#!/usr/bin/perl

use strict;  
use warnings;
use Math::BigInt;

use constant FAC_MAX => 200;

sub pretty_print;

my $three = {meth      => "init",
             value     => Math::BigInt->new(3), 
             old_value => Math::BigInt->new(3)};

my @list  = ($three);
my %done  = map {$_->{value} => $_} @list;

while (@list) {
    my $node = shift @list;
    
    for my $meth (qw /bsqrt bfac/) {
        next if $meth eq 'bfac' && $node->{value} > FAC_MAX;
        my $new_value = $node->{value}->copy->$meth;
        unless ($done{$new_value}) {
            my $new = {meth      => $meth,
                       value     => $new_value,
                       old_value => $node->{value}};
            push @list, $new;
            $done{$new_value} = $new;
        }
    }
    
    my $c = 0;
    $done{$_} && $c++ for 1 .. 10;
    if ($c >= 10) {last}
}

for my $n (1 .. 10) {
    printf "%2d = ", $n;
    pretty_print $done{$n};
    print "\n";  
}

             
sub pretty_print {
    my $n = shift;
    my $m = $n->{meth};
    if ($m eq "bsqrt") {
        print "|sqrt(";
        pretty_print($done{$n->{old_value}});
        print ")|";
    }
    elsif ($m eq "bfac") {
        print "(";
        pretty_print($done{$n->{old_value}});
        print ")!";
    }
    elsif ($m eq "init") {
        print $n->{value}
    }
    else {die}
}
     
__END__
 1 = |sqrt(3)|
 2 = |sqrt((3)!)|
 3 = 3
 4 = |sqrt(|sqrt(|sqrt(|sqrt(|sqrt(|sqrt((|sqrt(|sqrt(|sqrt(|sqrt(|sqrt(|sqrt(|sqrt(((|sqrt(|sqrt(((3)!)!)|)|)!)!)|)|)|)|)|)|)|)!)|)|)|)|)|)|
 5 = |sqrt(|sqrt(((3)!)!)|)|
 6 = (3)!
 7 = |sqrt(|sqrt(|sqrt(|sqrt(|sqrt(|sqrt((|sqrt(|sqrt(|sqrt(|sqrt((|sqrt(((3)!)!)|)!)|)|)|)|)!)|)|)|)|)|)|
 8 = |sqrt(|sqrt((|sqrt(|sqrt(|sqrt(|sqrt(|sqrt(|sqrt((|sqrt(|sqrt(|sqrt(|sqrt((|sqrt(((3)!)!)|)!)|)|)|)|)!)|)|)|)|)|)|)!)|)|
 9 = |sqrt(|sqrt(|sqrt(|sqrt(|sqrt((|sqrt(|sqrt(|sqrt(|sqrt(|sqrt(|sqrt(|sqrt(|sqrt((|sqrt((|sqrt(|sqrt((|sqrt(|sqrt(|sqrt(|sqrt(|sqrt(|sqrt((|sqrt(|sqrt(|sqrt(|sqrt((|sqrt(((3)!)!)|)!)|)|)|)|)!)|)|)|)|)|)|)!)|)|)!)|)!)|)|)|)|)|)|)|)|)!)|)|)|)|)|
10 = |sqrt((|sqrt(|sqrt(((3)!)!)|)|)!)|