If you are trying to solve a permutation problem (which is what it sounds like to me), I think you are applying yourself incorrectly. Rather than figuring out how to map your answer to their answer, your parser should generate a canonical read that will necessarily map to yours. For your simple case, it might look like:

my $eq = {op => '+',
          terms => [1,
                    'x',
                    ],
         };
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where a canonical sorting algorithm is used to order terms. Note that, if you want to accept complex expressions, the sort is non-trivial since you'll need to rank complex references, so cmp won't be enough. This type of format also supports nested operations:

my $eq = {op => '+',
          terms => [{  op => '*',
                       terms => [3,
                                 'z',
                                ],
                    },
                    'x',
                    ],
         };
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You can then do a recursive descent in order to check equivalence. Of course, this isn't going to help you with distributivity.