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We develop a very simple compensated scheme for computing very accurate Givens rotations. The approach is significantly more straightforward than the one in borges2021fast, and the derivation leads to a very satisfying algorithm whereby a naively computed Givens rotation can be used to construct a correction to itself. It is also seen that this scheme continues to provide high accuracy even when built on a hypoteneuse calculation that is of lesser accuracy.
Carlos F. Borges (2024) studied this question.