  Reduced precision floating point arithmetic is now routinely deployed in
numerical weather forecasting over short timescales. However the applicability
of these reduced precision techniques to longer timescale climate simulations -
especially those which seek to describe a dynamical, changing climate - remains
unclear. We investigate this question by deploying a global atmospheric, coarse
resolution model known as SPEEDY to simulate a changing climate system subject
to increased $\text{CO}_2$ concentrations, over a 100 year timescale. Whilst
double precision is typically the operational standard for climate modelling,
we find that reduced precision solutions (Float32, Float16) are sufficiently
accurate. Rounding the finite precision floats stochastically, rather than
using the more common ``round-to-nearest" technique, notably improves the
performance of the reduced precision solutions. Over 100 years the mean bias
error (MBE) in the global mean surface temperature (precipitation) relative to
the double precision solution is $+2 \times 10^{-4}$K ($-8 \times 10^{-5}$
mm/6hr) at single precision and $-3.5\times 10^{-2}$ K($-1 \times 10^{-2}$
mm/6hr) at half precision, whilst the inclusion of stochastic rounding reduced
the half precision error to +1.8 $\times 10^{-2}$ K ($-8 \times10^{-4}$
mm/6hr). By examining the resultant climatic distributions that arise after 100
years, the difference in the expected value of the global surface temperature,
relative to the double precision solution is $\leq 5 \times 10^{-3}$ K and for
precipitation $8 \times 10^{-4}$ mm/6h when numerically integrating at half
precision with stochastic rounding. Areas of the model which notably improve
due to the inclusion of stochastic over deterministic rounding are also
explored and discussed. [abridged]
