One way to simplify your test is to get rid of the extra map:

Usage() && exit if grep{not exists $args->{$_}}@mandatory;
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An alternate scenario could involve Getopt::Declare, which allows you to specify if an option is required or not (negating the need for this particular test). For example:

#!/usr/bin/perl -w
use strict;
use Getopt::Declare;

my $args = Getopt::Declare->new(<<'SPEC');
    -x <x:n>        x coord (real) [required]
    -y <y:n>        y coord (real) [required]
    -z <z:n>        z coord (real)
                        {print "Processing in three dimensions\n"}
SPEC

#rest of program...

__END__
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Running the above without -x or -y options causes an error message regarding the missing option. For free you also get -help which will print out a usage statement (for free you'll also get a slight speed hit on loading the module). Note: there are hard tabs between the option specs on the left and the option descriptions on the right (see the docs for formal specification requirements).