  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.
