source: trunk/sys/libm/s_log1p.c @ 1

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1
2/* @(#)s_log1p.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/* double log1p(double x)
15 *
16 * Method :                 
17 *   1. Argument Reduction: find k and f such that
18 *                      1+x = 2^k * (1+f),
19 *         where  sqrt(2)/2 < 1+f < sqrt(2) .
20 *
21 *      Note. If k=0, then f=x is exact. However, if k!=0, then f
22 *      may not be representable exactly. In that case, a correction
23 *      term is need. Let u=1+x rounded. Let c = (1+x)-u, then
24 *      log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
25 *      and add back the correction term c/u.
26 *      (Note: when x > 2**53, one can simply return log(x))
27 *
28 *   2. Approximation of log1p(f).
29 *      Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
30 *               = 2s + 2/3 s**3 + 2/5 s**5 + .....,
31 *               = 2s + s*R
32 *      We use a special Reme algorithm on [0,0.1716] to generate
33 *      a polynomial of degree 14 to approximate R The maximum error
34 *      of this polynomial approximation is bounded by 2**-58.45. In
35 *      other words,
36 *                      2      4      6      8      10      12      14
37 *          R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s  +Lp6*s  +Lp7*s
38 *      (the values of Lp1 to Lp7 are listed in the program)
39 *      and
40 *          |      2          14          |     -58.45
41 *          | Lp1*s +...+Lp7*s    -  R(z) | <= 2
42 *          |                             |
43 *      Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
44 *      In order to guarantee error in log below 1ulp, we compute log
45 *      by
46 *              log1p(f) = f - (hfsq - s*(hfsq+R)).
47 *     
48 *      3. Finally, log1p(x) = k*ln2 + log1p(f). 
49 *                           = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
50 *         Here ln2 is split into two floating point number:
51 *                      ln2_hi + ln2_lo,
52 *         where n*ln2_hi is always exact for |n| < 2000.
53 *
54 * Special cases:
55 *      log1p(x) is NaN with signal if x < -1 (including -INF) ;
56 *      log1p(+INF) is +INF; log1p(-1) is -INF with signal;
57 *      log1p(NaN) is that NaN with no signal.
58 *
59 * Accuracy:
60 *      according to an error analysis, the error is always less than
61 *      1 ulp (unit in the last place).
62 *
63 * Constants:
64 * The hexadecimal values are the intended ones for the following
65 * constants. The decimal values may be used, provided that the
66 * compiler will convert from decimal to binary accurately enough
67 * to produce the hexadecimal values shown.
68 *
69 * Note: Assuming log() return accurate answer, the following
70 *       algorithm can be used to compute log1p(x) to within a few ULP:
71 *     
72 *              u = 1+x;
73 *              if(u==1.0) return x ; else
74 *                         return log(u)*(x/(u-1.0));
75 *
76 *       See HP-15C Advanced Functions Handbook, p.193.
77 */
78
79#include <libm/fdlibm.h>
80
81#ifdef __STDC__
82static const double
83#else
84static double
85#endif
86ln2_hi  =  6.93147180369123816490e-01,  /* 3fe62e42 fee00000 */
87ln2_lo  =  1.90821492927058770002e-10,  /* 3dea39ef 35793c76 */
88two54   =  1.80143985094819840000e+16,  /* 43500000 00000000 */
89Lp1 = 6.666666666666735130e-01,  /* 3FE55555 55555593 */
90Lp2 = 3.999999999940941908e-01,  /* 3FD99999 9997FA04 */
91Lp3 = 2.857142874366239149e-01,  /* 3FD24924 94229359 */
92Lp4 = 2.222219843214978396e-01,  /* 3FCC71C5 1D8E78AF */
93Lp5 = 1.818357216161805012e-01,  /* 3FC74664 96CB03DE */
94Lp6 = 1.531383769920937332e-01,  /* 3FC39A09 D078C69F */
95Lp7 = 1.479819860511658591e-01;  /* 3FC2F112 DF3E5244 */
96
97static double zero = 0.0;
98
99#ifdef __STDC__
100        double log1p(double x)
101#else
102        double log1p(x)
103        double x;
104#endif
105{
106        double hfsq,f,c = 0,s,z,R,u;
107        int k,hx,n0,hu,ax;
108
109        n0 = (*((int*)&two54)>>30)^1;   /* high word index */
110        hx = *(n0+(int*)&x);            /* high word of x */
111        ax = hx&0x7fffffff;
112
113        k = 1;
114        if (hx < 0x3FDA827A) {                  /* x < 0.41422  */
115            if(ax>=0x3ff00000) {                /* x <= -1.0 */
116                if(x==-1.0) return -two54/zero; /* log1p(-1)=+inf */
117                else return (x-x)/(x-x);        /* log1p(x<-1)=NaN */
118            }
119            if(ax<0x3e200000) {                 /* |x| < 2**-29 */
120                if(two54+x>zero                 /* raise inexact */
121                    &&ax<0x3c900000)            /* |x| < 2**-54 */
122                    return x;
123                else
124                    return x - x*x*0.5;
125            }
126            if(hx>0||hx<=((int)0xbfd2bec3)) {
127                k=0;f=x;hu=1;}  /* -0.2929<x<0.41422 */
128        } 
129        if (hx >= 0x7ff00000) return x+x;
130        if(k!=0) {
131            if(hx<0x43400000) {
132                u  = 1.0+x; 
133                hu = *(n0+(int*)&u);            /* high word of u */
134                k  = (hu>>20)-1023;
135                c  = (k>0)? 1.0-(u-x):x-(u-1.0);/* correction term */
136                c /= u;
137            } else {
138                u  = x;
139                hu = *(n0+(int*)&u);            /* high word of u */
140                k  = (hu>>20)-1023;
141                c  = 0;
142            }
143            hu &= 0x000fffff;
144            if(hu<0x6a09e) {
145                *(n0+(int*)&u) = hu|0x3ff00000; /* normalize u */
146            } else {
147                k += 1; 
148                *(n0+(int*)&u) = hu|0x3fe00000; /* normalize u/2 */
149                hu = (0x00100000-hu)>>2;
150            }
151            f = u-1.0;
152        }
153        hfsq=0.5*f*f;
154        if(hu==0) {     /* |f| < 2**-20 */
155            if(f==zero) 
156              {
157                if(k==0) 
158                  return zero; 
159                else 
160                  {
161                    c += k*ln2_lo; 
162                    return k*ln2_hi+c;
163                  }
164              }
165            R = hfsq*(1.0-0.66666666666666666*f);
166            if(k==0) return f-R; else
167                     return k*ln2_hi-((R-(k*ln2_lo+c))-f);
168        }
169        s = f/(2.0+f); 
170        z = s*s;
171        R = z*(Lp1+z*(Lp2+z*(Lp3+z*(Lp4+z*(Lp5+z*(Lp6+z*Lp7))))));
172        if(k==0) return f-(hfsq-s*(hfsq+R)); else
173                 return k*ln2_hi-((hfsq-(s*(hfsq+R)+(k*ln2_lo+c)))-f);
174}
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