We present a symbolic perturbation scheme for arbitrary polynomial geometric predicates which combines the benefits of Emiris and Canny’s simple randomized linear perturbation scheme with Yap’s multiple infinitesimal scheme for general predicates. Like the randomized scheme, our method accepts black box polynomial functions as input. For nonmaliciously chosen predicates, our method is as fast as the linear scheme, scaling reasonably with the degree of the polynomial even for fully degenerate input. Like Yap’s scheme, the computed sign is deterministic, never requiring an algorithmic restart (assuming a high quality pseudorandom generator), and works for arbitrary predicates with no knowledge of their structure. We also apply our technique to exactly or nearly exactly rounded constructions that work correctly for degenerate input, using L’Hôpital’s rule to compute the necessary singular limits. We provide an open source prototype implementation including example algorithms for Delaunay triangulation and Boolean operations on polygons and circular arcs in the plane.