The machine learning explosion has created a prominent trend in modern computer hardware towards low precision floating-point operations. In response, there have been growing efforts to use low and mixed precision in general scientific computing. One important area that has received limited exploration is time-integration methods, which are used for solving differential equations that are ubiquitous in science and engineering applications. In this work, we develop two new approaches for leveraging mixed precision in exponential time integration methods. The first approach is based on a reformulation of the exponential Rosenbrock--Euler method allowing for low precision computations in matrix exponentials independent of the particular algorithm for matrix exponentiation. The second approach is based on an inexact and incomplete Arnoldi procedure in Krylov approximation methods for computing matrix exponentials and is agnostic to the chosen integration method. We show that both approaches improve accuracy compared to using purely low precision and offer better efficiency than using only double precision when solving an advection-diffusion-reaction partial differential equation.

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