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Copyright 2000, 2001, 2002, 2004 Free Software Foundation, Inc. This file is part of the GNU MP Library. The GNU MP Library is free software; you can redistribute it and/or modify it under the terms of the GNU Lesser General Public License as published by the Free Software Foundation; either version 2.1 of the License, or (at your option) any later version. The GNU MP Library is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License for more details. You should have received a copy of the GNU Lesser General Public License along with the GNU MP Library; see the file COPYING.LIB. If not, write to the Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA. GMP SPEED MEASURING AND PARAMETER TUNING The programs in this directory are for knowledgeable users who want to measure GMP routines on their machine, and perhaps tweak some settings or identify things that can be improved. The programs here are tools, not ready to run solutions. Nothing is built in a normal "make all", but various Makefile targets described below exist. Relatively few systems and CPUs have been tested, so be sure to verify that results are sensible before relying on them. MISCELLANEOUS NOTES --enable-assert Don't configure with --enable-assert, since the extra code added by assertion checking may influence measurements. Direct mapped caches Some effort has been made to accommodate CPUs with direct mapped caches, by putting data blocks more or less contiguously on the stack. But this will depend on TMP_ALLOC using alloca, and even then it may or may not be enough. FreeBSD 4.2 i486 getrusage This getrusage seems to be a bit doubtful, it looks like it's microsecond accurate, but sometimes ru_utime remains unchanged after a time of many microseconds has elapsed. It'd be good to detect this in the time.c initializations, but for now the suggestion is to pretend it doesn't exist. ./configure ac_cv_func_getrusage=no NetBSD 1.4.1 m68k macintosh time base On this system it's been found getrusage often goes backwards, making it unusable (time.c getrusage_backwards_p detects this). gettimeofday sometimes doesn't update atomically when it crosses a 1 second boundary. Not sure what to do about this. Expect possible intermittent failures. SCO OpenUNIX 8 /etc/hw /etc/hw takes about a second to return the cpu frequency, which suggests perhaps it's measuring each time it runs. If this is annoying when running the speed program repeatedly then set a GMP_CPU_FREQUENCY environment variable (see TIME BASE section below). Low resolution timebase Parameter tuning can be very time consuming if the only timebase available is a 10 millisecond clock tick, to the point of being unusable. This is currently the case on VAX and ARM systems. PARAMETER TUNING The "tuneup" program runs some tests designed to find the best settings for various thresholds, like MUL_KARATSUBA_THRESHOLD. Its output can be put into gmp-mparam.h. The program is built and run with make tune If the thresholds indicated are grossly different from the values in the selected gmp-mparam.h then there may be a performance boost in applicable size ranges by changing gmp-mparam.h accordingly. Be sure to do a full reconfigure and rebuild to get any newly set thresholds to take effect. A partial rebuild is enough sometimes, but a fresh configure and make is certain to be correct. If a CPU has specific tuned parameters coming from a gmp-mparam.h in one of the mpn subdirectories then the values from "make tune" should be similar. But check that the configured CPU is right and there are no machine specific effects causing a difference. It's hoped the compiler and options used won't have too much effect on thresholds, since for most CPUs they ultimately come down to comparisons between assembler subroutines. Missing out on the longlong.h macros by not using gcc will probably have an effect. Some thresholds produced by the tune program are merely single values chosen from what's a range of sizes where two algorithms are pretty much the same speed. When this happens the program is likely to give somewhat different values on successive runs. This is noticeable on the toom3 thresholds for instance. SPEED PROGRAM The "speed" program can be used for measuring and comparing various routines, and producing tables of data or gnuplot graphs. Compile it with make speed (Or on DOS systems "make speed.exe".) Here are some examples