Given an elementary function with algebraic branch cuts, we show how to decide which sheet of the Riemann surface we are on at any given point. We do this by establishing a correspondence between the Cylindrical Algebraic Decomposition (CAD) of the complex plane defined by the branch cuts and a finite subset of sheets of the Riemann surface. The key advantage is that we no longer have to deal with the difficult `constant problem', yet we can work completely symbolically. This also allows us to handle a larger class of expressions than in previously reported work by the authors.