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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[] = "@(#)log.c 5.3 (Berkeley) 6/30/88";
25: #endif /* not lint */
26:
27: /* LOG(X)
28: * RETURN THE LOGARITHM OF x
29: * DOUBLE PRECISION (VAX D FORMAT 56 bits or IEEE DOUBLE 53 BITS)
30: * CODED IN C BY K.C. NG, 1/19/85;
31: * REVISED BY K.C. NG on 2/7/85, 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: * 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(x) = k*ln2 + log(1+f). (Here n*ln2 will be stored
58: * in two floating point number: n*ln2hi + n*ln2lo, n*ln2hi is exact
59: * since the last 20 bits of ln2hi is 0.)
60: *
61: * Special cases:
62: * log(x) is NaN with signal if x < 0 (including -INF) ;
63: * log(+INF) is +INF; log(0) is -INF with signal;
64: * log(NaN) is that NaN with no signal.
65: *
66: * Accuracy:
67: * log(x) returns the exact log(x) nearly rounded. In a test run with
68: * 1,536,000 random arguments on a VAX, the maximum observed error was
69: * .826 ulps (units in the last place).
70: *
71: * Constants:
72: * The hexadecimal values are the intended ones for the following constants.
73: * The decimal values may be used, provided that the compiler will convert
74: * from decimal to binary accurately enough to produce the hexadecimal values
75: * shown.
76: */
77:
78: #if defined(vax)||defined(tahoe) /* VAX D format */
79: #include <errno.h>
80: #ifdef vax
81: #define _0x(A,B) 0x/**/A/**/B
82: #else /* vax */
83: #define _0x(A,B) 0x/**/B/**/A
84: #endif /* vax */
85: /* static double */
86: /* ln2hi = 6.9314718055829871446E-1 , Hex 2^ 0 * .B17217F7D00000 */
87: /* ln2lo = 1.6465949582897081279E-12 , Hex 2^-39 * .E7BCD5E4F1D9CC */
88: /* sqrt2 = 1.4142135623730950622E0 ; Hex 2^ 1 * .B504F333F9DE65 */
89: static long ln2hix[] = { _0x(7217,4031), _0x(0000,f7d0)};
90: static long ln2lox[] = { _0x(bcd5,2ce7), _0x(d9cc,e4f1)};
91: static long sqrt2x[] = { _0x(04f3,40b5), _0x(de65,33f9)};
92: #define ln2hi (*(double*)ln2hix)
93: #define ln2lo (*(double*)ln2lox)
94: #define sqrt2 (*(double*)sqrt2x)
95: #else /* defined(vax)||defined(tahoe) */
96: static double
97: ln2hi = 6.9314718036912381649E-1 , /*Hex 2^ -1 * 1.62E42FEE00000 */
98: ln2lo = 1.9082149292705877000E-10 , /*Hex 2^-33 * 1.A39EF35793C76 */
99: sqrt2 = 1.4142135623730951455E0 ; /*Hex 2^ 0 * 1.6A09E667F3BCD */
100: #endif /* defined(vax)||defined(tahoe) */
101:
102: double log(x)
103: double x;
104: {
105: static double zero=0.0, negone= -1.0, half=1.0/2.0;
106: double logb(),scalb(),copysign(),log__L(),s,z,t;
107: int k,n,finite();
108:
109: #if !defined(vax)&&!defined(tahoe)
110: if(x!=x) return(x); /* x is NaN */
111: #endif /* !defined(vax)&&!defined(tahoe) */
112: if(finite(x)) {
113: if( x > zero ) {
114:
115: /* argument reduction */
116: k=logb(x); x=scalb(x,-k);
117: if(k == -1022) /* subnormal no. */
118: {n=logb(x); x=scalb(x,-n); k+=n;}
119: if(x >= sqrt2 ) {k += 1; x *= half;}
120: x += negone ;
121:
122: /* compute log(1+x) */
123: s=x/(2+x); t=x*x*half;
124: z=k*ln2lo+s*(t+log__L(s*s));
125: x += (z - t) ;
126:
127: return(k*ln2hi+x);
128: }
129: /* end of if (x > zero) */
130:
131: else {
132: #if defined(vax)||defined(tahoe)
133: extern double infnan();
134: if ( x == zero )
135: return (infnan(-ERANGE)); /* -INF */
136: else
137: return (infnan(EDOM)); /* NaN */
138: #else /* defined(vax)||defined(tahoe) */
139: /* zero argument, return -INF with signal */
140: if ( x == zero )
141: return( negone/zero );
142:
143: /* negative argument, return NaN with signal */
144: else
145: return ( zero / zero );
146: #endif /* defined(vax)||defined(tahoe) */
147: }
148: }
149: /* end of if (finite(x)) */
150: /* NOTREACHED if defined(vax)||defined(tahoe) */
151:
152: /* log(-INF) is NaN with signal */
153: else if (x<0)
154: return(zero/zero);
155:
156: /* log(+INF) is +INF */
157: else return(x);
158:
159: }
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