of how to use it. Check the code for all the options. Draw a graph of mpn_mul_n, stepping through sizes by 10 or a factor of 1.05 (whichever is greater). ./speed -s 10-5000 -t 10 -f 1.05 -P foo mpn_mul_n gnuplot foo.gnuplot Compare mpn_add_n and an mpn_lshift by 1, showing times in cycles and showing under mpn_lshift the difference between it and mpn_add_n. ./speed -s 1-40 -c -d mpn_add_n mpn_lshift.1 Using option -c for times in cycles is interesting but normally only necessary when looking carefully at assembler subroutines. You might think it would always give an integer value, but this doesn't happen in practice, probably due to overheads in the time measurements. In the free-form output the "#" symbol against a measurement means the corresponding routine is fastest at that size. This is a convenient visual cue when comparing different routines. The graph data files <name>.data don't get this since it would upset gnuplot or other data viewers. TIME BASE The time measuring method is determined in time.c, based on what the configured host has available. A cycle counter is preferred, possibly supplemented by another method if the counter has a limited range. A microsecond accurate getrusage() or gettimeofday() will work quite well too. The cycle counters (except possibly on alpha) and gettimeofday() will depend on the machine being otherwise idle, or rather on other jobs not stealing CPU time from the measuring program. Short routines (those that complete within a timeslice) should work even on a busy machine. Some trouble is taken by speed_measure() in common.c to avoid ill effects from sporadic interrupts, or other intermittent things (like cron waking up every minute). But generally an idle machine will be necessary to be certain of consistent results. The CPU frequency is needed to convert between cycles and seconds, or for when a cycle counter is supplemented by getrusage() etc. The speed program will convert as necessary according to the output format requested. The tune program will work with either cycles or seconds. freq.c knows how to get the frequency on some systems, or can measure a cycle counter against gettimeofday() or getrusage(), but when that fails, or needs to be overridden, an environment variable GMP_CPU_FREQUENCY can be used (in Hertz). For example in "bash" on a 650 MHz machine, export GMP_CPU_FREQUENCY=650e6 A high precision time base makes it possible to get accurate measurements in a shorter time. EXAMPLE COMPARISONS - VARIOUS Here are some ideas for things that can be done with the speed program. There's always going to be a certain amount of overhead in the time measurements, due to reading the time base, and in the loop that runs a routine enough times to get a reading of the desired precision. Noop functions taking various arguments are available to measure this. The "overhead" printed by the speed program each time in its intro is the "noop" routine, but note that this is just for information, it isn't deducted from the times printed or anything. ./speed -s 1 noop noop_wxs noop_wxys To see how many cycles per limb a routine is taking, look at the time increase when the size increments, using option -D. This avoids fixed overheads in the measuring. Also, remember many of the assembler routines have unrolled loops, so it might be necessary to compare times at, say, 16, 32, 48, 64 etc to see what the unrolled part is taking, as opposed to any finishing off. ./speed -s 16-64 -t 16 -C -D mpn_add_n The -C option on its own gives cycles per limb, but is really only useful at big sizes where fixed overheads are small compared to the code doing the real work. Remember of course memory caching and/or page swapping will affect results at large sizes. ./speed -s 500000 -C mpn_add_n Once a calculation stops fitting in the CPU data cache, it's going to start taking longer. Exactly where this happens depends on the cache priming in the measuring routines, and on what sort of "least recently used" the hardware does. Here's an example for a CPU with a 16kbyte L1 data cache and 32-bit limb, showing a suddenly steeper curve for mpn_add_n at about 2000 limbs. ./speed -s 1-4000 -t 5 -f 1.02 -P foo mpn_add_n gnuplot foo.gnuplot When a routine has an unrolled loop for, say, multiples of 8 limbs and then an ordinary loop for the remainder, it can happen that it's actually faster to do an operation on, say, 8 limbs than it is on 7 limbs. The following draws a