DOCSLIB.ORG

Development of Elementary Mathematics Functions in an Avionics Context

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Development of Elementary Mathematics Functions in an Avionics Context Fakultät für Maschinenwesen Lehrstuhl für Flugsystemdynamik Development of Elementary Mathematics Functions in an Avionics Context Dipl.-Ing. Kajetan Nürnberger Vollständiger Abdruck der von der Fakultät für Maschinenwesen der Technischen Universität München zur Erlangung des akademischen Grades eines Doktor-Ingenieurs (Dr.-Ing.) genehmigten Dissertation. Vorsitzender: Prof. Dr.-Ing. Manfred Hajek Prüfer der Dissertation: 1. Prof. Dr.-Ing. Florian Holzapfel 2. Prof. Dr. Marco Caccamo Die Dissertation wurde am 12.04.2019 bei der Technischen Universität München eingereicht und durch die Fakultät für Maschinenwesen am 29.09.2019 angenommen. Acknowledgement This thesis is the final output after working for five years at the Institute of Flight System Dynamics at TU Munich plus some long evenings after leaving the university. The time at the institute was one of the most inspiring times of my life. I did not only gain many new insights in the area of my research, but also in many related fields as handling of projects or communicating in complex organizations. Although working at the institute was a really interesting and often quite funny time, there were also parts that were challenging. Without the support of various people, writing this thesis would not have been possible. Therefore, I would like to express my gratitude to all of these here. First of all, I would like to thank Professor Florian Holzapfel. He gave me the chance to work at the institute on really interesting technology. During the time at the institute, I had the chance to gain insights into different research fields. I appreciate a lot that Florian gave me the possibility to freely decide on the research area this thesis is dealing with. A big thank you to all my colleagues at the institute who contributed to the flight control model which served as a real-world example for the research work presented in this thesis. Special thanks to our coffee group at various locations. During the coffee breaks, several interesting discussions took place. In case I was suck at a problem, this brake gave the necessary distraction. Really important for me is that during the time at the institute, I met a lot of new colleagues who became good friends in the meantime. I would like to express my gratitude to Professor Marco Caccamo for taking the task of the second examiner shortly after he joined TU Munich. Thanks also to Professor Manfred Hajek who acted as the chairman of the committee after the original chairman has left the TU Munich. Thanks to everyone who gave comments on the work which showed me the right spots where further improvement was necessary. Finally, I want to thank my family who always supported me during the time the thesis was constructed. Without their support, it would not have been possible to finalize the thesis. Page iii Abstract Model based development approaches enable an early test of the developed algorithm in a simulation environment. To get the system to the actual target platform, auto code generation can be used. As the auto generated code relies on external components, the target behavior can deviate from the behavior in the simulation, although the automatically generated code does not contain any error. This problem is addressed in this thesis for a flight control system where the external components are elementary math functions. The main challenges addressed during the development of the math functions are an efficient execution and a formal verification of the algorithms. The efficient implementation can reduce the overall latency of the control loop and, therefore, lead to a better overall performance. The formal verification mainly addresses the precision of the implementation. Many available library implementations lack of such a precision proof. In case it is available the goal is to show the accuracy up to the last bit, which is not desired for an embedded system. To reach the goal of an efficient formally verified implementation, the input ranges of the elementary math functions are adapted. This is motivated based on functional considerations and verified by a runtime error analysis of the integrated software. In order to successfully execute this analysis with no false alarms, the elementary math functions are replaced by specific stubs in the analysis. These stubs abstract the behavior of the original function in a way that sound result with a low number of false alarms is achieved in the runtime error analysis. The validity of all properties of the stubs are shown via formal proofs along with the verification of the reached precision. In order to reach a high precision in an efficient manner advanced polynomial approximations are used and executed using double precision floating-point arithmetic. To ensure the portability between the target and the development host architecture, hardware specific functions are omitted. In case it is not possible to omit functions custom to the hardware, these are captured in a separate abstraction layer to ensure an easy portability. Approximations are given for the trigonometric functions, the square root and the power function with integer exponents. The proposed algorithms can completely be implemented on any processor supporting double precision floating-point arithmetic, as they do not rely on special hardware features. Different approximation methods are evaluated and the actually selected implementation is based on the worst case execution time. To determine the worst case timing a static analysis method is