Elementary Functions and Approximate Computing Jean-Michel Muller
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Elementary Functions: Towards Automatically Generated, Efficient
Elementary functions : towards automatically generated, efficient, and vectorizable implementations Hugues De Lassus Saint-Genies To cite this version: Hugues De Lassus Saint-Genies. Elementary functions : towards automatically generated, efficient, and vectorizable implementations. Other [cs.OH]. Université de Perpignan, 2018. English. NNT : 2018PERP0010. tel-01841424 HAL Id: tel-01841424 https://tel.archives-ouvertes.fr/tel-01841424 Submitted on 17 Jul 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Délivré par l’Université de Perpignan Via Domitia Préparée au sein de l’école doctorale 305 – Énergie et Environnement Et de l’unité de recherche DALI – LIRMM – CNRS UMR 5506 Spécialité: Informatique Présentée par Hugues de Lassus Saint-Geniès [email protected] Elementary functions: towards automatically generated, efficient, and vectorizable implementations Version soumise aux rapporteurs. Jury composé de : M. Florent de Dinechin Pr. INSA Lyon Rapporteur Mme Fabienne Jézéquel MC, HDR UParis 2 Rapporteur M. Marc Daumas Pr. UPVD Examinateur M. Lionel Lacassagne Pr. UParis 6 Examinateur M. Daniel Menard Pr. INSA Rennes Examinateur M. Éric Petit Ph.D. Intel Examinateur M. David Defour MC, HDR UPVD Directeur M. Guillaume Revy MC UPVD Codirecteur À la mémoire de ma grand-mère Françoise Lapergue et de Jos Perrot, marin-pêcheur bigouden. -
What Is a Closed-Form Number? Timothy Y. Chow the American
What is a Closed-Form Number? Timothy Y. Chow The American Mathematical Monthly, Vol. 106, No. 5. (May, 1999), pp. 440-448. Stable URL: http://links.jstor.org/sici?sici=0002-9890%28199905%29106%3A5%3C440%3AWIACN%3E2.0.CO%3B2-6 The American Mathematical Monthly is currently published by Mathematical Association of America. Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/journals/maa.html. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is an independent not-for-profit organization dedicated to and preserving a digital archive of scholarly journals. For more information regarding JSTOR, please contact [email protected]. http://www.jstor.org Mon May 14 16:45:04 2007 What Is a Closed-Form Number? Timothy Y. Chow 1. INTRODUCTION. When I was a high-school student, I liked giving exact answers to numerical problems whenever possible. If the answer to a problem were 2/7 or n-6 or arctan 3 or el/" I would always leave it in that form instead of giving a decimal approximation. -
Modified Fast Inverse Square Root and Square Root Approximation
computation Article Modified Fast Inverse Square Root and Square Root Approximation Algorithms: The Method of Switching Magic Constants Leonid V. Moroz 1, Volodymyr V. Samotyy 2,3,* and Oleh Y. Horyachyy 1 1 Information Technologies Security Department, Lviv Polytechnic National University, 79013 Lviv, Ukraine; [email protected] (L.V.M.); [email protected] (O.Y.H.) 2 Automation and Information Technologies Department, Cracow University of Technology, 31155 Cracow, Poland 3 Information Security Management Department, Lviv State University of Life Safety, 79007 Lviv, Ukraine * Correspondence: [email protected] Abstract: Many low-cost platforms that support floating-point arithmetic, such as microcontrollers and field-programmable gate arrays, do not include fast hardware or software methods for calculating the square root and/or reciprocal square root. Typically, such functions are implemented using direct lookup tables or polynomial approximations, with a subsequent application of the Newton– Raphson method. Other, more complex solutions include high-radix digit-recurrence and bipartite or multipartite table-based methods. In contrast, this article proposes a simple modification of the fast inverse square root method that has high accuracy and relatively low latency. Algorithms are given in C/C++ for single- and double-precision numbers in the IEEE 754 format for both square root and reciprocal square root functions. These are based on the switching of magic constants in the Citation: Moroz, L.V.; Samotyy, V.V.; initial approximation, depending on the input interval of the normalized floating-point numbers, in Horyachyy, O.Y. Modified Fast order to minimize the maximum relative error on each subinterval after the first iteration—giving Inverse Square Root and Square Root 13 correct bits of the result. -
Elementary Functions and Their Inverses*
