kashiの日記

変なベンチマーク?

a = 123;

for (j = 0; j< 1000000000; j++) {
        a = (a * a - a) / (a + a);
}

        a = (a - 1) /2

x = 0.5; // 0.4 < x < 0.6

for (i=0; i<1000000000; i++) {
        x = 1 / (x * (x - 1)) + 4.6;
}

いろんな言語でベンチマーク

C++ (gcc)

#include <iostream>

int main()
{
        int i;
        double x;

        x = 0.5; // 0.4 < x < 0.6

        for (i=0; i<1000000000; i++) {
                x = 1 / (x * (x - 1)) + 4.6;
        }

        std::cout << x << std::endl;
}

$ c++ --version
c++ (Ubuntu 9.3.0-17ubuntu1~20.04) 9.3.0
Copyright (C) 2019 Free Software Foundation, Inc.
This is free software; see the source for copying conditions.  There is NO
warranty; not even for MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.

$ c++ -O3 bench.cc
$ time ./a.out 
0.487308

real    0m6.065s
user    0m5.964s
sys     0m0.000s

C++ (clang)

$ clang++ --version
clang version 10.0.0-4ubuntu1 
Target: x86_64-pc-linux-gnu
Thread model: posix
InstalledDir: /usr/bin
$ clang++ -O3 bench.cc
$ time ./a.out 
0.487308

real    0m5.964s
user    0m5.859s
sys     0m0.008s

Java

class bench {
	public static void main(String[] args)
	{
		int i;
		double x;
		x = 0.5;
		long startTime = System.nanoTime();
		for (i=0; i<1000000000; i++) {
			x = 1 / (x * (x - 1)) + 4.6;
		}
		long endTime = System.nanoTime();
		System.out.println(x);
		System.out.println((endTime - startTime) / 1000000000.);
	}
}

$ java --version
openjdk 11.0.13 2021-10-19
OpenJDK Runtime Environment (build 11.0.13+8-Ubuntu-0ubuntu1.20.04)
OpenJDK 64-Bit Server VM (build 11.0.13+8-Ubuntu-0ubuntu1.20.04, mixed mode, sharing)
$ javac bench.java
$ java bench 
0.48730753067792154
5.940579694

fortran

program bench
	implicit none
	double precision :: x
	integer :: i
	x = 0.5d0
	do i=1,1000000000
		x = 1 / (x * (x - 1)) + 4.6d0
	end do
	print *, x
end program bench

$ gfortran --version
GNU Fortran (Ubuntu 9.3.0-17ubuntu1~20.04) 9.3.0
Copyright (C) 2019 Free Software Foundation, Inc.
This is free software; see the source for copying conditions.  There is NO
warranty; not even for MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.

$ gfortran -O3 bench.f90
$ time ./a.out 
  0.48730753067792154     

real    0m6.154s
user    0m6.045s
sys     0m0.004s

nim

var x = 0.5
for i in countup(1, 1000000000):
  x = 1 / (x * (x-1)) + 4.6
echo(x)

$ nim --version
Nim Compiler Version 1.0.6 [Linux: amd64]
Compiled at 2020-02-27
Copyright (c) 2006-2019 by Andreas Rumpf

active boot switches: -d:release
$ nim c -d:release bench.nim
Hint: used config file '/etc/nim/nim.cfg' [Conf]
Hint: system [Processing]
Hint: widestrs [Processing]
Hint: io [Processing]
Hint: bench [Processing]
CC: stdlib_formatfloat.nim
CC: stdlib_io.nim
CC: stdlib_system.nim
CC: bench.nim
Hint:  [Link]
Hint: operation successful (14486 lines compiled; 3.525 sec total; 15.961MiB pea
kmem; Release Build) [SuccessX]
$ time ./bench 
0.4873075306779215

real    0m6.029s
user    0m5.912s
sys     0m0.000s

julia

function hoge(x)
        for i=1:1000000000
                x = 1 / (x * (x-1)) + 4.6
        end
        return x
end

@time print(hoge(0.5))

