Simplification has been long recognised to be a fundamental problem within computer algebra\cite{Moses71}. However, even for the class of elementary functions, it has not been resolved in a satisfactory way. Algorithms were presented in \cite{BradfordDavenport2002,Beaumontetal2003} to solve this problem, and it was seen that both methods had their own strengths and weaknesses. Also, not all functions could be handled by either of the methods alone. The current paper continues this line of development by combining the two methods, and reporting on progress made with the various sub-algorithms involved.