1 /*	$OpenBSD: e_log2l.c,v 1.2 2013/11/12 20:35:19 martynas Exp $	*/
2 
3 /*
4  * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
5  *
6  * Permission to use, copy, modify, and distribute this software for any
7  * purpose with or without fee is hereby granted, provided that the above
8  * copyright notice and this permission notice appear in all copies.
9  *
10  * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
11  * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
12  * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
13  * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
14  * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
15  * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
16  * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
17  */
18 
19 /*							log2l.c
20  *
21  *	Base 2 logarithm, long double precision
22  *
23  *
24  *
25  * SYNOPSIS:
26  *
27  * long double x, y, log2l();
28  *
29  * y = log2l( x );
30  *
31  *
32  *
33  * DESCRIPTION:
34  *
35  * Returns the base 2 logarithm of x.
36  *
37  * The argument is separated into its exponent and fractional
38  * parts.  If the exponent is between -1 and +1, the (natural)
39  * logarithm of the fraction is approximated by
40  *
41  *     log(1+x) = x - 0.5 x**2 + x**3 P(x)/Q(x).
42  *
43  * Otherwise, setting  z = 2(x-1)/x+1),
44  *
45  *     log(x) = z + z**3 P(z)/Q(z).
46  *
47  *
48  *
49  * ACCURACY:
50  *
51  *                      Relative error:
52  * arithmetic   domain     # trials      peak         rms
53  *    IEEE      0.5, 2.0     30000      9.8e-20     2.7e-20
54  *    IEEE     exp(+-10000)  70000      5.4e-20     2.3e-20
55  *
56  * In the tests over the interval exp(+-10000), the logarithms
57  * of the random arguments were uniformly distributed over
58  * [-10000, +10000].
59  *
60  * ERROR MESSAGES:
61  *
62  * log singularity:  x = 0; returns -INFINITY
63  * log domain:       x < 0; returns NAN
64  */
65 
66 #include <math.h>
67 
68 #include "math_private.h"
69 
70 /* Coefficients for ln(1+x) = x - x**2/2 + x**3 P(x)/Q(x)
71  * 1/sqrt(2) <= x < sqrt(2)
72  * Theoretical peak relative error = 6.2e-22
73  */
74 static long double P[] = {
75  4.9962495940332550844739E-1L,
76  1.0767376367209449010438E1L,
77  7.7671073698359539859595E1L,
78  2.5620629828144409632571E2L,
79  4.2401812743503691187826E2L,
80  3.4258224542413922935104E2L,
81  1.0747524399916215149070E2L,
82 };
83 static long double Q[] = {
84 /* 1.0000000000000000000000E0,*/
85  2.3479774160285863271658E1L,
86  1.9444210022760132894510E2L,
87  7.7952888181207260646090E2L,
88  1.6911722418503949084863E3L,
89  2.0307734695595183428202E3L,
90  1.2695660352705325274404E3L,
91  3.2242573199748645407652E2L,
92 };
93 
94 /* Coefficients for log(x) = z + z^3 P(z^2)/Q(z^2),
95  * where z = 2(x-1)/(x+1)
96  * 1/sqrt(2) <= x < sqrt(2)
97  * Theoretical peak relative error = 6.16e-22
98  */
99 static long double R[4] = {
100  1.9757429581415468984296E-3L,
101 -7.1990767473014147232598E-1L,
102  1.0777257190312272158094E1L,
103 -3.5717684488096787370998E1L,
104 };
105 static long double S[4] = {
106 /* 1.00000000000000000000E0L,*/
107 -2.6201045551331104417768E1L,
108  1.9361891836232102174846E2L,
109 -4.2861221385716144629696E2L,
110 };
111 /* log2(e) - 1 */
112 #define LOG2EA 4.4269504088896340735992e-1L
113 
114 #define SQRTH 0.70710678118654752440L
115 
116 long double
117 log2l(long double x)
118 {
119 volatile long double z;
120 long double y;
121 int e;
122 
123 if( isnan(x) )
124 	return(x);
125 if( x == INFINITY )
126 	return(x);
127 /* Test for domain */
128 if( x <= 0.0L )
129 	{
130 	if( x == 0.0L )
131 		return( -INFINITY );
132 	else
133 		return( NAN );
134 	}
135 
136 /* separate mantissa from exponent */
137 
138 /* Note, frexp is used so that denormal numbers
139  * will be handled properly.
140  */
141 x = frexpl( x, &e );
142 
143 
144 /* logarithm using log(x) = z + z**3 P(z)/Q(z),
145  * where z = 2(x-1)/x+1)
146  */
147 if( (e > 2) || (e < -2) )
148 {
149 if( x < SQRTH )
150 	{ /* 2( 2x-1 )/( 2x+1 ) */
151 	e -= 1;
152 	z = x - 0.5L;
153 	y = 0.5L * z + 0.5L;
154 	}
155 else
156 	{ /*  2 (x-1)/(x+1)   */
157 	z = x - 0.5L;
158 	z -= 0.5L;
159 	y = 0.5L * x  + 0.5L;
160 	}
161 x = z / y;
162 z = x*x;
163 y = x * ( z * __polevll( z, R, 3 ) / __p1evll( z, S, 3 ) );
164 goto done;
165 }
166 
167 
168 /* logarithm using log(1+x) = x - .5x**2 + x**3 P(x)/Q(x) */
169 
170 if( x < SQRTH )
171 	{
172 	e -= 1;
173 	x = ldexpl( x, 1 ) - 1.0L; /*  2x - 1  */
174 	}
175 else
176 	{
177 	x = x - 1.0L;
178 	}
179 z = x*x;
180 y = x * ( z * __polevll( x, P, 6 ) / __p1evll( x, Q, 7 ) );
181 y = y - ldexpl( z, -1 );   /* -0.5x^2 + ... */
182 
183 done:
184 
185 /* Multiply log of fraction by log2(e)
186  * and base 2 exponent by 1
187  *
188  * ***CAUTION***
189  *
190  * This sequence of operations is critical and it may
191  * be horribly defeated by some compiler optimizers.
192  */
193 z = y * LOG2EA;
194 z += x * LOG2EA;
195 z += y;
196 z += x;
197 z += e;
198 return( z );
199 }
200