ELEMENTARYFUNCTIONS AND THEIR INVERSES* BY J. F. RITT The chief item of this paper is the determination of all elementary functions whose inverses are elementary. The elementary functions are understood here to be those which are obtained in a finite number of steps by performing algebraic operations and taking exponentials and logarithms. For instance, the function tan \cf — log,(l + Vz)] + [«*+ log arc sin*]1« is elementary. We prove that if F(z) and its inverse are both elementary, there exist n functions Vt(*)j <hiz), •', <Pniz), where each <piz) with an odd index is algebraic, and each (p(z) with an even index is either e? or log?, such that F(g) = <pn(pn-f-<Pn9i(z) each q>i(z)ii<ît) being substituted for z in ^>iiiz). That every F(z) of this type has an elementary inverse is obvious. It remains to develop a method for recognizing whether a given elementary function can be reduced to the above form for F(.z). How to test fairly simple functions will be evident from the details of our proofs. For the immediate present, we let the general question stand. The present paper is an addition to Liouville's work of almost a century ago on the classification of the elementary functions, on the possibility of effecting integrations in finite terms, and on the impossibility of solving certain differentia] equations, and certain transcendental equations, in finite terms.!" Free use is made here of the ingenious methods of Liouville. * Presented to the Society, October 25, 1924. -(•Journal de l'Ecole Polytechnique, vol. -
December 2010 VOLUME 35 NUMBER 6
December 2010 VOLUME 35 NUMBER 6 OPINION Musings 2 RikR FaR ow SECURITY Introducing Capsicum: Practical Capabilities for UNIX 7 Robe Rt N.M. watsoN, JoNathaN THE USENIX MAGAZINE aNdeRsoN, beN LauRie, aNd kRis keNNaway The Nocebo Effect on the Web: An Analysis of Fake Anti-Virus Distribution 18 Moheeb abu Ra Jab, Lucas baLLaRd, PaNayiotis MavRoMMatis, NieLs PRovos, aNd XiN Zhao Vulnerable Compliance 26 d aN GeeR Overcoming an Untrusted Computing Base: Detecting and Removing Malicious Hardware Automatically 31 Matthew hicks, MuRPh FiNNicuM, saMueL t. kiNG, MiLo M.k. MaRtiN, aNd JoNathaN M. sMith C OLUMNS Practical Perl Tools: Family Man 42 d avid N. bLaNk-edeLMaN Pete’s All Things Sun: Comparing Solaris to RedHat Enterprise and AIX 48 Petee R ba R GaLviN iVoyeur: Ganglia on the Brain 54 d ave JosePhseN /dev/random 58 Robe Rt G. Fe RReLL BOEO K reVI WS Book Reviews 60 El iZabeth Zwicky et a L. useni X NOTES Thankso t Our Volunteers 64 Ell u ie yo ng CN O FERENCES 19th USENIX Security Symposium Reports 67 Report on the 5th USENIX Workshop on Hot Topics in Security 97 Report on the 1st USENIX Workshop on Health Security and Privacy 103 Report on the 4th USENIX Workshop on Offensive Technologies 112 Report on the New Security Paradigms Workshop 117 The Advanced Computing Systems Association dec10covers.indd 1 11.17.10 1:28 PM Upcoming Events 9th USENIX CoNfErence oN fIlE aNd StoragE 2011 USENIX fEdEratEd CoNfErences Week techNologies (FASt ’11) j une 12–17, 2011, portland, OR, uSa Sponsored by USENIX in cooperation with ACM SIGOPS EventS inclUdE: -
Arxiv:1911.10319V2 [Math.CA] 29 Dec 2019 Roswl Egvni H Etscin.Atraiey Ecnuse Can We Alternatively, Sections
Elementary hypergeometric functions, Heun functions, and moments of MKZ operators Ana-Maria Acu1a, Ioan Rasab aLucian Blaga University of Sibiu, Department of Mathematics and Informatics, Str. Dr. I. Ratiu, No.5-7, RO-550012 Sibiu, Romania, e-mail: [email protected] bTechnical University of Cluj-Napoca, Faculty of Automation and Computer Science, Department of Mathematics, Str. Memorandumului nr. 28 Cluj-Napoca, Romania, e-mail: [email protected] Abstract We consider some hypergeometric functions and prove that they are elementary functions. Con- sequently, the second order moments of Meyer-K¨onig and Zeller type operators are elementary functions. The higher order moments of these operators are expressed in terms of elementary func- tions and polylogarithms. Other applications are concerned with the expansion of certain Heun functions in series or finite sums of elementary hypergeometric functions. Keywords: hypergeometric functions, elementary functions, Meyer-K¨onig and Zeller type operators, polylogarithms; Heun functions 2010 MSC: 33C05, 33C90, 33E30, 41A36 1. Introduction This paper is devoted to some families of elementary hypergeometric functions, with applica- tions to the moments of Meyer-K¨onig and Zeller type operators and to the expansion of certain Heun functions in series or finite sums of elementary hypergeometric functions. In [4], J.A.H. Alkemade proved that the second order moment of the Meyer-K¨onig and Zeller operators can be expressed as 2 2 x(1 − x) Mn(e2; x)= x + 2F1(1, 2; n + 2; x), x ∈ [0, 1). (1.1) arXiv:1911.10319v2 [math.CA] 29 Dec 2019 n +1 r Here er(t) := t , t ∈ [0, 1], r ≥ 0, and 2F1(a,b; c; x) denotes the hypergeometric function. -
Agents for Games and Simulations
