1 /* $OpenBSD: s_log1pl.c,v 1.3 2013/11/12 20:35:19 martynas Exp $ */
4 * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
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.
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.
21 * Relative error logarithm
22 * Natural logarithm of 1+x, long double precision
28 * long double x, y, log1pl();
36 * Returns the base e (2.718...) logarithm of 1+x.
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
42 * log(1+x) = x - 0.5 x^2 + x^3 P(x)/Q(x).
44 * Otherwise, setting z = 2(x-1)/x+1),
46 * log(x) = z + z^3 P(z)/Q(z).
53 * arithmetic domain # trials peak rms
54 * IEEE -1.0, 9.0 100000 8.2e-20 2.5e-20
58 * log singularity: x-1 = 0; returns -INFINITY
59 * log domain: x-1 < 0; returns NAN
64 #include "math_private.h"
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
71 static const 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
,
80 static const long double Q
[] = {
81 /* 1.0000000000000000000000E0,*/
82 1.5062909083469192043167E1L
,
83 8.3047565967967209469434E1L
,
84 2.2176239823732856465394E2L
,
85 3.0909872225312059774938E2L
,
86 2.1642788614495947685003E2L
,
87 6.0118660497603843919306E1L
,
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
96 static const long double R
[4] = {
97 1.9757429581415468984296E-3L,
98 -7.1990767473014147232598E-1L,
99 1.0777257190312272158094E1L
,
100 -3.5717684488096787370998E1L
,
102 static const long double S
[4] = {
103 /* 1.00000000000000000000E0L,*/
104 -2.6201045551331104417768E1L
,
105 1.9361891836232102174846E2L
,
106 -4.2861221385716144629696E2L
,
108 static const long double C1
= 6.9314575195312500000000E-1L;
109 static const long double C2
= 1.4286068203094172321215E-6L;
111 #define SQRTH 0.70710678118654752440L
114 log1pl(long double xm1
)
121 if( xm1
== INFINITY
)
128 /* Test for domain errors. */
137 /* Separate mantissa from exponent.
138 Use frexp so that denormal numbers will be handled properly. */
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) )
146 { /* 2( 2x-1 )/( 2x+1 ) */
152 { /* 2 (x-1)/(x+1) */
159 z
= x
* ( z
* __polevll( z
, (void *)R
, 3 ) / __p1evll( z
, (void *)S
, 3 ) );
167 /* logarithm using log(1+x) = x - .5x**2 + x**3 P(x)/Q(x) */
185 y
= x
* ( z
* __polevll( x
, (void *)P
, 6 ) / __p1evll( x
, (void *)Q
, 6 ) );