Several formalizations of floating-point arithmetic have been designed for the Coq system, a generic proof assistant. Their different purposes have favored some specific applications: program verification, high-level properties, automation. Based on our experience using and/or developing these libraries, we have built a new system that is meant to encompass the other ones in a unified framework. It offers a multi-radix and multi-precision formalization for various floating- and fixed-point formats. This fresh setting has been the occasion for reevaluating known properties and generalizing them. This paper presents the Flocq system: a library easy to use, suitable for automation yet high-level and generic.