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Re^2: is it prime?

by tobyink (Abbot)
on May 06, 2012 at 19:41 UTC ( #969167=note: print w/ replies, xml ) Need Help??


in reply to Re: is it prime?
in thread is it prime?

One speedy way is to use the Sieve of Eratosthenes.

Speedy? Benchmarking mine against yours...

NUMBER : TOBYINK JOHNGG NUMBER : RESULT TIME RESULT TIME 75 : NO 0.000065 NO 0.000215 169 : NO 0.000057 NO 0.000649 229 : YES 0.000070 YES 0.000888 367 : YES 0.000073 YES 0.001437 369 : NO 0.000052 NO 0.000805 8794 : NO 0.000051 NO 0.008682 9227 : YES 0.000113 YES 0.031170 10807 : NO 0.000131 NO 0.047135 11939 : YES 0.000121 YES 0.045032 14803 : NO 0.000122 NO 0.054764 19937 : YES 0.000143 YES 0.070692 19938 : NO 0.000040 NO 0.020510 39783 : NO 0.000057 NO 0.065555 47083 : NO 0.000230 NO 0.170125 199933 : YES 0.000346 YES 0.778786

For some of those higher numbers, mine is about 2000 times faster than yours. The sieve is efficient if you need to generate a list of all prime numbers below x, but very slow as a mechanism for determining whether a given number is prime.

Benchmark script:

use 5.010; use strict; use Carp qw/croak/; use Time::HiRes qw/time/; sub isPrime1 (_) { my $num = shift; return if $num == 1; # 1 is not prime croak "usage: isPrime(NATURAL NUMBER)" unless $num =~ /^[1-9][0-9]*$/; for my $div (2 .. sqrt $num) { return if $num % $div == 0; } return 1; } sub isPrime2 { my $toTest = shift; my $sqrtLimit = sqrt $toTest; my $sieve = q{}; vec( $sieve, 0 , 1 ) = 1; vec( $sieve, 1 , 1 ) = 1; vec( $sieve, $toTest, 1 ) = 0; my $marker = 1; while ( $marker < $sqrtLimit ) { my $possPrime = $marker + 1; $possPrime ++ while vec( $sieve, $possPrime, 1 ); my $fill = 2 * $possPrime; while ( $fill <= $toTest ) { vec( $sieve, $fill, 1 ) = 1; $fill += $possPrime; } last if vec( $sieve, $toTest, 1 ); $marker = $possPrime; } return not vec($sieve, $toTest, 1); } say q[ NUMBER : TOBYINK JOHNGG ]; say q[ NUMBER : RESULT TIME RESULT TIME ]; for my $num (qw(75 169 229 367 369 8794 9227 10807 11939 14803 19937 1 +9938 39783 47083 199933)) { printf('%8d : ', $num); timeit($num, $_) for \&isPrime1, \&isPrime2; print "\n"; } sub timeit { my ($n, $f) = @_; my $start = time(); print($f->($n) ? "YES " : "NO "); my $end = time(); printf '%.06f ', ($end - $start); }
perl -E'sub Monkey::do{say$_,for@_,do{($monkey=[caller(0)]->[3])=~s{::}{ }and$monkey}}"Monkey say"->Monkey::do'


Comment on Re^2: is it prime?
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Re^3: is it prime?
by johngg (Abbot) on May 09, 2012 at 09:33 UTC

    I'd only ever used Eratosthenes' Sieve to find a list of all primes up to a limit and, stupidly, assumed that it would be equally good at testing whether a single number was prime. Silly me!

    You have shown your method to be much faster but there's still room for improvement. You reject all even numbers the first time through your loop so there's no point in using even numbers again as divisor. Instead, take the evens test out of the loop then employ a C-style loop to test only odd divisors.

    The code. I've tweaked things to return 0 if not prime in order to make testing easier.

    The benchmark results.

