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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[] = "@(#)log1p.c 5.5 (Berkeley) 6/1/90";
27: #endif /* not lint */
28:
29: /* LOG1P(x)
30: * RETURN THE LOGARITHM OF 1+x
31: * DOUBLE PRECISION (VAX D FORMAT 56 bits, IEEE DOUBLE 53 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, 3/24/85, 4/16/85.
34: *
35: * Required system supported functions:
36: * scalb(x,n)
37: * copysign(x,y)
38: * logb(x)
39: * finite(x)
40: *
41: * Required kernel function:
42: * log__L(z)
43: *
44: * Method :
45: * 1. Argument Reduction: find k and f such that
46: * 1+x = 2^k * (1+f),
47: * where sqrt(2)/2 < 1+f < sqrt(2) .
48: *
49: * 2. Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
50: * = 2s + 2/3 s**3 + 2/5 s**5 + .....,
51: * log(1+f) is computed by
52: *
53: * log(1+f) = 2s + s*log__L(s*s)
54: * where
55: * log__L(z) = z*(L1 + z*(L2 + z*(... (L6 + z*L7)...)))
56: *
57: * See log__L() for the values of the coefficients.
58: *
59: * 3. Finally, log(1+x) = k*ln2 + log(1+f).
60: *
61: * Remarks 1. In step 3 n*ln2 will be stored in two floating point numbers
62: * n*ln2hi + n*ln2lo, where ln2hi is chosen such that the last
63: * 20 bits (for VAX D format), or the last 21 bits ( for IEEE
64: * double) is 0. This ensures n*ln2hi is exactly representable.
65: * 2. In step 1, f may not be representable. A correction term c
66: * for f is computed. It follows that the correction term for
67: * f - t (the leading term of log(1+f) in step 2) is c-c*x. We
68: * add this correction term to n*ln2lo to attenuate the error.
69: *
70: *
71: * Special cases:
72: * log1p(x) is NaN with signal if x < -1; log1p(NaN) is NaN with no signal;
73: * log1p(INF) is +INF; log1p(-1) is -INF with signal;
74: * only log1p(0)=0 is exact for finite argument.
75: *
76: * Accuracy:
77: * log1p(x) returns the exact log(1+x) nearly rounded. In a test run
78: * with 1,536,000 random arguments on a VAX, the maximum observed
79: * error was .846 ulps (units in 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: #include <errno.h>
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(sqrt2, 1.4142135623730950622E0 ,04f3,40b5,de65,33f9, 1, .B504F333F9DE65)
94:
95: ic(ln2hi, 6.9314718036912381649E-1, -1, 1.62E42FEE00000)
96: ic(ln2lo, 1.9082149292705877000E-10, -33, 1.A39EF35793C76)
97: ic(sqrt2, 1.4142135623730951455E0, 0, 1.6A09E667F3BCD)
98:
99: #ifdef vccast
100: #define ln2hi vccast(ln2hi)
101: #define ln2lo vccast(ln2lo)
102: #define sqrt2 vccast(sqrt2)
103: #endif
104:
105: double log1p(x)
106: double x;
107: {
108: const static double zero=0.0, negone= -1.0, one=1.0,
109: half=1.0/2.0, small=1.0E-20; /* 1+small == 1 */
110: double z,s,t,c;
111: int k;
112:
113: #if !defined(vax)&&!defined(tahoe)
114: if(x!=x) return(x); /* x is NaN */
115: #endif /* !defined(vax)&&!defined(tahoe) */
116:
117: if(finite(x)) {
118: if( x > negone ) {
119:
120: /* argument reduction */
121: if(copysign(x,one)<small) return(x);
122: k=logb(one+x); z=scalb(x,-k); t=scalb(one,-k);
123: if(z+t >= sqrt2 )
124: { k += 1 ; z *= half; t *= half; }
125: t += negone; x = z + t;
126: c = (t-x)+z ; /* correction term for x */
127:
128: /* compute log(1+x) */
129: s = x/(2+x); t = x*x*half;
130: c += (k*ln2lo-c*x);
131: z = c+s*(t+log__L(s*s));
132: x += (z - t) ;
133:
134: return(k*ln2hi+x);
135: }
136: /* end of if (x > negone) */
137:
138: else {
139: #if defined(vax)||defined(tahoe)
140: if ( x == negone )
141: return (infnan(-ERANGE)); /* -INF */
142: else
143: return (infnan(EDOM)); /* NaN */
144: #else /* defined(vax)||defined(tahoe) */
145: /* x = -1, return -INF with signal */
146: if ( x == negone ) return( negone/zero );
147:
148: /* negative argument for log, return NaN with signal */
149: else return ( zero / zero );
150: #endif /* defined(vax)||defined(tahoe) */
151: }
152: }
153: /* end of if (finite(x)) */
154:
155: /* log(-INF) is NaN */
156: else if(x<0)
157: return(zero/zero);
158:
159: /* log(+INF) is INF */
160: else return(x);
161: }
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