use strict;
use warnings;

sub is_valid_expr {
   use re 'eval';
   local our ($skip, $term, $expr);
   $skip = qr! \s* !x;
   $term = qr! $skip [a-zA-Z]+
             | $skip [1-9][0-9]*
             | $skip \( (??{ $expr }) $skip \) !x;
   $expr = qr! $term (?: $skip [-+*/] $term )* !x;
   return $_[0] =~ / ^ $expr $skip \z /x;
}

foreach (
   '(a + 3)',
   '(3 * 4)+(b + x)',
   '(5 - a)*z',
   '3 + 2',

   '!3 + 2',
   '3 + 2!',

   '3 a',
   '3 3',
   '3 * * 3',

   '2 - 3 * 4',  # Preceedence test.
   '2 - 3 + 4',  # Associativity test.
) {
   print(is_valid_expr($_)
      ? "$_ is valid\n"
      : "$_ is not valid\n"
   );
}

##</code><code>##

sub is_valid_expr {
   local our $expr;
   $expr = qr!(?:\s*[a-zA-Z]+|\s*[1-9][0-9]*|\s*\((??{$expr})\s*\))(?:\s*[-+*/](?:\s*[a-zA-Z]+|\s*[1-9][0-9]*|\s*\((??{$expr})\s*\)))*!;
   return $_[0] =~ /^$expr\s*\z/;
}