used. As the analysis during the early development phase is limited to a small function of the overall software, it is assumed that the function is available in the cache memory for the analysis. The validity of this assumption is later shown in an execution time analysis of the complete software. Here it is shown that locking the elementary math functions to the cache statically brings a benefit of the analyzed worst case time and the measured maximum and average execution time. With the use of the WCET analysis results further parts of the software can be identified as candidates for a cache locking. By statically locking these parts, the WCET, the measured maximum and average execution time can be further reduced. Page v Contents and Lists Table of Contents List of Figures ........................................................................................................................ xi List of Tables ....................................................................................................................... xiii List of Acronyms ................................................................................................................... xv List of Symbols ................................................................................................................... xvii 1 Introduction .................................................................................................................... 1 1.1 Motivation and Background ..................................................................................... 1 1.2 State of the Art ........................................................................................................ 5 1.3 Objectives ............................................................................................................... 9 1.4 Contributions ..........................................................................................................10 1.5 Outline ....................................................................................................................11 2 Fundamentals ...............................................................................................................13 2.1 Nomenclature .........................................................................................................13 2.2 Floating-Point Arithmetic ........................................................................................14 2.3 Worst Case Execution Time ...................................................................................17 3 Demonstration Environment Description .......................................................................21 3.1 Flight Control Computer Hardware Overview .........................................................21 3.2 Application Software Overview ...............................................................................22 3.3 Model-Based Development Approach ....................................................................23 3.4 Base Software ........................................................................................................25 3.5 Software Statistics ..................................................................................................26 4 Library Development Process........................................................................................29 4.1 Workflow ................................................................................................................29 4.2 Architecture and Implementation Considerations ...................................................35 4.3 Relation to Certification Aspects .............................................................................38 5 Trigonometric Functions ................................................................................................41 5.1 Sine and Cosine .....................................................................................................41 5.1.1 Input Restrictions ............................................................................................41
Load more

Recommended publications
  • On Various Ways to Split a Floating-Point Number
    On various ways to split a floating-point number Claude-Pierre Jeannerod Jean-Michel Muller Paul Zimmermann Inria, CNRS, ENS Lyon, Universit´ede Lyon, Universit´ede Lorraine France ARITH-25 June 2018 -1- Splitting a floating-point number X = ? round(x) frac(x) 0 2 First bit of x = ufp(x) (for scalings) X All products are s computed exactly 2 2 X with one FP p 2 2 k a b multiplication a + b ! 2 k + k X (Dekker product) 2 2 X Dekker product (1971) -2- Splitting a floating-point number absolute splittings (e.g., bxc), vs relative splittings (e.g., most significant bits, splitting of the significands for multiplication); In each "bin", the sum is no bit manipulations of the computed exactly binary representations (would result in less portable Matlab program in a paper by programs) ! onlyFP Zielke and Drygalla (2003), operations. analysed and improved by Rump, Ogita, and Oishi (2008), reproducible summation, by Demmel & Nguyen. -3- Notation and preliminary definitions IEEE-754 compliant FP arithmetic with radix β, precision p, and extremal exponents emin and emax; F = set of FP numbers. x 2 F can be written M x = · βe ; βp−1 p M, e 2 Z, with jMj < β and emin 6 e 6 emax, and jMj maximum under these constraints; significand of x: M · β−p+1; RN = rounding to nearest with some given tie-breaking rule (assumed to be either \to even" or \to away", as in IEEE 754-2008); -4- Notation and preliminary definitions Definition 1 (classical ulp) The unit in the last place of t 2 R is ( βblogβ jt|c−p+1 if jtj βemin , ulp(t) = > βemin−p+1 otherwise.
