/* $NetBSD: dtv_math.c,v 1.5 2011/08/09 01:42:24 jmcneill Exp $ */ /*- * Copyright (c) 2011 Alan Barrett * 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 __KERNEL_RCSID(0, "$NetBSD: dtv_math.c,v 1.5 2011/08/09 01:42:24 jmcneill Exp $"); #include #include #include #include /* * 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 #include #include 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 */