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use warnings;
use strict;
...
    }
    exit if ++$denom > 999;
}

$ perl fract.pl 3.14159265358979323
3/1             err=0.141592653589793
...
311/99          err=0.000178512175651679
333/106         err=8.32196275291075e-05
355/113         err=2.66764189404967e-07

52163/16604             err=2.66213257216208e-07