source: trunk/sys/libm/e_log.c @ 1

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1
2/* @(#)e_log.c 5.1 93/09/24 */
3/*
4 * ====================================================
5 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
6 *
7 * Developed at SunPro, a Sun Microsystems, Inc. business.
8 * Permission to use, copy, modify, and distribute this
9 * software is freely granted, provided that this notice
10 * is preserved.
11 * ====================================================
12 */
13
14/* __ieee754_log(x)
15 * Return the logrithm of x
16 *
17 * Method :                 
18 *   1. Argument Reduction: find k and f such that
19 *                      x = 2^k * (1+f),
20 *         where  sqrt(2)/2 < 1+f < sqrt(2) .
21 *
22 *   2. Approximation of log(1+f).
23 *      Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
24 *               = 2s + 2/3 s**3 + 2/5 s**5 + .....,
25 *               = 2s + s*R
26 *      We use a special Reme algorithm on [0,0.1716] to generate
27 *      a polynomial of degree 14 to approximate R The maximum error
28 *      of this polynomial approximation is bounded by 2**-58.45. In
29 *      other words,
30 *                      2      4      6      8      10      12      14
31 *          R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s  +Lg6*s  +Lg7*s
32 *      (the values of Lg1 to Lg7 are listed in the program)
33 *      and
34 *          |      2          14          |     -58.45
35 *          | Lg1*s +...+Lg7*s    -  R(z) | <= 2
36 *          |                             |
37 *      Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
38 *      In order to guarantee error in log below 1ulp, we compute log
39 *      by
40 *              log(1+f) = f - s*(f - R)        (if f is not too large)
41 *              log(1+f) = f - (hfsq - s*(hfsq+R)).     (better accuracy)
42 *     
43 *      3. Finally,  log(x) = k*ln2 + log(1+f). 
44 *                          = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
45 *         Here ln2 is split into two floating point number:
46 *                      ln2_hi + ln2_lo,
47 *         where n*ln2_hi is always exact for |n| < 2000.
48 *
49 * Special cases:
50 *      log(x) is NaN with signal if x < 0 (including -INF) ;
51 *      log(+INF) is +INF; log(0) is -INF with signal;
52 *      log(NaN) is that NaN with no signal.
53 *
54 * Accuracy:
55 *      according to an error analysis, the error is always less than
56 *      1 ulp (unit in the last place).
57 *
58 * Constants:
59 * The hexadecimal values are the intended ones for the following
60 * constants. The decimal values may be used, provided that the
61 * compiler will convert from decimal to binary accurately enough
62 * to produce the hexadecimal values shown.
63 */
64
65#include <libm/fdlibm.h>
66
67#ifdef __STDC__
68static const double
69#else
70static double
71#endif
72ln2_hi  =  6.93147180369123816490e-01,  /* 3fe62e42 fee00000 */
73ln2_lo  =  1.90821492927058770002e-10,  /* 3dea39ef 35793c76 */
74two54   =  1.80143985094819840000e+16,  /* 43500000 00000000 */
75Lg1 = 6.666666666666735130e-01,  /* 3FE55555 55555593 */
76Lg2 = 3.999999999940941908e-01,  /* 3FD99999 9997FA04 */
77Lg3 = 2.857142874366239149e-01,  /* 3FD24924 94229359 */
78Lg4 = 2.222219843214978396e-01,  /* 3FCC71C5 1D8E78AF */
79Lg5 = 1.818357216161805012e-01,  /* 3FC74664 96CB03DE */
80Lg6 = 1.531383769920937332e-01,  /* 3FC39A09 D078C69F */
81Lg7 = 1.479819860511658591e-01;  /* 3FC2F112 DF3E5244 */
82
83static double zero   =  0.0;
84
85#ifdef __STDC__
86        double __ieee754_log(double x)
87#else
88        double __ieee754_log(x)
89        double x;
90#endif
91{
92        double hfsq,f,s,z,R,w,t1,t2,dk;
93        int k,hx,n0,i,j;
94        unsigned lx;
95
96        n0 = (*((int*)&two54)>>30)^1;   /* high word index */
97        hx = *(n0+(int*)&x);            /* high word of x */
98        lx = *(1-n0+(int*)&x);          /* low  word of x */
99
100        k=0;
101        if (hx < 0x00100000) {                  /* x < 2**-1022  */
102            if (((hx&0x7fffffff)|lx)==0) 
103                return -two54/zero;             /* log(+-0)=-inf */
104            if (hx<0) return (x-x)/zero;        /* log(-#) = NaN */
105            k -= 54; x *= two54; /* subnormal number, scale up x */
106            hx = *(n0+(int*)&x);                /* high word of x */
107        } 
108        if (hx >= 0x7ff00000) return x+x;
109        k += (hx>>20)-1023;
110        hx &= 0x000fffff;
111        i = (hx+0x95f64)&0x100000;
112        *(n0+(int*)&x) = hx|(i^0x3ff00000);     /* normalize x or x/2 */
113        k += (i>>20);
114        f = x-1.0;
115        if((0x000fffff&(2+hx))<3) {     /* |f| < 2**-20 */
116            if(f==zero)
117            { 
118              if(k==0) 
119                return zero;
120            }
121            else 
122            {
123              dk=(double)k;
124              return dk*ln2_hi+dk*ln2_lo;
125            }
126            R = f*f*(0.5-0.33333333333333333*f);
127            if(k==0)
128              return f-R; 
129            else
130              {
131                dk=(double)k;
132                return dk*ln2_hi-((R-dk*ln2_lo)-f);
133              }
134        }
135        s = f/(2.0+f); 
136        dk = (double)k;
137        z = s*s;
138        i = hx-0x6147a;
139        w = z*z;
140        j = 0x6b851-hx;
141        t1= w*(Lg2+w*(Lg4+w*Lg6)); 
142        t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7))); 
143        i |= j;
144        R = t2+t1;
145        if(i>0) {
146            hfsq=0.5*f*f;
147            if(k==0) return f-(hfsq-s*(hfsq+R)); else
148                     return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f);
149        } else {
150            if(k==0) return f-s*(f-R); else
151                     return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f);
152        }
153}
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