$ julia --version
julia version 1.4.1
$ julia bench.jl
0.48730753067792154  6.103540 seconds (500.26 k allocations: 22.774 MiB)

python

import time

def hoge(x, n):
	for i in range(n):
		x = 1 / (x * (x - 1)) + 4.6;
	return x;

s = time.perf_counter()
print(hoge(0.5, 1000000000))
e = time.perf_counter()
print(e - s)

$ python --version
Python 3.8.10
$ python bench.py
0.48730753067792154
75.22909699298907

python + numba

from numba import jit
import time

@jit
def hoge(x, n):
	for i in range(n):
		x = 1 / (x * (x - 1)) + 4.6;
	return x;

s = time.perf_counter()
print(hoge(0.5, 1000000000))
e = time.perf_counter()
print(e - s)

$ python bench2.py
0.48730753067792154
6.189143533993047

perl

$x = 0.5;
for ($i = 0; $i < 1000000000; $i++) {
	$x = 1 / ($x * ($x - 1)) + 4.6;
}
print $x;

$ perl --version

This is perl 5, version 30, subversion 0 (v5.30.0) built for x86_64-linux-gnu-thread-multi
(with 50 registered patches, see perl -V for more detail)

Copyright 1987-2019, Larry Wall

Perl may be copied only under the terms of either the Artistic License or the
GNU General Public License, which may be found in the Perl 5 source kit.

Complete documentation for Perl, including FAQ lists, should be found on
this system using "man perl" or "perldoc perl".  If you have access to the
Internet, point your browser at http://www.perl.org/, the Perl Home Page.

$ time perl bench.pl
0.487307530677922
real    1m26.594s
user    1m25.330s
sys     0m0.029s

ruby

x = 0.5
for i in 1..1000000000
	x = 1 / (x * (x - 1)) + 4.6
end
print x, "\n"

$ ruby --version
ruby 2.7.0p0 (2019-12-25 revision 647ee6f091) [x86_64-linux-gnu]
$ time ruby bench.rb 
0.48730753067792154

real    1m13.829s
user    1m12.629s
sys     0m0.052s

lua

x = 0.5
for i=1,1000000000 do
        x = 1 / (x * (x - 1)) + 4.6
end
print(x)

$ lua -v
Lua 5.3.3  Copyright (C) 1994-2016 Lua.org, PUC-Rio
$ time lua bench.lua
0.48730753067792

real    0m36.561s
user    0m36.033s
sys     0m0.009s

luajit

$ luajit -v
LuaJIT 2.1.0-beta3 -- Copyright (C) 2005-2017 Mike Pall. http://luajit.org/
$ time luajit bench.lua
0.48730753067792

real    0m5.958s
user    0m5.863s
sys     0m0.001s

awk

BEGIN {
	x = 0.5;
	for (i=1; i<=1000000000; i++) {
		x = 1 / (x * (x - 1)) + 4.6;
	}
	print x;
}

$ awk --version
GNU Awk 5.0.1, API: 2.0 (GNU MPFR 4.0.2, GNU MP 6.2.0)
Copyright (C) 1989, 1991-2019 Free Software Foundation.

This program is free software; you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation; either version 3 of the License, or
(at your option) any later version.

This program 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 General Public License for more details.

You should have received a copy of the GNU General Public License
along with this program. If not, see http://www.gnu.org/licenses/.
$ time awk -f bench.awk
0.487308

real    1m37.501s
user    1m35.837s
sys     0m0.095s

php

<?php
$x = 0.5;
for ($i=1; $i<=1000000000; $i++) {
	$x = 1/($x * ($x - 1)) + 4.6;
}
echo $x, "\n";
?>

$ php --version
PHP 7.4.3 (cli) (built: Oct 25 2021 18:20:54) ( NTS )
Copyright (c) The PHP Group
Zend Engine v3.4.0, Copyright (c) Zend Technologies
    with Zend OPcache v7.4.3, Copyright (c), by Zend Technologies
$ time php bench.php
0.48730753067792

real    0m16.971s
user    0m16.491s
sys     0m0.048s

octave

1;
function x = hoge(x)
	for i=1:1000000000
		x = 1 / (x*(x-1)) +4.6;
	end
end
tic; x = hoge(0.5); toc
x

$ octave --version
GNU Octave, version 5.2.0
Copyright (C) 2020 John W. Eaton and others.
This is free software; see the source code for copying conditions.
There is ABSOLUTELY NO WARRANTY; not even for MERCHANTABILITY or
FITNESS FOR A PARTICULAR PURPOSE.