graph of mpn_sub_n, to see whether times smoothly increase with size. ./speed -s 1-100 -c -P foo mpn_sub_n gnuplot foo.gnuplot If mpn_lshift and mpn_rshift have special case code for shifts by 1, it ought to be faster (or at least not slower) than shifting by, say, 2 bits. ./speed -s 1-200 -c mpn_rshift.1 mpn_rshift.2 An mpn_lshift by 1 can be done by mpn_add_n adding a number to itself, and if the lshift isn't faster there's an obvious improvement that's possible. ./speed -s 1-200 -c mpn_lshift.1 mpn_add_n_self On some CPUs (AMD K6 for example) an "in-place" mpn_add_n where the destination is one of the sources is faster than a separate destination. Here's an example to see this. ".1" selects dst==src1 for mpn_add_n (and mpn_sub_n), for other values see speed.h SPEED_ROUTINE_MPN_BINARY_N_CALL. ./speed -s 1-200 -c mpn_add_n mpn_add_n.1 The gmp manual points out that divisions by powers of two should be done using a right shift because it'll be significantly faster than an actual division. The following shows by what factor mpn_rshift is faster than mpn_divrem_1, using division by 32 as an example. ./speed -s 10-20 -r mpn_rshift.5 mpn_divrem_1.32 EXAMPLE COMPARISONS - MULTIPLICATION mul_basecase takes a ".<r>" parameter which is the first (larger) size parameter. For example to show speeds for 20x1 up to 20x15 in cycles, ./speed -s 1-15 -c mpn_mul_basecase.20 mul_basecase with no parameter does an NxN multiply, so for example to show speeds in cycles for 1x1, 2x2, 3x3, etc, up to 20x20, in cycles, ./speed -s 1-20 -c mpn_mul_basecase sqr_basecase is implemented by a "triangular" method on most CPUs, making it up to twice as fast as mul_basecase. In practice loop overheads and the products on the diagonal mean it falls short of this. Here's an example running the two and showing by what factor an NxN mul_basecase is slower than an NxN sqr_basecase. (Some versions of sqr_basecase only allow sizes below SQR_KARATSUBA_THRESHOLD, so if it crashes at that point don't worry.) ./speed -s 1-20 -r mpn_sqr_basecase mpn_mul_basecase The technique described above with -CD for showing the time difference in cycles per limb between two size operations can be done on an NxN mul_basecase using -E to change the basis for the size increment to N*N. For instance a 20x20 operation is taken to be doing 400 limbs, and a 16x16 doing 256 limbs. The following therefore shows the per crossproduct speed of mul_basecase and sqr_basecase at around 20x20 limbs. ./speed -s 16-20 -t 4 -CDE mpn_mul_basecase mpn_sqr_basecase Of course sqr_basecase isn't really doing NxN crossproducts, but it can be interesting to compare it to mul_basecase as if it was. For sqr_basecase the -F option can be used to base the deltas on N*(N+1)/2 operations, which is the triangular products sqr_basecase does. For example, ./speed -s 16-20 -t 4 -CDF mpn_sqr_basecase Both -E and -F are preliminary and might change. A consistent approach to using them when claiming certain per crossproduct or per triangularproduct speeds hasn't really been established, but the increment between speeds in the range karatsuba will call seems sensible, that being k to k/2. For instance, if the karatsuba threshold was 20 for the multiply and 30 for the square, ./speed -s 10-20 -t 10 -CDE mpn_mul_basecase ./speed -s 15-30 -t 15 -CDF mpn_sqr_basecase EXAMPLE COMPARISONS - MALLOC The gmp manual recommends application programs avoid excessive initializing and clearing of mpz_t variables (and mpq_t and mpf_t too). Every new variable will at a minimum go through an init, a realloc for its first store, and finally a clear. Quite how long that takes depends on the C library. The following compares an mpz_init/realloc/clear to a 10 limb mpz_add. Don't be surprised if the mallocing is quite slow. ./speed -s 10 -c mpz_init_realloc_clear mpz_add On some systems malloc and free are much slower when dynamic linked. The speed-dynamic program can be used to see this. For example the following measures malloc/free, first static then dynamic. ./speed -s 10 -c malloc_free ./speed-dynamic -s 10 -c malloc_free Of course a real world program has big problems if it's doing so many mallocs and frees that it gets slowed down by a dynamic linked malloc. EXAMPLE COMPARISONS - STRING CONVERSIONS mpn_get_str does a binary to string conversion. The base is specified with a ".