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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[] = "@(#)log.c 4.5 (Berkeley) 8/21/85";
16: #endif not lint
17:
18: /* LOG(X)
19: * RETURN THE LOGARITHM OF x
20: * DOUBLE PRECISION (VAX D FORMAT 56 bits or IEEE DOUBLE 53 BITS)
21: * CODED IN C BY K.C. NG, 1/19/85;
22: * REVISED BY K.C. NG on 2/7/85, 3/7/85, 3/24/85, 4/16/85.
23: *
24: * Required system supported functions:
25: * scalb(x,n)
26: * copysign(x,y)
27: * logb(x)
28: * finite(x)
29: *
30: * Required kernel function:
31: * log__L(z)
32: *
33: * Method :
34: * 1. Argument Reduction: find k and f such that
35: * x = 2^k * (1+f),
36: * where sqrt(2)/2 < 1+f < sqrt(2) .
37: *
38: * 2. Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
39: * = 2s + 2/3 s**3 + 2/5 s**5 + .....,
40: * log(1+f) is computed by
41: *
42: * log(1+f) = 2s + s*log__L(s*s)
43: * where
44: * log__L(z) = z*(L1 + z*(L2 + z*(... (L6 + z*L7)...)))
45: *
46: * See log__L() for the values of the coefficients.
47: *
48: * 3. Finally, log(x) = k*ln2 + log(1+f). (Here n*ln2 will be stored
49: * in two floating point number: n*ln2hi + n*ln2lo, n*ln2hi is exact
50: * since the last 20 bits of ln2hi is 0.)
51: *
52: * Special cases:
53: * log(x) is NaN with signal if x < 0 (including -INF) ;
54: * log(+INF) is +INF; log(0) is -INF with signal;
55: * log(NaN) is that NaN with no signal.
56: *
57: * Accuracy:
58: * log(x) returns the exact log(x) nearly rounded. In a test run with
59: * 1,536,000 random arguments on a VAX, the maximum observed error was
60: * .826 ulps (units in 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: #include <errno.h>
71:
72: /* double static */
73: /* ln2hi = 6.9314718055829871446E-1 , Hex 2^ 0 * .B17217F7D00000 */
74: /* ln2lo = 1.6465949582897081279E-12 , Hex 2^-39 * .E7BCD5E4F1D9CC */
75: /* sqrt2 = 1.4142135623730950622E0 ; Hex 2^ 1 * .B504F333F9DE65 */
76: static long ln2hix[] = { 0x72174031, 0x0000f7d0};
77: static long ln2lox[] = { 0xbcd52ce7, 0xd9cce4f1};
78: static long sqrt2x[] = { 0x04f340b5, 0xde6533f9};
79: #define ln2hi (*(double*)ln2hix)
80: #define ln2lo (*(double*)ln2lox)
81: #define sqrt2 (*(double*)sqrt2x)
82: #else /* IEEE double */
83: double static
84: ln2hi = 6.9314718036912381649E-1 , /*Hex 2^ -1 * 1.62E42FEE00000 */
85: ln2lo = 1.9082149292705877000E-10 , /*Hex 2^-33 * 1.A39EF35793C76 */
86: sqrt2 = 1.4142135623730951455E0 ; /*Hex 2^ 0 * 1.6A09E667F3BCD */
87: #endif
88:
89: double log(x)
90: double x;
91: {
92: static double zero=0.0, negone= -1.0, half=1.0/2.0;
93: double logb(),scalb(),copysign(),log__L(),s,z,t;
94: int k,n,finite();
95:
96: #ifndef VAX
97: if(x!=x) return(x); /* x is NaN */
98: #endif
99: if(finite(x)) {
100: if( x > zero ) {
101:
102: /* argument reduction */
103: k=logb(x); x=scalb(x,-k);
104: if(k == -1022) /* subnormal no. */
105: {n=logb(x); x=scalb(x,-n); k+=n;}
106: if(x >= sqrt2 ) {k += 1; x *= half;}
107: x += negone ;
108:
109: /* compute log(1+x) */
110: s=x/(2+x); t=x*x*half;
111: z=k*ln2lo+s*(t+log__L(s*s));
112: x += (z - t) ;
113:
114: return(k*ln2hi+x);
115: }
116: /* end of if (x > zero) */
117:
118: else {
119: #ifdef VAX
120: extern double infnan();
121: if ( x == zero )
122: return (infnan(-ERANGE)); /* -INF */
123: else
124: return (infnan(EDOM)); /* NaN */
125: #else /* IEEE double */
126: /* zero argument, return -INF with signal */
127: if ( x == zero )
128: return( negone/zero );
129:
130: /* negative argument, return NaN with signal */
131: else
132: return ( zero / zero );
133: #endif
134: }
135: }
136: /* end of if (finite(x)) */
137: /* NOT REACHED ifdef VAX */
138:
139: /* log(-INF) is NaN with signal */
140: else if (x<0)
141: return(zero/zero);
142:
143: /* log(+INF) is +INF */
144: else return(x);
145:
146: }
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