1 /* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_log2l.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 * Base 2 logarithm, long double precision
19 *
20 *
21 * SYNOPSIS:
22 *
23 * long double x, y, log2l();
24 *
25 * y = log2l( x );
26 *
27 *
28 * DESCRIPTION:
29 *
30 * Returns the base 2 logarithm of x.
31 *
32 * The argument is separated into its exponent and fractional
33 * parts. If the exponent is between -1 and +1, the (natural)
34 * logarithm of the fraction is approximated by
35 *
36 * log(1+x) = x - 0.5 x**2 + x**3 P(x)/Q(x).
37 *
38 * Otherwise, setting z = 2(x-1)/x+1),
39 *
40 * log(x) = z + z**3 P(z)/Q(z).
41 *
42 *
43 * ACCURACY:
44 *
45 * Relative error:
46 * arithmetic domain # trials peak rms
47 * IEEE 0.5, 2.0 30000 9.8e-20 2.7e-20
48 * IEEE exp(+-10000) 70000 5.4e-20 2.3e-20
49 *
50 * In the tests over the interval exp(+-10000), the logarithms
51 * of the random arguments were uniformly distributed over
52 * [-10000, +10000].
53 */
54 55 #include "libm.h"
56 57 #if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
58 long double log2l(long double x)
59 {
60 return log2(x);
61 }
62 #elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
63 /* Coefficients for ln(1+x) = x - x**2/2 + x**3 P(x)/Q(x)
64 * 1/sqrt(2) <= x < sqrt(2)
65 * Theoretical peak relative error = 6.2e-22
66 */
67 static const long double P[] = {
68 4.9962495940332550844739E-1L,
69 1.0767376367209449010438E1L,
70 7.7671073698359539859595E1L,
71 2.5620629828144409632571E2L,
72 4.2401812743503691187826E2L,
73 3.4258224542413922935104E2L,
74 1.0747524399916215149070E2L,
75 };
76 static const long double Q[] = {
77 /* 1.0000000000000000000000E0,*/
78 2.3479774160285863271658E1L,
79 1.9444210022760132894510E2L,
80 7.7952888181207260646090E2L,
81 1.6911722418503949084863E3L,
82 2.0307734695595183428202E3L,
83 1.2695660352705325274404E3L,
84 3.2242573199748645407652E2L,
85 };
86 87 /* Coefficients for log(x) = z + z^3 P(z^2)/Q(z^2),
88 * where z = 2(x-1)/(x+1)
89 * 1/sqrt(2) <= x < sqrt(2)
90 * Theoretical peak relative error = 6.16e-22
91 */
92 static const long double R[4] = {
93 1.9757429581415468984296E-3L,
94 -7.1990767473014147232598E-1L,
95 1.0777257190312272158094E1L,
96 -3.5717684488096787370998E1L,
97 };
98 static const long double S[4] = {
99 /* 1.00000000000000000000E0L,*/
100 -2.6201045551331104417768E1L,
101 1.9361891836232102174846E2L,
102 -4.2861221385716144629696E2L,
103 };
104 /* log2(e) - 1 */
105 #define LOG2EA 4.4269504088896340735992e-1L
106 107 #define SQRTH 0.70710678118654752440L
108 109 long double log2l(long double x)
110 {
111 long double y, z;
112 int e;
113 114 if (isnan(x))
115 return x;
116 if (x == INFINITY)
117 return x;
118 if (x <= 0.0) {
119 if (x == 0.0)
120 return -1/(x*x); /* -inf with divbyzero */
121 return 0/0.0f; /* nan with invalid */
122 }
123 124 /* separate mantissa from exponent */
125 /* Note, frexp is used so that denormal numbers
126 * will be handled properly.
127 */
128 x = frexpl(x, &e);
129 130 /* logarithm using log(x) = z + z**3 P(z)/Q(z),
131 * where z = 2(x-1)/x+1)
132 */
133 if (e > 2 || e < -2) {
134 if (x < SQRTH) { /* 2(2x-1)/(2x+1) */
135 e -= 1;
136 z = x - 0.5;
137 y = 0.5 * z + 0.5;
138 } else { /* 2 (x-1)/(x+1) */
139 z = x - 0.5;
140 z -= 0.5;
141 y = 0.5 * x + 0.5;
142 }
143 x = z / y;
144 z = x*x;
145 y = x * (z * __polevll(z, R, 3) / __p1evll(z, S, 3));
146 goto done;
147 }
148 149 /* logarithm using log(1+x) = x - .5x**2 + x**3 P(x)/Q(x) */
150 if (x < SQRTH) {
151 e -= 1;
152 x = 2.0*x - 1.0;
153 } else {
154 x = x - 1.0;
155 }
156 z = x*x;
157 y = x * (z * __polevll(x, P, 6) / __p1evll(x, Q, 7));
158 y = y - 0.5*z;
159 160 done:
161 /* Multiply log of fraction by log2(e)
162 * and base 2 exponent by 1
163 *
164 * ***CAUTION***
165 *
166 * This sequence of operations is critical and it may
167 * be horribly defeated by some compiler optimizers.
168 */
169 z = y * LOG2EA;
170 z += x * LOG2EA;
171 z += y;
172 z += x;
173 z += e;
174 return z;
175 }
176 #elif LDBL_MANT_DIG == 113 && LDBL_MAX_EXP == 16384
177 // TODO: broken implementation to make things compile
178 long double log2l(long double x)
179 {
180 return log2(x);
181 }
182 #endif
183