Ultimately Fast Accurate Summation

Abstract

Abstract. We present two new algorithms FastAccSum and FastPrecSum, one to compute a faithful rounding of the sum of floating-point numbers and the other for a result "as if" computed in K-fold precision. Faithful rounding means the computed result either is one of the immediate floating-point neighbors of the exact result or is equal to the exact sum if this is a floating-point number. The algorithms are based on our previous algorithms AccSum and PrecSum and improve them by up to 25%. The first algorithm adapts to the condition number of the sum; i.e., the computing time is proportional to the difficulty of the problem. The second algorithm does not need extra memory, and the computing time depends only on the number of summands and K. Both algorithms are the fastest known in terms of flops. They allow good instruction-level parallelism so that they are also fast in terms of measured computing time. The algorithms require only standard floating-point addition, subtraction, and multiplication in one working precision, for example, double precision.

On Various Ways to Split a Floating-Point Number

“…The first claim is a direct consequence of [15,Lemma 3.6]. The second claim then follows from the fact that Algorithm 7 is Algorithm 6 applied to 2 p x.…”

“…In particular, for β = 2 and p = 53, this yields the "magic number" C = 2 52 + 2 51 mentioned before, together with the range |x| 2 51 . Also, for β = 10 and p = 16 (decimal64 IEEE-754 format), the largest range for nonnegative inputs is x 9·10 15 , obtained with C = 10 15 ; and the largest range for signed inputs is |x| 4.5 · 10 15 , obtained with C = 5.5 · 10 15 .…”

“…For computing sign(x) · ufp(x), we can use the following algorithm, introduced by Rump in [15]. These solutions, however, raise the following issues.…”

Efficiency of Reproducible Level 1 BLAS

“…Algorithms AccSum [12] and FastAccSum [11] also rely on error-free transformations of the entry vector. They split the summands, relatively to max |p i | and n, such that their higher order parts are then exactly accumulated.…”

Numerical reproducibility for the parallel reduction on multi- and many-core architectures

“…However, as we mentioned already, this solution solely increases the accuracy of basic operations, but it neither reaches bit-accurate results nor is efficient for large precisions. Therefore, some recent works focus on hybrid solutions that store the sum as floating-point numbers of fixed exponent [26,27] without completely avoiding the previous drawbacks. These algorithms are mainly sequential and are not suitable for parallelization.…”