1/* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_logl.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 *      Natural logarithm, long double precision
19 *
20 *
21 * SYNOPSIS:
22 *
23 * long double x, y, logl();
24 *
25 * y = logl( x );
26 *
27 *
28 * DESCRIPTION:
29 *
30 * Returns the base e (2.718...) 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) = log(1+z/2) - log(1-z/2) = 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    150000      8.71e-20    2.75e-20
48 *    IEEE     exp(+-10000) 100000      5.39e-20    2.34e-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
55#include "libm.h"
56
57#if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
58long double logl(long double x)
59{
60	return log(x);
61}
62#elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
63/* Coefficients for log(1+x) = x - x**2/2 + x**3 P(x)/Q(x)
64 * 1/sqrt(2) <= x < sqrt(2)
65 * Theoretical peak relative error = 2.32e-20
66 */
67static const long double P[] = {
68 4.5270000862445199635215E-5L,
69 4.9854102823193375972212E-1L,
70 6.5787325942061044846969E0L,
71 2.9911919328553073277375E1L,
72 6.0949667980987787057556E1L,
73 5.7112963590585538103336E1L,
74 2.0039553499201281259648E1L,
75};
76static const long double Q[] = {
77/* 1.0000000000000000000000E0,*/
78 1.5062909083469192043167E1L,
79 8.3047565967967209469434E1L,
80 2.2176239823732856465394E2L,
81 3.0909872225312059774938E2L,
82 2.1642788614495947685003E2L,
83 6.0118660497603843919306E1L,
84};
85
86/* Coefficients for log(x) = z + z^3 P(z^2)/Q(z^2),
87 * where z = 2(x-1)/(x+1)
88 * 1/sqrt(2) <= x < sqrt(2)
89 * Theoretical peak relative error = 6.16e-22
90 */
91static const long double R[4] = {
92 1.9757429581415468984296E-3L,
93-7.1990767473014147232598E-1L,
94 1.0777257190312272158094E1L,
95-3.5717684488096787370998E1L,
96};
97static const long double S[4] = {
98/* 1.00000000000000000000E0L,*/
99-2.6201045551331104417768E1L,
100 1.9361891836232102174846E2L,
101-4.2861221385716144629696E2L,
102};
103static const long double C1 = 6.9314575195312500000000E-1L;
104static const long double C2 = 1.4286068203094172321215E-6L;
105
106#define SQRTH 0.70710678118654752440L
107
108long double logl(long double x)
109{
110	long double y, z;
111	int e;
112
113	if (isnan(x))
114		return x;
115	if (x == INFINITY)
116		return x;
117	if (x <= 0.0) {
118		if (x == 0.0)
119			return -1/(x*x); /* -inf with divbyzero */
120		return 0/0.0f; /* nan with invalid */
121	}
122
123	/* separate mantissa from exponent */
124	/* Note, frexp is used so that denormal numbers
125	 * will be handled properly.
126	 */
127	x = frexpl(x, &e);
128
129	/* logarithm using log(x) = z + z**3 P(z)/Q(z),
130	 * where z = 2(x-1)/(x+1)
131	 */
132	if (e > 2 || e < -2) {
133		if (x < SQRTH) {  /* 2(2x-1)/(2x+1) */
134			e -= 1;
135			z = x - 0.5;
136			y = 0.5 * z + 0.5;
137		} else {  /*  2 (x-1)/(x+1)   */
138			z = x - 0.5;
139			z -= 0.5;
140			y = 0.5 * x  + 0.5;
141		}
142		x = z / y;
143		z = x*x;
144		z = x * (z * __polevll(z, R, 3) / __p1evll(z, S, 3));
145		z = z + e * C2;
146		z = z + x;
147		z = z + e * C1;
148		return z;
149	}
150
151	/* logarithm using log(1+x) = x - .5x**2 + x**3 P(x)/Q(x) */
152	if (x < SQRTH) {
153		e -= 1;
154		x = 2.0*x - 1.0;
155	} else {
156		x = x - 1.0;
157	}
158	z = x*x;
159	y = x * (z * __polevll(x, P, 6) / __p1evll(x, Q, 6));
160	y = y + e * C2;
161	z = y - 0.5*z;
162	/* Note, the sum of above terms does not exceed x/4,
163	 * so it contributes at most about 1/4 lsb to the error.
164	 */
165	z = z + x;
166	z = z + e * C1; /* This sum has an error of 1/2 lsb. */
167	return z;
168}
169#elif LDBL_MANT_DIG == 113 && LDBL_MAX_EXP == 16384
170// TODO: broken implementation to make things compile
171long double logl(long double x)
172{
173	return log(x);
174}
175#endif
176