Tight Interval Inclusions with Compensated Algorithms

Abstract

Compensated algorithms consist in computing the rounding errors of individual operations and then adding them later on to the computed result. This makes it possible to increase the accuracy of the computed result efficiently. Computing the rounding error of an individual operation is possible through the use of a so-called error-free transformation. In this article, we show that it is possible to use compensated algorithms for having tight interval inclusions. We study compensated algorithms for summation, dot product and polynomial evaluation. We prove that the use of directed rounding makes it possible to get narrow inclusions with compensated algorithms. This is due to the fact that error-free transformations are no more exact but still sufficiently accurate to improve the numerical quality of results.

Towards a correctly-rounded and fast power function in binary64 arithmetic

“…It is shown in [24, Theorem 1] that the corresponding approximation error satisfies |h + ℓ − (a + b)| ≤ 2 −105 |h| and that it is zero when the exponent difference between a and b does not exceed 53. (See also [10] for a weaker bound.) Also, it is shown in [8,Theorem 3] that the middle computation t ← •(h − a) is always exact, whatever the rounding mode.…”

Alternative Split Functions and Dekker’s Product

“…This can be of interest on systems on which changing the rounding mode is an expensive operation, or even an impossible one (for instance, with the GCC compiler, the directed roundings are not supported correctly unless the -frounding-math switch is provided; but the implementation of this switch is incomplete, so that in practice, GCC currently assumes that the same rounding mode is used everywhere 1 ). Moreover, directed roundings are heavily used in interval arithmetic or stochastic arithmetic [12].…”