Computing correctly rounded integer powers in floating-point arithmetic

Supplementary material to “ Accelerating Correctly Rounded Floating - Point Division When the Divisor is Known in Advance ”

• If m > n, the exact quotient of two n-bit numbers cannot be an m-bit number. • Let x, y ∈ Mn. x = y ⇒ |x/y − 1| ≥ 2−n. We call a breakpoint a value z where the rounding changes, that is, if t1 and t2 are real numbers satisfying t1 < z < t2 and ◦t is the rounding mode, then ◦t(t1) < ◦t(t2). For “directed” rounding modes (i.e., towards +∞, −∞ or 0), the breakpoints are the FP numbers. For round...

Computing Convex Hull in a Floating Point Arithmetic

We present a numerically stable and time and space complexity optimal algorithm for constructing a convex hull for a set of points on a plane. In contrast to already existing numerically stable algorithms which return only an approximate hull, our algorithm constructs a polygon that is truly convex. The algorithm is simple and easy to implement.

Floating-Point Arithmetic

Towards Correctly Rounded Transcendentals

The Table Maker’s Dilemma is the problem of always getting correctly rounded results when computing the elementary functions. After a brief presentation of this problem, we present new developments that have helped us to solve this problem for the double-precision exponential function in a small domain. These new results show that this problem can be solved, at least for the double-precision fo...

Precision Arithmetic: A New Floating-Point Arithmetic

A new deterministic floating-point arithmetic called precision arithmetic is developed to track precision for arithmetic calculations. It uses a novel rounding scheme to avoid the excessive rounding error propagation of conventional floating-point arithmetic. Unlike interval arithmetic, its uncertainty tracking is based on statistics and the central limit theorem, with a much tighter bounding r...