Error Bounds for Computer Arithmetics

Abstract

This note summarizes recent progress in error bounds for compound operations performed in some computer arithmetic. Given a general set of real numbers together with some operations satisfying the first standard model, we identify three types A, B, and C of weak sufficient assumptions implying new results and sharper error estimates. Those include linearized error estimates in the number of operations, faithfully rounded and reproducible results. All types of assumptions are satisfied for an IEEE-754 p-digit base-β floating-point arithmetic.

A fast parallel high-precision summation algorithm based on AccSumK

“…Note that ( 5) is valid for any order of addition in the summation. Recently, it was shown that γ n can be replaced by nu, and the restriction on n can be removed [12,13,14,15]. The algorithm AccSumK is based on the error-free transformations of addition and/or multiplication of two floating-point numbers.…”

$a \cdot(x\cdot\ x)$ or $(a\cdot x)\cdot x?$

“…is bounded by u/µ(x). This allows one for instance to show that the relative error due to rounding is bounded by u/(1 + u) [4], and to show very tight bounds on the relative errors of Floating-Point operations [3], [6]. We assume that all intermediate calculations are performed in the same format.…”

“…We start by computing (a • x) • x: the relative error of that calculation was studied in Section I-B, with bounds given for the various subintervals in Table II. The relative error of the last operation is bounded by u/µ(ax 3 ), i.e., it is given by (6). As previously, we need to order the comparison constants of Table II and ( 6).…”

“…The relative error of the last operation is bounded by u/µ(ax 3 ), i.e., it is given by (6). As previously, we need to order the comparison constants of Table II and ( 6).…”

“…The number u = 2 −p = 1 2 ulp(1) denotes the roundoff error unit. If x is a nonzero real number, with 2 k ≤ |x| < 2 k+1 (i.e., 2 k is what Rump defines as ufp(x) [6]), µ(x) = x/2 k denotes the "infinite-precision significand" of x. Barring underflow or overflow, the relative error due to rounding to nearest a nonzero real number x, i.e.,…”