1 /*	$OpenBSD: e_logl.c,v 1.3 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 /*							logl.c
20  *
21  *	Natural logarithm, long double precision
22  *
23  *
24  *
25  * SYNOPSIS:
26  *
27  * long double x, y, logl();
28  *
29  * y = logl( x );
30  *
31  *
32  *
33  * DESCRIPTION:
34  *
35  * Returns the base e (2.718...) 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    150000      8.71e-20    2.75e-20
54  *    IEEE     exp(+-10000) 100000      5.39e-20    2.34e-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 -INFINITY
63  * log domain:       x < 0; returns NAN
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 = 2.32e-20
73  */
74 static long double P[] = {
75  4.5270000862445199635215E-5L,
76  4.9854102823193375972212E-1L,
77  6.5787325942061044846969E0L,
78  2.9911919328553073277375E1L,
79  6.0949667980987787057556E1L,
80  5.7112963590585538103336E1L,
81  2.0039553499201281259648E1L,
82 };
83 static long double Q[] = {
84 /* 1.0000000000000000000000E0,*/
85  1.5062909083469192043167E1L,
86  8.3047565967967209469434E1L,
87  2.2176239823732856465394E2L,
88  3.0909872225312059774938E2L,
89  2.1642788614495947685003E2L,
90  6.0118660497603843919306E1L,
91 };
92 
93 /* Coefficients for log(x) = z + z^3 P(z^2)/Q(z^2),
94  * where z = 2(x-1)/(x+1)
95  * 1/sqrt(2) <= x < sqrt(2)
96  * Theoretical peak relative error = 6.16e-22
97  */
98 
99 static long double R[4] = {
100  1.9757429581415468984296E-3L,
101 -7.1990767473014147232598E-1L,
102  1.0777257190312272158094E1L,
103 -3.5717684488096787370998E1L,
104 };
105 static long double S[4] = {
106 /* 1.00000000000000000000E0L,*/
107 -2.6201045551331104417768E1L,
108  1.9361891836232102174846E2L,
109 -4.2861221385716144629696E2L,
110 };
111 static const long double C1 = 6.9314575195312500000000E-1L;
112 static const long double C2 = 1.4286068203094172321215E-6L;
113 
114 #define SQRTH 0.70710678118654752440L
115 
116 long double
117 logl(long double x)
118 {
119 long double y, z;
120 int e;
121 
122 if( isnan(x) )
123 	return(x);
124 if( x == INFINITY )
125 	return(x);
126 /* Test for domain */
127 if( x <= 0.0L )
128 	{
129 	if( x == 0.0L )
130 		return( -INFINITY );
131 	else
132 		return( NAN );
133 	}
134 
135 /* separate mantissa from exponent */
136 
137 /* Note, frexp is used so that denormal numbers
138  * will be handled properly.
139  */
140 x = frexpl( x, &e );
141 
142 /* logarithm using log(x) = z + z**3 P(z)/Q(z),
143  * where z = 2(x-1)/x+1)
144  */
145 if( (e > 2) || (e < -2) )
146 {
147 if( x < SQRTH )
148 	{ /* 2( 2x-1 )/( 2x+1 ) */
149 	e -= 1;
150 	z = x - 0.5L;
151 	y = 0.5L * z + 0.5L;
152 	}
153 else
154 	{ /*  2 (x-1)/(x+1)   */
155 	z = x - 0.5L;
156 	z -= 0.5L;
157 	y = 0.5L * x  + 0.5L;
158 	}
159 x = z / y;
160 z = x*x;
161 z = x * ( z * __polevll( z, R, 3 ) / __p1evll( z, S, 3 ) );
162 z = z + e * C2;
163 z = z + x;
164 z = z + e * C1;
165 return( z );
166 }
167 
168 
169 /* logarithm using log(1+x) = x - .5x**2 + x**3 P(x)/Q(x) */
170 
171 if( x < SQRTH )
172 	{
173 	e -= 1;
174 	x = ldexpl( x, 1 ) - 1.0L; /*  2x - 1  */
175 	}
176 else
177 	{
178 	x = x - 1.0L;
179 	}
180 z = x*x;
181 y = x * ( z * __polevll( x, P, 6 ) / __p1evll( x, Q, 6 ) );
182 y = y + e * C2;
183 z = y - ldexpl( z, -1 );   /*  y - 0.5 * z  */
184 /* Note, the sum of above terms does not exceed x/4,
185  * so it contributes at most about 1/4 lsb to the error.
186  */
187 z = z + x;
188 z = z + e * C1; /* This sum has an error of 1/2 lsb. */
189 return( z );
190 }
191