A p×p bit fraction model of binary floating point division and extremal rounding cases
✍ Scribed by David W. Matula; Lee D. McFearin
- Book ID
- 104325433
- Publisher
- Elsevier Science
- Year
- 2003
- Tongue
- English
- Weight
- 295 KB
- Volume
- 291
- Category
- Article
- ISSN
- 0304-3975
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✦ Synopsis
We introduce the ordered series Fp×p of irreducible p × p bit fractions as a model of p-bit precision binary oating point division. We employ and extend results from the number theoretic literature on Farey fractions and continued fractions to provide a foundation for generation and analysis of the series Fp×p. An algorithm for ordered on-the-y enumeration of a consecutive subsequence of Fp×p over a selected interval is introduced which requires only a couple of integer additions and=or subtractions per p × p bit fraction enumerated.
We characterize two extremal rounding boundary sets, RNp, respectively RDp, of irreducible p × p bit fractions over the standard binade [1; 2) whose 2 p+O(1) member fractions have rational values that are each comparably close to a boundary for rounding to a normalized p-bit oating point number by round-to-nearest, respectively, by a directed rounding. A transformation is shown allowing either set RNp; RDp, to be simply computed from the other. We determine properties of these extremal rounding boundary sets RNp; RDp, and describe their use in the testing of oating point division implementations.