    [Show full text]
  • Floating Point Arithmetic
    Systems Architecture Lecture 14: Floating Point Arithmetic Jeremy R. Johnson Anatole D. Ruslanov William M. Mongan Some or all figures from Computer Organization and Design: The Hardware/Software Approach, Third Edition, by David Patterson and John Hennessy, are copyrighted material (COPYRIGHT 2004 MORGAN KAUFMANN PUBLISHERS, INC. ALL RIGHTS RESERVED). Lec 14 Systems Architecture 1 Introduction • Objective: To provide hardware support for floating point arithmetic. To understand how to represent floating point numbers in the computer and how to perform arithmetic with them. Also to learn how to use floating point arithmetic in MIPS. • Approximate arithmetic – Finite Range – Limited Precision • Topics – IEEE format for single and double precision floating point numbers – Floating point addition and multiplication – Support for floating point computation in MIPS Lec 14 Systems Architecture 2 Distribution of Floating Point Numbers e = -1 e = 0 e = 1 • 3 bit mantissa 1.00 X 2^(-1) = 1/2 1.00 X 2^0 = 1 1.00 X 2^1 = 2 1.01 X 2^(-1) = 5/8 1.01 X 2^0 = 5/4 1.01 X 2^1 = 5/2 • exponent {-1,0,1} 1.10 X 2^(-1) = 3/4 1.10 X 2^0 = 3/2 1.10 X 2^1= 3 1.11 X 2^(-1) = 7/8 1.11 X 2^0 = 7/4 1.11 X 2^1 = 7/2 0 1 2 3 Lec 14 Systems Architecture 3 Floating Point • An IEEE floating point representation consists of – A Sign Bit (no surprise) – An Exponent (“times 2 to the what?”) – Mantissa (“Significand”), which is assumed to be 1.xxxxx (thus, one bit of the mantissa is implied as 1) – This is called a normalized representation • So a mantissa = 0 really is interpreted to be 1.0, and a mantissa of all 1111 is interpreted to be 1.1111 • Special cases are used to represent denormalized mantissas (true mantissa = 0), NaN, etc., as will be discussed.
    [Show full text]
  • Floating Point Square Root Under HUB Format
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Repositorio Institucional Universidad de Málaga Floating Point Square Root under HUB Format Julio Villalba-Moreno Javier Hormigo Dept. of Computer Architecture Dept. of Computer Architecture University of Malaga University of Malaga Malaga, SPAIN Malaga, SPAIN [email protected] [email protected] Abstract—Unit-Biased (HUB) is an emerging format based on maintaining the same accuracy, the area cost and delay of FIR shifting the representation line of the binary numbers by half filter implementations has been drastically reduced in [3], and unit in the last place. The HUB format is specially relevant similarly for the QR decomposition in [4]. for computers where rounding to nearest is required because it is performed simply by truncation. From a hardware point In [5] the authors carry out an analysis of using HUB format of view, the circuits implementing this representation save both for floating point adders, multipliers and converters from a area and time since rounding does not involve any carry propa- quantitative point of view. Experimental studies demonstrate gation. Designs to perform the four basic operations have been that HUB format keeps the same accuracy as conventional proposed under HUB format recently. Nevertheless, the square format for the aforementioned units, simultaneously improving root operation has not been confronted yet. In this paper we present an architecture to carry out the square root operation area, speed and power consumption (14% speed-up, 38% less under HUB format for floating point numbers. The results of area and 26% less power for single precision floating point this work keep supporting the fact that the HUB representation adder, 17% speed-up, 22% less area and slightly less power for involves simpler hardware than its conventional counterpart for the floating point multiplier).
    [Show full text]
  • AC Library for Emulating Low-Precision Arithmetic Fasi
    CPFloat: A C library for emulating low-precision arithmetic Fasi, Massimiliano and Mikaitis, Mantas 2020 MIMS EPrint: 2020.22 Manchester Institute for Mathematical Sciences School of Mathematics The University of Manchester Reports available from: http://eprints.maths.manchester.ac.uk/ And by contacting: The MIMS Secretary School of Mathematics The University of Manchester Manchester, M13 9PL, UK ISSN 1749-9097 CPFloat: A C library for emulating low-precision arithmetic∗ Massimiliano Fasi y Mantas Mikaitis z Abstract Low-precision floating-point arithmetic can be simulated via software by executing each arith- metic operation in hardware and rounding the result to the desired number of significant bits. For IEEE-compliant formats, rounding requires only standard mathematical library functions, but handling subnormals, underflow, and overflow demands special attention, and numerical errors can cause mathematically correct formulae to behave incorrectly in finite arithmetic. Moreover, the ensuing algorithms are not necessarily efficient, as the library functions these techniques build upon are typically designed to handle a broad range of cases and may not be optimized for the specific needs of rounding algorithms. CPFloat is a C library that of- fers efficient routines for rounding arrays of binary32 and binary64 numbers to lower precision. The software exploits the bit level representation of the underlying formats and performs only low-level bit manipulation and integer arithmetic, without relying on costly library calls. In numerical experiments the new techniques bring a considerable speedup (typically one order of magnitude or more) over existing alternatives in C, C++, and MATLAB. To the best of our knowledge, CPFloat is currently the most efficient and complete library for experimenting with custom low-precision floating-point arithmetic available in any language.
