1/* @(#)s_log1p.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#include <sys/cdefs.h>
14__FBSDID("$FreeBSD$");
15
16/* double log1p(double x)
17 *
18 * Method :
19 *   1. Argument Reduction: find k and f such that
20 *			1+x = 2^k * (1+f),
21 *	   where  sqrt(2)/2 < 1+f < sqrt(2) .
22 *
23 *      Note. If k=0, then f=x is exact. However, if k!=0, then f
24 *	may not be representable exactly. In that case, a correction
25 *	term is need. Let u=1+x rounded. Let c = (1+x)-u, then
26 *	log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
27 *	and add back the correction term c/u.
28 *	(Note: when x > 2**53, one can simply return log(x))
29 *
30 *   2. Approximation of log1p(f).
31 *	Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
32 *		 = 2s + 2/3 s**3 + 2/5 s**5 + .....,
33 *	     	 = 2s + s*R
34 *      We use a special Reme algorithm on [0,0.1716] to generate
35 * 	a polynomial of degree 14 to approximate R The maximum error
36 *	of this polynomial approximation is bounded by 2**-58.45. In
37 *	other words,
38 *		        2      4      6      8      10      12      14
39 *	    R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s  +Lp6*s  +Lp7*s
40 *  	(the values of Lp1 to Lp7 are listed in the program)
41 *	and
42 *	    |      2          14          |     -58.45
43 *	    | Lp1*s +...+Lp7*s    -  R(z) | <= 2
44 *	    |                             |
45 *	Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
46 *	In order to guarantee error in log below 1ulp, we compute log
47 *	by
48 *		log1p(f) = f - (hfsq - s*(hfsq+R)).
49 *
50 *	3. Finally, log1p(x) = k*ln2 + log1p(f).
51 *		 	     = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
52 *	   Here ln2 is split into two floating point number:
53 *			ln2_hi + ln2_lo,
54 *	   where n*ln2_hi is always exact for |n| < 2000.
55 *
56 * Special cases:
57 *	log1p(x) is NaN with signal if x < -1 (including -INF) ;
58 *	log1p(+INF) is +INF; log1p(-1) is -INF with signal;
59 *	log1p(NaN) is that NaN with no signal.
60 *
61 * Accuracy:
62 *	according to an error analysis, the error is always less than
63 *	1 ulp (unit in the last place).
64 *
65 * Constants:
66 * The hexadecimal values are the intended ones for the following
67 * constants. The decimal values may be used, provided that the
68 * compiler will convert from decimal to binary accurately enough
69 * to produce the hexadecimal values shown.
70 *
71 * Note: Assuming log() return accurate answer, the following
72 * 	 algorithm can be used to compute log1p(x) to within a few ULP:
73 *
74 *		u = 1+x;
75 *		if(u==1.0) return x ; else
76 *			   return log(u)*(x/(u-1.0));
77 *
78 *	 See HP-15C Advanced Functions Handbook, p.193.
79 */
80
81#include <float.h>
82
83#include "math.h"
84#include "math_private.h"
85
86static const double
87ln2_hi  =  6.93147180369123816490e-01,	/* 3fe62e42 fee00000 */
88ln2_lo  =  1.90821492927058770002e-10,	/* 3dea39ef 35793c76 */
89two54   =  1.80143985094819840000e+16,  /* 43500000 00000000 */
90Lp1 = 6.666666666666735130e-01,  /* 3FE55555 55555593 */
91Lp2 = 3.999999999940941908e-01,  /* 3FD99999 9997FA04 */
92Lp3 = 2.857142874366239149e-01,  /* 3FD24924 94229359 */
93Lp4 = 2.222219843214978396e-01,  /* 3FCC71C5 1D8E78AF */
94Lp5 = 1.818357216161805012e-01,  /* 3FC74664 96CB03DE */
95Lp6 = 1.531383769920937332e-01,  /* 3FC39A09 D078C69F */
96Lp7 = 1.479819860511658591e-01;  /* 3FC2F112 DF3E5244 */
97
98static const double zero = 0.0;
99
100double
101log1p(double x)
102{
103	double hfsq,f,c,s,z,R,u;
104	int32_t k,hx,hu,ax;
105
106	GET_HIGH_WORD(hx,x);
107	ax = hx&0x7fffffff;
108
109	k = 1;
110	if (hx < 0x3FDA827A) {			/* 1+x < sqrt(2)+ */
111	    if(ax>=0x3ff00000) {		/* x <= -1.0 */
112		if(x==-1.0) return -two54/zero; /* log1p(-1)=+inf */
113		else return (x-x)/(x-x);	/* log1p(x<-1)=NaN */
114	    }
115	    if(ax<0x3e200000) {			/* |x| < 2**-29 */
116		if(two54+x>zero			/* raise inexact */
117	            &&ax<0x3c900000) 		/* |x| < 2**-54 */
118		    return x;
119		else
120		    return x - x*x*0.5;
121	    }
122	    if(hx>0||hx<=((int32_t)0xbfd2bec4)) {
123		k=0;f=x;hu=1;}		/* sqrt(2)/2- <= 1+x < sqrt(2)+ */
124	}
125	if (hx >= 0x7ff00000) return x+x;
126	if(k!=0) {
127	    if(hx<0x43400000) {
128		STRICT_ASSIGN(double,u,1.0+x);
129		GET_HIGH_WORD(hu,u);
130	        k  = (hu>>20)-1023;
131	        c  = (k>0)? 1.0-(u-x):x-(u-1.0);/* correction term */
132		c /= u;
133	    } else {
134		u  = x;
135		GET_HIGH_WORD(hu,u);
136	        k  = (hu>>20)-1023;
137		c  = 0;
138	    }
139	    hu &= 0x000fffff;
140	    /*
141	     * The approximation to sqrt(2) used in thresholds is not
142	     * critical.  However, the ones used above must give less
143	     * strict bounds than the one here so that the k==0 case is
144	     * never reached from here, since here we have committed to
145	     * using the correction term but don't use it if k==0.
146	     */
147	    if(hu<0x6a09e) {			/* u ~< sqrt(2) */
148	        SET_HIGH_WORD(u,hu|0x3ff00000);	/* normalize u */
149	    } else {
150	        k += 1;
151		SET_HIGH_WORD(u,hu|0x3fe00000);	/* normalize u/2 */
152	        hu = (0x00100000-hu)>>2;
153	    }
154	    f = u-1.0;
155	}
156	hfsq=0.5*f*f;
157	if(hu==0) {	/* |f| < 2**-20 */
158	    if(f==zero) {
159		if(k==0) {
160		    return zero;
161		} else {
162		    c += k*ln2_lo;
163		    return k*ln2_hi+c;
164		}
165	    }
166	    R = hfsq*(1.0-0.66666666666666666*f);
167	    if(k==0) return f-R; else
168	    	     return k*ln2_hi-((R-(k*ln2_lo+c))-f);
169	}
170 	s = f/(2.0+f);
171	z = s*s;
172	R = z*(Lp1+z*(Lp2+z*(Lp3+z*(Lp4+z*(Lp5+z*(Lp6+z*Lp7))))));
173	if(k==0) return f-(hfsq-s*(hfsq+R)); else
174		 return k*ln2_hi-((hfsq-(s*(hfsq+R)+(k*ln2_lo+c)))-f);
175}
176