1 
2 /* @(#)s_log1p.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 /* double ieee_log1p(double x)
15  *
16  * Method :
17  *   1. Argument Reduction: find k and f such that
18  *			1+x = 2^k * (1+f),
19  *	   where  ieee_sqrt(2)/2 < 1+f < ieee_sqrt(2) .
20  *
21  *      Note. If k=0, then f=x is exact. However, if k!=0, then f
22  *	may not be representable exactly. In that case, a correction
23  *	term is need. Let u=1+x rounded. Let c = (1+x)-u, then
24  *	log(1+x) - ieee_log(u) ~ c/u. Thus, we proceed to compute ieee_log(u),
25  *	and add back the correction term c/u.
26  *	(Note: when x > 2**53, one can simply return ieee_log(x))
27  *
28  *   2. Approximation of ieee_log1p(f).
29  *	Let s = f/(2+f) ; based on ieee_log(1+f) = ieee_log(1+s) - ieee_log(1-s)
30  *		 = 2s + 2/3 s**3 + 2/5 s**5 + .....,
31  *	     	 = 2s + s*R
32  *      We use a special Reme algorithm on [0,0.1716] to generate
33  * 	a polynomial of degree 14 to approximate R The maximum error
34  *	of this polynomial approximation is bounded by 2**-58.45. In
35  *	other words,
36  *		        2      4      6      8      10      12      14
37  *	    R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s  +Lp6*s  +Lp7*s
38  *  	(the values of Lp1 to Lp7 are listed in the program)
39  *	and
40  *	    |      2          14          |     -58.45
41  *	    | Lp1*s +...+Lp7*s    -  R(z) | <= 2
42  *	    |                             |
43  *	Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
44  *	In order to guarantee error in log below 1ulp, we compute log
45  *	by
46  *		log1p(f) = f - (hfsq - s*(hfsq+R)).
47  *
48  *	3. Finally, ieee_log1p(x) = k*ln2 + ieee_log1p(f).
49  *		 	     = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
50  *	   Here ln2 is split into two floating point number:
51  *			ln2_hi + ln2_lo,
52  *	   where n*ln2_hi is always exact for |n| < 2000.
53  *
54  * Special cases:
55  *	log1p(x) is NaN with signal if x < -1 (including -INF) ;
56  *	log1p(+INF) is +INF; ieee_log1p(-1) is -INF with signal;
57  *	log1p(NaN) is that NaN with no signal.
58  *
59  * Accuracy:
60  *	according to an error analysis, the error is always less than
61  *	1 ulp (unit in the last place).
62  *
63  * Constants:
64  * The hexadecimal values are the intended ones for the following
65  * constants. The decimal values may be used, provided that the
66  * compiler will convert from decimal to binary accurately enough
67  * to produce the hexadecimal values shown.
68  *
69  * Note: Assuming ieee_log() return accurate answer, the following
70  * 	 algorithm can be used to compute ieee_log1p(x) to within a few ULP:
71  *
72  *		u = 1+x;
73  *		if(u==1.0) return x ; else
74  *			   return ieee_log(u)*(x/(u-1.0));
75  *
76  *	 See HP-15C Advanced Functions Handbook, p.193.
77  */
78 
79 #include "fdlibm.h"
80 
81 #ifdef __STDC__
82 static const double
83 #else
84 static double
85 #endif
86 ln2_hi  =  6.93147180369123816490e-01,	/* 3fe62e42 fee00000 */
87 ln2_lo  =  1.90821492927058770002e-10,	/* 3dea39ef 35793c76 */
88 two54   =  1.80143985094819840000e+16,  /* 43500000 00000000 */
89 Lp1 = 6.666666666666735130e-01,  /* 3FE55555 55555593 */
90 Lp2 = 3.999999999940941908e-01,  /* 3FD99999 9997FA04 */
91 Lp3 = 2.857142874366239149e-01,  /* 3FD24924 94229359 */
92 Lp4 = 2.222219843214978396e-01,  /* 3FCC71C5 1D8E78AF */
93 Lp5 = 1.818357216161805012e-01,  /* 3FC74664 96CB03DE */
94 Lp6 = 1.531383769920937332e-01,  /* 3FC39A09 D078C69F */
95 Lp7 = 1.479819860511658591e-01;  /* 3FC2F112 DF3E5244 */
96 
97 static double zero = 0.0;
98 
99 #ifdef __STDC__
ieee_log1p(double x)100 	double ieee_log1p(double x)
101 #else
102 	double ieee_log1p(x)
103 	double x;
104 #endif
105 {
106 	double hfsq,f,c,s,z,R,u;
107 	int k,hx,hu,ax;
108 
109 	hx = __HI(x);		/* high word of x */
110 	ax = hx&0x7fffffff;
111 
112 	k = 1;
113 	if (hx < 0x3FDA827A) {			/* x < 0.41422  */
114 	    if(ax>=0x3ff00000) {		/* x <= -1.0 */
115 		if(x==-1.0) return -two54/zero; /* ieee_log1p(-1)=+inf */
116 		else return (x-x)/(x-x);	/* ieee_log1p(x<-1)=NaN */
117 	    }
118 	    if(ax<0x3e200000) {			/* |x| < 2**-29 */
119 		if(two54+x>zero			/* raise inexact */
120 	            &&ax<0x3c900000) 		/* |x| < 2**-54 */
121 		    return x;
122 		else
123 		    return x - x*x*0.5;
124 	    }
125 	    if(hx>0||hx<=((int)0xbfd2bec3)) {
126 		k=0;f=x;hu=1;}	/* -0.2929<x<0.41422 */
127 	}
128 	if (hx >= 0x7ff00000) return x+x;
129 	if(k!=0) {
130 	    if(hx<0x43400000) {
131 		u  = 1.0+x;
132 	        hu = __HI(u);		/* high word of u */
133 	        k  = (hu>>20)-1023;
134 	        c  = (k>0)? 1.0-(u-x):x-(u-1.0);/* correction term */
135 		c /= u;
136 	    } else {
137 		u  = x;
138 	        hu = __HI(u);		/* high word of u */
139 	        k  = (hu>>20)-1023;
140 		c  = 0;
141 	    }
142 	    hu &= 0x000fffff;
143 	    if(hu<0x6a09e) {
144 	        __HI(u) = hu|0x3ff00000;	/* normalize u */
145 	    } else {
146 	        k += 1;
147 	        __HI(u) = hu|0x3fe00000;	/* normalize u/2 */
148 	        hu = (0x00100000-hu)>>2;
149 	    }
150 	    f = u-1.0;
151 	}
152 	hfsq=0.5*f*f;
153 	if(hu==0) {	/* |f| < 2**-20 */
154 	    if(f==zero) if(k==0) return zero;
155 			else {c += k*ln2_lo; return k*ln2_hi+c;}
156 	    R = hfsq*(1.0-0.66666666666666666*f);
157 	    if(k==0) return f-R; else
158 	    	     return k*ln2_hi-((R-(k*ln2_lo+c))-f);
159 	}
160  	s = f/(2.0+f);
161 	z = s*s;
162 	R = z*(Lp1+z*(Lp2+z*(Lp3+z*(Lp4+z*(Lp5+z*(Lp6+z*Lp7))))));
163 	if(k==0) return f-(hfsq-s*(hfsq+R)); else
164 		 return k*ln2_hi-((hfsq-(s*(hfsq+R)+(k*ln2_lo+c)))-f);
165 }
166