Basic building blocks for a triple-double intermediate format

Abstract

The implementation of correctly rounded elementary functions needs high intermediate accuracy before final rounding. This accuracy can be provided by (pseudo-) expansions of size three, i.e. a triple-double format. The report presents all basic operators for such a format. Triple-double numbers can be redundant. A renormalization procedure is presented and proven. Elementary functions' implementations need addition and multiplication sequences. These operators must take operands in double, double-double and triple-double format. The results must be accordingly in one of the formats. Several procedures are presented. Proofs are given for their accuracy bounds. Intermediate triple-double results must finally be correctly rounded to double precision. Two effective rounding sequences are presented, one for round-to-nearest mode, one for the directed rounding modes. Their complete proofs constitute half of the report.