1/*
2 * ====================================================
3 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
4 *
5 * Developed at SunPro, a Sun Microsystems, Inc. business.
6 * Permission to use, copy, modify, and distribute this
7 * software is freely granted, provided that this notice
8 * is preserved.
9 * ====================================================
10 */
11
12/* double log1p(double x)
13 *
14 * Method :
15 *   1. Argument Reduction: find k and f such that
16 *			1+x = 2^k * (1+f),
17 *	   where  sqrt(2)/2 < 1+f < sqrt(2) .
18 *
19 *      Note. If k=0, then f=x is exact. However, if k!=0, then f
20 *	may not be representable exactly. In that case, a correction
21 *	term is need. Let u=1+x rounded. Let c = (1+x)-u, then
22 *	log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
23 *	and add back the correction term c/u.
24 *	(Note: when x > 2**53, one can simply return log(x))
25 *
26 *   2. Approximation of log1p(f).
27 *	Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
28 *		 = 2s + 2/3 s**3 + 2/5 s**5 + .....,
29 *	     	 = 2s + s*R
30 *      We use a special Reme algorithm on [0,0.1716] to generate
31 * 	a polynomial of degree 14 to approximate R The maximum error
32 *	of this polynomial approximation is bounded by 2**-58.45. In
33 *	other words,
34 *		        2      4      6      8      10      12      14
35 *	    R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s  +Lp6*s  +Lp7*s
36 *  	(the values of Lp1 to Lp7 are listed in the program)
37 *	and
38 *	    |      2          14          |     -58.45
39 *	    | Lp1*s +...+Lp7*s    -  R(z) | <= 2
40 *	    |                             |
41 *	Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
42 *	In order to guarantee error in log below 1ulp, we compute log
43 *	by
44 *		log1p(f) = f - (hfsq - s*(hfsq+R)).
45 *
46 *	3. Finally, log1p(x) = k*ln2 + log1p(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 *	log1p(x) is NaN with signal if x < -1 (including -INF) ;
54 *	log1p(+INF) is +INF; log1p(-1) is -INF with signal;
55 *	log1p(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 * Note: Assuming log() return accurate answer, the following
68 * 	 algorithm can be used to compute log1p(x) to within a few ULP:
69 *
70 *		u = 1+x;
71 *		if(u==1.0) return x ; else
72 *			   return log(u)*(x/(u-1.0));
73 *
74 *	 See HP-15C Advanced Functions Handbook, p.193.
75 */
76
77#include <float.h>
78
79#include "math.h"
80#include "math_private.h"
81
82static const double
83ln2_hi  =  6.93147180369123816490e-01,	/* 3fe62e42 fee00000 */
84ln2_lo  =  1.90821492927058770002e-10,	/* 3dea39ef 35793c76 */
85two54   =  1.80143985094819840000e+16,  /* 43500000 00000000 */
86Lp1 = 6.666666666666735130e-01,  /* 3FE55555 55555593 */
87Lp2 = 3.999999999940941908e-01,  /* 3FD99999 9997FA04 */
88Lp3 = 2.857142874366239149e-01,  /* 3FD24924 94229359 */
89Lp4 = 2.222219843214978396e-01,  /* 3FCC71C5 1D8E78AF */
90Lp5 = 1.818357216161805012e-01,  /* 3FC74664 96CB03DE */
91Lp6 = 1.531383769920937332e-01,  /* 3FC39A09 D078C69F */
92Lp7 = 1.479819860511658591e-01;  /* 3FC2F112 DF3E5244 */
93
94static const double zero = 0.0;
95static volatile double vzero = 0.0;
96
97double
98log1p(double x)
99{
100	double hfsq,f,c,s,z,R,u;
101	int32_t k,hx,hu,ax;
102
103	GET_HIGH_WORD(hx,x);
104	ax = hx&0x7fffffff;
105
106	k = 1;
107	if (hx < 0x3FDA827A) {			/* 1+x < sqrt(2)+ */
108	    if(ax>=0x3ff00000) {		/* x <= -1.0 */
109		if(x==-1.0) return -two54/vzero; /* log1p(-1)=+inf */
110		else return (x-x)/(x-x);	/* log1p(x<-1)=NaN */
111	    }
112	    if(ax<0x3e200000) {			/* |x| < 2**-29 */
113		if(two54+x>zero			/* raise inexact */
114	            &&ax<0x3c900000) 		/* |x| < 2**-54 */
115		    return x;
116		else
117		    return x - x*x*0.5;
118	    }
119	    if(hx>0||hx<=((int32_t)0xbfd2bec4)) {
120		k=0;f=x;hu=1;}		/* sqrt(2)/2- <= 1+x < sqrt(2)+ */
121	}
122	if (hx >= 0x7ff00000) return x+x;
123	if(k!=0) {
124	    if(hx<0x43400000) {
125		STRICT_ASSIGN(double,u,1.0+x);
126		GET_HIGH_WORD(hu,u);
127	        k  = (hu>>20)-1023;
128	        c  = (k>0)? 1.0-(u-x):x-(u-1.0);/* correction term */
129		c /= u;
130	    } else {
131		u  = x;
132		GET_HIGH_WORD(hu,u);
133	        k  = (hu>>20)-1023;
134		c  = 0;
135	    }
136	    hu &= 0x000fffff;
137	    /*
138	     * The approximation to sqrt(2) used in thresholds is not
139	     * critical.  However, the ones used above must give less
140	     * strict bounds than the one here so that the k==0 case is
141	     * never reached from here, since here we have committed to
142	     * using the correction term but don't use it if k==0.
143	     */
144	    if(hu<0x6a09e) {			/* u ~< sqrt(2) */
145	        SET_HIGH_WORD(u,hu|0x3ff00000);	/* normalize u */
146	    } else {
147	        k += 1;
148		SET_HIGH_WORD(u,hu|0x3fe00000);	/* normalize u/2 */
149	        hu = (0x00100000-hu)>>2;
150	    }
151	    f = u-1.0;
152	}
153	hfsq=0.5*f*f;
154	if(hu==0) {	/* |f| < 2**-20 */
155	    if(f==zero) {
156		if(k==0) {
157		    return zero;
158		} else {
159		    c += k*ln2_lo;
160		    return k*ln2_hi+c;
161		}
162	    }
163	    R = hfsq*(1.0-0.66666666666666666*f);
164	    if(k==0) return f-R; else
165	    	     return k*ln2_hi-((R-(k*ln2_lo+c))-f);
166	}
167 	s = f/(2.0+f);
168	z = s*s;
169	R = z*(Lp1+z*(Lp2+z*(Lp3+z*(Lp4+z*(Lp5+z*(Lp6+z*Lp7))))));
170	if(k==0) return f-(hfsq-s*(hfsq+R)); else
171		 return k*ln2_hi-((hfsq-(s*(hfsq+R)+(k*ln2_lo+c)))-f);
172}
173
174#if (LDBL_MANT_DIG == 53)
175__weak_reference(log1p, log1pl);
176#endif
177