IEEE Transactions on Computers - Special issue on computer arithmetic
Improving Goldschmidt Division, Square Root, and Square Root Reciprocal
IEEE Transactions on Computers - Special issue on computer arithmetic
A p × p bit fraction model of binary floating point division and extremal rounding cases
Theoretical Computer Science - Real numbers and computers
The abc conjecture and correctly rounded reciprocal square roots
Theoretical Computer Science - Algebraic and numerical algorithm
Searching Worst Cases of a One-Variable Function Using Lattice Reduction
IEEE Transactions on Computers
Techniques and tools for implementing IEEE 754 floating-point arithmetic on VLIW integer processors
Proceedings of the 4th International Workshop on Parallel and Symbolic Computation
Optimizing correctly-rounded reciprocal square roots for embedded VLIW cores
Asilomar'09 Proceedings of the 43rd Asilomar conference on Signals, systems and computers
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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 root 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 2p+O(1) is usually sufficient. Algorithms are given 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.