Here's how I would do it:

use warnings;
use strict;

my $number = shift;

my $error = $number+1;
my $denom = 1;

while ($error) {
    my $num = int($number * $denom + .5);
    if (abs($num/$denom - $number) < $error) {
        $error = abs($num/$denom - $number);
        print "$num/$denom\t\terr=$error\n";
    }
    exit if ++$denom > 999;
}
[download]

Sample output:

$ perl fract.pl 3.14159265358979323
3/1             err=0.141592653589793
13/4            err=0.108407346410207
16/5            err=0.0584073464102071
19/6            err=0.0250740130768734
22/7            err=0.00126448926734968
179/57          err=0.00124177639681067
201/64          err=0.000967653589793116
223/71          err=0.000747583167258092
245/78          err=0.000567012564152147
267/85          err=0.000416183001557879
289/92          err=0.000288305763706198
311/99          err=0.000178512175651679
333/106         err=8.32196275291075e-05
355/113         err=2.66764189404967e-07
[download]

That last (355/113) is actually an extremely good value; you don't get better until:

52163/16604             err=2.66213257216208e-07
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