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
67
68 /* Coefficients for ln(1+x) = x - x**2/2 + x**3 P(x)/Q(x)
69 * 1/sqrt(2) <= x < sqrt(2)
70 * Theoretical peak relative error = 6.2e-22
71 */
72 static long double P[] = {
73 4.9962495940332550844739E-1L,
74 1.0767376367209449010438E1L,
75 7.7671073698359539859595E1L,
76 2.5620629828144409632571E2L,
77 4.2401812743503691187826E2L,
78 3.4258224542413922935104E2L,
79 1.0747524399916215149070E2L,
80 };
81 static long double Q[] = {
82 /* 1.0000000000000000000000E0,*/
83 2.3479774160285863271658E1L,
84 1.9444210022760132894510E2L,
85 7.7952888181207260646090E2L,
86 1.6911722418503949084863E3L,
87 2.0307734695595183428202E3L,
88 1.2695660352705325274404E3L,
89 3.2242573199748645407652E2L,
90 };
91
92 /* Coefficients for log(x) = z + z^3 P(z^2)/Q(z^2),
93 * where z = 2(x-1)/(x+1)
94 * 1/sqrt(2) <= x < sqrt(2)
95 * Theoretical peak relative error = 6.16e-22
96 */
97 static long double R[4] = {
98 1.9757429581415468984296E-3L,
99 -7.1990767473014147232598E-1L,
100 1.0777257190312272158094E1L,
101 -3.5717684488096787370998E1L,
102 };
103 static long double S[4] = {
104 /* 1.00000000000000000000E0L,*/
105 -2.6201045551331104417768E1L,
106 1.9361891836232102174846E2L,
107 -4.2861221385716144629696E2L,
108 };
109 /* log2(e) - 1 */
110 #define LOG2EA 4.4269504088896340735992e-1L
111
112 #define SQRTH 0.70710678118654752440L
113
114 long double
log2l(long double x)115 log2l(long double x)
116 {
117 volatile long double z;
118 long double y;
119 int e;
120
121 if( isnan(x) )
122 return(x + x);
123 /* Test for domain */
124 if( x <= 0.0L )
125 {
126 if( x == 0.0L )
127 return __math_divzerol(1);
128 else
129 return __math_invalidl(x);
130 }
131
132 if( isinf(x) )
133 return(x);
134
135 /* separate mantissa from exponent */
136
137 /* Note, frexp is used so that denormal numbers
138 * will be handled properly.
139 */
140 x = frexpl( x, &e );
141
142
143 /* logarithm using log(x) = z + z**3 P(z)/Q(z),
144 * where z = 2(x-1)/x+1)
145 */
146 if( (e > 2) || (e < -2) )
147 {
148 if( x < SQRTH )
149 { /* 2( 2x-1 )/( 2x+1 ) */
150 e -= 1;
151 z = x - 0.5L;
152 y = 0.5L * z + 0.5L;
153 }
154 else
155 { /* 2 (x-1)/(x+1) */
156 z = x - 0.5L;
157 z -= 0.5L;
158 y = 0.5L * x + 0.5L;
159 }
160 x = z / y;
161 z = x*x;
162 y = x * ( z * __polevll( z, R, 3 ) / __p1evll( z, S, 3 ) );
163 goto done;
164 }
165
166
167 /* logarithm using log(1+x) = x - .5x**2 + x**3 P(x)/Q(x) */
168
169 if( x < SQRTH )
170 {
171 e -= 1;
172 x = ldexpl( x, 1 ) - 1.0L; /* 2x - 1 */
173 }
174 else
175 {
176 x = x - 1.0L;
177 }
178 z = x*x;
179 y = x * ( z * __polevll( x, P, 6 ) / __p1evll( x, Q, 7 ) );
180 y = y - ldexpl( z, -1 ); /* -0.5x^2 + ... */
181
182 done:
183
184 /* Multiply log of fraction by log2(e)
185 * and base 2 exponent by 1
186 *
187 * ***CAUTION***
188 *
189 * This sequence of operations is critical and it may
190 * be horribly defeated by some compiler optimizers.
191 */
192 z = y * LOG2EA;
193 z += x * LOG2EA;
194 z += y;
195 z += x;
196 z += e;
197 return( z );
198 }
199
