Section: New Results

Arithmetic Algorithms

Binary floating-point operators for VLIW integer processors

C.-P. Jeannerod and J. Jourdan-Lu [35] proposed software implementations of sinf, cosf and sincosf over [-pi/4, pi/4] that have proven 1-ulp accuracy and whose respective latencies on STMicroelectronics' ST231 VLIW integer processor are 19, 18 and 19 cycles. To get such performances they introduced a novel algorithm for simultaneous sine and cosine that combines univariate and bivariate polynomial evaluation schemes.

In the same context, C.-P. Jeannerod, J. Jourdan-Lu and C. Monat (STMicroelectronics Compilation Expertise Center, Grenoble) [36] studied the implementation of custom (i.e., specialized, fused, or simultaneous) operators, and provided qualitative evidence of the benefits of supporting such operators in addition to the five basic ones: this allows to be up to 4.2x faster on individual calls, and up to 1.59x faster on DSP kernels and benchmarks.

Error bounds for complex floating-point division with an FMA

Assuming that a fused multiply-add (FMA) instruction is available, C.-P. Jeannerod, N. Louvet and J.-M. Muller [37] obtained sharp error bounds for various alternatives to Kahan's 2 by 2 determinant algorithm. Combining such alternatives with Kahan's original scheme leads to componentwise-accurate algorithms for complex floating-point division, and for these algorithms sharp or reasonably sharp error bounds were also obtained.

Computation of correctly-rounded sums

P. Kornerup (U. of Southern Denmark), V. Lefèvre and J.-M. Muller [19] have shown that among the set of the algorithms with no comparisons performing only floating-point additions/subtractions, the 2Sum algorithm introduced by Knuth is minimal, both in terms of number of operations and depth of the dependency graph. They also prove that under reasonable conditions, an algorithm performing only round-to-nearest additions/subtractions cannot compute the round-to-nearest sum of at least three floating-point numbers. They also present new results about the computation of the correctly-rounded sum of three floating-point numbers.

Comparison between binary64 and decimal64 floating-point numbers

N. Brisebarre, C. Lauter (U. Paris 6), M. Mezzarobba and J.-M. Muller [27] introduce an algorithm that allows one to quickly compare a binary64 floating-point (FP) number and a decimal64 FP number, assuming the “binary encoding” of the decimal formats specified by the IEEE 754-2008 standard for FP arithmetic is used. It is a two-step algorithm: a first pass, based on the exponents only, makes it possible to quickly eliminate most cases, then when the first pass does not suffice, a more accurate second pass is required. They provide an implementation of several variants of their algorithm, and compare them.