1 /* $OpenBSD: e_log10l.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 /* log10l.c
20 *
21 * Common logarithm, long double precision
22 *
23 *
24 *
25 * SYNOPSIS:
26 *
27 * long double x, y, log10l();
28 *
29 * y = log10l( x );
30 *
31 *
32 *
33 * DESCRIPTION:
34 *
35 * Returns the base 10 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 logarithm
39 * 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 9.0e-20 2.6e-20
54 * IEEE exp(+-10000) 30000 6.0e-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 MINLOG
63 * log domain: x < 0; returns MINLOG
64 */
65
66 #include <math.h>
67
68 #include "math_private.h"
69
70 /* Coefficients for log(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
100 static long double R[4] = {
101 1.9757429581415468984296E-3L,
102 -7.1990767473014147232598E-1L,
103 1.0777257190312272158094E1L,
104 -3.5717684488096787370998E1L,
105 };
106 static long double S[4] = {
107 /* 1.00000000000000000000E0L,*/
108 -2.6201045551331104417768E1L,
109 1.9361891836232102174846E2L,
110 -4.2861221385716144629696E2L,
111 };
112 /* log10(2) */
113 #define L102A 0.3125L
114 #define L102B -1.1470004336018804786261e-2L
115 /* log10(e) */
116 #define L10EA 0.5L
117 #define L10EB -6.5705518096748172348871e-2L
118
119 #define SQRTH 0.70710678118654752440L
120
121 long double
log10l(long double x)122 log10l(long double x)
123 {
124 long double y;
125 volatile long double z;
126 int e;
127
128 if( isnan(x) )
129 return(x);
130 /* Test for domain */
131 if( x <= 0.0L )
132 {
133 if( x == 0.0L )
134 return (-1.0L / (x - x));
135 else
136 return (x - x) / (x - x);
137 }
138 if( x == INFINITY )
139 return(INFINITY);
140 /* separate mantissa from exponent */
141
142 /* Note, frexp is used so that denormal numbers
143 * will be handled properly.
144 */
145 x = frexpl( x, &e );
146
147
148 /* logarithm using log(x) = z + z**3 P(z)/Q(z),
149 * where z = 2(x-1)/x+1)
150 */
151 if( (e > 2) || (e < -2) )
152 {
153 if( x < SQRTH )
154 { /* 2( 2x-1 )/( 2x+1 ) */
155 e -= 1;
156 z = x - 0.5L;
157 y = 0.5L * z + 0.5L;
158 }
159 else
160 { /* 2 (x-1)/(x+1) */
161 z = x - 0.5L;
162 z -= 0.5L;
163 y = 0.5L * x + 0.5L;
164 }
165 x = z / y;
166 z = x*x;
167 y = x * ( z * __polevll( z, R, 3 ) / __p1evll( z, S, 3 ) );
168 goto done;
169 }
170
171
172 /* logarithm using log(1+x) = x - .5x**2 + x**3 P(x)/Q(x) */
173
174 if( x < SQRTH )
175 {
176 e -= 1;
177 x = ldexpl( x, 1 ) - 1.0L; /* 2x - 1 */
178 }
179 else
180 {
181 x = x - 1.0L;
182 }
183 z = x*x;
184 y = x * ( z * __polevll( x, P, 6 ) / __p1evll( x, Q, 7 ) );
185 y = y - ldexpl( z, -1 ); /* -0.5x^2 + ... */
186
187 done:
188
189 /* Multiply log of fraction by log10(e)
190 * and base 2 exponent by log10(2).
191 *
192 * ***CAUTION***
193 *
194 * This sequence of operations is critical and it may
195 * be horribly defeated by some compiler optimizers.
196 */
197 z = y * (L10EB);
198 z += x * (L10EB);
199 z += e * (L102B);
200 z += y * (L10EA);
201 z += x * (L10EA);
202 z += e * (L102A);
203
204 return( z );
205 }
206