Is there a reverse of Baker-Campbell-Hausdorff formula?

Question

The BCH formula says that if $\exp(A)\exp(B) = \exp(C)$ where $C$ is a series in nested commutators of $A$ and $B$, $$C = A + B + \frac{1}{2}[A,B] + \frac{1}{12}\Big(\big[A,[A,B]\big] + \big[B,[B,A]\big]\Big) + \cdots.$$ I was wondering if there was a way to read the equation backwards, where if you start with one exponential, $$\exp(A+B)$$ then is it it possible to get some expression with two (or more?) exponentials, for example $$\exp(A+B) = \exp(A)\exp(B)\exp(D).$$ I imagine that the extra $\exp(D)$ could very well be in front instead in the back, or it could be in the middle, or perhaps there are 3 extra factors that go in all 3 places. If there is, is there a specific formula for the extra factors in terms of $A$ and $B$?