Vectorised/Semi-Parallel Interval Multiplication

To date, two principle methods for the multiplication of two intervals have been proposed. Namely, the multiplica- tion of all four bounds and finding their minima/maxima; or by pre-processing the bounds and determining which mul- tiplicands to use based upon their signs. In either case, a minimum of four multiplications are required for complete coverage and special cases such as [0, 1] × [-, 1] can result in the less than enlightening [- , +]. This paper describes a new method of interval multiplication that never requires more than two multiplications, has no special cases and elegantly handles the above case. We continue by de- scribing reformulations of the brute-force and 9-case meth- ods which, through making use of SIMD technology, paral- lelise and vectorise their operation, ultimately allowing the complete removal of branching. We conclude with an analy- sis of the algorithms and their performance, compared with the two forementioned traditional techniques.