schlitt.info - php, photography and private stuff
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:Author: 		Tobias Schlitt
:Date:			Thu, 20 Nov 2008 12:34:46 +0100
:Revision: 		2
:Copyright:		CC by-nc-sa


=========================
Randomized Pi calculation
=========================

:Description:
    I don't know which is the most common way to calculate Pi in computer
    programs, but from the stochastics book a read for my recent stochastics
    exam, I have a randomized variation, which is quite cool I think. To show
    it, I implemented it in PHP:

I don't know which is the most common way to calculate Pi in computer
programs, but from the stochastics book a read for my recent stochastics exam,
I have a randomized variation, which is quite cool I think. To show it, I
implemented it in PHP:

::

    define( "ITERATIONS", 100 ); 
    $precision = pow( 10, (int) ini_get( "precision" ) - 5 );
    
    $hits = 0; 
    for ( $i = 0; $i < ITERATIONS; $i++ )
    {
        $x = mt_rand( 0, 2 * $precision ) / $precision - 1; 
        $y = mt_rand( 0, 2 * $precision ) / $precision - 1; 
        if ( ( $x * $x ) + ( $y * $y ) < 1 )
        {  
            $hits++;
        }  
    }
    echo "Pi is about " . ( ( 4 * $hits ) / ITERATIONS );

The basic assumption in this algorith is, that the generated coordinates are
rectangular distributed in the rect (-1/-1) - (1,1). For the numbers I
generate using mt_rand() this should be almost correct (note that the
generation is not part of the algorith itself, it only says, that you need
rectengular distributed points). For each of this generated points, a check is
performed, if the point is located inside the unit circle. If it is, a hit is
recorded. The relative frequence for this event is exactly the ratio between
the unit circle and the whole square, which means it is Pi/4.

The algorith is not really fast, but gives usable results for a large number
of iterations:

.. csv-table::

    "Iterations", "Pi estimation", "Time (sec.)"
    "100", "3.2", "0.000174045562744"
    "1000", "3.128", "0.00145411491394"
    "10000", "3.1436", "0.0180418491364"
    "100000", "3.13876", "0.16107583046"
    "1000000", "3.143652", "1.64533686638"
    "10000000", "3.1423896", "18.0556662083"

**Update**, *2007-04-02*: Note, that **you should never use this algorithm to
calculate Pi in PHP**! There is the predefined constant M_PI, as well as the
function pi() to retrieve a value with the precision specified in your
php.ini! This is just a cool algorith and it's really slow (in comparison of
using the constant)!


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   vim: et syn=rst tw=79

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Comments
========

- Joshua E Cook at Mon, 02 Apr 2007 19:17:26 +0200

  Why not use the constant M_PI or the builtin function pi()?  I don't think
  it would be common to need more precision than what they provide.

- Toby at Mon, 02 Apr 2007 22:40:54 +0200

  That is, why I did not post this into my PHP category. :) For realworld
  you'd never want to calculate Pi yourself, but always use the predefined
  constants. This is just a very cool algorithm for this calculation. :)