In a Hamiltonian dynamical system, ‘area’ in the phase space of the system is conserved over time. Symplectic algorithms are numerical algorithms that solve for the time evolution of a Hamiltonian system and are explicitly designed to preserve areas in phase space. Due to their area-preserving properties, symplectic algorithms often demonstrate much higher accuracy over long time intervals than the equivalent non-symplectic numerical algorithms for Hamiltonian systems. However, symplectic algorithms are still susceptible to rounding error as they use finite-precision floating-point numbers in their computations. We describe a modified symplectic method due to Robert Skeel that takes rounding errors into account, and we mathematically verify the accuracy of this method for the case of the simple harmonic oscillator.