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1.1 root 1: /*
2: * Copyright (c) 1985 Regents of the University of California.
3: * All rights reserved.
4: *
5: * Redistribution and use in source and binary forms are permitted
6: * provided that: (1) source distributions retain this entire copyright
7: * notice and comment, and (2) distributions including binaries display
8: * the following acknowledgement: ``This product includes software
9: * developed by the University of California, Berkeley and its contributors''
10: * in the documentation or other materials provided with the distribution
11: * and in all advertising materials mentioning features or use of this
12: * software. Neither the name of the University nor the names of its
13: * contributors may be used to endorse or promote products derived
14: * from this software without specific prior written permission.
15: * THIS SOFTWARE IS PROVIDED ``AS IS'' AND WITHOUT ANY EXPRESS OR
16: * IMPLIED WARRANTIES, INCLUDING, WITHOUT LIMITATION, THE IMPLIED
17: * WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE.
18: *
19: * All recipients should regard themselves as participants in an ongoing
20: * research project and hence should feel obligated to report their
21: * experiences (good or bad) with these elementary function codes, using
22: * the sendbug(8) program, to the authors.
23: */
24:
25: #ifndef lint
26: static char sccsid[] = "@(#)expm1.c 5.5 (Berkeley) 6/1/90";
27: #endif /* not lint */
28:
29: /* EXPM1(X)
30: * RETURN THE EXPONENTIAL OF X MINUS ONE
31: * DOUBLE PRECISION (IEEE 53 BITS, VAX D FORMAT 56 BITS)
32: * CODED IN C BY K.C. NG, 1/19/85;
33: * REVISED BY K.C. NG on 2/6/85, 3/7/85, 3/21/85, 4/16/85.
34: *
35: * Required system supported functions:
36: * scalb(x,n)
37: * copysign(x,y)
38: * finite(x)
39: *
40: * Kernel function:
41: * exp__E(x,c)
42: *
43: * Method:
44: * 1. Argument Reduction: given the input x, find r and integer k such
45: * that
46: * x = k*ln2 + r, |r| <= 0.5*ln2 .
47: * r will be represented as r := z+c for better accuracy.
48: *
49: * 2. Compute EXPM1(r)=exp(r)-1 by
50: *
51: * EXPM1(r=z+c) := z + exp__E(z,c)
52: *
53: * 3. EXPM1(x) = 2^k * ( EXPM1(r) + 1-2^-k ).
54: *
55: * Remarks:
56: * 1. When k=1 and z < -0.25, we use the following formula for
57: * better accuracy:
58: * EXPM1(x) = 2 * ( (z+0.5) + exp__E(z,c) )
59: * 2. To avoid rounding error in 1-2^-k where k is large, we use
60: * EXPM1(x) = 2^k * { [z+(exp__E(z,c)-2^-k )] + 1 }
61: * when k>56.
62: *
63: * Special cases:
64: * EXPM1(INF) is INF, EXPM1(NaN) is NaN;
65: * EXPM1(-INF)= -1;
66: * for finite argument, only EXPM1(0)=0 is exact.
67: *
68: * Accuracy:
69: * EXPM1(x) returns the exact (exp(x)-1) nearly rounded. In a test run with
70: * 1,166,000 random arguments on a VAX, the maximum observed error was
71: * .872 ulps (units of the last place).
72: *
73: * Constants:
74: * The hexadecimal values are the intended ones for the following constants.
75: * The decimal values may be used, provided that the compiler will convert
76: * from decimal to binary accurately enough to produce the hexadecimal values
77: * shown.
78: */
79:
80: #include "mathimpl.h"
81:
82: vc(ln2hi, 6.9314718055829871446E-1 ,7217,4031,0000,f7d0, 0, .B17217F7D00000)
83: vc(ln2lo, 1.6465949582897081279E-12 ,bcd5,2ce7,d9cc,e4f1, -39, .E7BCD5E4F1D9CC)
84: vc(lnhuge, 9.4961163736712506989E1 ,ec1d,43bd,9010,a73e, 7, .BDEC1DA73E9010)
85: vc(invln2, 1.4426950408889634148E0 ,aa3b,40b8,17f1,295c, 1, .B8AA3B295C17F1)
86:
87: ic(ln2hi, 6.9314718036912381649E-1, -1, 1.62E42FEE00000)
88: ic(ln2lo, 1.9082149292705877000E-10, -33, 1.A39EF35793C76)
89: ic(lnhuge, 7.1602103751842355450E2, 9, 1.6602B15B7ECF2)
90: ic(invln2, 1.4426950408889633870E0, 0, 1.71547652B82FE)
91:
92: #ifdef vccast
93: #define ln2hi vccast(ln2hi)
94: #define ln2lo vccast(ln2lo)
95: #define lnhuge vccast(lnhuge)
96: #define invln2 vccast(invln2)
97: #endif
98:
99: double expm1(x)
100: double x;
101: {
102: const static double one=1.0, half=1.0/2.0;
103: double z,hi,lo,c;
104: int k;
105: #if defined(vax)||defined(tahoe)
106: static prec=56;
107: #else /* defined(vax)||defined(tahoe) */
108: static prec=53;
109: #endif /* defined(vax)||defined(tahoe) */
110:
111: #if !defined(vax)&&!defined(tahoe)
112: if(x!=x) return(x); /* x is NaN */
113: #endif /* !defined(vax)&&!defined(tahoe) */
114:
115: if( x <= lnhuge ) {
116: if( x >= -40.0 ) {
117:
118: /* argument reduction : x - k*ln2 */
119: k= invln2 *x+copysign(0.5,x); /* k=NINT(x/ln2) */
120: hi=x-k*ln2hi ;
121: z=hi-(lo=k*ln2lo);
122: c=(hi-z)-lo;
123:
124: if(k==0) return(z+exp__E(z,c));
125: if(k==1)
126: if(z< -0.25)
127: {x=z+half;x +=exp__E(z,c); return(x+x);}
128: else
129: {z+=exp__E(z,c); x=half+z; return(x+x);}
130: /* end of k=1 */
131:
132: else {
133: if(k<=prec)
134: { x=one-scalb(one,-k); z += exp__E(z,c);}
135: else if(k<100)
136: { x = exp__E(z,c)-scalb(one,-k); x+=z; z=one;}
137: else
138: { x = exp__E(z,c)+z; z=one;}
139:
140: return (scalb(x+z,k));
141: }
142: }
143: /* end of x > lnunfl */
144:
145: else
146: /* expm1(-big#) rounded to -1 (inexact) */
147: if(finite(x))
148: { ln2hi+ln2lo; return(-one);}
149:
150: /* expm1(-INF) is -1 */
151: else return(-one);
152: }
153: /* end of x < lnhuge */
154:
155: else
156: /* expm1(INF) is INF, expm1(+big#) overflows to INF */
157: return( finite(x) ? scalb(one,5000) : x);
158: }
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