This is an algorithm that converges very quickly on pi. Precision allowing, the number of 'good' pi digits doubles per iteration of the algorithm.

#!/usr/bin/perl
use strict;

use constant ITERS => 6;

use Math::BigFloat;

my $x = Math::BigFloat->new("2.0")->fsqrt;
my $p = $x + 2;
my $y = $x->fsqrt;

for(1 .. ITERS) {
        my $xroot = $x->fsqrt;
        $x = ($xroot + 1/$xroot)/2;
        $xroot = $x->fsqrt;
        $p = $p * (($x + 1)/($y + 1));
        $y = (($y * $xroot) + (1/$xroot))/($y + 1);

        print "$p\n\n";
}

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In reply to Arithmetic-Geometric Mean calculation of pi. by rpc

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