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We motivate and propose an alternative criterion for the dynamical control of regularization in the context of the standard RLS algorithm. The proposed criterion explores the fact that in finite precision the numerical solution of a regularized linear system of equations may be closer to the analytical (unknown) solution of the original (unregularized) system than the numerical solution of the latter. We develop a measure of accuracy for such solutions and use it to automatically adjust the regularization parameter via a simple feedback mechanism. In order to keep computational complexity low, regularization is implemented indirectly via dithering of the input signal. Simulations show that the proposed criterion can effectively react, and compensate for large condition numbers, the precision available and unnecessarily large levels of regularization.