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 
86 static const double
87 ln2_hi  =  6.93147180369123816490e-01,	/* 3fe62e42 fee00000 */
88 ln2_lo  =  1.90821492927058770002e-10,	/* 3dea39ef 35793c76 */
89 two54   =  1.80143985094819840000e+16,  /* 43500000 00000000 */
90 Lp1 = 6.666666666666735130e-01,  /* 3FE55555 55555593 */
91 Lp2 = 3.999999999940941908e-01,  /* 3FD99999 9997FA04 */
92 Lp3 = 2.857142874366239149e-01,  /* 3FD24924 94229359 */
93 Lp4 = 2.222219843214978396e-01,  /* 3FCC71C5 1D8E78AF */
94 Lp5 = 1.818357216161805012e-01,  /* 3FC74664 96CB03DE */
95 Lp6 = 1.531383769920937332e-01,  /* 3FC39A09 D078C69F */
96 Lp7 = 1.479819860511658591e-01;  /* 3FC2F112 DF3E5244 */
97 
98 static const double zero = 0.0;
99 static volatile double vzero = 0.0;
100 
101 double
log1p(double x)102 log1p(double x)
103 {
104 	double hfsq,f,c,s,z,R,u;
105 	int32_t k,hx,hu,ax;
106 
107 	GET_HIGH_WORD(hx,x);
108 	ax = hx&0x7fffffff;
109 
110 	k = 1;
111 	if (hx < 0x3FDA827A) {			/* 1+x < sqrt(2)+ */
112 	    if(ax>=0x3ff00000) {		/* x <= -1.0 */
113 		if(x==-1.0) return -two54/vzero; /* log1p(-1)=+inf */
114 		else return (x-x)/(x-x);	/* log1p(x<-1)=NaN */
115 	    }
116 	    if(ax<0x3e200000) {			/* |x| < 2**-29 */
117 		if(two54+x>zero			/* raise inexact */
118 	            &&ax<0x3c900000) 		/* |x| < 2**-54 */
119 		    return x;
120 		else
121 		    return x - x*x*0.5;
122 	    }
123 	    if(hx>0||hx<=((int32_t)0xbfd2bec4)) {
124 		k=0;f=x;hu=1;}		/* sqrt(2)/2- <= 1+x < sqrt(2)+ */
125 	}
126 	if (hx >= 0x7ff00000) return x+x;
127 	if(k!=0) {
128 	    if(hx<0x43400000) {
129 		STRICT_ASSIGN(double,u,1.0+x);
130 		GET_HIGH_WORD(hu,u);
131 	        k  = (hu>>20)-1023;
132 	        c  = (k>0)? 1.0-(u-x):x-(u-1.0);/* correction term */
133 		c /= u;
134 	    } else {
135 		u  = x;
136 		GET_HIGH_WORD(hu,u);
137 	        k  = (hu>>20)-1023;
138 		c  = 0;
139 	    }
140 	    hu &= 0x000fffff;
141 	    /*
142 	     * The approximation to sqrt(2) used in thresholds is not
143 	     * critical.  However, the ones used above must give less
144 	     * strict bounds than the one here so that the k==0 case is
145 	     * never reached from here, since here we have committed to
146 	     * using the correction term but don't use it if k==0.
147 	     */
148 	    if(hu<0x6a09e) {			/* u ~< sqrt(2) */
149 	        SET_HIGH_WORD(u,hu|0x3ff00000);	/* normalize u */
150 	    } else {
151 	        k += 1;
152 		SET_HIGH_WORD(u,hu|0x3fe00000);	/* normalize u/2 */
153 	        hu = (0x00100000-hu)>>2;
154 	    }
155 	    f = u-1.0;
156 	}
157 	hfsq=0.5*f*f;
158 	if(hu==0) {	/* |f| < 2**-20 */
159 	    if(f==zero) {
160 		if(k==0) {
161 		    return zero;
162 		} else {
163 		    c += k*ln2_lo;
164 		    return k*ln2_hi+c;
165 		}
166 	    }
167 	    R = hfsq*(1.0-0.66666666666666666*f);
168 	    if(k==0) return f-R; else
169 	    	     return k*ln2_hi-((R-(k*ln2_lo+c))-f);
170 	}
171  	s = f/(2.0+f);
172 	z = s*s;
173 	R = z*(Lp1+z*(Lp2+z*(Lp3+z*(Lp4+z*(Lp5+z*(Lp6+z*Lp7))))));
174 	if(k==0) return f-(hfsq-s*(hfsq+R)); else
175 		 return k*ln2_hi-((hfsq-(s*(hfsq+R)+(k*ln2_lo+c)))-f);
176 }
177 
178 #if (LDBL_MANT_DIG == 53)
179 __weak_reference(log1p, log1pl);
180 #endif
181