Related papers: Twofold fast arithmetic
The error function of real argument can be uniformly approximated to a given accuracy by a single closed-form expression for the whole variable range either in terms of addition, multiplication, division, and square root operations only, or…
The recent hardware trend towards reduced precision computing has reignited the interest in numerical techniques that can be used to enhance the accuracy of floating point operations beyond what is natively supported for basic arithmetic…
We describe an approximate rational arithmetic with round-off errors (both absolute and relative) controlled by the user. The rounding procedure is based on the continued fraction expansion of real numbers. Results of computer experiments…
Floating-point arithmetic is error-prone and unintuitive. Floating-point debuggers instrument programs to monitor floating-point arithmetic at run time and flag numerical issues. They estimate residues, i.e., the difference between actual…
A rapidly convergent series, based on Taylor expansion of the imaginary part of the complex error function, is presented for highly accurate approximation of the Voigt/complex error function with small imaginary argument (Y less than 0.1).…
Ootomo, Ozaki, and Yokota [Int. J. High Perform. Comput. Appl., 38 (2024), p. 297-313] have proposed a strategy to recast a floating-point matrix multiplication in terms of integer matrix products. The factors A and B are split into integer…
Floating point arithmetic allows us to use a finite machine, the digital computer, to reach conclusions about models based on continuous mathematics. In this article we work in the other direction, that is, we present examples in which…
Static analysis by abstract interpretation aims at automatically proving properties of computer programs. To do this, an over-approximation of program semantics, defined as the least fixpoint of a system of semantic equations, must be…
Numerical codes that require arbitrary precision floating point (APFP) numbers for their core computation are dominated by elementary arithmetic operations due to the super-linear complexity of multiplication in the number of mantissa bits.…
This paper introduces a computationally efficient technique for estimating high-resolution Doppler blood flow from an ultrafast ultrasound image sequence. More precisely, it consists in a new fast alternating minimization algorithm that…
Acceleration in symbolic verification consists in computing the exact effect of some control-flow loops in order to speed up the iterative fix-point computation of reachable states. Even if no termination guarantee is provided in theory,…
We introduce two algorithms for accurately evaluating powers to a positive integer in floating-point arithmetic, assuming a fused multiply-add (fma) instruction is available. We show that our log-time algorithm always produce…
Approximate integer programming is the following: For a convex body $K \subseteq \mathbb{R}^n$, either determine whether $K \cap \mathbb{Z}^n$ is empty, or find an integer point in the convex body scaled by $2$ from its center of gravity…
We propose a new instruction (FPADDRE) that computes the round-off error in floating-point addition. We explain how this instruction benefits high-precision arithmetic operations in applications where double precision is not sufficient.…
We provide tools to help automate the error analysis of algorithms that evaluate simple functions over the floating-point numbers. The aim is to obtain tight relative error bounds for these algorithms, expressed as a function of the unit…
Renewed interest in mixed-precision algorithms has emerged due to growing data capacity and bandwidth concerns, as well as the advancement of GPUs, which enable significant speedup for low precision arithmetic. In light of this, we propose…
A pervasive approach in scientific computing is to express the solution to a given problem as the limit of a sequence of vectors or other mathematical objects. In many situations these sequences are generated by slowly converging iterative…
This paper introduces some methods to determine the simultaneous approximation constants of a class of well approximable numbers $\zeta_{1},\zeta_{2},...,\zeta_{k}$. The approach relies on results on the connection between the set of all…
Solving the floating-point equation $x \otimes y = z$, where $x$, $y$ and $z$ belong to floating-point intervals, is a common task in automated reasoning for which no efficient algorithm is known in general. We show that it can be solved by…
We present a MATLAB function for the numerical evaluation of the Faddeyeva function w(z). The function is based on a newly developed accurate algorithm. In addition to its higher accuracy, the software provides a flexible accuracy vs…