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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[] = "@(#)log1p.c 1.3 (Berkeley) 8/21/85";
16: #endif not lint
17:
18: /* LOG1P(x)
19: * RETURN THE LOGARITHM OF 1+x
20: * DOUBLE PRECISION (VAX D FORMAT 56 bits, IEEE DOUBLE 53 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, 3/24/85, 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: * 1+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(1+x) = k*ln2 + log(1+f).
49: *
50: * Remarks 1. In step 3 n*ln2 will be stored in two floating point numbers
51: * n*ln2hi + n*ln2lo, where ln2hi is chosen such that the last
52: * 20 bits (for VAX D format), or the last 21 bits ( for IEEE
53: * double) is 0. This ensures n*ln2hi is exactly representable.
54: * 2. In step 1, f may not be representable. A correction term c
55: * for f is computed. It follows that the correction term for
56: * f - t (the leading term of log(1+f) in step 2) is c-c*x. We
57: * add this correction term to n*ln2lo to attenuate the error.
58: *
59: *
60: * Special cases:
61: * log1p(x) is NaN with signal if x < -1; log1p(NaN) is NaN with no signal;
62: * log1p(INF) is +INF; log1p(-1) is -INF with signal;
63: * only log1p(0)=0 is exact for finite argument.
64: *
65: * Accuracy:
66: * log1p(x) returns the exact log(1+x) nearly rounded. In a test run
67: * with 1,536,000 random arguments on a VAX, the maximum observed
68: * error was .846 ulps (units in the last place).
69: *
70: * Constants:
71: * The hexadecimal values are the intended ones for the following constants.
72: * The decimal values may be used, provided that the compiler will convert
73: * from decimal to binary accurately enough to produce the hexadecimal values
74: * shown.
75: */
76:
77: #ifdef VAX /* VAX D format */
78: #include <errno.h>
79:
80: /* double static */
81: /* ln2hi = 6.9314718055829871446E-1 , Hex 2^ 0 * .B17217F7D00000 */
82: /* ln2lo = 1.6465949582897081279E-12 , Hex 2^-39 * .E7BCD5E4F1D9CC */
83: /* sqrt2 = 1.4142135623730950622E0 ; Hex 2^ 1 * .B504F333F9DE65 */
84: static long ln2hix[] = { 0x72174031, 0x0000f7d0};
85: static long ln2lox[] = { 0xbcd52ce7, 0xd9cce4f1};
86: static long sqrt2x[] = { 0x04f340b5, 0xde6533f9};
87: #define ln2hi (*(double*)ln2hix)
88: #define ln2lo (*(double*)ln2lox)
89: #define sqrt2 (*(double*)sqrt2x)
90: #else /* IEEE double */
91: double static
92: ln2hi = 6.9314718036912381649E-1 , /*Hex 2^ -1 * 1.62E42FEE00000 */
93: ln2lo = 1.9082149292705877000E-10 , /*Hex 2^-33 * 1.A39EF35793C76 */
94: sqrt2 = 1.4142135623730951455E0 ; /*Hex 2^ 0 * 1.6A09E667F3BCD */
95: #endif
96:
97: double log1p(x)
98: double x;
99: {
100: static double zero=0.0, negone= -1.0, one=1.0,
101: half=1.0/2.0, small=1.0E-20; /* 1+small == 1 */
102: double logb(),copysign(),scalb(),log__L(),z,s,t,c;
103: int k,finite();
104:
105: #ifndef VAX
106: if(x!=x) return(x); /* x is NaN */
107: #endif
108:
109: if(finite(x)) {
110: if( x > negone ) {
111:
112: /* argument reduction */
113: if(copysign(x,one)<small) return(x);
114: k=logb(one+x); z=scalb(x,-k); t=scalb(one,-k);
115: if(z+t >= sqrt2 )
116: { k += 1 ; z *= half; t *= half; }
117: t += negone; x = z + t;
118: c = (t-x)+z ; /* correction term for x */
119:
120: /* compute log(1+x) */
121: s = x/(2+x); t = x*x*half;
122: c += (k*ln2lo-c*x);
123: z = c+s*(t+log__L(s*s));
124: x += (z - t) ;
125:
126: return(k*ln2hi+x);
127: }
128: /* end of if (x > negone) */
129:
130: else {
131: #ifdef VAX
132: extern double infnan();
133: if ( x == negone )
134: return (infnan(-ERANGE)); /* -INF */
135: else
136: return (infnan(EDOM)); /* NaN */
137: #else /* IEEE double */
138: /* x = -1, return -INF with signal */
139: if ( x == negone ) return( negone/zero );
140:
141: /* negative argument for log, return NaN with signal */
142: else return ( zero / zero );
143: #endif
144: }
145: }
146: /* end of if (finite(x)) */
147:
148: /* log(-INF) is NaN */
149: else if(x<0)
150: return(zero/zero);
151:
152: /* log(+INF) is INF */
153: else return(x);
154: }
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