Refinement of a newton reciprocal algorithm for arbitrary precision numbers

For arbitrary precision numbers, reciprocal computing algorithms based on Newton iteration is asymptotically the fastest. In this work we provide a refined algorithm based on the Newton reciprocal algorithm by Brent and Zimmermann in their MCA book. The key techniques used in the refinement are D1 balancing, clear specification, remainder operation, and economical multiplication. The refined algorithm is more general, and gives exact and unique result. Numerical results show that these improvements are made without the cost of time efficiency. There is still the potential to further improve the efficiency by utilizing a short multiplication algorithm.