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1.1 root 1: /*
2: * Copyright (c) 1985 Regents of the University of California.
3: *
4: * Use and reproduction of this software are granted in accordance with
5: * the terms and conditions specified in the Berkeley Software License
6: * Agreement (in particular, this entails acknowledgement of the programs'
7: * source, and inclusion of this notice) with the additional understanding
8: * that all recipients should regard themselves as participants in an
9: * ongoing research project and hence should feel obligated to report
10: * their experiences (good or bad) with these elementary function codes,
11: * using "sendbug 4bsd-bugs@BERKELEY", to the authors.
12: */
13:
14: #ifndef lint
15: static char sccsid[] = "@(#)expm1.c 1.2 (Berkeley) 8/21/85";
16: #endif not lint
17:
18: /* EXPM1(X)
19: * RETURN THE EXPONENTIAL OF X MINUS ONE
20: * DOUBLE PRECISION (IEEE 53 BITS, VAX D FORMAT 56 BITS)
21: * CODED IN C BY K.C. NG, 1/19/85;
22: * REVISED BY K.C. NG on 2/6/85, 3/7/85, 3/21/85, 4/16/85.
23: *
24: * Required system supported functions:
25: * scalb(x,n)
26: * copysign(x,y)
27: * finite(x)
28: *
29: * Kernel function:
30: * exp__E(x,c)
31: *
32: * Method:
33: * 1. Argument Reduction: given the input x, find r and integer k such
34: * that
35: * x = k*ln2 + r, |r| <= 0.5*ln2 .
36: * r will be represented as r := z+c for better accuracy.
37: *
38: * 2. Compute EXPM1(r)=exp(r)-1 by
39: *
40: * EXPM1(r=z+c) := z + exp__E(z,c)
41: *
42: * 3. EXPM1(x) = 2^k * ( EXPM1(r) + 1-2^-k ).
43: *
44: * Remarks:
45: * 1. When k=1 and z < -0.25, we use the following formula for
46: * better accuracy:
47: * EXPM1(x) = 2 * ( (z+0.5) + exp__E(z,c) )
48: * 2. To avoid rounding error in 1-2^-k where k is large, we use
49: * EXPM1(x) = 2^k * { [z+(exp__E(z,c)-2^-k )] + 1 }
50: * when k>56.
51: *
52: * Special cases:
53: * EXPM1(INF) is INF, EXPM1(NaN) is NaN;
54: * EXPM1(-INF)= -1;
55: * for finite argument, only EXPM1(0)=0 is exact.
56: *
57: * Accuracy:
58: * EXPM1(x) returns the exact (exp(x)-1) nearly rounded. In a test run with
59: * 1,166,000 random arguments on a VAX, the maximum observed error was
60: * .872 ulps (units of the last place).
61: *
62: * Constants:
63: * The hexadecimal values are the intended ones for the following constants.
64: * The decimal values may be used, provided that the compiler will convert
65: * from decimal to binary accurately enough to produce the hexadecimal values
66: * shown.
67: */
68:
69: #ifdef VAX /* VAX D format */
70: /* double static */
71: /* ln2hi = 6.9314718055829871446E-1 , Hex 2^ 0 * .B17217F7D00000 */
72: /* ln2lo = 1.6465949582897081279E-12 , Hex 2^-39 * .E7BCD5E4F1D9CC */
73: /* lnhuge = 9.4961163736712506989E1 , Hex 2^ 7 * .BDEC1DA73E9010 */
74: /* invln2 = 1.4426950408889634148E0 ; Hex 2^ 1 * .B8AA3B295C17F1 */
75: static long ln2hix[] = { 0x72174031, 0x0000f7d0};
76: static long ln2lox[] = { 0xbcd52ce7, 0xd9cce4f1};
77: static long lnhugex[] = { 0xec1d43bd, 0x9010a73e};
78: static long invln2x[] = { 0xaa3b40b8, 0x17f1295c};
79: #define ln2hi (*(double*)ln2hix)
80: #define ln2lo (*(double*)ln2lox)
81: #define lnhuge (*(double*)lnhugex)
82: #define invln2 (*(double*)invln2x)
83: #else /* IEEE double */
84: double static
85: ln2hi = 6.9314718036912381649E-1 , /*Hex 2^ -1 * 1.62E42FEE00000 */
86: ln2lo = 1.9082149292705877000E-10 , /*Hex 2^-33 * 1.A39EF35793C76 */
87: lnhuge = 7.1602103751842355450E2 , /*Hex 2^ 9 * 1.6602B15B7ECF2 */
88: invln2 = 1.4426950408889633870E0 ; /*Hex 2^ 0 * 1.71547652B82FE */
89: #endif
90:
91: double expm1(x)
92: double x;
93: {
94: double static one=1.0, half=1.0/2.0;
95: double scalb(), copysign(), exp__E(), z,hi,lo,c;
96: int k,finite();
97: #ifdef VAX
98: static prec=56;
99: #else /* IEEE double */
100: static prec=53;
101: #endif
102: #ifndef VAX
103: if(x!=x) return(x); /* x is NaN */
104: #endif
105:
106: if( x <= lnhuge ) {
107: if( x >= -40.0 ) {
108:
109: /* argument reduction : x - k*ln2 */
110: k= invln2 *x+copysign(0.5,x); /* k=NINT(x/ln2) */
111: hi=x-k*ln2hi ;
112: z=hi-(lo=k*ln2lo);
113: c=(hi-z)-lo;
114:
115: if(k==0) return(z+exp__E(z,c));
116: if(k==1)
117: if(z< -0.25)
118: {x=z+half;x +=exp__E(z,c); return(x+x);}
119: else
120: {z+=exp__E(z,c); x=half+z; return(x+x);}
121: /* end of k=1 */
122:
123: else {
124: if(k<=prec)
125: { x=one-scalb(one,-k); z += exp__E(z,c);}
126: else if(k<100)
127: { x = exp__E(z,c)-scalb(one,-k); x+=z; z=one;}
128: else
129: { x = exp__E(z,c)+z; z=one;}
130:
131: return (scalb(x+z,k));
132: }
133: }
134: /* end of x > lnunfl */
135:
136: else
137: /* expm1(-big#) rounded to -1 (inexact) */
138: if(finite(x))
139: { ln2hi+ln2lo; return(-one);}
140:
141: /* expm1(-INF) is -1 */
142: else return(-one);
143: }
144: /* end of x < lnhuge */
145:
146: else
147: /* expm1(INF) is INF, expm1(+big#) overflows to INF */
148: return( finite(x) ? scalb(one,5000) : x);
149: }
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