First International Workshop on Agents for Games and Simulations Budapest May 11, 2009 i Table of Contents Preface iii Avi Rosenfeld and Sarit Kraus. Modeling Agents through Bounded Rationality Theories 1 Jean-Pierre Briot, Alessandro Sordoni, Eurico Vasconcelos, Gustavo Melo, Marta de Azevedo Irving and Isabelle Alvarez. Design of a Decision Maker Agent for a Distributed Role Playing Game - Experience of the SimParc Project 16 Michael Köster, Peter Novák, David Mainzer and Bernd Fuhrmann. Two Case Studies for Jazzyk BSM 31 Joost Westra, Hado van Hasselt, Frank Dignum and Virginia Dignum. Adaptive serious games using agent organizations 46 D.W.F. van Krevelen. Intelligent Agent Modeling as Serious Game 61 Gustavo Aranda, Vicent Botti and Carlos Carrascosa. The MMOG Layer: MMOG based on MAS 75 Ivan M. Monteiro and Luis O. Alvares. A Teamwork Infrastructure for Computer Games with Real-Time Requirements 90 Barry Silverman, Deepthi Chandrasekaran, Nathan Weyer, David Pietrocola, Robert Might and Ransom Weaver. NonKin Village: A Training Game for Learning Cultural Terrain and Sustainable Counter-Insurgent Operations 106 Mei Yii Lim, Joao Dias, Ruth Aylett and Ana Paiva. Intelligent NPCs for Education Role Play Game 117 Derek J. Sollenberger and Munindar P. Singh. Architecture for Affective Social Games 129 Jakub Gemrot, Rudolf Kadlec, Michal Bída, Ondřej Burkert, Radek Píbil, Jan Havlíček, Juraj Šimlovič, Radim Vansa, Michal Štolba, Lukáš Zemčák and Cyril Brom. Pogamut 3 Can Assist Developers in Building AI for Their Videogame Agents 144 Daniel Castro Silva, Ricardo Silva, Luís Paulo Reis and Eugénio Oliveira. Agent-Based Aircraft Control Strategies in a Simulated Environment 149 ii Preface Multi Agent System research offers a promising technology to implement cognitive intelligent Non Playing Characters. -
The Mathematical-Function Computation Handbook Nelson H.F
The Mathematical-Function Computation Handbook Nelson H.F. Beebe The Mathematical-Function Computation Handbook Programming Using the MathCW Portable Software Library Nelson H.F. Beebe Department of Mathematics University of Utah Salt Lake City, UT USA ISBN 978-3-319-64109-6 ISBN 978-3-319-64110-2 (eBook) DOI 10.1007/978-3-319-64110-2 Library of Congress Control Number: 2017947446 © Springer International Publishing AG 2017 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Dedication This book and its software are dedicated to three Williams: Cody, Kahan, and Waite. -
ABSTRACT LOHMEYER, EDWIN LLOYD. Unstable Aesthetics
ABSTRACT LOHMEYER, EDWIN LLOYD. Unstable Aesthetics: The Game Engine and Art Modifications (Under the direction of Dr. Andrew Johnston). This dissertation examines episodes in the history of video game modding between 1995 and 2010, situated around the introduction of the game engine as a software framework for developing three-dimensional gamespaces. These modifications made to existing software and hardware were an aesthetic practice used by programmers and artists to explore the relationship between abstraction, the materiality of game systems, and our phenomenal engagement with digital media. The contemporary artists that I highlight—JODI, Cory Arcangel, Orhan Kipcak, Julian Oliver, and Tom Betts—gravitated toward modding because it allowed them to unveil the technical processes of the engine underneath layers of the game’s familiar interface, in turn, recalibrating conventional play into sensual experiences of difference, uncertainty, and the new. From an engagement with abstract forms, they employed modding techniques to articulate new modes of aesthetic participation through an affective encounter with altered game systems. Furthermore, they used abstraction, the very strangeness of the mod’s formal elements, to reveal our habitual interactions with video games by destabilizing conventional gamespaces through sensory modalities of apperception and proprioception. In considering the imbrication of technics and aesthetics in game engines, this work aims to resituate modding practices within a dynamic and more inclusive understanding -
INTEGRATION STRATEGIES 1. Introduction It Should Be Clear Now
