1/* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_log10l.c */
2/*
3 * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
4 *
5 * Permission to use, copy, modify, and distribute this software for any
6 * purpose with or without fee is hereby granted, provided that the above
7 * copyright notice and this permission notice appear in all copies.
8 *
9 * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
10 * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
11 * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
12 * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
13 * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
14 * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
15 * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
16 */
17/*
18 * Common logarithm, long double precision
19 *
20 *
21 * SYNOPSIS:
22 *
23 * long double x, y, log10l();
24 *
25 * y = log10l( x );
26 *
27 *
28 * DESCRIPTION:
29 *
30 * Returns the base 10 logarithm of x.
31 *
32 * The argument is separated into its exponent and fractional
33 * parts. If the exponent is between -1 and +1, the logarithm
34 * of the fraction is approximated by
35 *
36 * log(1+x) = x - 0.5 x**2 + x**3 P(x)/Q(x).
37 *
38 * Otherwise, setting z = 2(x-1)/x+1),
39 *
40 * log(x) = z + z**3 P(z)/Q(z).
41 *
42 *
43 * ACCURACY:
44 *
45 * Relative error:
46 * arithmetic domain # trials peak rms
47 * IEEE 0.5, 2.0 30000 9.0e-20 2.6e-20
48 * IEEE exp(+-10000) 30000 6.0e-20 2.3e-20
49 *
50 * In the tests over the interval exp(+-10000), the logarithms
51 * of the random arguments were uniformly distributed over
52 * [-10000, +10000].
53 *
54 * ERROR MESSAGES:
55 *
56 * log singularity: x = 0; returns MINLOG
57 * log domain: x < 0; returns MINLOG
58 */
59
60#include "libm.h"
61
62#if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
63long double log10l(long double x)
64{
65 return log10(x);
66}
67#elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
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 */
72static const 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};
81static const 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 */
97static const long double R[4] = {
98 1.9757429581415468984296E-3L,
99-7.1990767473014147232598E-1L,
100 1.0777257190312272158094E1L,
101-3.5717684488096787370998E1L,
102};
103static const long double S[4] = {
104/* 1.00000000000000000000E0L,*/
105-2.6201045551331104417768E1L,
106 1.9361891836232102174846E2L,
107-4.2861221385716144629696E2L,
108};
109/* log10(2) */
110#define L102A 0.3125L
111#define L102B -1.1470004336018804786261e-2L
112/* log10(e) */
113#define L10EA 0.5L
114#define L10EB -6.5705518096748172348871e-2L
115
116#define SQRTH 0.70710678118654752440L
117
118long double log10l(long double x)
119{
120 long double y, z;
121 int e;
122
123 if (isnan(x))
124 return x;
125 if(x <= 0.0) {
126 if(x == 0.0)
127 return -1.0 / (x*x);
128 return (x - x) / 0.0;
129 }
130 if (x == INFINITY)
131 return INFINITY;
132 /* separate mantissa from exponent */
133 /* Note, frexp is used so that denormal numbers
134 * will be handled properly.
135 */
136 x = frexpl(x, &e);
137
138 /* logarithm using log(x) = z + z**3 P(z)/Q(z),
139 * where z = 2(x-1)/x+1)
140 */
141 if (e > 2 || e < -2) {
142 if (x < SQRTH) { /* 2(2x-1)/(2x+1) */
143 e -= 1;
144 z = x - 0.5;
145 y = 0.5 * z + 0.5;
146 } else { /* 2 (x-1)/(x+1) */
147 z = x - 0.5;
148 z -= 0.5;
149 y = 0.5 * x + 0.5;
150 }
151 x = z / y;
152 z = x*x;
153 y = x * (z * __polevll(z, R, 3) / __p1evll(z, S, 3));
154 goto done;
155 }
156
157 /* logarithm using log(1+x) = x - .5x**2 + x**3 P(x)/Q(x) */
158 if (x < SQRTH) {
159 e -= 1;
160 x = 2.0*x - 1.0;
161 } else {
162 x = x - 1.0;
163 }
164 z = x*x;
165 y = x * (z * __polevll(x, P, 6) / __p1evll(x, Q, 7));
166 y = y - 0.5*z;
167
168done:
169 /* Multiply log of fraction by log10(e)
170 * and base 2 exponent by log10(2).
171 *
172 * ***CAUTION***
173 *
174 * This sequence of operations is critical and it may
175 * be horribly defeated by some compiler optimizers.
176 */
177 z = y * (L10EB);
178 z += x * (L10EB);
179 z += e * (L102B);
180 z += y * (L10EA);
181 z += x * (L10EA);
182 z += e * (L102A);
183 return z;
184}
185#elif LDBL_MANT_DIG == 113 && LDBL_MAX_EXP == 16384
186// TODO: broken implementation to make things compile
187long double log10l(long double x)
188{
189 return log10(x);
190}
191#endif