<r>" parameter, or decimal by default. Power of 2 bases are much faster than general bases. The following compares decimal and hex for instance. ./speed -s 1-20 -c mpn_get_str mpn_get_str.16 Smaller bases need more divisions to split a given size number, and so are slower. The following compares base 3 and base 9. On small operands 9 will be nearly twice as fast, though at bigger sizes this reduces since in the current implementation both divide repeatedly by 3^20 (or 3^40 for 64 bit limbs) and those divisions come to dominate. ./speed -s 1-20 -cr mpn_get_str.3 mpn_get_str.9 mpn_set_str does a string to binary conversion. The base is specified with a ".<r>" parameter, or decimal by default. Power of 2 bases are faster than general bases on large conversions. ./speed -s 1-512 -f 2 -c mpn_set_str.8 mpn_set_str.10 mpn_set_str also has some special case code for decimal which is a bit faster than the general case, basically by giving the compiler a chance to optimize some multiplications by 10. ./speed -s 20-40 -c mpn_set_str.9 mpn_set_str.10 mpn_set_str.11 EXAMPLE COMPARISONS - GCDs mpn_gcd_1 has a threshold for when to reduce using an initial x%y when both x and y are single limbs. This isn't tuned currently, but a value can be established by a measurement like ./speed -s 10-32 mpn_gcd_1.10 This runs src[0] from 10 to 32 bits, and y fixed at 10 bits. If the div threshold is high, say 31 so it's effectively disabled then a 32x10 bit gcd is done by nibbling away at the 32-bit operands bit-by-bit. When the threshold is small, say 1 bit, then an initial x%y is done to reduce it to a 10x10 bit operation. The threshold in mpn/generic/gcd_1.c or the various assembler implementations can be tweaked up or down until there's no more speedups on interesting combinations of sizes. Note that this affects only a 1x1 limb operation and so isn't very important. (An Nx1 limb operation always does an initial modular reduction, using mpn_mod_1 or mpn_modexact_1_odd.) SPEED PROGRAM EXTENSIONS Potentially lots of things could be made available in the program, but it's been left at only the things that have actually been wanted and are likely to be reasonably useful in the future. Extensions should be fairly easy to make though. speed-ext.c is an example, in a style that should suit one-off tests, or new code fragments under development. many.pl is a script for generating a new speed program supplemented with alternate versions of the standard routines. It can be used for measuring experimental code, or for comparing different implementations that exist within a CPU family. THRESHOLD EXAMINING The speed program can be used to examine the speeds of different algorithms to check the tune program has done the right thing. For example to examine the karatsuba multiply threshold, ./speed -s 5-40 mpn_mul_basecase mpn_kara_mul_n When examining the toom3 threshold, remember it depends on the karatsuba threshold, so the right karatsuba threshold needs to be compiled into the library first. The tune program uses specially recompiled versions of mpn/mul_n.c etc for this reason, but the speed program simply uses the normal libgmp.la. Note further that the various routines may recurse into themselves on sizes far enough above applicable thresholds. For example, mpn_kara_mul_n will recurse into itself on sizes greater than twice the compiled-in MUL_KARATSUBA_THRESHOLD. When doing the above comparison between mul_basecase and kara_mul_n what's probably of interest is mul_basecase versus a kara_mul_n that does one level of Karatsuba then calls to mul_basecase, but this only happens on sizes less than twice the compiled MUL_KARATSUBA_THRESHOLD. A larger value for that setting can be compiled-in to avoid the problem if necessary. The same applies to toom3 and DC, though in a trickier fashion. There are some upper limits on some of the thresholds, arising from arrays dimensioned according to a threshold (mpn_mul_n), or asm code with certain sized displacements (some x86 versions of sqr_basecase). So putting huge values for the thresholds, even just for testing, may fail. FUTURE Make a program to check the time base is working properly, for small and large measurements. Make it able to test each available method, including perhaps the apparent resolution of each. Make a general mechanism for specifying operand overlap, and a syntax like maybe "mpn_add_n.dst=src2" to select it. Some measuring routines do this sort of thing with the "r" parameter currently. ---------------- Local variables: mode: text fill-column: 76 End: