13.3 Examples

The single-precision algorithm is given by the following sequence of operations. The spacing of the steps is intended to denote groups of operations that may be executed in parallel. Note that the rounding mode is used for all intermediate steps and the input mode is used for the final step.

The following properties of the function *rcp24* have been verified by exhaustive computation.

The next lemma, which has been similarly verified, will be used in conjunction with Lemma 13.2.2 in the proof of Theorem 13.1.

PROOF: According to Lemma 13.1.2, we need only establish the inequalities (ii) and (i) , where is defined as in the lemma.

Let

and |

By Lemma 13.3.1, . By Lemma 13.2.1 (under the substitutions of for both and , for both and , and for ), and . It is easily verified by direct computation that and . The required inequality (ii) then follows from Lemmas 13.2.2 and 13.3.2.

Let . By Lemma 13.1.1,

The double-precision algorithm follows:

Note that in this case, the initial approximation is based on a truncation of the denominator, which
increases its relative error:^{13.2}

PROOF: Let . Then and by Lemma 13.3.3,

The relative error of the final reciprocal approximation must be computed explicitly in 1027 cases:

PROOF: Applying Lemma 13.1.2 once again, we need only establish the two inequalities (ii) and (i) .

Let

and |

Let . Then and by Lemma 13.3.3,

Let . Direct computation yields , and (i) again follows from Lemma 13.1.3.

David Russinoff 2017-08-01