Specifically, here's an example of a program trapping the "die", and continuing if desired. Note the eval which ambrus and cowboy mentioned, wrapping the entire while clause:

use strict;
use warnings;

# Globals
my $num = 1_000_000_000;

# Signal handling
$SIG{__DIE__} = sub { trap_otherwise_fatals() };

# Main program
sure_to_fail();
print "Continuing anyway\n";
# ... more code here if desired ...

# Subroutines
sub trap_otherwise_fatals {
    print "** Would have died at num = $num **\n";
}

sub sure_to_fail {
    eval {
        while (1) {
            my $log = log($num);
            printf "Log(%12d) = %10.6f\n", $num, $log;
            $num = $log;
        }
    }
}
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If you want to capture an error in such a way to continue execution, $SIG{__DIE__} won't work, you have to use eval{} instead.

Try using eval{} around the code that may die. You can even wrap your entire program in an eval if need be.

I use this under mod perl to trap any 500 errors that may occur alerting the relevant people that something is wrong, as well as other eval's around code that may die, but if it does, I can deal with the error and continue.

{
    my $realdie = \&CORE::GLOBAL::die;
    local *CORE::GLOBAL::die = sub {
        $realdie->(@_) if $something;
        print LOG @_
    };
    ## do_stuff
}
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