1 /* $OpenBSD: s_log1pl.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 /* 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
63
64 /* Coefficients for log(1+x) = x - x^2 / 2 + x^3 P(x)/Q(x)
65 * 1/sqrt(2) <= x < sqrt(2)
66 * Theoretical peak relative error = 2.32e-20
67 */
68
69 static long double P[] = {
70 4.5270000862445199635215E-5L,
71 4.9854102823193375972212E-1L,
72 6.5787325942061044846969E0L,
73 2.9911919328553073277375E1L,
74 6.0949667980987787057556E1L,
75 5.7112963590585538103336E1L,
76 2.0039553499201281259648E1L,
77 };
78 static long double Q[] = {
79 /* 1.0000000000000000000000E0,*/
80 1.5062909083469192043167E1L,
81 8.3047565967967209469434E1L,
82 2.2176239823732856465394E2L,
83 3.0909872225312059774938E2L,
84 2.1642788614495947685003E2L,
85 6.0118660497603843919306E1L,
86 };
87
88 /* Coefficients for log(x) = z + z^3 P(z^2)/Q(z^2),
89 * where z = 2(x-1)/(x+1)
90 * 1/sqrt(2) <= x < sqrt(2)
91 * Theoretical peak relative error = 6.16e-22
92 */
93
94 static long double R[4] = {
95 1.9757429581415468984296E-3L,
96 -7.1990767473014147232598E-1L,
97 1.0777257190312272158094E1L,
98 -3.5717684488096787370998E1L,
99 };
100 static long double S[4] = {
101 /* 1.00000000000000000000E0L,*/
102 -2.6201045551331104417768E1L,
103 1.9361891836232102174846E2L,
104 -4.2861221385716144629696E2L,
105 };
106 static const long double C1 = 6.9314575195312500000000E-1L;
107 static const long double C2 = 1.4286068203094172321215E-6L;
108
109 #define SQRTH 0.70710678118654752440L
110
111 long double
log1pl(long double xm1)112 log1pl(long double xm1)
113 {
114 long double x, y, z;
115 int e;
116
117 if( isnan(xm1) )
118 return(xm1 + xm1);
119 if( xm1 == (long double) INFINITY )
120 return(xm1);
121 if(xm1 == 0.0l)
122 return(xm1);
123
124 x = xm1 + 1.0L;
125
126 /* Test for domain errors. */
127 if( x <= 0.0L )
128 {
129 if( x == 0.0L )
130 return __math_divzerol(1);
131 else
132 return __math_invalidl(xm1);
133 }
134
135 /* Separate mantissa from exponent.
136 Use frexp so that denormal numbers will be handled properly. */
137 x = frexpl( x, &e );
138
139 /* logarithm using log(x) = z + z^3 P(z)/Q(z),
140 where z = 2(x-1)/x+1) */
141 if( (e > 2) || (e < -2) )
142 {
143 if( x < SQRTH )
144 { /* 2( 2x-1 )/( 2x+1 ) */
145 e -= 1;
146 z = x - 0.5L;
147 y = 0.5L * z + 0.5L;
148 }
149 else
150 { /* 2 (x-1)/(x+1) */
151 z = x - 0.5L;
152 z -= 0.5L;
153 y = 0.5L * x + 0.5L;
154 }
155 x = z / y;
156 z = x*x;
157 z = x * ( z * __polevll( z, R, 3 ) / __p1evll( z, S, 3 ) );
158 z = z + e * C2;
159 z = z + x;
160 z = z + e * C1;
161 return( z );
162 }
163
164
165 /* logarithm using log(1+x) = x - .5x**2 + x**3 P(x)/Q(x) */
166
167 if( x < SQRTH )
168 {
169 e -= 1;
170 if (e != 0)
171 x = 2.0l * x - 1.0L;
172 else
173 x = xm1;
174 }
175 else
176 {
177 if (e != 0)
178 x = x - 1.0L;
179 else
180 x = xm1;
181 }
182 z = x*x;
183 y = x * ( z * __polevll( x, P, 6 ) / __p1evll( x, Q, 6 ) );
184 y = y + e * C2;
185 z = y - 0.5l * z;
186 z = z + x;
187 z = z + e * C1;
188 return( z );
189 }
190
