k_log.h revision 97409 
1/* @(#)e_log.c 5.1 93/09/24 */
2/*
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5 *
6 * Developed at SunPro, a Sun Microsystems, Inc. business.
7 * Permission to use, copy, modify, and distribute this
8 * software is freely granted, provided that this notice
9 * is preserved.
10 * ====================================================
11 */
12
13#ifndef lint
14static char rcsid[] = "$FreeBSD: head/lib/msun/src/e_log.c 97409 2002-05-28 17:51:46Z alfred $";
15#endif
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
85	double __generic___ieee754_log(double x)
86{
87	double hfsq,f,s,z,R,w,t1,t2,dk;
88	int32_t k,hx,i,j;
89	u_int32_t lx;
90
91	EXTRACT_WORDS(hx,lx,x);
92
93	k=0;
94	if (hx < 0x00100000) {			/* x < 2**-1022  */
95	    if (((hx&0x7fffffff)|lx)==0)
96		return -two54/zero;		/* log(+-0)=-inf */
97	    if (hx<0) return (x-x)/zero;	/* log(-#) = NaN */
98	    k -= 54; x *= two54; /* subnormal number, scale up x */
99	    GET_HIGH_WORD(hx,x);
100	}
101	if (hx >= 0x7ff00000) return x+x;
102	k += (hx>>20)-1023;
103	hx &= 0x000fffff;
104	i = (hx+0x95f64)&0x100000;
105	SET_HIGH_WORD(x,hx|(i^0x3ff00000));	/* normalize x or x/2 */
106	k += (i>>20);
107	f = x-1.0;
108	if((0x000fffff&(2+hx))<3) {	/* |f| < 2**-20 */
109	    if(f==zero) if(k==0) return zero;  else {dk=(double)k;
110				 return dk*ln2_hi+dk*ln2_lo;}
111	    R = f*f*(0.5-0.33333333333333333*f);
112	    if(k==0) return f-R; else {dk=(double)k;
113	    	     return dk*ln2_hi-((R-dk*ln2_lo)-f);}
114	}
115 	s = f/(2.0+f);
116	dk = (double)k;
117	z = s*s;
118	i = hx-0x6147a;
119	w = z*z;
120	j = 0x6b851-hx;
121	t1= w*(Lg2+w*(Lg4+w*Lg6));
122	t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
123	i |= j;
124	R = t2+t1;
125	if(i>0) {
126	    hfsq=0.5*f*f;
127	    if(k==0) return f-(hfsq-s*(hfsq+R)); else
128		     return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f);
129	} else {
130	    if(k==0) return f-s*(f-R); else
131		     return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f);
132	}
133}
134