1 /* $NetBSD: dtv_math.c,v 1.5 2011/08/09 01:42:24 jmcneill Exp $ */
2
3 /*-
4 * Copyright (c) 2011 Alan Barrett <apb@NetBSD.org>
5 * All rights reserved.
6 *
7 * Redistribution and use in source and binary forms, with or without
8 * modification, are permitted provided that the following conditions
9 * are met:
10 * 1. Redistributions of source code must retain the above copyright
11 * notice, this list of conditions and the following disclaimer.
12 * 2. Redistributions in binary form must reproduce the above copyright
13 * notice, this list of conditions and the following disclaimer in the
14 * documentation and/or other materials provided with the distribution.
15 *
16 * THIS SOFTWARE IS PROVIDED BY THE NETBSD FOUNDATION, INC. AND CONTRIBUTORS
17 * ``AS IS'' AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
18 * TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
19 * PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE FOUNDATION OR CONTRIBUTORS
20 * BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
21 * CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
22 * SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
23 * INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
24 * CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
25 * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
26 * POSSIBILITY OF SUCH DAMAGE.
27 */
28
29 #include <sys/cdefs.h>
30 __KERNEL_RCSID(0, "$NetBSD: dtv_math.c,v 1.5 2011/08/09 01:42:24 jmcneill Exp $");
31
32 #include <sys/types.h>
33 #include <sys/bitops.h>
34 #include <sys/module.h>
35
36 #include <dev/dtv/dtv_math.h>
37
38 /*
39 * dtv_intlog10 -- return an approximation to log10(x) * 1<<24,
40 * using integer arithmetic.
41 *
42 * As a special case, returns 0 when x == 0. The mathematical
43 * result is -infinity.
44 *
45 * This function uses 0.5 + x/2 - 1/x as an approximation to
46 * log2(x) for x in the range [1.0, 2.0], and scales the input value
47 * to fit this range. The resulting error is always better than
48 * 0.2%.
49 *
50 * Here's a table of the desired and actual results, as well
51 * as the absolute and relative errors, for several values of x.
52 *
53 * x desired actual err_abs err_rel
54 * 0 0 0 +0 +0.00000
55 * 1 0 0 +0 +0.00000
56 * 2 5050445 5050122 -323 -0.00006
57 * 3 8004766 7996348 -8418 -0.00105
58 * 4 10100890 10100887 -3 -0.00000
59 * 5 11726770 11741823 +15053 +0.00128
60 * 6 13055211 13046470 -8741 -0.00067
61 * 7 14178392 14158860 -19532 -0.00138
62 * 8 15151335 15151009 -326 -0.00002
63 * 9 16009532 16028061 +18529 +0.00116
64 * 10 16777216 16792588 +15372 +0.00092
65 * 11 17471670 17475454 +3784 +0.00022
66 * 12 18105656 18097235 -8421 -0.00047
67 * 13 18688868 18672077 -16791 -0.00090
68 * 14 19228837 19209625 -19212 -0.00100
69 * 15 19731537 19717595 -13942 -0.00071
70 * 16 20201781 20201774 -7 -0.00000
71 * 20 21827661 21842710 +15049 +0.00069
72 * 24 23156102 23147357 -8745 -0.00038
73 * 30 24781982 24767717 -14265 -0.00058
74 * 40 26878106 26893475 +15369 +0.00057
75 * 60 29832427 29818482 -13945 -0.00047
76 * 100 33554432 33540809 -13623 -0.00041
77 * 1000 50331648 50325038 -6610 -0.00013
78 * 10000 67108864 67125985 +17121 +0.00026
79 * 100000 83886080 83875492 -10588 -0.00013
80 * 1000000 100663296 100652005 -11291 -0.00011
81 * 10000000 117440512 117458739 +18227 +0.00016
82 * 100000000 134217728 134210175 -7553 -0.00006
83 * 1000000000 150994944 150980258 -14686 -0.00010
84 * 4294967295 161614248 161614192 -56 -0.00000
85 */
86 uint32_t
dtv_intlog10(uint32_t x)87 dtv_intlog10(uint32_t x)
88 {
89 uint32_t ilog2x;
90 uint32_t t;
91 uint32_t t1;
92
93 if (__predict_false(x == 0))
94 return 0;
95
96 /*
97 * find ilog2x = floor(log2(x)), as an integer in the range [0,31].