    Testing with value 2 which is a prime ok 1 - eratosthenes ok 2 - tobyink ok 3 - tobyinkPlus Testing with value 8357 which is not a prime ok 4 - eratosthenes ok 5 - tobyink ok 6 - tobyinkPlus Testing with value 9293 which is a prime ok 7 - eratosthenes ok 8 - tobyink ok 9 - tobyinkPlus Rate eratosthenes tobyink tobyinkPlus eratosthenes 0.154/s -- -100% -100% tobyink 1291/s 838266% -- -35% tobyinkPlus 1984/s 1288047% 54% -- 1..9

    I hope this is of interest.

    Cheers,

    JohnGG

      Necroposting, but maybe of interest still.

      First, no surprise, there are modules that will be much faster than doing it yourself, and easier overall. But let's take a look at pure Perl methods. We can improve the SoE method about 2-3x by using the simple vector sieve from RosettaCode. The sieve using a string is faster yet, but let's keep it simple since clearly this is not the right way to go about testing primality. I also included my version of the all, mod-2 wheel, and mod-6 wheel trial division methods. Although going to a mod-30 wheel can improve things a little more and still be reasonable code, I didn't include it. The mod-6 wheel runs about 2x faster than Toby's original code.

      For modules, some choices include:

      • Math::Prime::Util        scads faster than anything else, especially for large inputs. Needs Math::Prime::Util::GMP to be very fast for bigints, though it works fine without. I'm biased, being the author.
      • Math::Prime::XS           trial division with a mod-30 wheel in XS. Very fast for small numbers, slows down rapidly, no bigint support.
      • Math::Pari                  reasonably fast, supports bigints. Since it is based on the ancient Pari 2.1, be careful as it will sometimes give you incorrect results.
      • Math::Primality           works, but quite slow as it is designed for bigints
      • Math::Prime::FastSieve  if your range is small (e.g. under 1000M) and you have lots of numbers to test, this can work well by doing an efficient sieve in XS then do fast bit tests to determine primality.
      For generation, both Math::Prime::Util and Math::Prime::FastSieve should be much faster than trying to download a list from the net, as a previous suggestion mentioned, and probably faster than loading from disk as well.

      Code:

      Timing results:

      Validating tests on small primes Validating tests on small composites TOBYINK DIV2 DIV6 MPU NUMBER : RES TIME RES TIME RES TIME RES TIME 75 : N 0.000004 N 0.000002 N 0.000001 N 0.000002 169 : N 0.000002 N 0.000002 N 0.000001 N 0.000000 229 : Y 0.000004 Y 0.000001 Y 0.000001 Y 0.000000 367 : Y 0.000004 Y 0.000002 Y 0.000002 Y 0.000000 369 : N 0.000002 N 0.000001 N 0.000001 N 0.000000 8794 : N 0.000002 N 0.000001 N 0.000001 N 0.000000 9227 : Y 0.000008 Y 0.000005 Y 0.000004 Y 0.000000 10807 : N 0.000008 N 0.000005 N 0.000004 N 0.000000 11939 : Y 0.000009 Y 0.000005 Y 0.000005 Y 0.000000 14803 : N 0.000009 N 0.000005 N 0.000004 N 0.000000 19937 : Y 0.000011 Y 0.000006 Y 0.000005 Y 0.000001 19938 : N 0.000002 N 0.000001 N 0.000001 N 0.000001 39783 : N 0.000002 N 0.000001 N 0.000001 N 0.000000 47083 : N 0.000014 N 0.000008 N 0.000006 N 0.000000 199933 : Y 0.000030 Y 0.000017 Y 0.000014 Y 0.000008 75640351 : Y 0.000554 Y 0.000311 Y 0.000246 Y 0.000001 760149189769 : Y 0.056347 Y 0.031018 Y 0.024714 Y 0.000011 635921898906263 : Y 1.605829 Y 0.904163 Y 0.721130 Y 0.000004

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