Octave was configured for "x86_64-pc-linux-gnu".

Additional information about Octave is available at https://www.octave.org.

Please contribute if you find this software useful.
For more information, visit https://www.octave.org/get-involved.html

Read https://www.octave.org/bugs.html to learn how to submit bug reports.
$ octave bench.m
Elapsed time is 1434.74 seconds.
x =  0.48731

いろんな言語でベンチマークまとめ

• IEEE754には従っているようで、試したすべての言語で同じ結果が得られた。
• コンパイラ言語、もしくはJITありのインタプリタは大体6秒前後。
• JITなしのインタプリタは千差万別。地味にphpが速い、次にlua、有名インタプリタ勢は大体1分強。octaveはなにかおかしい。

倍精度以外で計算させてみる

単精度(float)

#include <iostream>

int main()
{
        int i;
        float x;

        x = 0.5f; // 0.4 < x < 0.6

        for (i=0; i<1000000000; i++) {
                x = 1.0f / (x * (x - 1.0f)) + 4.6f;
        }

        std::cout << x << std::endl;
}

$ c++ -O3 float32.cc 
$ time ./a.out 
0.596567

real    0m5.460s
user    0m5.371s
sys     0m0.004s

Intelの80bit拡張倍精度

#include <iostream>
#include <cmath>

int main()
{
	int i;
	_Float64x x;

	x = 0.5; // 0.4 < x < 0.6

	for (i=0; i<1000000000; i++) {
		x = 1 / (x * (x - 1)) + 4.6;
	}

	std::cout << x << std::endl;
}

$ c++ -O3 float64x.cc
$ time ./a.out 
0.447779

real    0m6.436s
user    0m6.325s
sys     0m0.008s

double-double (qdライブラリ)

#include <iostream>
#include <qd/qd_real.h>

int main()
{
	unsigned int oldcw;
	fpu_fix_start(&oldcw);

	int i;
	dd_real x;

	x = 0.5;
	for (i=0; i<1000000000; i++) {
		x = 1 / (x * (x - 1)) + 4.6;
	}

	std::cout << x << "\n";
	
	fpu_fix_end(&oldcw);
}

$ c++ -O3 qd.cc -lqd
$ time ./a.out 
5.343242e-01

real    0m48.568s
user    0m48.047s
sys     0m0.008s

double-double (kvライブラリ)

#include <kv/dd.hpp>

int main()
{
        int i;
        kv::dd x;

        x = 0.5; // 0.4 < x < 0.6

        for (i=0; i<1000000000; i++) {
                x = 1 / (x * (x - 1)) + 4.6;
        }

        std::cout << x << std::endl;
}

$ c++ -O3 dd.cc
$ time ./a.out 
0.599108

real    0m37.365s
user    0m36.737s
sys     0m0.012s

$ c++ -O3 -DKV_USE_TPFMA dd.cc
$ time ./a.out 
0.599108

real    0m30.161s
user    0m29.656s
sys     0m0.009s

float128

#include <iostream>
#include <boost/multiprecision/float128.hpp> 

int main()
{
	int i;
	boost::multiprecision::float128 x;

	x = 0.5; // 0.4 < x < 0.6

	for (i=0; i<1000000000; i++) {
		x = 1 / (x * (x - 1)) + 4.6;
	}

	std::cout << x << std::endl;
}

$ c++ -O3 float128.cc -lquadmath
$ time ./a.out 
0.547514

real    1m41.450s
user    1m39.683s
sys     0m0.028s

mpfr

#include <kv/mpfr.hpp>

int main()
{
        int i;
        kv::mpfr<53> x;

        x = 0.5; // 0.4 < x < 0.6

        for (i=0; i<1000000000; i++) {
                x = 1 / (x * (x - 1)) + 4.6;
        }

        std::cout << x << std::endl;
}

$ c++ -O3 mpfr53.cc -lmpfr
$ time ./a.out 
0.487308

real    3m54.608s
user    3m50.821s
sys     0m0.044s
$ c++ -O3 mpfr64.cc -lmpfr
$ time ./a.out 
0.447779

