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__
82static const double
83#else
84static double
85#endif
86ln2_hi  =  6.93147180369123816490e-01,	/* 3fe62e42 fee00000 */
87ln2_lo  =  1.90821492927058770002e-10,	/* 3dea39ef 35793c76 */
88two54   =  1.80143985094819840000e+16,  /* 43500000 00000000 */
89Lp1 = 6.666666666666735130e-01,  /* 3FE55555 55555593 */
90Lp2 = 3.999999999940941908e-01,  /* 3FD99999 9997FA04 */
91Lp3 = 2.857142874366239149e-01,  /* 3FD24924 94229359 */
92Lp4 = 2.222219843214978396e-01,  /* 3FCC71C5 1D8E78AF */
93Lp5 = 1.818357216161805012e-01,  /* 3FC74664 96CB03DE */
94Lp6 = 1.531383769920937332e-01,  /* 3FC39A09 D078C69F */
95Lp7 = 1.479819860511658591e-01;  /* 3FC2F112 DF3E5244 */
96
97static double zero = 0.0;
98
99#ifdef __STDC__
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