1 /* origin: OpenBSD /usr/src/lib/libm/src/ld80/s_log1pl.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 * Relative error logarithm
19 * Natural logarithm of 1+x, long double precision
20 *
21 *
22 * SYNOPSIS:
23 *
24 * long double x, y, log1pl();
25 *
26 * y = log1pl( x );
27 *
28 *
29 * DESCRIPTION:
30 *
31 * Returns the base e (2.718...) logarithm of 1+x.
32 *
33 * The argument 1+x is separated into its exponent and fractional
34 * parts. If the exponent is between -1 and +1, the logarithm
35 * of the fraction is approximated by
36 *
37 * log(1+x) = x - 0.5 x^2 + x^3 P(x)/Q(x).
38 *
39 * Otherwise, setting z = 2(x-1)/x+1),
40 *
41 * log(x) = z + z^3 P(z)/Q(z).
42 *
43 *
44 * ACCURACY:
45 *
46 * Relative error:
47 * arithmetic domain # trials peak rms
48 * IEEE -1.0, 9.0 100000 8.2e-20 2.5e-20
49 */
50 51 #include "libm.h"
52 53 #if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
54 long double log1pl(long double x)
55 {
56 return log1p(x);
57 }
58 #elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
59 /* Coefficients for log(1+x) = x - x^2 / 2 + x^3 P(x)/Q(x)
60 * 1/sqrt(2) <= x < sqrt(2)
61 * Theoretical peak relative error = 2.32e-20
62 */
63 static const long double P[] = {
64 4.5270000862445199635215E-5L,
65 4.9854102823193375972212E-1L,
66 6.5787325942061044846969E0L,
67 2.9911919328553073277375E1L,
68 6.0949667980987787057556E1L,
69 5.7112963590585538103336E1L,
70 2.0039553499201281259648E1L,
71 };
72 static const long double Q[] = {
73 /* 1.0000000000000000000000E0,*/
74 1.5062909083469192043167E1L,
75 8.3047565967967209469434E1L,
76 2.2176239823732856465394E2L,
77 3.0909872225312059774938E2L,
78 2.1642788614495947685003E2L,
79 6.0118660497603843919306E1L,
80 };
81 82 /* Coefficients for log(x) = z + z^3 P(z^2)/Q(z^2),
83 * where z = 2(x-1)/(x+1)
84 * 1/sqrt(2) <= x < sqrt(2)
85 * Theoretical peak relative error = 6.16e-22
86 */
87 static const long double R[4] = {
88 1.9757429581415468984296E-3L,
89 -7.1990767473014147232598E-1L,
90 1.0777257190312272158094E1L,
91 -3.5717684488096787370998E1L,
92 };
93 static const long double S[4] = {
94 /* 1.00000000000000000000E0L,*/
95 -2.6201045551331104417768E1L,
96 1.9361891836232102174846E2L,
97 -4.2861221385716144629696E2L,
98 };
99 static const long double C1 = 6.9314575195312500000000E-1L;
100 static const long double C2 = 1.4286068203094172321215E-6L;
101 102 #define SQRTH 0.70710678118654752440L
103 104 long double log1pl(long double xm1)
105 {
106 long double x, y, z;
107 int e;
108 109 if (isnan(xm1))
110 return xm1;
111 if (xm1 == INFINITY)
112 return xm1;
113 if (xm1 == 0.0)
114 return xm1;
115 116 x = xm1 + 1.0;
117 118 /* Test for domain errors. */
119 if (x <= 0.0) {
120 if (x == 0.0)
121 return -1/(x*x); /* -inf with divbyzero */
122 return 0/0.0f; /* nan with invalid */
123 }
124 125 /* Separate mantissa from exponent.
126 Use frexp so that denormal numbers will be handled properly. */
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 if (e > 2 || e < -2) {
132 if (x < SQRTH) { /* 2(2x-1)/(2x+1) */
133 e -= 1;
134 z = x - 0.5;
135 y = 0.5 * z + 0.5;
136 } else { /* 2 (x-1)/(x+1) */
137 z = x - 0.5;
138 z -= 0.5;
139 y = 0.5 * x + 0.5;
140 }
141 x = z / y;
142 z = x*x;
143 z = x * (z * __polevll(z, R, 3) / __p1evll(z, S, 3));
144 z = z + e * C2;
145 z = z + x;
146 z = z + e * C1;
147 return z;
148 }
149 150 /* logarithm using log(1+x) = x - .5x**2 + x**3 P(x)/Q(x) */
151 if (x < SQRTH) {
152 e -= 1;
153 if (e != 0)
154 x = 2.0 * x - 1.0;
155 else
156 x = xm1;
157 } else {
158 if (e != 0)
159 x = x - 1.0;
160 else
161 x = xm1;
162 }
163 z = x*x;
164 y = x * (z * __polevll(x, P, 6) / __p1evll(x, Q, 6));
165 y = y + e * C2;
166 z = y - 0.5 * z;
167 z = z + x;
168 z = z + e * C1;
169 return z;
170 }
171 #elif LDBL_MANT_DIG == 113 && LDBL_MAX_EXP == 16384
172 // TODO: broken implementation to make things compile
173 long double log1pl(long double x)
174 {
175 return log1p(x);
176 }
177 #endif
178