Siegfried M. Rump
Abstract
We show how an IEEE-754 conformant precision-p base-β arithmetic can be implemented based on some binary floating-point and/or integer arithmetic. This includes the four basic operations and square root subject to the five IEEE-754 rounding modes, namely the nearest roundings with roundTiesToEven and roundTiesToAway, the directed roundings downwards and upwards, as well as rounding towards zero. Exceptional values like ∞ of NaN are covered according to the IEEE-754 arithmetic standard.
The results of the precision-p base-β operations are computed using some underlying precision-q binary arithmetic. We distinguish two cases. When using a precision-q binary integer arithmetic, the base-β precision p is limited for all operations by β2p ≤ 2q, whereas using a precision-q binary floating-point arithmetic imposes stronger limits on the base-β precision, namely β2p ≤ 2q for addition and multiplication, β2p ≤ 2q-1 for division and β2p ≤ 2q-3 for the square root. Those limitations cannot be improved.
The algorithms are implemented in a Matlab/Octave flbeta-toolbox with the choice of using
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Index Terms
- IEEE-754 Precision-p base-β Arithmetic Implemented in Binary
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