dtv_math.c revision 1.4
1 /* $NetBSD: dtv_math.c,v 1.4 2011/07/16 22:41:59 apb Exp $ */ 2 3 /*- 4 * Copyright (c) 2011 Alan Barrett <apb (at) 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 1.4 2011/07/16 22:41:59 apb Exp $"); 31 32 #include <sys/types.h> 33 #include <sys/bitops.h> 34 35 #include <dev/dtv/dtvif.h> 36 37 /* 38 * dtv_intlog10 -- return an approximation to log10(x) * 1<<24, 39 * using integer arithmetic. 40 * 41 * As a special case, returns 0 when x == 0. The mathematical 42 * result is -infinity. 43 * 44 * This function uses 0.5 + x/2 - 1/x as an approximation to 45 * log2(x) for x in the range [1.0, 2.0], and scales the input value 46 * to fit this range. The resulting error is always better than 47 * 0.2%. 48 * 49 * Here's a table of the desired and actual results, as well 50 * as the absolute and relative errors, for several values of x. 51 * 52 * x desired actual err_abs err_rel 53 * 0 0 0 +0 +0.00000 54 * 1 0 0 +0 +0.00000 55 * 2 5050445 5050122 -323 -0.00006 56 * 3 8004766 7996348 -8418 -0.00105 57 * 4 10100890 10100887 -3 -0.00000 58 * 5 11726770 11741823 +15053 +0.00128 59 * 6 13055211 13046470 -8741 -0.00067 60 * 7 14178392 14158860 -19532 -0.00138 61 * 8 15151335 15151009 -326 -0.00002 62 * 9 16009532 16028061 +18529 +0.00116 63 * 10 16777216 16792588 +15372 +0.00092 64 * 11 17471670 17475454 +3784 +0.00022 65 * 12 18105656 18097235 -8421 -0.00047 66 * 13 18688868 18672077 -16791 -0.00090 67 * 14 19228837 19209625 -19212 -0.00100 68 * 15 19731537 19717595 -13942 -0.00071 69 * 16 20201781 20201774 -7 -0.00000 70 * 20 21827661 21842710 +15049 +0.00069 71 * 24 23156102 23147357 -8745 -0.00038 72 * 30 24781982 24767717 -14265 -0.00058 73 * 40 26878106 26893475 +15369 +0.00057 74 * 60 29832427 29818482 -13945 -0.00047 75 * 100 33554432 33540809 -13623 -0.00041 76 * 1000 50331648 50325038 -6610 -0.00013 77 * 10000 67108864 67125985 +17121 +0.00026 78 * 100000 83886080 83875492 -10588 -0.00013 79 * 1000000 100663296 100652005 -11291 -0.00011 80 * 10000000 117440512 117458739 +18227 +0.00016 81 * 100000000 134217728 134210175 -7553 -0.00006 82 * 1000000000 150994944 150980258 -14686 -0.00010 83 * 4294967295 161614248 161614192 -56 -0.00000 84 */ 85 uint32_t 86 dtv_intlog10(uint32_t x) 87 { 88 uint32_t ilog2x; 89 uint32_t t; 90 uint32_t t1; 91 92 if (__predict_false(x == 0)) 93 return 0; 94 95 /* 96 * find ilog2x = floor(log2(x)), as an integer in the range [0,31]. 97 */ 98 ilog2x = ilog2(x); 99 100 /* 101 * Set "t" to the result of shifting x left or right 102 * until the most significant bit that was actually set 103 * moves into the 1<<24 position. 104 * 105 * Now we can think of "t" as representing 106 * x / 2**(floor(log2(x))), 107 * as a fixed-point value with 8 integer bits and 24 fraction bits. 108 * 109 * This value is in the semi-closed interval [1.0, 2.0) 110 * when interpreting it as a fixed-point number, or in the 111 * interval [0x01000000, 0x01ffffff] when examining the 112 * underlying uint32_t representation. 113 */ 114 t = (ilog2x > 24 ? x >> (ilog2x - 24) : x << (24 - ilog2x)); 115 116 /* 117 * Calculate "t1 = 1 / t" in the 8.24 fixed-point format. 118 * This value is in the interval [0.5, 1.0] 119 * when interpreting it as a fixed-point number, or in the 120 * interval [0x00800000, 0x01000000] when examining the 121 * underlying uint32_t representation. 122 * 123 */ 124 t1 = ((uint64_t)1 << 48) / t; 125 126 /* 127 * Calculate "t = ilog2x + t/2 - t1 + 0.5" in the 8.24 128 * fixed-point format. 129 * 130 * If x is a power of 2, then t is now exactly equal to log2(x) 131 * when interpreting it as a fixed-point number, or exactly 132 * log2(x) << 24 when examining the underlying uint32_t 133 * representation. 134 * 135 * If x is not a power of 2, then t is the result of 136 * using the function x/2 - 1/x + 0.5 as an approximation for 137 * log2(x) for x in the range [1, 2], and scaling both the 138 * input and the result by the appropriate number of powers of 2. 139 */ 140 t = (ilog2x << 24) + (t >> 1) - t1 + (1 << 23); 141 142 /* 143 * Multiply t by log10(2) to get the final result. 144 * 145 * log10(2) is approximately 643/2136 We divide before 146 * multiplying to avoid overflow. 147 */ 148 return t / 2136 * 643; 149 } 150 151 #ifdef TEST_DTV_MATH 152 /* 153 * To test: 154 * cc -DTEST_DTV_MATH ./dtv_math.c -lm -o ./a.out && ./a.out 155 */ 156 157 #include <stdio.h> 158 #include <inttypes.h> 159 #include <math.h> 160 161 int 162 main(void) 163 { 164 uint32_t xlist[] = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 165 14, 15, 16, 20, 24, 30, 40, 60, 100, 1000, 10000, 166 100000, 1000000, 10000000, 100000000, 1000000000, 167 0xffffffff}; 168 int i; 169 170 printf("%11s %11s %11s %11s %s\n", 171 "x", "desired", "actual", "err_abs", "err_rel"); 172 for (i = 0; i < __arraycount(xlist); i++) 173 { 174 uint32_t x = xlist[i]; 175 uint32_t desired = (uint32_t)(log10((double)x) 176 * (double)(1<<24)); 177 uint32_t actual = dtv_intlog10(x); 178 int32_t err_abs = actual - desired; 179 double err_rel = (err_abs == 0 ? 0.0 180 : err_abs / (double)actual); 181 182 printf("%11"PRIu32" %11"PRIu32" %11"PRIu32 183 " %+11"PRId32" %+.5f\n", 184 x, desired, actual, err_abs, err_rel); 185 } 186 return 0; 187 } 188 189 #endif /* TEST_DTV_MATH */ 190
Indexes created Tue Sep 30 20:09:53 GMT 2025