High-Speed Function Approximation using a Minimax Quadratic Interpolator

A table-based method for high-speed function approximation in single-precision floating-point format is presented in this paper. Our focus is the approximation of reciprocal, square root, square root reciprocal, exponentials, logarithms, trigonometric functions, powering (with a fixed exponent p), or special functions. The algorithm presented here combines table look-up, an enhanced minimax quadratic approximation, and an efficient evaluation of the second-degree polynomial (using a specialized squaring unit, redundant arithmetic, and multioperand addition). The execution times and area costs of an architecture implementing our method are estimated, showing the achievement of the fast execution times of linear approximation methods and the reduced area requirements of other second-degree interpolation algorithms. Moreover, the use of an enhanced minimax approximation which, through an iterative process, takes into account the effect of rounding the polynomial coefficients to a finite size allows for a further reduction in the size of the look-up tables to be used, making our method very suitable for the implementation of an elementary function generator in state-of-the-art DSPs or graphics processing units (GPUs)

Solid state and molecular theory group Semiannual progress report

Use of scattered wave method to compute molecular wave functions, augmented plane wave method for energy band calculations, and Casimir invariants as invariant operators in Lie group

Modified Shepard interpolation of gas-surface potential energy surfaces with strict plane group symmetry and translational periodicity

A new formulation of modified Shepard interpolation of potential energy surface data for gas-surface reactions has been developed. The approach has been formulated for monoatomic or polyatomic adsorbates interacting with crystalline solid surfaces of any plane group symmetry. The interpolation obeys the two dimensional translational periodicity and plane group symmetry of the solid surface by construction. The interpolation remains continuous and smooth everywhere. The interpolation developed here is suitable for constructing potential energy surfaces by sampling classical trajectories using the Grow procedure. A model function has been used to demonstrate the method, showing the convergence of the classical gas-surface reaction probability

Survey of Floating-Point Software Arithmetics and Basic Library Mathematical Functions

Abstract Not Provided

Composite Iterative Algorithm and Architecture for q-th Root Calculation

An algorithm for the q-th root extraction, being q any integer, is presented in this paper. The algorithm is based on an optimized implementation of X^{1/q} by a sequence of parallel and/or overlapped operations: (1) reciprocal, (2) digit-recurrence logarithm, (3) left-to-right carry-free multiplication and (4) on-line exponential. A detailed error analysis and two architectures are proposed, for low precision q and for higher precision q. The execution time and hardware requirements are estimated for single and double precision floating-point computations for several radices; this helps to determine which radices result in the most efficient implementations. The architectures proposed improve the features of other architectures for q-th root extraction.Dans cet article, nous présentons un algorithme matériel pour l'extraction de la racine q-ième d'un nombre X, où q est un entier naturel non nul. Cet algorithme est basé sur une implantation optimisée de la fonction X^{1/q} par une séquence d'opérations parallèles et/ou superposées: (1) réciproque, (2) logarithme chiffre par chiffre, (3) multiplication de gauche-à-droite sans propagation de retenue et (4) exponentielle en ligne. Une analyse détaillée des erreurs et deux architectures sont proposées, pour q de basse précision et pour q de précision plus haute. Le temps d'exécution et les composants matériels à utiliser sont estimés pour des calculs en virgule flottante simple et double précision et pour plusieurs bases. Cette étude aide à déterminer quelles bases mènent aux implantations les plus efficaces. Les architectures proposées améliorent les caractéristiques d'architectures précédentes destinées à l'extraction des racines

A Fast and Low-Complexity Operator for the Computation of the Arctangent of a Complex Number

[EN] The computation of the arctangent of a complex number, i.e., the atan2 function, is frequently needed in hardware systems that could profit from an optimized operator. In this brief, we present a novel method to compute the atan2 function and a hardware architecture for its implementation. The method is based on a first stage that performs a coarse approximation of the atan2 function and a second stage that improves the output accuracy by means of a lookup table. We present results for fixed-point implementations in a field-programmable gate array device, all of them guaranteeing last-bit accuracy, which provide an advantage in latency, speed, and use of resources, when compared with well-established fixed-point options.This work was supported by the Spanish Ministerio de Economia y Competitividad and FEDER under Grant TEC2015-70858-C2-2-R.Torres Carot, V.; Valls Coquillat, J. (2017). A Fast and Low-Complexity Operator for the Computation of the Arctangent of a Complex Number. IEEE Transactions on Very Large Scale Integration (VLSI) Systems. 25(9):2663-2667. https://doi.org/10.1109/TVLSI.2017.2700519S2663266725