1,678 research outputs found
Improving Goldschmidt Division, Square Root and Square Root Reciprocal
The aim of this paper is to accelerate division, square root and square root reciprocal computations, when Goldschmidt method is used on a pipelined multiplier. This is done by replacing the last iteration by the addition of a correcting term that can be looked up during the early iterations. We describe several variants of the Goldschmidt algorithm assuming 4-cycle pipelined multiplier and discuss obtained number of cycles and error achieved. Extensions to other than 4-cycle multipliers are given.Le but de cet article est l'accĂ©lĂ©ration de la division, et du calcul de racines carrĂ©es et d'inverses de racines carrĂ©es lorsque la mĂ©thode de Goldschmidt est utilisĂ©e sur un multiplieur pipe-line. Nous faisons ceci en remplaçant la derniĂšre itĂ©ration par l'addition d'un terme de correction qui peut ĂȘtre dĂ©duit d'une lecture de table effectuĂ©e lors des premiĂšres itĂ©rations. Nous dĂ©crivons plusieurs variantes de l'algorithme obtenu en supposant un multiplieur Ă 4 Ă©tages de pipe-line, et donnons pour chaque variante l'erreur obtenue et le nombre de cycles de calcul. Des extensions de ce travail Ă des multiplieurs dont le nombre d'Ă©tages est diffĂ©rent sont prĂ©sentĂ©es
High-level algorithms for correctly-rounded reciprocal square roots
International audienceWe analyze two fast and accurate algorithms recently presented by Borges for computing in binary floating-point arithmetic (assuming that efficient and correctly-rounded FMA and square root are available). The first algorithm is based on the Newton-Raphson iteration, and the second one uses an order-3 iteration. We give attainable relative-error bounds for these two algorithms, build counterexamples showing that in very rare cases they do not provide a correctly-rounded result, and characterize precisely when such failures happen in IEEE 754 binary32 and binary64 arithmetics. We then give a generic (i.e., precision-independent) algorithm that always returns a correctly-rounded result, and show how it can be simplified and made more efficient in the important cases of binary32 and binary64
Accurate and Efficient Expression Evaluation and Linear Algebra
We survey and unify recent results on the existence of accurate algorithms
for evaluating multivariate polynomials, and more generally for accurate
numerical linear algebra with structured matrices. By "accurate" we mean that
the computed answer has relative error less than 1, i.e., has some correct
leading digits. We also address efficiency, by which we mean algorithms that
run in polynomial time in the size of the input. Our results will depend
strongly on the model of arithmetic: Most of our results will use the so-called
Traditional Model (TM). We give a set of necessary and sufficient conditions to
decide whether a high accuracy algorithm exists in the TM, and describe
progress toward a decision procedure that will take any problem and provide
either a high accuracy algorithm or a proof that none exists. When no accurate
algorithm exists in the TM, it is natural to extend the set of available
accurate operations by a library of additional operations, such as , dot
products, or indeed any enumerable set which could then be used to build
further accurate algorithms. We show how our accurate algorithms and decision
procedure for finding them extend to this case. Finally, we address other
models of arithmetic, and the relationship between (im)possibility in the TM
and (in)efficient algorithms operating on numbers represented as bit strings.Comment: 49 pages, 6 figures, 1 tabl
Avoiding double roundings in scaled Newton-Raphson division
Abstract-When performing divisions using Newton-Raphson (or similar) iterations on a processor with a floating-point fused multiply-add instruction, one must sometimes scale the iterations, to avoid over/underflow and/or loss of accuracy. This may lead to double-roundings, resulting in output values that may not be correctly rounded when the quotient falls in the subnormal range. We show how to avoid this problem
Study of the posit number system: a practical approach
The IEEE Standard for Floating-Point Arithmetic (IEEE 754) has been for decades the standard for floating-point arithmetic and is implemented in a vast majority of modern computer systems. Recently, a new number representation format called posit (Type III unum) introduced by John L. Gustafson â who claims this new format can provide higher accuracy using equal or less number of bits and simpler hardware than current standard â is proposed as an alternative to the now omnipresent IEEE 754 arithmetic.
In this Bachelor dissertation, the novel posit number format, its characteristics and properties â presented in literature â are analyzed and compared with the standard for floating-point numbers (floats). Based on the literature assertions, we focus on determining whether posits would be a good âdrop-in replacementâ for floats. With the help of Wolfram Mathematica and Python, different environments are created to compare the performance of IEEE 754 floating-point standard with Type III unum: posits. In order to get a more practical approach, first, we propose different numerical problems to compare the accuracy of both formats, including algebraic problems and numerical methods. Then, we focus on the possible use of posits in Deep Learning problems, such as training artificial Neural Networks or preforming low-precision inference on Convolutional Neural Networks. To conclude this work, we propose a low-level design for posit arithmetic multiplier using the FloPoCo tool to generate synthesizable VHDL code
The use of primitives in the calculation of radiative view factors
Compilations of radiative view factors (often in closed analytical form) are readily available in the open literature for commonly encountered geometries. For more complex three-dimensional (3D) scenarios, however, the effort required to solve the requisite multi-dimensional integrations needed to estimate a required view factor can be daunting to say the least. In such cases, a combination of finite element methods (where the geometry in question is sub-divided into a large number of uniform, often triangular, elements) and Monte Carlo Ray Tracing (MC-RT) has been developed, although frequently the software implementation is suitable only for a limited set of geometrical scenarios. Driven initially by a need to calculate the radiative heat transfer occurring within an operational fibre-drawing furnace, this research set out to examine options whereby MC-RT could be used to cost-effectively calculate any generic 3D radiative view factor using current vectorisation technologies
Proceedings of the 7th Conference on Real Numbers and Computers (RNC'7)
These are the proceedings of RNC7
- âŠ