real    5m16.806s
user    5m11.236s
sys     0m0.111s
$ c++ -O3 mpfr106.cc -lmpfr
$ time ./a.out 
0.576081

real    5m37.827s
user    5m32.035s
sys     0m0.111s
$ c++ -O3 mpfr113.cc -lmpfr
$ time ./a.out 
0.547514

real    5m42.499s
user    5m36.300s
sys     0m0.241s

倍精度以外の計算速度まとめ

変なベンチマークの力学系的性質

• dd-interval.pdf
• inflate.pdf

#include <kv/interval.hpp>
#include <kv/dd.hpp>
#include <kv/rdd.hpp>

int main(){
        kv::interval<kv::dd> x;
        std::cout.precision(34);

        x = std::numeric_limits<kv::dd>::max();

        std::cout << x << "\n";
        std::cout << sqrt(x) << "\n";
}

[1.340780792994259561100831764080293e+154,1.340780792994259672743259815885529e+154]

[1.340780792994259672743259815885478e+154,1.340780792994259672743259815885529e+154]

はじめに

精度保証せずに普通に計算

#include <iostream>
#include <cmath>

int main()
{
	double s, tmp;
	long long int i;
	std::cout.precision(17);

	s = 0;
	i = 1;
	while (true) {
		tmp = s + 1.0 / i;
		if (floor(tmp) != floor(s)) {
			std::cout << i-1 << ":" << s << "\n";
			std::cout << i << ":" << tmp << "\n";
		}
		s = tmp;
		i++;
	}
}

区間演算してみる

#include <kv/interval.hpp>
#include <kv/rdouble.hpp>
#include <kv/dd.hpp>
#include <kv/rdd.hpp>

typedef kv::interval<double> itv;
// typedef kv::interval< kv::dd > itv;

int main()
{
	itv s, tmp;
	long long int i;
	std::cout.precision(17);

	s = 0;
	i = 1;
	while (true) {
		tmp = s + 1.0 / itv(i);
		if (floor(tmp.lower()) != floor(tmp.upper())) {
			std::cout << "warning\n";
			std::cout << i-1 << ":" << s << "\n";
			std::cout << i << ":" << tmp << "\n";
			break;
		} else if (floor(tmp.lower()) != floor(s.lower())) {
			std::cout << i-1 << ":" << s << "\n";
			std::cout << i << ":" << tmp << "\n";
		}
		s = tmp;
		i++;
	}
}

double-double区間演算で頑張る

もうちょっと数学っぽく

#include <iostream>
#include <kv/interval.hpp>
#ifdef DD
#include <kv/dd.hpp>
#include <kv/rdd.hpp>
#endif
#include <kv/gamma.hpp>

#ifdef DD
typedef kv::interval<kv::dd> itv;
#define MAX 66
#else
typedef kv::interval<double> itv;
#define MAX 32
#endif

int main() {
	#ifdef DD
	std::cout.precision(34);
	#else
	std::cout.precision(17);
	#endif

	itv gamma, x, y;
	int i;

	gamma = - kv::digamma(itv(1.));

	for (i=2; i<=MAX; i++) {
		std::cout << i << "\n";

		x = i;
		y = floor(mid(exp(x - gamma) + 0.5));
		std::cout << y.lower() << "\n";
		std::cout <<  kv::digamma(y) + gamma << "\n";;
		std::cout <<  kv::digamma(y + 1) + gamma << "\n";;
	}
}

• test-rounding.ccで、非正規化数を強制的に0にする"flush to zero"モードになっていないかどうかをチェックするようにした。
• iccでコンパイルできるように修正。
• optimize.hppで、「端にない」区間で傾きが単調ならばそこに最小値は存在しない というチェックを入れる改良を行った。
• defint-newtoncotes.hpp, double-newtoncotes.hppで、最適な分割数の 推定がうまくできていなかったバグを修正。
• defint.hpp, defint-singular.hppで、急減少関数などの特定の被積分関数 に対してstep sizeがうまく計算できていなかった問題を修正。
• psa.hppの中のinv関数を、係数のオーバーフローが発生しづらいように改良。
• gamma.hppの中のdigamma関数の計算アルゴリズムをより高精度なものに変更。
• その他細かい修正。