# 13.2  Reciprocal Refinement

A refinement of a given reciprocal approximation may be derived in two steps:

Alternatively, as illustrated by the double precision algorithm of Section 13.3, an approximation may be computed from the preceding two approximations as follows. This results in lower accuracy but allows the two steps to be executed in parallel:

The following lemma may be applied to either of these computations.

(recip-refinement-1, recip-refinement-2) Assume that and . Let , and , where and . Let

and

Then (a) and (b) .

PROOF: Let and . Then , , and

where . Thus,

and

The inequality (b) of Lemma 13.2.1 provides a significantly reduced error bound for a refined reciprocal approximation as long as the bounds and for the earlier approximations and are large in comparison to . To establish the bound for the final approximation as required by Lemma 13.1.2, we shall use the inequality (a), pertaining to the corresponding unrounded value, in conjunction with the following additional lemma, which is a variation by Harrison [HAR00] of another result of Markstein [MAR90]. In practice, the application of this lemma involves explicitly checking a small number of excluded cases.

(harrison-lemma) Let be -exact. Assume that and . Let . Then , with the possible exceptions , .

PROOF: If , then implies and . Thus we may assume , and therefore . Consequently,

and it follows that . Since

and apart from the allowed exceptions, , we have

David Russinoff 2017-08-01