It may or may not contain all the variables I have functions for.

Herein lieth the nub of your problem. To avoid the "Use of uninitialized ..." you need to provide a default value for your variables, instead of just letting them be undef.

However, this is non-trivial in your case. I recommend detecting any variables used by your expression, and populating %value_hash with default values. Warning: untested code

foreach ($function_hash{$foo} =~ /\$(\w+)/g) {
    $value_hash{$1} = 0;  # default value
}
[download]

hth

--rW