Abstract: This paper presents upper bounds on the number of zeros and of ones after the rounding bit for algebraic functions. These functions include reciprocal, division, square root, and reciprocal square root, which have been considered in previous work. We here propose simpler proofs for the previously given bounds and generalize to all algebraic functions. We also determine cases for which the bound is achieved for square root. As is mentioned in the previous work, these bounds are useful for determining the precision required in the computation of approximations in order to be able to perform correct rounding. We consider here rounding to nearest, but the results can be easily extended to other rounding modes.