98 */
99 ilog2x = ilog2(x);
100
101 /*
102 * Set "t" to the result of shifting x left or right
103 * until the most significant bit that was actually set
104 * moves into the 1<<24 position.
105 *
106 * Now we can think of "t" as representing
107 * x / 2**(floor(log2(x))),
108 * as a fixed-point value with 8 integer bits and 24 fraction bits.
109 *
110 * This value is in the semi-closed interval [1.0, 2.0)
111 * when interpreting it as a fixed-point number, or in the
112 * interval [0x01000000, 0x01ffffff] when examining the
113 * underlying uint32_t representation.
114 */
115 t = (ilog2x > 24 ? x >> (ilog2x - 24) : x << (24 - ilog2x));
116
117 /*
118 * Calculate "t1 = 1 / t" in the 8.24 fixed-point format.
119 * This value is in the interval [0.5, 1.0]
120 * when interpreting it as a fixed-point number, or in the
121 * interval [0x00800000, 0x01000000] when examining the
122 * underlying uint32_t representation.
123 *
124 */
125 t1 = ((uint64_t)1 << 48) / t;
126
127 /*
128 * Calculate "t = ilog2x + t/2 - t1 + 0.5" in the 8.24
129 * fixed-point format.
130 *
131 * If x is a power of 2, then t is now exactly equal to log2(x)
132 * when interpreting it as a fixed-point number, or exactly
133 * log2(x) << 24 when examining the underlying uint32_t
134 * representation.
135 *
136 * If x is not a power of 2, then t is the result of
137 * using the function x/2 - 1/x + 0.5 as an approximation for
138 * log2(x) for x in the range [1, 2], and scaling both the
139 * input and the result by the appropriate number of powers of 2.
140 */
141 t = (ilog2x << 24) + (t >> 1) - t1 + (1 << 23);
142
143 /*
144 * Multiply t by log10(2) to get the final result.
145 *
146 * log10(2) is approximately 643/2136 We divide before
147 * multiplying to avoid overflow.
148 */
149 return t / 2136 * 643;
150 }
151
152 #ifdef _KERNEL
153 MODULE(MODULE_CLASS_MISC, dtv_math, NULL);
154
155 static int
dtv_math_modcmd(modcmd_t cmd,void * opaque)156 dtv_math_modcmd(modcmd_t cmd, void *opaque)
157 {
158 if (cmd == MODULE_CMD_INIT || cmd == MODULE_CMD_FINI)
159 return 0;
160 return ENOTTY;
161 }
162 #endif
163
164 #ifdef TEST_DTV_MATH
165 /*
166 * To test:
167 * cc -DTEST_DTV_MATH ./dtv_math.c -lm -o ./a.out && ./a.out
168 */
169
170 #include <stdio.h>
171 #include <inttypes.h>
172 #include <math.h>
173
174 int
main(void)175 main(void)
176 {
177 uint32_t xlist[] = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13,
178 14, 15, 16, 20, 24, 30, 40, 60, 100, 1000, 10000,
179 100000, 1000000, 10000000, 100000000, 1000000000,
180 0xffffffff};
181 int i;
182
183 printf("%11s %11s %11s %11s %s\n",
184 "x", "desired", "actual", "err_abs", "err_rel");
185 for (i = 0; i < __arraycount(xlist); i++)
186 {
187 uint32_t x = xlist[i];
188 uint32_t desired = (uint32_t)(log10((double)x)
189 * (double)(1<<24));
190 uint32_t actual = dtv_intlog10(x);
191 int32_t err_abs = actual - desired;
192 double err_rel = (err_abs == 0 ? 0.0
193 : err_abs / (double)actual);
194
195 printf("%11"PRIu32" %11"PRIu32" %11"PRIu32
196 " %+11"PRId32" %+.5f\n",
197 x, desired, actual, err_abs, err_rel);
198 }
199 return 0;
200 }
201
202 #endif /* TEST_DTV_MATH */
203