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/* log(x)
14 * Return the logarithm of x
15 *
16 * Method :
17 *   1. Argument Reduction: find k and f such that
18 *			x = 2^k * (1+f),
19 *	   where  sqrt(2)/2 < 1+f < sqrt(2) .
20 *
21 *   2. Approximation of log(1+f).
22 *	Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
23 *		 = 2s + 2/3 s**3 + 2/5 s**5 + .....,
24 *	     	 = 2s + s*R
25 *      We use a special Remes algorithm on [0,0.1716] to generate
26 * 	a polynomial of degree 14 to approximate R The maximum error
27 *	of this polynomial approximation is bounded by 2**-58.45. In
28 *	other words,
29 *		        2      4      6      8      10      12      14
30 *	    R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s  +Lg6*s  +Lg7*s
31 *  	(the values of Lg1 to Lg7 are listed in the program)
32 *	and
33 *	    |      2          14          |     -58.45
34 *	    | Lg1*s +...+Lg7*s    -  R(z) | <= 2
35 *	    |                             |
36 *	Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
37 *	In order to guarantee error in log below 1ulp, we compute log
38 *	by
39 *		log(1+f) = f - s*(f - R)	(if f is not too large)
40 *		log(1+f) = f - (hfsq - s*(hfsq+R)).	(better accuracy)
41 *
42 *	3. Finally,  log(x) = k*ln2 + log(1+f).
43 *			    = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
44 *	   Here ln2 is split into two floating point number:
45 *			ln2_hi + ln2_lo,
46 *	   where n*ln2_hi is always exact for |n| < 2000.
47 *
48 * Special cases:
49 *	log(x) is NaN with signal if x < 0 (including -INF) ;
50 *	log(+INF) is +INF; log(0) is -INF with signal;
51 *	log(NaN) is that NaN with no signal.
52 *
53 * Accuracy:
54 *	according to an error analysis, the error is always less than
55 *	1 ulp (unit in the last place).
56 *
57 * Constants:
58 * The hexadecimal values are the intended ones for the following
59 * constants. The decimal values may be used, provided that the
60 * compiler will convert from decimal to binary accurately enough
61 * to produce the hexadecimal values shown.
62 */
63
64#include <float.h>
65#include <math.h>
66
67#include "math_private.h"
68
69static const double
70ln2_hi  =  6.93147180369123816490e-01,	/* 3fe62e42 fee00000 */
71ln2_lo  =  1.90821492927058770002e-10,	/* 3dea39ef 35793c76 */
72two54   =  1.80143985094819840000e+16,  /* 43500000 00000000 */
73Lg1 = 6.666666666666735130e-01,  /* 3FE55555 55555593 */
74Lg2 = 3.999999999940941908e-01,  /* 3FD99999 9997FA04 */
75Lg3 = 2.857142874366239149e-01,  /* 3FD24924 94229359 */
76Lg4 = 2.222219843214978396e-01,  /* 3FCC71C5 1D8E78AF */
77Lg5 = 1.818357216161805012e-01,  /* 3FC74664 96CB03DE */
78Lg6 = 1.531383769920937332e-01,  /* 3FC39A09 D078C69F */
79Lg7 = 1.479819860511658591e-01;  /* 3FC2F112 DF3E5244 */
80
81static const double zero   =  0.0;
82
83double
84log(double x)
85{
86	double hfsq,f,s,z,R,w,t1,t2,dk;
87	int32_t k,hx,i,j;
88	u_int32_t lx;
89
90	EXTRACT_WORDS(hx,lx,x);
91
92	k=0;
93	if (hx < 0x00100000) {			/* x < 2**-1022  */
94	    if (((hx&0x7fffffff)|lx)==0)
95		return -two54/zero;		/* log(+-0)=-inf */
96	    if (hx<0) return (x-x)/zero;	/* log(-#) = NaN */
97	    k -= 54; x *= two54; /* subnormal number, scale up x */
98	    GET_HIGH_WORD(hx,x);
99	}
100	if (hx >= 0x7ff00000) return x+x;
101	k += (hx>>20)-1023;
102	hx &= 0x000fffff;
103	i = (hx+0x95f64)&0x100000;
104	SET_HIGH_WORD(x,hx|(i^0x3ff00000));	/* normalize x or x/2 */
105	k += (i>>20);
106	f = x-1.0;
107	if((0x000fffff&(2+hx))<3) {	/* |f| < 2**-20 */
108	    if(f==zero) if(k==0) return zero;  else {dk=(double)k;
109				 return dk*ln2_hi+dk*ln2_lo;}
110	    R = f*f*(0.5-0.33333333333333333*f);
111	    if(k==0) return f-R; else {dk=(double)k;
112	    	     return dk*ln2_hi-((R-dk*ln2_lo)-f);}
113	}
114 	s = f/(2.0+f);
115	dk = (double)k;
116	z = s*s;
117	i = hx-0x6147a;
118	w = z*z;
119	j = 0x6b851-hx;
120	t1= w*(Lg2+w*(Lg4+w*Lg6));
121	t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
122	i |= j;
123	R = t2+t1;
124	if(i>0) {
125	    hfsq=0.5*f*f;
126	    if(k==0) return f-(hfsq-s*(hfsq+R)); else
127		     return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f);
128	} else {
129	    if(k==0) return f-s*(f-R); else
130		     return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f);
131	}
132}
133DEF_STD(log);
134LDBL_MAYBE_CLONE(log);
135