    [Show full text]
  • Numerical Computation Guide
    Numerical Computation Guide A Sun Microsystems, Inc. Business 2550 Garcia Avenue Mountain View, CA 94043 U.S.A. Part No.: 801-7639-10 Revision A, August 1994 1994 Sun Microsystems, Inc. 2550 Garcia Avenue, Mountain View, California 94043-1100 U.S.A. All rights reserved. This product and related documentation are protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation. No part of this product or related documentation may be reproduced in any form by any means without prior written authorization of Sun and its licensors, if any. Portions of this product may be derived from the UNIX® and Berkeley 4.3 BSD systems, licensed from UNIX System Laboratories, Inc., a wholly owned subsidiary of Novell, Inc., and the University of California, respectively. Third-party font software in this product is protected by copyright and licensed from Sun’s font suppliers. RESTRICTED RIGHTS LEGEND: Use, duplication, or disclosure by the United States Government is subject to the restrictions set forth in DFARS 252.227-7013 (c)(1)(ii) and FAR 52.227-19. The product described in this manual may be protected by one or more U.S. patents, foreign patents, or pending applications. TRADEMARKS Sun, the Sun logo, Sun Microsystems, Sun Microsystems Computer Corporation, Solaris, are trademarks or registered trademarks of Sun Microsystems, Inc. in the U.S. and certain other countries. UNIX is a registered trademark of Novell, Inc., in the United States and other countries; X/Open Company, Ltd., is the exclusive licensor of such trademark. OPEN LOOK® is a registered trademark of Novell, Inc.
    [Show full text]
  • A Study of the Floating-Point Tuning Behaviour on the N-Body Problem
    A Study of the Floating-Point Tuning Behaviour on the N-body Problem Dorra Ben Khalifa1 and Matthieu Martel1;2 1 University of Perpignan, LAMPS laboratory, 52 Av. P. Alduy, Perpignan, France 2 Numalis, Cap Omega, Rond-point Benjamin Franklin, Montpellier, France fdorra.ben-khalifa, [email protected] Abstract. In this article, we apply a new methodology for precision tun- ing to the N-body problem. Our technique, implemented in a tool named POP, makes it possible to optimize the numerical data types of a pro- gram performing floating-point computations by taking into account the requested accuracy on the results. POP reduces the problem of finding the minimal number of bits needed for each variable of the program to an Integer Linear Problem (ILP) which can be optimally solved in one shot by a classical linear programming solver. The POP tool has been suc- cessfully tested on programs implementing several numerical algorithms coming from mathematical libraries and other applicative domains such as IoT. In this work, we demonstrate the efficiency of POP to tune the classical gravitational N-body problem by considering five bodies that interact under gravitational force from one another, subject to Newton's laws of motion. Results on the effect of POP in term of mixed-precision tuning of the N-body example are discussed. Keywords: Computer arithmetic, precision tuning, integer linear prob- lems, N-body problem, numerical accuracy. 1 Introduction Reducing the precision of floating-point data, also called precision tuning, pro- vides for many practitioners in High-Performance Computing (HPC) and related fields an opportunity for exploring trade-offs in accuracy and performance [16].
