1
2/* @(#)e_log.c 1.3 95/01/18 */
3/*
4 * ====================================================
5 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
6 *
7 * Developed at SunSoft, 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#include <sys/cdefs.h>
15__FBSDID("$FreeBSD$");
16
17/* __ieee754_log(x)
18 * Return the logrithm of x
19 *
20 * Method :
21 *   1. Argument Reduction: find k and f such that
22 *			x = 2^k * (1+f),
23 *	   where  sqrt(2)/2 < 1+f < sqrt(2) .
24 *
25 *   2. Approximation of log(1+f).
26 *	Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
27 *		 = 2s + 2/3 s**3 + 2/5 s**5 + .....,
28 *	     	 = 2s + s*R
29 *      We use a special Reme algorithm on [0,0.1716] to generate
30 * 	a polynomial of degree 14 to approximate R The maximum error
31 *	of this polynomial approximation is bounded by 2**-58.45. In
32 *	other words,
33 *		        2      4      6      8      10      12      14
34 *	    R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s  +Lg6*s  +Lg7*s
35 *  	(the values of Lg1 to Lg7 are listed in the program)
36 *	and
37 *	    |      2          14          |     -58.45
38 *	    | Lg1*s +...+Lg7*s    -  R(z) | <= 2
39 *	    |                             |
40 *	Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
41 *	In order to guarantee error in log below 1ulp, we compute log
42 *	by
43 *		log(1+f) = f - s*(f - R)	(if f is not too large)
44 *		log(1+f) = f - (hfsq - s*(hfsq+R)).	(better accuracy)
45 *
46 *	3. Finally,  log(x) = k*ln2 + log(1+f).
47 *			    = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
48 *	   Here ln2 is split into two floating point number:
49 *			ln2_hi + ln2_lo,
50 *	   where n*ln2_hi is always exact for |n| < 2000.
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 *	according to an error analysis, the error is always less than
59 *	1 ulp (unit in the last place).
60 *
61 * Constants:
62 * The hexadecimal values are the intended ones for the following
63 * constants. The decimal values may be used, provided that the
64 * compiler will convert from decimal to binary accurately enough
65 * to produce the hexadecimal values shown.
66 */
67
68#include "math.h"
69#include "math_private.h"
70
71static const double
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 const double zero   =  0.0;
84
85double
86__ieee754_log(double x)
87{
88	double hfsq,f,s,z,R,w,t1,t2,dk;
89	int32_t k,hx,i,j;
90	u_int32_t lx;
91
92	EXTRACT_WORDS(hx,lx,x);
93
94	k=0;
95	if (hx < 0x00100000) {			/* x < 2**-1022  */
96	    if (((hx&0x7fffffff)|lx)==0)
97		return -two54/zero;		/* log(+-0)=-inf */
98	    if (hx<0) return (x-x)/zero;	/* log(-#) = NaN */
99	    k -= 54; x *= two54; /* subnormal number, scale up x */
100	    GET_HIGH_WORD(hx,x);
101	}
102	if (hx >= 0x7ff00000) return x+x;
103	k += (hx>>20)-1023;
104	hx &= 0x000fffff;
105	i = (hx+0x95f64)&0x100000;
106	SET_HIGH_WORD(x,hx|(i^0x3ff00000));	/* normalize x or x/2 */
107	k += (i>>20);
108	f = x-1.0;
109	if((0x000fffff&(2+hx))<3) {	/* -2**-20 <= f < 2**-20 */
110	    if(f==zero) {
111		if(k==0) {
112		    return zero;
113		} else {
114		    dk=(double)k;
115		    return dk*ln2_hi+dk*ln2_lo;
116		}
117	    }
118	    R = f*f*(0.5-0.33333333333333333*f);
119	    if(k==0) return f-R; else {dk=(double)k;
120	    	     return dk*ln2_hi-((R-dk*ln2_lo)-f);}
121	}
122 	s = f/(2.0+f);
123	dk = (double)k;
124	z = s*s;
125	i = hx-0x6147a;
126	w = z*z;
127	j = 0x6b851-hx;
128	t1= w*(Lg2+w*(Lg4+w*Lg6));
129	t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
130	i |= j;
131	R = t2+t1;
132	if(i>0) {
133	    hfsq=0.5*f*f;
134	    if(k==0) return f-(hfsq-s*(hfsq+R)); else
135		     return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f);
136	} else {
137	    if(k==0) return f-s*(f-R); else
138		     return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f);
139	}
140}
141