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use List::Util qw( reduce );

...
    my ( $str ) = @_;
    reduce { mul2( $a, $b ) } "", split //, $str;
}

sub mul2 {
    my ( $str, $have_overflow ) = @_;
...

    ( $have_overflow ? 1 : '' ) . $ret;
}

sub mul2 {
    my ( $str, $have_overflow ) = @_;
...
        ( $c % 10 ) or ( length() ? 0 : '' );
    } reverse '', split //, $str;
}