1 /* $OpenBSD: s_log1pl.c,v 1.5 2017/01/21 08:29:13 krw 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 /* log1pl.c
20 *
21 * Relative error logarithm
22 * Natural logarithm of 1+x, long double precision
23 *
24 *
25 *
26 * SYNOPSIS:
27 *
28 * long double x, y, log1pl();
29 *
30 * y = log1pl( x );
31 *
32 *
33 *
34 * DESCRIPTION:
35 *
36 * Returns the base e (2.718...) logarithm of 1+x.
37 *
38 * The argument 1+x is separated into its exponent and fractional
39 * parts. If the exponent is between -1 and +1, the logarithm
40 * of the fraction is approximated by
41 *
42 * log(1+x) = x - 0.5 x^2 + x^3 P(x)/Q(x).
43 *
44 * Otherwise, setting z = 2(x-1)/x+1),
45 *
46 * log(x) = z + z^3 P(z)/Q(z).
47 *
48 *
49 *
50 * ACCURACY:
51 *
52 * Relative error:
53 * arithmetic domain # trials peak rms
54 * IEEE -1.0, 9.0 100000 8.2e-20 2.5e-20
55 *
56 * ERROR MESSAGES:
57 *
58 * log singularity: x-1 = 0; returns -INFINITY
59 * log domain: x-1 < 0; returns NAN
60 */
61
62 #include <math.h>
63
64 #include "math_private.h"
65
66 /* Coefficients for log(1+x) = x - x^2 / 2 + x^3 P(x)/Q(x)
67 * 1/sqrt(2) <= x < sqrt(2)
68 * Theoretical peak relative error = 2.32e-20
69 */
70
71 static long double P[] = {
72 4.5270000862445199635215E-5L,
73 4.9854102823193375972212E-1L,
74 6.5787325942061044846969E0L,
75 2.9911919328553073277375E1L,
76 6.0949667980987787057556E1L,
77 5.7112963590585538103336E1L,
78 2.0039553499201281259648E1L,
79 };
80 static long double Q[] = {
81 /* 1.0000000000000000000000E0,*/
82 1.5062909083469192043167E1L,
83 8.3047565967967209469434E1L,
84 2.2176239823732856465394E2L,
85 3.0909872225312059774938E2L,
86 2.1642788614495947685003E2L,
87 6.0118660497603843919306E1L,
88 };
89
90 /* Coefficients for log(x) = z + z^3 P(z^2)/Q(z^2),
91 * where z = 2(x-1)/(x+1)
92 * 1/sqrt(2) <= x < sqrt(2)
93 * Theoretical peak relative error = 6.16e-22
94 */
95
96 static long double R[4] = {
97 1.9757429581415468984296E-3L,
98 -7.1990767473014147232598E-1L,
99 1.0777257190312272158094E1L,
100 -3.5717684488096787370998E1L,
101 };
102 static long double S[4] = {
103 /* 1.00000000000000000000E0L,*/
104 -2.6201045551331104417768E1L,
105 1.9361891836232102174846E2L,
106 -4.2861221385716144629696E2L,
107 };
108 static const long double C1 = 6.9314575195312500000000E-1L;
109 static const long double C2 = 1.4286068203094172321215E-6L;
110
111 #define SQRTH 0.70710678118654752440L
112
113 long double
log1pl(long double xm1)114 log1pl(long double xm1)
115 {
116 long double x, y, z;
117 int e;
118
119 if( isnan(xm1) )
120 return(xm1);
121 if( xm1 == INFINITY )
122 return(xm1);
123 if(xm1 == 0.0)
124 return(xm1);
125
126 x = xm1 + 1.0L;
127
128 /* Test for domain errors. */
129 if( x <= 0.0L )
130 {
131 if( x == 0.0L )
132 return( -INFINITY );
133 else
134 return( NAN );
135 }
136
137 /* Separate mantissa from exponent.
138 Use frexp so that denormal numbers will be handled properly. */
139 x = frexpl( x, &e );
140
141 /* logarithm using log(x) = z + z^3 P(z)/Q(z),
142 where z = 2(x-1)/x+1) */
143 if( (e > 2) || (e < -2) )
144 {
145 if( x < SQRTH )
146 { /* 2( 2x-1 )/( 2x+1 ) */
147 e -= 1;
148 z = x - 0.5L;
149 y = 0.5L * z + 0.5L;
150 }
151 else
152 { /* 2 (x-1)/(x+1) */
153 z = x - 0.5L;
154 z -= 0.5L;
155 y = 0.5L * x + 0.5L;
156 }
157 x = z / y;
158 z = x*x;
159 z = x * ( z * __polevll( z, R, 3 ) / __p1evll( z, S, 3 ) );
160 z = z + e * C2;
161 z = z + x;
162 z = z + e * C1;
163 return( z );
164 }
165
166
167 /* logarithm using log(1+x) = x - .5x**2 + x**3 P(x)/Q(x) */
168
169 if( x < SQRTH )
170 {
171 e -= 1;
172 if (e != 0)
173 x = 2.0 * x - 1.0L;
174 else
175 x = xm1;
176 }
177 else
178 {
179 if (e != 0)
180 x = x - 1.0L;
181 else
182 x = xm1;
183 }
184 z = x*x;
185 y = x * ( z * __polevll( x, P, 6 ) / __p1evll( x, Q, 6 ) );
186 y = y + e * C2;
187 z = y - 0.5 * z;
188 z = z + x;
189 z = z + e * C1;
190 return( z );
191 }
192 DEF_STD(log1pl);
193