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An efficient rounding boundary test for pow(x,y) in double precision

Christoph Quirin Lauter 1 Vincent Lefèvre 1
1 ARENAIRE - Computer arithmetic
Inria Grenoble - Rhône-Alpes, LIP - Laboratoire de l'Informatique du Parallélisme
Abstract : The correct rounding of the function pow: (x,y) → x^y is currently based on Ziv's iterative approximation process. In order to ensure its termination, cases when x^y falls on a rounding boundary must be filtered out. Such rounding boundaries are floating-point numbers and midpoints between two consecutive floating-point numbers. Detecting rounding boundaries for pow is a difficult problem. Previous approaches use repeated square root extraction followed by repeated square and multiply. This article presents a new rounding boundary test for pow in double precision which reduces this to a few comparisons with pre-computed constants. These constants are deduced from worst cases for the Table Maker's Dilemma, searched over a small subset of the input domain. This is a novel use of such worst-case bounds. The resulting algorithm has been designed for a fast-on-average correctly rounded implementation of pow, considering the scarcity of rounding boundary cases. It does not stall average computations for rounding boundary detection. The article includes its correctness proof and experimental results.
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Submitted on : Thursday, April 7, 2011 - 11:19:10 AM
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Christoph Quirin Lauter, Vincent Lefèvre. An efficient rounding boundary test for pow(x,y) in double precision. IEEE Transactions on Computers, Institute of Electrical and Electronics Engineers, 2009, 58 (2), pp.197-207. ⟨10.1109/TC.2008.202⟩. ⟨inria-00583988⟩



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