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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 the above copyright notice and this paragraph are
7: * duplicated in all such forms and that any documentation,
8: * advertising materials, and other materials related to such
9: * distribution and use acknowledge that the software was developed
10: * by the University of California, Berkeley. The name of the
11: * University may not be used to endorse or promote products derived
12: * from this software without specific prior written permission.
13: * THIS SOFTWARE IS PROVIDED ``AS IS'' AND WITHOUT ANY EXPRESS OR
14: * IMPLIED WARRANTIES, INCLUDING, WITHOUT LIMITATION, THE IMPLIED
15: * WARRANTIES OF MERCHANTIBILITY AND FITNESS FOR A PARTICULAR PURPOSE.
16: *
17: * All recipients should regard themselves as participants in an ongoing
18: * research project and hence should feel obligated to report their
19: * experiences (good or bad) with these elementary function codes, using
20: * the sendbug(8) program, to the authors.
21: */
22:
23: #ifndef lint
24: static char sccsid[] = "@(#)log1p.c 5.3 (Berkeley) 6/30/88";
25: #endif /* not lint */
26:
27: /* LOG1P(x)
28: * RETURN THE LOGARITHM OF 1+x
29: * DOUBLE PRECISION (VAX D FORMAT 56 bits, IEEE DOUBLE 53 BITS)
30: * CODED IN C BY K.C. NG, 1/19/85;
31: * REVISED BY K.C. NG on 2/6/85, 3/7/85, 3/24/85, 4/16/85.
32: *
33: * Required system supported functions:
34: * scalb(x,n)
35: * copysign(x,y)
36: * logb(x)
37: * finite(x)
38: *
39: * Required kernel function:
40: * log__L(z)
41: *
42: * Method :
43: * 1. Argument Reduction: find k and f such that
44: * 1+x = 2^k * (1+f),
45: * where sqrt(2)/2 < 1+f < sqrt(2) .
46: *
47: * 2. Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
48: * = 2s + 2/3 s**3 + 2/5 s**5 + .....,
49: * log(1+f) is computed by
50: *
51: * log(1+f) = 2s + s*log__L(s*s)
52: * where
53: * log__L(z) = z*(L1 + z*(L2 + z*(... (L6 + z*L7)...)))
54: *
55: * See log__L() for the values of the coefficients.
56: *
57: * 3. Finally, log(1+x) = k*ln2 + log(1+f).
58: *
59: * Remarks 1. In step 3 n*ln2 will be stored in two floating point numbers
60: * n*ln2hi + n*ln2lo, where ln2hi is chosen such that the last
61: * 20 bits (for VAX D format), or the last 21 bits ( for IEEE
62: * double) is 0. This ensures n*ln2hi is exactly representable.
63: * 2. In step 1, f may not be representable. A correction term c
64: * for f is computed. It follows that the correction term for
65: * f - t (the leading term of log(1+f) in step 2) is c-c*x. We
66: * add this correction term to n*ln2lo to attenuate the error.
67: *
68: *
69: * Special cases:
70: * log1p(x) is NaN with signal if x < -1; log1p(NaN) is NaN with no signal;
71: * log1p(INF) is +INF; log1p(-1) is -INF with signal;
72: * only log1p(0)=0 is exact for finite argument.
73: *
74: * Accuracy:
75: * log1p(x) returns the exact log(1+x) nearly rounded. In a test run
76: * with 1,536,000 random arguments on a VAX, the maximum observed
77: * error was .846 ulps (units in the last place).
78: *
79: * Constants:
80: * The hexadecimal values are the intended ones for the following constants.
81: * The decimal values may be used, provided that the compiler will convert
82: * from decimal to binary accurately enough to produce the hexadecimal values
83: * shown.
84: */
85:
86: #if defined(vax)||defined(tahoe) /* VAX D format */
87: #include <errno.h>
88: #ifdef vax
89: #define _0x(A,B) 0x/**/A/**/B
90: #else /* vax */
91: #define _0x(A,B) 0x/**/B/**/A
92: #endif /* vax */
93: /* static double */
94: /* ln2hi = 6.9314718055829871446E-1 , Hex 2^ 0 * .B17217F7D00000 */
95: /* ln2lo = 1.6465949582897081279E-12 , Hex 2^-39 * .E7BCD5E4F1D9CC */
96: /* sqrt2 = 1.4142135623730950622E0 ; Hex 2^ 1 * .B504F333F9DE65 */
97: static long ln2hix[] = { _0x(7217,4031), _0x(0000,f7d0)};
98: static long ln2lox[] = { _0x(bcd5,2ce7), _0x(d9cc,e4f1)};
99: static long sqrt2x[] = { _0x(04f3,40b5), _0x(de65,33f9)};
100: #define ln2hi (*(double*)ln2hix)
101: #define ln2lo (*(double*)ln2lox)
102: #define sqrt2 (*(double*)sqrt2x)
103: #else /* defined(vax)||defined(tahoe) */
104: static double
105: ln2hi = 6.9314718036912381649E-1 , /*Hex 2^ -1 * 1.62E42FEE00000 */
106: ln2lo = 1.9082149292705877000E-10 , /*Hex 2^-33 * 1.A39EF35793C76 */
107: sqrt2 = 1.4142135623730951455E0 ; /*Hex 2^ 0 * 1.6A09E667F3BCD */
108: #endif /* defined(vax)||defined(tahoe) */
109:
110: double log1p(x)
111: double x;
112: {
113: static double zero=0.0, negone= -1.0, one=1.0,
114: half=1.0/2.0, small=1.0E-20; /* 1+small == 1 */
115: double logb(),copysign(),scalb(),log__L(),z,s,t,c;
116: int k,finite();
117:
118: #if !defined(vax)&&!defined(tahoe)
119: if(x!=x) return(x); /* x is NaN */
120: #endif /* !defined(vax)&&!defined(tahoe) */
121:
122: if(finite(x)) {
123: if( x > negone ) {
124:
125: /* argument reduction */
126: if(copysign(x,one)<small) return(x);
127: k=logb(one+x); z=scalb(x,-k); t=scalb(one,-k);
128: if(z+t >= sqrt2 )
129: { k += 1 ; z *= half; t *= half; }
130: t += negone; x = z + t;
131: c = (t-x)+z ; /* correction term for x */
132:
133: /* compute log(1+x) */
134: s = x/(2+x); t = x*x*half;
135: c += (k*ln2lo-c*x);
136: z = c+s*(t+log__L(s*s));
137: x += (z - t) ;
138:
139: return(k*ln2hi+x);
140: }
141: /* end of if (x > negone) */
142:
143: else {
144: #if defined(vax)||defined(tahoe)
145: extern double infnan();
146: if ( x == negone )
147: return (infnan(-ERANGE)); /* -INF */
148: else
149: return (infnan(EDOM)); /* NaN */
150: #else /* defined(vax)||defined(tahoe) */
151: /* x = -1, return -INF with signal */
152: if ( x == negone ) return( negone/zero );
153:
154: /* negative argument for log, return NaN with signal */
155: else return ( zero / zero );
156: #endif /* defined(vax)||defined(tahoe) */
157: }
158: }
159: /* end of if (finite(x)) */
160:
161: /* log(-INF) is NaN */
162: else if(x<0)
163: return(zero/zero);
164:
165: /* log(+INF) is INF */
166: else return(x);
167: }
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