INTEGRATION STRATEGIES 1. Introduction It should be clear now that it is a whole lot easier to differentiate a function than integrating one. Typically the formula used to determine the derivative of a function is apparent. With integration, the appropriate formula won't necessarily be obvious. So far we have reviewed each of the basic techniques of integration: with substitutions (ch. 5.5, 7.2, 7.3 ), integration by parts (ch. 7.1), and partial fractions (ch 7.4). More realistically, integrals will be given outside of the context of a textbook chapter and the challenge is to identify which formula(s) to use. We will review several integrals at random and suggest strategies for determining which formula to evaluate the integral. These strategies will only be helpful if one has a knowledge of basic integration formulas. Listed in the table below are an expanded list of the formulas for commonly used integrals. Many of these can be derived using basic principles it will be helpful to memorize these. In lieu of formulas 19 and 20 one could use partial fractions and trigonometric substitutions respectively. Figure 1. Table of Integration Formulas Constant of integra- tion have been omitted. 1 2 INTEGRATION STRATEGIES 2. Strategies If an integral does not fit into this basic list of integration formulas, try the following strategy: (1) Simplify the Integrand if Possible: Algebraic manipulation or trigono- metric identities can simplify the integrand and allow for integration in its new form. For example, p p p R x(1 + x)dx = R ( x + x)dx; R tan θ R sin θ 2 R R sec2 θ dθ = cos θ cos θdθ = sin θ cos θdθ = frac12 sin 2θdθ; R (sin x + cos x)2dx = R (sin2 x + 2 sin x cos x + cos2 x)dx = R (1 + 2 sin x cos x)dx; (2) Look for an Obvious Substitution: Try substitutions u = g(x) in the integrand for which the differential du = g0(x)dx also appears in the integran. -
Survey of the Fast Computation of the Reciprocal Square Root Through Integer Operations on Floating-Point Values
Survey of the Fast Computation of the Reciprocal Square Root through Integer Operations on Floating-Point Values Thomas Nelson University of Massachusetts Lowell Published July 27, 2017 Abstract Finding a value's reciprocal square root has many uses in vector-based calculations, but arithmetic calculations of finding a square root and performing division are too computationally expensive for typical large-scale real-time performance constraints. This survey examines the \Fast Inverse Square Root" algorithm and explores the techniques of its implementation through examination of the Newton-Raphson Method of Approximation and the magic-number 0x5f3759df which allow for the reciprocal square root to be calculated with only multiplication and subtraction operations. 1 Overview p1 The reciprocal of the square root of a value, x , also called the \inverse square root" is necessary for vector calculations which are instrumental in common 3D-rendering applications. Due to the time-sensitive nature of these applications, there has been a great deal of development over the past 20 years in providing fast approximations of this reciprocal value. While hardware manufacturers began including instruction-set level methods of performing this calculation in 1999, it has taken some time for these to become widespread in end-user machines [5]. An algorithm to quickly find this reciprocal, known as the \Fast Inverse Square Root" algorithm, was popularized and publicized in the late 90s as a general-use solution. 1 Fast Computation of the Reciprocal Square Root THE ALGORITHM This survey explores this \Fast Inverse Square Root" algorithm in detail, providing a full examination of the mathematical formula it employs as well as a derivation of the notorious magic number 0x5f3759df found therein. -
Antiderivatives Basic Integration Formulas Structural Type Formulas
Antiderivatives Definition 1 (Antiderivative). If F 0(x) = f(x) we call F an antideriv- ative of f. Definition 2 (Indefinite Integral). If F is an antiderivative of f, then R f(x) dx = F (x) + c is called the (general) Indefinite Integral of f, where c is an arbitrary constant. The indefinite integral of a function represents every possible antideriv- ative, since it has been shown that if two functions have the same de- rivative on an interval then they differ by a constant on that interval. Terminology: When we write R f(x) dx, f(x) is referred to as the in- tegrand. Basic Integration Formulas As with differentiation, there are two types of formulas, formulas for the integrals of specific functions and structural type formulas. Each formula for the derivative of a specific function corresponds to a formula for the derivative of an elementary function. The following table lists integration formulas side by side with the corresponding differentiation formulas. Z xn+1 d xn dx = if n 6= −1 (xn) = nxn−1 n + 1 dx Z d sin x dx = − cos x + c (cos x) = − sin x dx Z d cos x dx = sin x + c (sin x) = cos x dx Z d sec2 x dx = tan x + c (tan x) = sec2 x dx Z d ex dx = ex + c (ex) = ex dx Z 1 d 1 dx = ln x + c (ln x) = x dx x Z d k dx = kx + c (kx) = k dx Structural Type Formulas We may integrate term-by-term: R kf(x) dx = k R f(x) dx 1 2 R f(x) ± g(x) dx = R f(x) dx ± R g(x) dx In plain language, the integral of a constant times a function equals the constant times the derivative of the function and the derivative of a sum or difference is equal to the sum or difference of the derivatives.