    [Show full text]
  • Customizing Floating-Point Units for Fpgas: Area-Performance-Standard
    Customizing floating-point units for FPGAs: Area-performance-standard trade-offs Pedro Echeverría *, Marisa López-Vallejo Departamento de Ingeniería Electrónica, Universidad Politécnica de Madrid, Spain ABSTRACT The high integration density of current nanometer technologies allows the implementation of complex floating-point applications in a single FPGA. In this work the intrinsic complexity of floating-point oper­ ators is addressed targeting configurable devices and making design decisions providing the most suit­ Keywords: able performance-standard compliance trade-offs. A set of floating-point libraries composed of adder/ Floating-point arithmetic subtracter, multiplier, divisor, square root, exponential, logarithm and power function are presented. FPGAs Each library has been designed taking into account special characteristics of current FPGAs, and with this Library of operators purpose we have adapted the IEEE floating-point standard (software-oriented) to a custom FPGA-ori- High performance ented format. Extended experimental results validate the design decisions made and prove the usefulness of reducing the format complexity. 1. Introduction impossible or makes the design cycle extremely long. Thus, the availability of complete and fully characterized floating-point Current deep sub-micron technologies allow manufacturing of libraries targeting FPGAs has become a must. FPGAs with extraordinary logic density and speed. The initial chal­ The IEEE standard for binary floating-point arithmetic was con­ lenges related to FPGAs programmability and large interconnection ceived to be implemented through custom VLSI units developed for capacitances (poor performance, low logic density and high power microprocessors. However, if the target hardware is an FPGA, the dissipation) have been overcome while providing attractive low internal architecture of these operators must be highly optimized cost and flexibility [1].
    [Show full text]
  • Floating Point
    Contents Articles Floating point 1 Positional notation 22 References Article Sources and Contributors 32 Image Sources, Licenses and Contributors 33 Article Licenses License 34 Floating point 1 Floating point In computing, floating point describes a method of representing an approximation of a real number in a way that can support a wide range of values. The numbers are, in general, represented approximately to a fixed number of significant digits (the significand) and scaled using an exponent. The base for the scaling is normally 2, 10 or 16. The typical number that can be represented exactly is of the form: Significant digits × baseexponent The idea of floating-point representation over intrinsically integer fixed-point numbers, which consist purely of significand, is that An early electromechanical programmable computer, expanding it with the exponent component achieves greater range. the Z3, included floating-point arithmetic (replica on display at Deutsches Museum in Munich). For instance, to represent large values, e.g. distances between galaxies, there is no need to keep all 39 decimal places down to femtometre-resolution (employed in particle physics). Assuming that the best resolution is in light years, only the 9 most significant decimal digits matter, whereas the remaining 30 digits carry pure noise, and thus can be safely dropped. This represents a savings of 100 bits of computer data storage. Instead of these 100 bits, much fewer are used to represent the scale (the exponent), e.g. 8 bits or 2 A diagram showing a representation of a decimal decimal digits. Given that one number can encode both astronomic floating-point number using a mantissa and an and subatomic distances with the same nine digits of accuracy, but exponent.
    [Show full text]
  • ERC Advanced Grant Research Proposal (Part B2)
    ERC Advanced Grant Research Proposal (Part B2) CORE-MATH: Ensuring Correctly Rounded Mathematical Functions Principal Investigator (PI): Dr Paul Zimmermann PI's host institution: Inria, France Proposal full title: Ensuring Correctly Rounded Mathematical Functions Proposal short name: CORE-MATH Project duration: 60 months Targeted Review Panel: PE6 (Computer Science and Informatics) In 1985, the IEEE 754 standard defined for the first time what the result of a computa- tion on floating-point numbers should be. Today, any program using floating-point additions, subtractions, multiplications and divisions yields bit-to-bit identical results, whatever the hard- ware, compiler, or operating system. This is because IEEE 754 requires the best possible result for these operations, called correct rounding. However, most scientific or industrial applications, like the Large Hadron Collider software, developed by thousands of physicists and engineers over two decades, or Karplus' equation J(φ) = A cos2 φ + B cos φ + C in nuclear magnetic resonance spectroscopy, also require the evaluation of various mathematical functions: sine, exponential, logarithm, etc. These functions are provided by a mathematical library, which does not always provide correct rounding. As a resulting effect, a program using such mathematical functions might yield wrong results, which could have disastrous consequences [25]. Moreover, these results might differ depending on the mathematical library, hardware, compiler or operating system. We strongly believe it is the right time to fix that numerical reproducibility issue once and for all. CORE-MATH will provide new numerical algorithms to evaluate mathematical functions, which will always yield correct rounding (i.e., the best possible result) with speed comparable to the best libraries currently available.
