about summary refs log tree commit diff
path: root/src/math/logl.c
blob: ffd8103813eade25cc51b70944b93d78a49b8cf8 (plain) (blame)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
/* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_logl.c */
/*
 * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
 *
 * Permission to use, copy, modify, and distribute this software for any
 * purpose with or without fee is hereby granted, provided that the above
 * copyright notice and this permission notice appear in all copies.
 *
 * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
 * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
 * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
 * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
 * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
 * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
 * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
 */
/*
 *      Natural logarithm, long double precision
 *
 *
 * SYNOPSIS:
 *
 * long double x, y, logl();
 *
 * y = logl( x );
 *
 *
 * DESCRIPTION:
 *
 * Returns the base e (2.718...) logarithm of x.
 *
 * The argument is separated into its exponent and fractional
 * parts.  If the exponent is between -1 and +1, the logarithm
 * of the fraction is approximated by
 *
 *     log(1+x) = x - 0.5 x**2 + x**3 P(x)/Q(x).
 *
 * Otherwise, setting  z = 2(x-1)/x+1),
 *
 *     log(x) = z + z**3 P(z)/Q(z).
 *
 *
 * ACCURACY:
 *
 *                      Relative error:
 * arithmetic   domain     # trials      peak         rms
 *    IEEE      0.5, 2.0    150000      8.71e-20    2.75e-20
 *    IEEE     exp(+-10000) 100000      5.39e-20    2.34e-20
 *
 * In the tests over the interval exp(+-10000), the logarithms
 * of the random arguments were uniformly distributed over
 * [-10000, +10000].
 *
 * ERROR MESSAGES:
 *
 * log singularity:  x = 0; returns -INFINITY
 * log domain:       x < 0; returns NAN
 */

#include "libm.h"

#if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
long double logl(long double x)
{
	return log(x);
}
#elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
/* Coefficients for log(1+x) = x - x**2/2 + x**3 P(x)/Q(x)
 * 1/sqrt(2) <= x < sqrt(2)
 * Theoretical peak relative error = 2.32e-20
 */
static const long double P[] = {
 4.5270000862445199635215E-5L,
 4.9854102823193375972212E-1L,
 6.5787325942061044846969E0L,
 2.9911919328553073277375E1L,
 6.0949667980987787057556E1L,
 5.7112963590585538103336E1L,
 2.0039553499201281259648E1L,
};
static const long double Q[] = {
/* 1.0000000000000000000000E0,*/
 1.5062909083469192043167E1L,
 8.3047565967967209469434E1L,
 2.2176239823732856465394E2L,
 3.0909872225312059774938E2L,
 2.1642788614495947685003E2L,
 6.0118660497603843919306E1L,
};

/* Coefficients for log(x) = z + z^3 P(z^2)/Q(z^2),
 * where z = 2(x-1)/(x+1)
 * 1/sqrt(2) <= x < sqrt(2)
 * Theoretical peak relative error = 6.16e-22
 */
static const long double R[4] = {
 1.9757429581415468984296E-3L,
-7.1990767473014147232598E-1L,
 1.0777257190312272158094E1L,
-3.5717684488096787370998E1L,
};
static const long double S[4] = {
/* 1.00000000000000000000E0L,*/
-2.6201045551331104417768E1L,
 1.9361891836232102174846E2L,
-4.2861221385716144629696E2L,
};
static const long double C1 = 6.9314575195312500000000E-1L;
static const long double C2 = 1.4286068203094172321215E-6L;

#define SQRTH 0.70710678118654752440L

long double logl(long double x)
{
	long double y, z;
	int e;

	if (isnan(x))
		return x;
	if (x == INFINITY)
		return x;
	if (x <= 0.0) {
		if (x == 0.0)
			return -INFINITY;
		return NAN;
	}

	/* separate mantissa from exponent */
	/* Note, frexp is used so that denormal numbers
	 * will be handled properly.
	 */
	x = frexpl(x, &e);

	/* logarithm using log(x) = z + z**3 P(z)/Q(z),
	 * where z = 2(x-1)/x+1)
	 */
	if (e > 2 || e < -2) {
		if (x < SQRTH) {  /* 2(2x-1)/(2x+1) */
			e -= 1;
			z = x - 0.5;
			y = 0.5 * z + 0.5;
		} else {  /*  2 (x-1)/(x+1)   */
			z = x - 0.5;
			z -= 0.5;
			y = 0.5 * x  + 0.5;
		}
		x = z / y;
		z = x*x;
		z = x * (z * __polevll(z, R, 3) / __p1evll(z, S, 3));
		z = z + e * C2;
		z = z + x;
		z = z + e * C1;
		return z;
	}

	/* logarithm using log(1+x) = x - .5x**2 + x**3 P(x)/Q(x) */
	if (x < SQRTH) {
		e -= 1;
		x = 2.0*x - 1.0;
	} else {
		x = x - 1.0;
	}
	z = x*x;
	y = x * (z * __polevll(x, P, 6) / __p1evll(x, Q, 6));
	y = y + e * C2;
	z = y - 0.5*z;
	/* Note, the sum of above terms does not exceed x/4,
	 * so it contributes at most about 1/4 lsb to the error.
	 */
	z = z + x;
	z = z + e * C1; /* This sum has an error of 1/2 lsb. */
	return z;
}
#endif