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Lower precision arithmetic can improve the throughput of adaptive filters, while requiring less hardware resources and less power. Such benefits are crucial for adaptive filters, especially for IoT and wearable applications. In order to apply lower precision arithmetic to adaptive filters, a clear rounding error analysis framework is required, since lower precision arithmetic can degrade the filter performance. Previously, rounding error analyses of adaptive filters were based on forward error analysis. This limited the descriptiveness of rounding error impact on adaptive filter performance in relation to other external variables such as measurement noise, regularisation, and numerical stability of an algorithm. To overcome such limitations, we first present a new backward error analysis framework for adaptive Recursive Least Squares (RLS) filters. Our framework transforms finite precision arithmetic adaptive filters into exact arithmetic adaptive filters with the input data corrupted by rounding error noise that is additive to measurement noise. Findings throughout our backward error analysis framework can provide a guide on how to apply lower precision arithmetic to adaptive filters: (i) the magnitudes of the rounding error noise depend on the numerical stability of the implementation algorithm, arithmetic precision, and regularisation, (ii) the rounding error noise is independently additive to measurement noise, (iii) a higher regularisation is recommended for lower precision arithmetic adaptive filters, and (iv) adaptive filters using lower precision arithmetic have equivalent filter performance to those using higher precision if the magnitudes of rounding error noise are lower than measurement noise.