    [Show full text]
  • Ulps and Relative Error
    2017 IEEE 24th Symposium on Computer Arithmetic ULPs and Relative Error Marius Cornea Intel Corporation Hillsboro, OR, USA [email protected] The paper establishes several simple, but useful relationships much more convenient. Combining the two methods might between ulp (unit in the last place) errors and the corresponding represent the best solution for such proofs, especially when we relative errors. These can be used when converting between the need to analyze sequences of multiplications, divisions, and in two types of errors, ensuring that the least amount of information some cases also fused multiply-add operations. A good is lost in the process. The properties presented here were already example can be found in [3]. The present paper aims to make useful in IEEE conformance proofs for iterative division and this process easier. square root algorithms, and should be so again to numerical analysts both in ‘hand’ proofs of error bounds for floating-point computations, and in automated tools which carry out, or support The IEEE Standard 754-2008 for Floating-Point Arithmetic deriving such proofs. In most cases, the properties shown herein [4] defines both the format of binary floating-point numbers, establish tighter bounds than found in the literature. They also and the value of an ulp (named quantum in the standard). We provide ‘conversion’ rules of finer granularity, at floating-point will use the same definition of floating-point numbers in this value level instead of binade level, and take into account the special paper, however we will consider the integer exponent as being conditions which occur at binade ends.
    [Show full text]
  • Low Power Floating-Point Multiplication and Squaring Units with Shared Circuitry
    Low Power Floating-Point Multiplication and Squaring Units with Shared Circuitry Jason Moore, Mitchell A. Thornton, and David W. Matula Department of Computer Science and Engineering Southern Methodist University Dallas, TX Abstract—An architecture for a combinational floating point architecture presented here is capable of utilizing any full- multiplier and squarer is described for the purpose of producing precision squaring implementation as a core design. a low power floating point square with small area requirements. The floating-point multiplier and squarer architecture are II. BACKGROUND compared to demonstrate the power advantage of the squarer. The multiplier and squarer are combined into one circuit in The IEEE 754 floating point standard [1] can be order to take advantage of the squarer power improvements with interpreted as follows: a minimal increase in area. Shared circuitry among the units 0. 2 0 provides justification for inclusion of a dedicated squarer since a 0 small amount of additional circuitry is required and the power 255 0 savings for squaring computations is significant as compared to the use of a general-purpose multiplier to generate a squared 1 1.2 0 255 value. Conceptually, floating point multiplication is performed Keywords: Floating-point, squarer, combined multiplier and through addition of the exponents and an accompanying squarer multiplication of the significands with normalization governing the value of the product exponent. If both a and b I. INTRODUCTION are normalized, the equation,2 2 The floating point squarer design presented in this paper is 2, is all that is needed to calculate the floating-point a full precision squarer.
    [Show full text]
  • Revisiting" What Every Computer Scientist Should Know About
    Computing and Software for Big Science manuscript No. (will be inserted by the editor) Revisiting “What Every Computer Scientist Should Know About Floating-point Arithmetic” Vincent Lafage Received: Abstract The differences between the sets in which We note that this standard emerged in a rich environ- ideal arithmetics takes place and the sets of floating ment of incompatible formats, and is now supported on point numbers are outlined. A set of classical problems all current major microprocessor architectures. Yet all in correct numerical evaluation is presented, to increase this is but modeling the numbers. Our familiarity with the awareness of newcomers to the field. A self-defense, numbers typically used for computational purpose, i.e. prophylactic approach to numerical computation is ad- the real numbers, may then blur our understanding of vocated. floating point numbers. We will first discuss the common floating point rep- resentation of real numbers in a computer and list the differences between the properties of the numbers and 1 Introduction the properties of their representation. We will then il- lustrate common examples where a naive implementa- Precision, and its commoditization, particularly due to tion of the computation can lead to unexpectedly large the plummeting of its cost and because of its repro- uncertainties. In the next part, we delineate some ap- ducibility, has been the base for our industrial societies proaches to understand, contain or mitigate these ac- since the mid XVIIIth century. The numerical preci- curacy problems. We then conclude on the need for a sion went even faster and farther than in mechanical broader awareness about floating point accuracy issue.
    [Show full text]