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;
        }
    }
}
[download]