# Sum in counting satisfying assignments

Is there a polynomial-time algorithm that computes the sum of two boolean formulas, such that, (#SUM(F,G) = #F + #G), the output satisfying assignments equals the sum of the satisfying assignments of each formula?

The reference that have been working with is the Computational Complexity: A Modern Approach book, in which the idea of setting it as a CNF-SAT problem to sure it is #P complete, thus the number of satisfying assignments for f(x) is equal to the number of certificates for x. I understand that maybe am approaching it wrong but the most similar theorem found is the Valiant and can not seem to find arithmetic related counting complexity.

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$$(x \land F) \lor (\lnot x \land G)$$