Let's say that I have a multi-variate arithmetical expression $A(x_1, \ldots, x_n)$ that uses addition and multiplication operations and is also in a very simple form of sums of products, e.g., $A_1(x_1, x_2) = x_1\cdot x_1 + x_1\cdot x_2$.
Let's also say that the size of an expression is the number of addition/multiplication operators that appear in that expression. For example, size of $A_1(x_1, x_2)$ is three.
Now, I want to find another expression (also only using addition and/or multiplication) that is equivalent to the given expression $A$ and is of the minimum size. For example, formula $A_2(x_1, x_2) = x_1\cdot(x_1+x_2)$ of size two would be the answer to input $A_1$.
My question is about the complexity of finding $A_2$ from $A_1$. The decision version of this problem (does there exist such $A_2$ of size at most $k$) is obviously in NP but is it NP-complete?
If yes, then do we have any good approximation algorithm for it? Or other heuristic algorithms?
If not, what is the best known algorithm to solve it?