An Efficient Rounding Boundary Test for pow(x, y) in Double Precision
The correct rounding of the function pow: (x, y) vertical bar -> 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 paper presents a new rounding-boundary test for pow in double precision, which reduces this to a few comparisons with precomputed 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. This paper includes its correctness proof and experimental results.
correct rounding floating-point arithmetic power function