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 /* double log1p(double x)
14  *
15  * Method :
16  *   1. Argument Reduction: find k and f such that
17  *			1+x = 2^k * (1+f),
18  *	   where  sqrt(2)/2 < 1+f < sqrt(2) .
19  *
20  *      Note. If k=0, then f=x is exact. However, if k!=0, then f
21  *	may not be representable exactly. In that case, a correction
22  *	term is need. Let u=1+x rounded. Let c = (1+x)-u, then
23  *	log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
24  *	and add back the correction term c/u.
25  *	(Note: when x > 2**53, one can simply return log(x))
26  *
27  *   2. Approximation of log1p(f).
28  *	Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
29  *		 = 2s + 2/3 s**3 + 2/5 s**5 + .....,
30  *	     	 = 2s + s*R
31  *      We use a special Remes algorithm on [0,0.1716] to generate
32  * 	a polynomial of degree 14 to approximate R The maximum error
33  *	of this polynomial approximation is bounded by 2**-58.45. In
34  *	other words,
35  *		        2      4      6      8      10      12      14
36  *	    R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s  +Lp6*s  +Lp7*s
37  *  	(the values of Lp1 to Lp7 are listed in the program)
38  *	and
39  *	    |      2          14          |     -58.45
40  *	    | Lp1*s +...+Lp7*s    -  R(z) | <= 2
41  *	    |                             |
42  *	Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
43  *	In order to guarantee error in log below 1ulp, we compute log
44  *	by
45  *		log1p(f) = f - (hfsq - s*(hfsq+R)).
46  *
47  *	3. Finally, log1p(x) = k*ln2 + log1p(f).
48  *		 	     = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
49  *	   Here ln2 is split into two floating point number:
50  *			ln2_hi + ln2_lo,
51  *	   where n*ln2_hi is always exact for |n| < 2000.
52  *
53  * Special cases:
54  *	log1p(x) is NaN with signal if x < -1 (including -INF) ;
55  *	log1p(+INF) is +INF; log1p(-1) is -INF with signal;
56  *	log1p(NaN) is that NaN with no signal.
57  *
58  * Accuracy:
59  *	according to an error analysis, the error is always less than
60  *	1 ulp (unit in the last place).
61  *
62  * Constants:
63  * The hexadecimal values are the intended ones for the following
64  * constants. The decimal values may be used, provided that the
65  * compiler will convert from decimal to binary accurately enough
66  * to produce the hexadecimal values shown.
67  *
68  * Note: Assuming log() return accurate answer, the following
69  * 	 algorithm can be used to compute log1p(x) to within a few ULP:
70  *
71  *		u = 1+x;
72  *		if(u==1.0) return x ; else
73  *			   return log(u)*(x/(u-1.0));
74  *
75  *	 See HP-15C Advanced Functions Handbook, p.193.
76  */
77 
78 #include <float.h>
79 #include <math.h>
80 
81 #include "math_private.h"
82 
83 static const double
84 ln2_hi  =  6.93147180369123816490e-01,	/* 3fe62e42 fee00000 */
85 ln2_lo  =  1.90821492927058770002e-10,	/* 3dea39ef 35793c76 */
86 two54   =  1.80143985094819840000e+16,  /* 43500000 00000000 */
87 Lp1 = 6.666666666666735130e-01,  /* 3FE55555 55555593 */
88 Lp2 = 3.999999999940941908e-01,  /* 3FD99999 9997FA04 */
89 Lp3 = 2.857142874366239149e-01,  /* 3FD24924 94229359 */
90 Lp4 = 2.222219843214978396e-01,  /* 3FCC71C5 1D8E78AF */
91 Lp5 = 1.818357216161805012e-01,  /* 3FC74664 96CB03DE */
92 Lp6 = 1.531383769920937332e-01,  /* 3FC39A09 D078C69F */
93 Lp7 = 1.479819860511658591e-01;  /* 3FC2F112 DF3E5244 */
94 
95 static const double zero = 0.0;
96 
97 double
98 log1p(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) {			/* x < 0.41422  */
108 	    if(ax>=0x3ff00000) {		/* x <= -1.0 */
109 		if(x==-1.0) return -two54/zero; /* 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)0xbfd2bec3)) {
120 		k=0;f=x;hu=1;}	/* -0.2929<x<0.41422 */
121 	}
122 	if (hx >= 0x7ff00000) return x+x;
123 	if(k!=0) {
124 	    if(hx<0x43400000) {
125 		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 	    if(hu<0x6a09e) {
138 	        SET_HIGH_WORD(u,hu|0x3ff00000);	/* normalize u */
139 	    } else {
140 	        k += 1;
141 		SET_HIGH_WORD(u,hu|0x3fe00000);	/* normalize u/2 */
142 	        hu = (0x00100000-hu)>>2;
143 	    }
144 	    f = u-1.0;
145 	}
146 	hfsq=0.5*f*f;
147 	if(hu==0) {	/* |f| < 2**-20 */
148 	    if(f==zero) if(k==0) return zero;
149 			else {c += k*ln2_lo; return k*ln2_hi+c;}
150 	    R = hfsq*(1.0-0.66666666666666666*f);
151 	    if(k==0) return f-R; else
152 	    	     return k*ln2_hi-((R-(k*ln2_lo+c))-f);
153 	}
154  	s = f/(2.0+f);
155 	z = s*s;
156 	R = z*(Lp1+z*(Lp2+z*(Lp3+z*(Lp4+z*(Lp5+z*(Lp6+z*Lp7))))));
157 	if(k==0) return f-(hfsq-s*(hfsq+R)); else
158 		 return k*ln2_hi-((hfsq-(s*(hfsq+R)+(k*ln2_lo+c)))-f);
159 }
160 
161 #if	LDBL_MANT_DIG == DBL_MANT_DIG
162 __strong_alias(log1pl, log1p);
163 #endif	/* LDBL_MANT_DIG == DBL_MANT_DIG */
164