1 /* $NetBSD: n_expm1.c,v 1.8 2013/11/24 18:50:58 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[] = "@(#)expm1.c 8.1 (Berkeley) 6/4/93";
34 #endif
35 #endif /* not lint */
36
37 /* EXPM1(X)
38 * RETURN THE EXPONENTIAL OF X MINUS ONE
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, 3/7/85, 3/21/85, 4/16/85.
42 *
43 * Required system supported functions:
44 * scalb(x,n)
45 * copysign(x,y)
46 * finite(x)
47 *
48 * Kernel function:
49 * exp__E(x,c)
50 *
51 * Method:
52 * 1. Argument Reduction: given the input x, find r and integer k such
53 * that
54 * x = k*ln2 + r, |r| <= 0.5*ln2 .
55 * r will be represented as r := z+c for better accuracy.
56 *
57 * 2. Compute EXPM1(r)=exp(r)-1 by
58 *
59 * EXPM1(r=z+c) := z + exp__E(z,c)
60 *
61 * 3. EXPM1(x) = 2^k * ( EXPM1(r) + 1-2^-k ).
62 *
63 * Remarks:
64 * 1. When k=1 and z < -0.25, we use the following formula for
65 * better accuracy:
66 * EXPM1(x) = 2 * ( (z+0.5) + exp__E(z,c) )
67 * 2. To avoid rounding error in 1-2^-k where k is large, we use
68 * EXPM1(x) = 2^k * { [z+(exp__E(z,c)-2^-k )] + 1 }
69 * when k>56.
70 *
71 * Special cases:
72 * EXPM1(INF) is INF, EXPM1(NaN) is NaN;
73 * EXPM1(-INF)= -1;
74 * for finite argument, only EXPM1(0)=0 is exact.
75 *
76 * Accuracy:
77 * EXPM1(x) returns the exact (exp(x)-1) nearly rounded. In a test run with
78 * 1,166,000 random arguments on a VAX, the maximum observed error was
79 * .872 ulps (units of the last place).
80 *
81 * Constants:
82 * The hexadecimal values are the intended ones for the following constants.
83 * The decimal values may be used, provided that the compiler will convert
84 * from decimal to binary accurately enough to produce the hexadecimal values
85 * shown.
86 */
87
88 #define _LIBM_STATIC
89 #include "mathimpl.h"
90
91 vc(ln2hi, 6.9314718055829871446E-1 ,7217,4031,0000,f7d0, 0, .B17217F7D00000)
92 vc(ln2lo, 1.6465949582897081279E-12 ,bcd5,2ce7,d9cc,e4f1, -39, .E7BCD5E4F1D9CC)
93 vc(lnhuge, 9.4961163736712506989E1 ,ec1d,43bd,9010,a73e, 7, .BDEC1DA73E9010)
94 vc(invln2, 1.4426950408889634148E0 ,aa3b,40b8,17f1,295c, 1, .B8AA3B295C17F1)
95
96 ic(ln2hi, 6.9314718036912381649E-1, -1, 1.62E42FEE00000)
97 ic(ln2lo, 1.9082149292705877000E-10, -33, 1.A39EF35793C76)
98 ic(lnhuge, 7.1602103751842355450E2, 9, 1.6602B15B7ECF2)
99 ic(invln2, 1.4426950408889633870E0, 0, 1.71547652B82FE)
100
101 #ifdef vccast
102 #define ln2hi vccast(ln2hi)
103 #define ln2lo vccast(ln2lo)
104 #define lnhuge vccast(lnhuge)
105 #define invln2 vccast(invln2)
106 #endif
107
108 #if defined(__vax__)||defined(tahoe)
109 #define PREC 56
110 #else /* defined(__vax__)||defined(tahoe) */
111 #define PREC 53
112 #endif /* defined(__vax__)||defined(tahoe) */
113
114 float
expm1f(float x)115 expm1f(float x)
116 {
117 return (float)expm1(x);
118 }
119
120 double
expm1(double x)121 expm1(double x)
122 {
123 static const double one=1.0, half=1.0/2.0;
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
131 if( x <= lnhuge ) {
132 if( x >= -40.0 ) {
133
134 /* argument reduction : x - k*ln2 */
135 k= invln2 *x+copysign(0.5,x); /* k=NINT(x/ln2) */
136 hi=x-k*ln2hi ;
137 z=hi-(lo=k*ln2lo);
138 c=(hi-z)-lo;
139
140 if(k==0) return(z+__exp__E(z,c));
141 if(k==1)
142 if(z< -0.25)
143 {x=z+half;x +=__exp__E(z,c); return(x+x);}
144 else
145 {z+=__exp__E(z,c); x=half+z; return(x+x);}
146 /* end of k=1 */
147
148 else {
149 if(k<=PREC)
150 { x=one-scalb(one,-k); z += __exp__E(z,c);}
151 else if(k<100)
152 { x = __exp__E(z,c)-scalb(one,-k); x+=z; z=one;}
153 else
154 { x = __exp__E(z,c)+z; z=one;}
155
156 return (scalb(x+z,k));
157 }
158 }
159 /* end of x > lnunfl */
160
161 else
162 /* expm1(-big#) rounded to -1 (inexact) */
163 if(finite(x))
164 { c=ln2hi+ln2lo; return(-one);} /* ??? -ragge */
165
166 /* expm1(-INF) is -1 */
167 else return(-one);
168 }
169 /* end of x < lnhuge */
170
171 else
172 /* expm1(INF) is INF, expm1(+big#) overflows to INF */
173 return( finite(x) ? scalb(one,5000) : x);
174 }
175