Reduced-precision floating-point arithmetic is increasingly supported by modern hardware and has opened up opportunities to improve performance and energy efficiency. At the same time, solvers for hyperbolic partial differential equations (PDEs), such as the shallow water equations, are very sensitive to numerical errors and therefore represent a critical test case. This thesis has a dual focus: First, it compiles and systematizes the mathematical formulations of five widely used approximate Riemann solvers – Rusanov, HLLC, Osher, F-Wave and Augmented Riemann – providing a unified reference of their theoretical structure. Second, this thesis experimentally evaluates these five solvers when executed in low or mixed floating-point precision. Benchmark experiments reveal that half-precision often leads to overflow, catastrophic cancellation or severe loss of accuracy, while single and double precision remain stable. Mixed-precision schemes, in which sensitive quantities (e.g., wave speeds, water depth) are computed in higher precision, while other state variables are stored in lower precision, provide a promising compromise: they maintain stability and accuracy while reducing computational cost. These findings highlight both the pitfalls of naive low-precision use and the potential of carefully designed mixed-precision strategies for future use in the large-scale geophysical simulations.