We report on a model for CGAL’s AlgebraicKernelWithAnalysis_d_2 concept which refines AlgebraicKernel_d_2. Our implementation handles bivariate polynomials in full generality, ie, with no restriction on their degree. Moreover, it allows both integers and nested square-root numbers as coefficient type. The Curve analysis and Curve pair analysis required by the concept are realized using recent work of Eigenwillig, Kerber and Wolpert (’Fast and exact geometric analysis…’, ISSAC 2007 and’Exact and efficient 2D-Arrangements…’. SODA 2008). The consequent use of certified numerical methods leads to significant speed-ups without spoiling exactness. We present benchmark results about the performance of several key methods.