/* $NetBSD: dtv_math.c,v 1.5 2011/08/09 01:42:24 jmcneill Exp $ */
/*-
* Copyright (c) 2011 Alan Barrett <apb@NetBSD.org>
* All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
* 1. Redistributions of source code must retain the above copyright
* notice, this list of conditions and the following disclaimer.
* 2. Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in the
* documentation and/or other materials provided with the distribution.
*
* THIS SOFTWARE IS PROVIDED BY THE NETBSD FOUNDATION, INC. AND CONTRIBUTORS
* ``AS IS'' AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
* TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
* PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE FOUNDATION OR CONTRIBUTORS
* BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
* CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
* SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
* INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
* CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
* ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
* POSSIBILITY OF SUCH DAMAGE.
*/
#include <sys/cdefs.h>
__KERNEL_RCSID(0, "$NetBSD: dtv_math.c,v 1.5 2011/08/09 01:42:24 jmcneill Exp $");
#include <sys/types.h>
#include <sys/bitops.h>
#include <sys/module.h>
#include <dev/dtv/dtv_math.h>
/*
* dtv_intlog10 -- return an approximation to log10(x) * 1<<24,
* using integer arithmetic.
*
* As a special case, returns 0 when x == 0. The mathematical
* result is -infinity.
*
* This function uses 0.5 + x/2 - 1/x as an approximation to
* log2(x) for x in the range [1.0, 2.0], and scales the input value
* to fit this range. The resulting error is always better than
* 0.2%.
*
* Here's a table of the desired and actual results, as well
* as the absolute and relative errors, for several values of x.
*
* x desired actual err_abs err_rel
* 0 0 0 +0 +0.00000
* 1 0 0 +0 +0.00000
* 2 5050445 5050122 -323 -0.00006
* 3 8004766 7996348 -8418 -0.00105
* 4 10100890 10100887 -3 -0.00000
* 5 11726770 11741823 +15053 +0.00128
* 6 13055211 13046470 -8741 -0.00067
* 7 14178392 14158860 -19532 -0.00138
* 8 15151335 15151009 -326 -0.00002
* 9 16009532 16028061 +18529 +0.00116
* 10 16777216 16792588 +15372 +0.00092
* 11 17471670 17475454 +3784 +0.00022
* 12 18105656 18097235 -8421 -0.00047
* 13 18688868 18672077 -16791 -0.00090
* 14 19228837 19209625 -19212 -0.00100
* 15 19731537 19717595 -13942 -0.00071
* 16 20201781 20201774 -7 -0.00000
* 20 21827661 21842710 +15049 +0.00069
* 24 23156102 23147357 -8745 -0.00038
* 30 24781982 24767717 -14265 -0.00058
* 40 26878106 26893475 +15369 +0.00057
* 60 29832427 29818482 -13945 -0.00047
* 100 33554432 33540809 -13623 -0.00041
* 1000 50331648 50325038 -6610 -0.00013
* 10000 67108864 67125985 +17121 +0.00026
* 100000 83886080 83875492 -10588 -0.00013
* 1000000 100663296 100652005 -11291 -0.00011
* 10000000 117440512 117458739 +18227 +0.00016
* 100000000 134217728 134210175 -7553 -0.00006
* 1000000000 150994944 150980258 -14686 -0.00010
* 4294967295 161614248 161614192 -56 -0.00000
*/
uint32_t
dtv_intlog10(uint32_t x)
{
uint32_t ilog2x;
uint32_t t;
uint32_t t1;
if (__predict_false(x == 0))
return 0;
/*
* find ilog2x = floor(log2(x)), as an integer in the range [0,31].
*/
ilog2x = ilog2(x);
/*
* Set "t" to the result of shifting x left or right
* until the most significant bit that was actually set
* moves into the 1<<24 position.
*
* Now we can think of "t" as representing
* x / 2**(floor(log2(x))),
* as a fixed-point value with 8 integer bits and 24 fraction bits.
*
* This value is in the semi-closed interval [1.0, 2.0)
* when interpreting it as a fixed-point number, or in the
* interval [0x01000000, 0x01ffffff] when examining the
* underlying uint32_t representation.
*/
t = (ilog2x > 24 ? x >> (ilog2x - 24) : x << (24 - ilog2x));
/*
* Calculate "t1 = 1 / t" in the 8.24 fixed-point format.
* This value is in the interval [0.5, 1.0]
* when interpreting it as a fixed-point number, or in the
* interval [0x00800000, 0x01000000] when examining the
* underlying uint32_t representation.
*
*/
t1 = ((uint64_t)1 << 48) / t;
/*
* Calculate "t = ilog2x + t/2 - t1 + 0.5" in the 8.24
* fixed-point format.
*
* If x is a power of 2, then t is now exactly equal to log2(x)
* when interpreting it as a fixed-point number, or exactly
* log2(x) << 24 when examining the underlying uint32_t
* representation.
*
* If x is not a power of 2, then t is the result of
* using the function x/2 - 1/x + 0.5 as an approximation for
* log2(x) for x in the range [1, 2], and scaling both the
* input and the result by the appropriate number of powers of 2.
*/
t = (ilog2x << 24) + (t >> 1) - t1 + (1 << 23);
/*
* Multiply t by log10(2) to get the final result.
*
* log10(2) is approximately 643/2136 We divide before
* multiplying to avoid overflow.
*/
return t / 2136 * 643;
}
#ifdef _KERNEL
MODULE(MODULE_CLASS_MISC, dtv_math, NULL);
static int
dtv_math_modcmd(modcmd_t cmd, void *opaque)
{
if (cmd == MODULE_CMD_INIT || cmd == MODULE_CMD_FINI)
return 0;
return ENOTTY;
}
#endif
#ifdef TEST_DTV_MATH
/*
* To test:
* cc -DTEST_DTV_MATH ./dtv_math.c -lm -o ./a.out && ./a.out
*/
#include <stdio.h>
#include <inttypes.h>
#include <math.h>
int
main(void)
{
uint32_t xlist[] = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13,
14, 15, 16, 20, 24, 30, 40, 60, 100, 1000, 10000,
100000, 1000000, 10000000, 100000000, 1000000000,
0xffffffff};
int i;
printf("%11s %11s %11s %11s %s\n",
"x", "desired", "actual", "err_abs", "err_rel");
for (i = 0; i < __arraycount(xlist); i++)
{
uint32_t x = xlist[i];
uint32_t desired = (uint32_t)(log10((double)x)
* (double)(1<<24));
uint32_t actual = dtv_intlog10(x);
int32_t err_abs = actual - desired;
double err_rel = (err_abs == 0 ? 0.0
: err_abs / (double)actual);
printf("%11"PRIu32" %11"PRIu32" %11"PRIu32
" %+11"PRId32" %+.5f\n",
x, desired, actual, err_abs, err_rel);
}
return 0;
}
#endif /* TEST_DTV_MATH */