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 <float.h>
69 
70 #include "math.h"
71 #include "math_private.h"
72 
73 static const double
74 ln2_hi  =  6.93147180369123816490e-01,	/* 3fe62e42 fee00000 */
75 ln2_lo  =  1.90821492927058770002e-10,	/* 3dea39ef 35793c76 */
76 two54   =  1.80143985094819840000e+16,  /* 43500000 00000000 */
77 Lg1 = 6.666666666666735130e-01,  /* 3FE55555 55555593 */
78 Lg2 = 3.999999999940941908e-01,  /* 3FD99999 9997FA04 */
79 Lg3 = 2.857142874366239149e-01,  /* 3FD24924 94229359 */
80 Lg4 = 2.222219843214978396e-01,  /* 3FCC71C5 1D8E78AF */
81 Lg5 = 1.818357216161805012e-01,  /* 3FC74664 96CB03DE */
82 Lg6 = 1.531383769920937332e-01,  /* 3FC39A09 D078C69F */
83 Lg7 = 1.479819860511658591e-01;  /* 3FC2F112 DF3E5244 */
84 
85 static const double zero   =  0.0;
86 static volatile double vzero = 0.0;
87 
88 double
__ieee754_log(double x)89 __ieee754_log(double x)
90 {
91 	double hfsq,f,s,z,R,w,t1,t2,dk;
92 	int32_t k,hx,i,j;
93 	u_int32_t lx;
94 
95 	EXTRACT_WORDS(hx,lx,x);
96 
97 	k=0;
98 	if (hx < 0x00100000) {			/* x < 2**-1022  */
99 	    if (((hx&0x7fffffff)|lx)==0)
100 		return -two54/vzero;		/* log(+-0)=-inf */
101 	    if (hx<0) return (x-x)/zero;	/* log(-#) = NaN */
102 	    k -= 54; x *= two54; /* subnormal number, scale up x */
103 	    GET_HIGH_WORD(hx,x);
104 	}
105 	if (hx >= 0x7ff00000) return x+x;
106 	k += (hx>>20)-1023;
107 	hx &= 0x000fffff;
108 	i = (hx+0x95f64)&0x100000;
109 	SET_HIGH_WORD(x,hx|(i^0x3ff00000));	/* normalize x or x/2 */
110 	k += (i>>20);
111 	f = x-1.0;
112 	if((0x000fffff&(2+hx))<3) {	/* -2**-20 <= f < 2**-20 */
113 	    if(f==zero) {
114 		if(k==0) {
115 		    return zero;
116 		} else {
117 		    dk=(double)k;
118 		    return dk*ln2_hi+dk*ln2_lo;
119 		}
120 	    }
121 	    R = f*f*(0.5-0.33333333333333333*f);
122 	    if(k==0) return f-R; else {dk=(double)k;
123 	    	     return dk*ln2_hi-((R-dk*ln2_lo)-f);}
124 	}
125  	s = f/(2.0+f);
126 	dk = (double)k;
127 	z = s*s;
128 	i = hx-0x6147a;
129 	w = z*z;
130 	j = 0x6b851-hx;
131 	t1= w*(Lg2+w*(Lg4+w*Lg6));
132 	t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
133 	i |= j;
134 	R = t2+t1;
135 	if(i>0) {
136 	    hfsq=0.5*f*f;
137 	    if(k==0) return f-(hfsq-s*(hfsq+R)); else
138 		     return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f);
139 	} else {
140 	    if(k==0) return f-s*(f-R); else
141 		     return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f);
142 	}
143 }
144 
145 #if (LDBL_MANT_DIG == 53)
146 __weak_reference(log, logl);
147 #endif
148