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Accurate forecasts of weather and climate will become increasingly important as the world adapts to anthropogenic climatic change. Forecasts' accuracy is limited by the computer power available to forecast centres, which determines the maximum resolution, ensemble size and complexity of atmospheric models. Furthermore, faster supercomputers are increasingly energy-hungry and unaffordable to run. In this thesis, a new means of making computer simulations more efficient is presented that could lead to more accurate forecasts without increasing computational costs. This 'scale-selective reduced precision' technique builds on previous work that shows that weather models can be run with almost all real numbers represented in 32 bit precision or lower without any impact on forecast accuracy, challenging the paradigm that 64 bits of numerical precision are necessary for sufficiently accurate computations. The observational and model errors inherent in weather and climate simulations, combined with the sensitive dependence on initial conditions of the atmosphere and atmospheric models, renders such high precision unnecessary, especially at small scales. The 'scale-selective' technique introduced here therefore represents smaller, less influential scales of motion with less precision. Experiments are described in which reduced precision is emulated on conventional hardware and applied to three models of increasing complexity. In a three-scale extension of the Lorenz '96 toy model, it is demonstrated that high resolution scale-dependent precision forecasts are more accurate than low resolution high-precision forecasts of a similar computational cost. A spectral model based on the Surface Quasi-Geostrophic Equations is used to determine a power law describing how low precision can be safely reduced as a function of spatial scale; and experiments using four historical test-cases in an open-source version of the real-world Integrated Forecasting System demonstrate that a similar power law holds for the spectral part of this model. It is concluded that the scale-selective approach could be beneficially employed to optimally balance forecast cost and accuracy if utilised on real reduced precision hardware.