1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
|
/*
* Copyright (c) 1985, 1993
* The Regents of the University of California. All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
* 1. Redistributions of source code must retain the above copyright
* notice, this list of conditions and the following disclaimer.
* 2. Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in the
* documentation and/or other materials provided with the distribution.
* 3. All advertising materials mentioning features or use of this software
* must display the following acknowledgement:
* This product includes software developed by the University of
* California, Berkeley and its contributors.
* 4. Neither the name of the University nor the names of its contributors
* may be used to endorse or promote products derived from this software
* without specific prior written permission.
*
* THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
* ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
* ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
* FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
* DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
* OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
* LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
* OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
* SUCH DAMAGE.
*/
#ifndef lint
static char sccsid[] = "@(#)expm1.c 8.1 (Berkeley) 6/4/93";
#endif /* not lint */
/* EXPM1(X)
* RETURN THE EXPONENTIAL OF X MINUS ONE
* DOUBLE PRECISION (IEEE 53 BITS, VAX D FORMAT 56 BITS)
* CODED IN C BY K.C. NG, 1/19/85;
* REVISED BY K.C. NG on 2/6/85, 3/7/85, 3/21/85, 4/16/85.
*
* Required system supported functions:
* scalb(x,n)
* copysign(x,y)
* finite(x)
*
* Kernel function:
* exp__E(x,c)
*
* Method:
* 1. Argument Reduction: given the input x, find r and integer k such
* that
* x = k*ln2 + r, |r| <= 0.5*ln2 .
* r will be represented as r := z+c for better accuracy.
*
* 2. Compute EXPM1(r)=exp(r)-1 by
*
* EXPM1(r=z+c) := z + exp__E(z,c)
*
* 3. EXPM1(x) = 2^k * ( EXPM1(r) + 1-2^-k ).
*
* Remarks:
* 1. When k=1 and z < -0.25, we use the following formula for
* better accuracy:
* EXPM1(x) = 2 * ( (z+0.5) + exp__E(z,c) )
* 2. To avoid rounding error in 1-2^-k where k is large, we use
* EXPM1(x) = 2^k * { [z+(exp__E(z,c)-2^-k )] + 1 }
* when k>56.
*
* Special cases:
* EXPM1(INF) is INF, EXPM1(NaN) is NaN;
* EXPM1(-INF)= -1;
* for finite argument, only EXPM1(0)=0 is exact.
*
* Accuracy:
* EXPM1(x) returns the exact (exp(x)-1) nearly rounded. In a test run with
* 1,166,000 random arguments on a VAX, the maximum observed error was
* .872 ulps (units of the last place).
*
* Constants:
* The hexadecimal values are the intended ones for the following constants.
* The decimal values may be used, provided that the compiler will convert
* from decimal to binary accurately enough to produce the hexadecimal values
* shown.
*/
#include "mathimpl.h"
vc(ln2hi, 6.9314718055829871446E-1 ,7217,4031,0000,f7d0, 0, .B17217F7D00000)
vc(ln2lo, 1.6465949582897081279E-12 ,bcd5,2ce7,d9cc,e4f1, -39, .E7BCD5E4F1D9CC)
vc(lnhuge, 9.4961163736712506989E1 ,ec1d,43bd,9010,a73e, 7, .BDEC1DA73E9010)
vc(invln2, 1.4426950408889634148E0 ,aa3b,40b8,17f1,295c, 1, .B8AA3B295C17F1)
ic(ln2hi, 6.9314718036912381649E-1, -1, 1.62E42FEE00000)
ic(ln2lo, 1.9082149292705877000E-10, -33, 1.A39EF35793C76)
ic(lnhuge, 7.1602103751842355450E2, 9, 1.6602B15B7ECF2)
ic(invln2, 1.4426950408889633870E0, 0, 1.71547652B82FE)
#ifdef vccast
#define ln2hi vccast(ln2hi)
#define ln2lo vccast(ln2lo)
#define lnhuge vccast(lnhuge)
#define invln2 vccast(invln2)
#endif
extern double scalb();
double expm1(x)
double x;
{
const static double one=1.0, half=1.0/2.0;
double z,hi,lo,c;
int k;
#if defined(vax)||defined(tahoe)
static prec=56;
#else /* defined(vax)||defined(tahoe) */
static prec=53;
#endif /* defined(vax)||defined(tahoe) */
#if !defined(vax)&&!defined(tahoe)
if(x!=x) return(x); /* x is NaN */
#endif /* !defined(vax)&&!defined(tahoe) */
if( x <= lnhuge ) {
if( x >= -40.0 ) {
/* argument reduction : x - k*ln2 */
k= invln2 *x+copysign(0.5,x); /* k=NINT(x/ln2) */
hi=x-k*ln2hi ;
z=hi-(lo=k*ln2lo);
c=(hi-z)-lo;
if(k==0) return(z+__exp__E(z,c));
if(k==1)
if(z< -0.25)
{x=z+half;x +=__exp__E(z,c); return(x+x);}
else
{z+=__exp__E(z,c); x=half+z; return(x+x);}
/* end of k=1 */
else {
if(k<=prec)
{ x=one-scalb(one,-k); z += __exp__E(z,c);}
else if(k<100)
{ x = __exp__E(z,c)-scalb(one,-k); x+=z; z=one;}
else
{ x = __exp__E(z,c)+z; z=one;}
return (scalb(x+z,k));
}
}
/* end of x > lnunfl */
else
/* expm1(-big#) rounded to -1 (inexact) */
if(finite(x))
{ ln2hi+ln2lo; return(-one);}
/* expm1(-INF) is -1 */
else return(-one);
}
/* end of x < lnhuge */
else
/* expm1(INF) is INF, expm1(+big#) overflows to INF */
return( finite(x) ? scalb(one,5000) : x);
}
|
