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
67
68 /* Coefficients for log(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
98 static long double R[4] = {
99 1.9757429581415468984296E-3L,
100 -7.1990767473014147232598E-1L,
101 1.0777257190312272158094E1L,
102 -3.5717684488096787370998E1L,
103 };
104 static long double S[4] = {
105 /* 1.00000000000000000000E0L,*/
106 -2.6201045551331104417768E1L,
107 1.9361891836232102174846E2L,
108 -4.2861221385716144629696E2L,
109 };
110 /* log10(2) */
111 #define L102A 0.3125L
112 #define L102B -1.1470004336018804786261e-2L
113 /* log10(e) */
114 #define L10EA 0.5L
115 #define L10EB -6.5705518096748172348871e-2L
116
117 #define SQRTH 0.70710678118654752440L
118
119 long double
log10l(long double x)120 log10l(long double x)
121 {
122 long double y;
123 volatile long double z;
124 int e;
125
126 if( isnan(x) )
127 return(x + x);
128 /* Test for domain */
129 if( x <= 0.0L )
130 {
131 if( x == 0.0L )
132 return __math_divzerol(1);
133 else
134 return __math_invalidl(x);
135 }
136 if( x == (long double) INFINITY )
137 return((long double) INFINITY);
138 /* separate mantissa from exponent */
139
140 /* Note, frexp is used so that denormal numbers
141 * will be handled properly.
142 */
143 x = frexpl( x, &e );
144
145
146 /* logarithm using log(x) = z + z**3 P(z)/Q(z),
147 * where z = 2(x-1)/x+1)
148 */
149 if( (e > 2) || (e < -2) )
150 {
151 if( x < SQRTH )
152 { /* 2( 2x-1 )/( 2x+1 ) */
153 e -= 1;
154 z = x - 0.5L;
155 y = 0.5L * z + 0.5L;
156 }
157 else
158 { /* 2 (x-1)/(x+1) */
159 z = x - 0.5L;
160 z -= 0.5L;
161 y = 0.5L * x + 0.5L;
162 }
163 x = z / y;
164 z = x*x;
165 y = x * ( z * __polevll( z, R, 3 ) / __p1evll( z, S, 3 ) );
166 goto done;
167 }
168
169
170 /* logarithm using log(1+x) = x - .5x**2 + x**3 P(x)/Q(x) */
171
172 if( x < SQRTH )
173 {
174 e -= 1;
175 x = ldexpl( x, 1 ) - 1.0L; /* 2x - 1 */
176 }
177 else
178 {
179 x = x - 1.0L;
180 }
181 z = x*x;
182 y = x * ( z * __polevll( x, P, 6 ) / __p1evll( x, Q, 7 ) );
183 y = y - ldexpl( z, -1 ); /* -0.5x^2 + ... */
184
185 done:
186
187 /* Multiply log of fraction by log10(e)
188 * and base 2 exponent by log10(2).
189 *
190 * ***CAUTION***
191 *
192 * This sequence of operations is critical and it may
193 * be horribly defeated by some compiler optimizers.
194 */
195 z = y * (L10EB);
196 z += x * (L10EB);
197 z += e * (L102B);
198 z += y * (L10EA);
199 z += x * (L10EA);
200 z += e * (L102A);
201
202 return( z );
203 }
204
