On infinitely precise rounding for division, square root, reciprocal and square root reciprocal

Quotients, reciprocals, square roots and square root reciprocals all have the property that infinitely precise p-bit rounded results for p-bit input operands can be obtained from approximate results of bounded accuracy. We investigate lower bounds on the number of bits of an approximation accurate to a unit in the last place sufficient to guarantee that correct round and sticky bits can be determined. Known lower bounds for quotients and square roots are given and/or sharpened, and a new lower bound for root reciprocals is proved. Specifically for reciprocals, quotients and square roots, tight bounds of order 2p+O(1) are presented. For infinitely precise rounding of the root reciprocal, a lower bound can be found at 3p+O(1), but exhaustive testing for small sizes of the operand suggests that in practice (2+ε)p for small ε is usually sufficient. Algorithms can be designed for obtaining the round and sticky bits based on the bit pattern of an approximation computed to the required accuracy. We show that some improvement of the known lower bound for reciprocals and division is achievable at the cost of somewhat more complex hardware for rounding. Tests for the exactness of the quotient and square root are also provided