2 * Copyright (c) 1985, 1993
3 * The Regents of the University of California. All rights reserved.
5 * Redistribution and use in source and binary forms, with or without
6 * modification, are permitted provided that the following conditions
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.
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
35 static char sccsid
[] = "@(#)expm1.c 8.1 (Berkeley) 6/4/93";
39 * RETURN THE EXPONENTIAL OF X MINUS ONE
40 * DOUBLE PRECISION (IEEE 53 BITS, VAX D FORMAT 56 BITS)
41 * CODED IN C BY K.C. NG, 1/19/85;
42 * REVISED BY K.C. NG on 2/6/85, 3/7/85, 3/21/85, 4/16/85.
44 * Required system supported functions:
53 * 1. Argument Reduction: given the input x, find r and integer k such
55 * x = k*ln2 + r, |r| <= 0.5*ln2 .
56 * r will be represented as r := z+c for better accuracy.
58 * 2. Compute EXPM1(r)=exp(r)-1 by
60 * EXPM1(r=z+c) := z + exp__E(z,c)
62 * 3. EXPM1(x) = 2^k * ( EXPM1(r) + 1-2^-k ).
65 * 1. When k=1 and z < -0.25, we use the following formula for
67 * EXPM1(x) = 2 * ( (z+0.5) + exp__E(z,c) )
68 * 2. To avoid rounding error in 1-2^-k where k is large, we use
69 * EXPM1(x) = 2^k * { [z+(exp__E(z,c)-2^-k )] + 1 }
73 * EXPM1(INF) is INF, EXPM1(NaN) is NaN;
75 * for finite argument, only EXPM1(0)=0 is exact.
78 * EXPM1(x) returns the exact (exp(x)-1) nearly rounded. In a test run with
79 * 1,166,000 random arguments on a VAX, the maximum observed error was
80 * .872 ulps (units of the last place).
83 * The hexadecimal values are the intended ones for the following constants.
84 * The decimal values may be used, provided that the compiler will convert
85 * from decimal to binary accurately enough to produce the hexadecimal values
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
)
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
)
102 #define ln2hi vccast(ln2hi)
103 #define ln2lo vccast(ln2lo)
104 #define lnhuge vccast(lnhuge)
105 #define invln2 vccast(invln2)
111 const static double one
=1.0, half
=1.0/2.0;
114 #if defined(vax)||defined(tahoe)
116 #else /* defined(vax)||defined(tahoe) */
118 #endif /* defined(vax)||defined(tahoe) */
120 #if !defined(vax)&&!defined(tahoe)
121 if(x
!=x
) return(x
); /* x is NaN */
122 #endif /* !defined(vax)&&!defined(tahoe) */
127 /* argument reduction : x - k*ln2 */
128 k
= invln2
*x
+copysign(0.5,x
); /* k=NINT(x/ln2) */
133 if(k
==0) return(z
+__exp__E(z
,c
));
136 {x
=z
+half
;x
+=__exp__E(z
,c
); return(x
+x
);}
138 {z
+=__exp__E(z
,c
); x
=half
+z
; return(x
+x
);}
143 { x
=one
-scalb(one
,-k
); z
+= __exp__E(z
,c
);}
145 { x
= __exp__E(z
,c
)-scalb(one
,-k
); x
+=z
; z
=one
;}
147 { x
= __exp__E(z
,c
)+z
; z
=one
;}
149 return (scalb(x
+z
,k
));
152 /* end of x > lnunfl */
155 /* expm1(-big#) rounded to -1 (inexact) */
157 { ln2hi
+ln2lo
; return(-one
);}
159 /* expm1(-INF) is -1 */
162 /* end of x < lnhuge */
165 /* expm1(INF) is INF, expm1(+big#) overflows to INF */
166 return( finite(x
) ? scalb(one
,5000) : x
);
170 weak_alias (__expm1
, expm1
)