Today, GPUs represent an important hardware development platform for many problems in dynamical systems, where massive parallel computations are needed. Beside that, many numerical studies of chaotic dynamical systems require a computing precision higher than common oating point (FP) formats. One such application is locating invariant sets for chaotic dynamical systems. In particular, we focus on rigorously proving the existence of stable periodic orbits for the Hénon map for parameter values close to the classical ones. For that, we present a multiple-precision floating-point arithmetic library in CUDA programming language for the NVIDIA GPU platform. Our library extends the precision using so-called FP expansions, where a number is represented as the unevaluated sum of standard machine precision FP numbers. This format offers the advantage of using directly available and highly optimized hardware FP operations. We generalize algorithms used by multiple-precisions libraries such as Bailey's QD, or the analogue GPU version, GQD.