic(p4, -1.6533902205465251539E-6, -20, -1.BBD41C5D26BF1)
117 ic(p5, 4.1381367970572384604E-8, -25, 1.6376972BEA4D0)
118 ic(ln2hi, 6.9314718036912381649E-1, -1, 1.62E42FEE00000)
119 ic(ln2lo, 1.9082149292705877000E-10,-33, 1.A39EF35793C76)
120 ic(lnhuge, 7.1602103751842355450E2, 9, 1.6602B15B7ECF2)
121 ic(lntiny,-7.5137154372698068983E2, 9, -1.77AF8EBEAE354)
122 ic(invln2, 1.4426950408889633870E0, 0, 1.71547652B82FE)
123
124 double
exp(double x)125 exp(double x)
126 {
127 double z,hi,lo,c;
128 int k;
129
130 #if !defined(__vax__)&&!defined(tahoe)
131 if(x!=x) return(x); /* x is NaN */
132 #endif /* !defined(__vax__)&&!defined(tahoe) */
133 if( x <= lnhuge ) {
134 if( x >= lntiny ) {
135
136 /* argument reduction : x --> x - k*ln2 */
137
138 k=invln2*x+copysign(0.5,x); /* k=NINT(x/ln2) */
139
140 /* express x-k*ln2 as hi-lo and let x=hi-lo rounded */
141
142 hi=x-k*ln2hi;
143 x=hi-(lo=k*ln2lo);
144
145 /* return 2^k*[1+x+x*c/(2+c)] */
146 z=x*x;
147 c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5))));
148 return scalb(1.0+(hi-(lo-(x*c)/(2.0-c))),k);
149
150 }
151 /* end of x > lntiny */
152
153 else
154 /* exp(-big#) underflows to zero */
155 if(finite(x)) return(scalb(1.0,-5000));
156
157 /* exp(-INF) is zero */
158 else return(0.0);
159 }
160 /* end of x < lnhuge */
161
162 else
163 /* exp(INF) is INF, exp(+big#) overflows to INF */
164 return( finite(x) ? scalb(1.0,5000) : x);
165 }
166
167 float
expf(float x)168 expf(float x)
169 {
170 return(exp((double)x));
171 }
172
173 /* returns exp(r = x + c) for |c| < |x| with no overlap. */
174
175 double
__exp__D(double x,double c)176 __exp__D(double x, double c)
177 {
178 double z,hi,lo;
179 int k;
180
181 #if !defined(__vax__)&&!defined(tahoe)
182 if (x!=x) return(x); /* x is NaN */
183 #endif /* !defined(__vax__)&&!defined(tahoe) */
184 if ( x <= lnhuge ) {
185 if ( x >= lntiny ) {
186
187 /* argument reduction : x --> x - k*ln2 */
188 z = invln2*x;
189 k = z + copysign(.5, x);
190
191 /* express (x+c)-k*ln2 as hi-lo and let x=hi-lo rounded */
192
193 hi=(x-k*ln2hi); /* Exact. */
194 x= hi - (lo = k*ln2lo-c);
195 /* return 2^k*[1+x+x*c/(2+c)] */
196 z=x*x;
197 c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5))));
198 c = (x*c)/(2.0-c);
199
200 return scalb(1.+(hi-(lo - c)), k);
201 }
202 /* end of x > lntiny */
203
204 else
205 /* exp(-big#) underflows to zero */
206 if(finite(x)) return(scalb(1.0,-5000));
207
208 /* exp(-INF) is zero */
209 else return(0.0);
210 }
211 /* end of x < lnhuge */
212
213 else
214 /* exp(INF) is INF, exp(+big#) overflows to INF */
215 return( finite(x) ? scalb(1.0,5000) : x);
216 }
217