The paper describes in detail a symbolic-numeric strategy to compute floating point Puiseux series defined by a bivariate polynomial. The key idea is to perform modular reduction, that is, working modulo a suitably chosen prime \(p\). The symbolic part of the computation ensures that a later numerical stage has a sufficiently precise input. In Sections 2 and 3, details of Puiseux series expansion are presented such that the paper serves as reference for such computations. In Section 4 the Newton polygon technique is refined to be applicable in the algorithm ``RNPuiseux'', the main result of this paper. To this end the data structure of ``polygon trees'' is defined and examined. Section 5 contains proofs that a good choice of \(p\) is possible, and Section 6 discuss various aspects of the algorithm, such as the size of the chosen prime \(p\).