Call a function $h: \Bbb R^2 \to \Bbb R$ simple if there are functions $f,g: \Bbb R \to \Bbb R $ such that $h(x,y) = f(x)g(y)$. In fancy words, $h$ factors under the tensor product.
Is the function $h(x,y) = \sin(xy)$ equal to a finite sum of simple functions?
The Taylor expansion of $h$ gives an infinite sum of simple functions equal to $h$. Testing for simplicity can be done by diving by the partial derivatives and checking if the result is independent of one variable.