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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 /* __ieee754_log(x)
15  * Return the logrithm of x
16  *
17  * Method :
18  *   1. Argument Reduction: find k and f such that
19  *			x = 2^k * (1+f),
20  *	   where  ieee_sqrt(2)/2 < 1+f < ieee_sqrt(2) .
21  *
22  *   2. Approximation of ieee_log(1+f).
23  *	Let s = f/(2+f) ; based on ieee_log(1+f) = ieee_log(1+s) - ieee_log(1-s)
24  *		 = 2s + 2/3 s**3 + 2/5 s**5 + .....,
25  *	     	 = 2s + s*R
26  *      We use a special Reme algorithm on [0,0.1716] to generate
27  * 	a polynomial of degree 14 to approximate R The maximum error
28  *	of this polynomial approximation is bounded by 2**-58.45. In
29  *	other words,
30  *		        2      4      6      8      10      12      14
31  *	    R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s  +Lg6*s  +Lg7*s
32  *  	(the values of Lg1 to Lg7 are listed in the program)
33  *	and
34  *	    |      2          14          |     -58.45
35  *	    | Lg1*s +...+Lg7*s    -  R(z) | <= 2
36  *	    |                             |
37  *	Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
38  *	In order to guarantee error in log below 1ulp, we compute log
39  *	by
40  *		log(1+f) = f - s*(f - R)	(if f is not too large)
41  *		log(1+f) = f - (hfsq - s*(hfsq+R)).	(better accuracy)
42  *
43  *	3. Finally,  ieee_log(x) = k*ln2 + ieee_log(1+f).
44  *			    = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
45  *	   Here ln2 is split into two floating point number:
46  *			ln2_hi + ln2_lo,
47  *	   where n*ln2_hi is always exact for |n| < 2000.
48  *
49  * Special cases:
50  *	log(x) is NaN with signal if x < 0 (including -INF) ;
51  *	log(+INF) is +INF; ieee_log(0) is -INF with signal;
52  *	log(NaN) is that NaN with no signal.
53  *
54  * Accuracy:
55  *	according to an error analysis, the error is always less than
56  *	1 ulp (unit in the last place).
57  *
58  * Constants:
59  * The hexadecimal values are the intended ones for the following
60  * constants. The decimal values may be used, provided that the
61  * compiler will convert from decimal to binary accurately enough
62  * to produce the hexadecimal values shown.
63  */
64 
65 #include "fdlibm.h"
66 
67 #ifdef __STDC__
68 static const double
69 #else
70 static double
71 #endif
72 ln2_hi  =  6.93147180369123816490e-01,	/* 3fe62e42 fee00000 */
73 ln2_lo  =  1.90821492927058770002e-10,	/* 3dea39ef 35793c76 */
74 two54   =  1.80143985094819840000e+16,  /* 43500000 00000000 */
75 Lg1 = 6.666666666666735130e-01,  /* 3FE55555 55555593 */
76 Lg2 = 3.999999999940941908e-01,  /* 3FD99999 9997FA04 */
77 Lg3 = 2.857142874366239149e-01,  /* 3FD24924 94229359 */
78 Lg4 = 2.222219843214978396e-01,  /* 3FCC71C5 1D8E78AF */
79 Lg5 = 1.818357216161805012e-01,  /* 3FC74664 96CB03DE */
80 Lg6 = 1.531383769920937332e-01,  /* 3FC39A09 D078C69F */
81 Lg7 = 1.479819860511658591e-01;  /* 3FC2F112 DF3E5244 */
82 
83 static double zero   =  0.0;
84 
85 #ifdef __STDC__
__ieee754_log(double x)86 	double __ieee754_log(double x)
87 #else
88 	double __ieee754_log(x)
89 	double x;
90 #endif
91 {
92 	double hfsq,f,s,z,R,w,t1,t2,dk;
93 	int k,hx,i,j;
94 	unsigned lx;
95 
96 	hx = __HI(x);		/* high word of x */
97 	lx = __LO(x);		/* low  word of x */
98 
99 	k=0;
100 	if (hx < 0x00100000) {			/* x < 2**-1022  */
101 	    if (((hx&0x7fffffff)|lx)==0)
102 		return -two54/zero;		/* ieee_log(+-0)=-inf */
103 	    if (hx<0) return (x-x)/zero;	/* ieee_log(-#) = NaN */
104 	    k -= 54; x *= two54; /* subnormal number, scale up x */
105 	    hx = __HI(x);		/* high word of x */
106 	}
107 	if (hx >= 0x7ff00000) return x+x;
108 	k += (hx>>20)-1023;
109 	hx &= 0x000fffff;
110 	i = (hx+0x95f64)&0x100000;
111 	__HI(x) = hx|(i^0x3ff00000);	/* normalize x or x/2 */
112 	k += (i>>20);
113 	f = x-1.0;
114 	if((0x000fffff&(2+hx))<3) {	/* |f| < 2**-20 */
115 	    if(f==zero) if(k==0) return zero;  else {dk=(double)k;
116 				 return dk*ln2_hi+dk*ln2_lo;}
117 	    R = f*f*(0.5-0.33333333333333333*f);
118 	    if(k==0) return f-R; else {dk=(double)k;
119 	    	     return dk*ln2_hi-((R-dk*ln2_lo)-f);}
120 	}
121  	s = f/(2.0+f);
122 	dk = (double)k;
123 	z = s*s;
124 	i = hx-0x6147a;
125 	w = z*z;
126 	j = 0x6b851-hx;
127 	t1= w*(Lg2+w*(Lg4+w*Lg6));
128 	t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
129 	i |= j;
130 	R = t2+t1;
131 	if(i>0) {
132 	    hfsq=0.5*f*f;
133 	    if(k==0) return f-(hfsq-s*(hfsq+R)); else
134 		     return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f);
135 	} else {
136 	    if(k==0) return f-s*(f-R); else
137 		     return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f);
138 	}
139 }
140