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

Formal verification of a fully IEEE compliant floating point unit

In this thesis we describe the formal verification of a fully IEEE compliant floating point unit (FPU). The hardware is verified on the gate-level against a formalization of the IEEE standard. The verification is performed using the theorem proving system PVS. The FPU supports both single and double precision floating point numbers, normal and denormal numbers, all four IEEE rounding modes, and exceptions as required by the standard. Beside the verification of the combinatorial correctness of the FPUs we pipeline the FPUs to allow the integration into an out-of-order processor. We formally define the correctness criterion the pipelines must obey in order to work properly within the processor. We then describe a new methodology based on combining model checking and theorem proving for the verification of the pipelines.Die vorliegende Arbeit behandelt die formale Verifikation einer vollständig IEEE konformen Floating Point Unit (FPU). Die Hardware wird auf Gatter-Ebene gegen eine Formalisierung des IEEE Standards verifiziert. Zur Verifikation wird das Beweis-System PVS benutzt. Die FPU unterstützt Fließkommazahlen mit einfacher und doppelter Genauigkeit, normale und denormale Zahlen, alle vier Rundungsmodi und alle Exception-Signale. Neben der Verifikation der kombinatorischen Schaltkreise werden die FPUs gepipelined, um sie in einen Out-of-order Prozessor zu integrieren. Die Korrektheits- Kriterien, die die gepipelineten FPUs befolgen müssen, um im Prozessor korrekt zu arbeiten, werden formal definiert. Es wird eine neue Methode zur Verifikation solcher Pipelines beschrieben. Die Methode beruht auf der Kombination von Model-Checking und Theorem-Proving

Correctly rounded multiplication by arbitrary precision constants

We introduce an algorithm for multiplying a floating-point number $x$ by a constant $C$ that is not exactly representable in floating-point arithmetic. Our algorithm uses a multiplication and a fused multiply accumulate instruction. We give methods for checking whether, for a given value of $C$ and a given floating-point format, our algorithm returns a correctly rounded result for any $x$. When it does not, our methods give the values $x$ for which the multiplication is not correctly rounded.Nous proposons un algorithme permettant de multiplier un nombre virgule flottante x par une constante C qui n’est pas exactement représentable en virgule flottante.Notre algorithme nécessite la disponibilité d’une instruction “multiplication-accumulation”. Nous donnons des méthodes pour tester si,pour une constante C et un format virgule flottante donnés, notre algorithme donnera un arrondi correct pour toutes les valeurs de x.Quand ce n’est pas le cas,nos méthodes permettent de connaître toutes les valeurs de x pour lesquelles la multiplication par C n’est pas arrondie correctement

Refresher course in maths and a project on numerical modeling done in twos

These lecture notes accompany a refresher course in applied mathematics with a focus on numerical concepts (Part I), numerical linear algebra (Part II), numerical analysis, Fourier series and Fourier transforms (Part III), and differential equations (Part IV). Several numerical projects for group work are provided in Part V. In these projects, the tasks are threefold: mathematical modeling, algorithmic design, and implementation. Therein, it is important to draw interpretations of the obtained results and provide measures (Parts I-IV) how to build confidence into numerical findings such intuition, error analysis, convergence analysis, and comparison to manufactured solutions. Both authors have been jointly teaching over several years this class and bring in a unique mixture of their respective teaching and research fields

Rigorous numerical approaches in electronic structure theory

Electronic structure theory concerns the description of molecular properties according to the postulates of quantum mechanics. For practical purposes, this is realized entirely through numerical computation, the scope of which is constrained by computational costs that increases rapidly with the size of the system. The significant progress made in this field over the past decades have been facilitated in part by the willingness of chemists to forego some mathematical rigour in exchange for greater efficiency. While such compromises allow large systems to be computed feasibly, there are lingering concerns over the impact that these compromises have on the quality of the results that are produced. This research is motivated by two key issues that contribute to this loss of quality, namely i) the numerical errors accumulated due to the use of finite precision arithmetic and the application of numerical approximations, and ii) the reliance on iterative methods that are not guaranteed to converge to the correct solution. Taking the above issues in consideration, the aim of this thesis is to explore ways to perform electronic structure calculations with greater mathematical rigour, through the application of rigorous numerical methods. Of which, we focus in particular on methods based on interval analysis and deterministic global optimization. The Hartree-Fock electronic structure method will be used as the subject of this study due to its ubiquity within this domain. We outline an approach for placing rigorous bounds on numerical error in Hartree-Fock computations. This is achieved through the application of interval analysis techniques, which are able to rigorously bound and propagate quantities affected by numerical errors. Using this approach, we implement a program called Interval Hartree-Fock. Given a closed-shell system and the current electronic state, this program is able to compute rigorous error bounds on quantities including i) the total energy, ii) molecular orbital energies, iii) molecular orbital coefficients, and iv) derived electronic properties. Interval Hartree-Fock is adapted as an error analysis tool for studying the impact of numerical error in Hartree-Fock computations. It is used to investigate the effect of input related factors such as system size and basis set types on the numerical accuracy of the Hartree-Fock total energy. Consideration is also given to the impact of various algorithm design decisions. Examples include the application of different integral screening thresholds, the variation between single and double precision arithmetic in two-electron integral evaluation, and the adjustment of interpolation table granularity. These factors are relevant to both the usage of conventional Hartree-Fock code, and the development of Hartree-Fock code optimized for novel computing devices such as graphics processing units. We then present an approach for solving the Hartree-Fock equations to within a guaranteed margin of error. This is achieved by treating the Hartree-Fock equations as a non-convex global optimization problem, which is then solved using deterministic global optimization. The main contribution of this work is the development of algorithms for handling quantum chemistry specific expressions such as the one and two-electron integrals within the deterministic global optimization framework. This approach was implemented as an extension to an existing open source solver. Proof of concept calculations are performed for a variety of problems within Hartree-Fock theory, including those in i) point energy calculation, ii) geometry optimization, iii) basis set optimization, and iv) excited state calculation. Performance analyses of these calculations are also presented and discussed

Correctness proofs outline for Newton-Raphson based floating-point divide and square root algorithms

This paper describes a study of a class of algorithms for the floating-point divide and square root operations, based on the Newton-Raphson iterative method. The two main goals were: (1) Proving the IEEE correctness of these iterative floating-point algorithms, i.e. compliance with the IEEE-754 standard for binary floating-point operations [1]. The focus was on software driven iterative algorithms, instead of the hardware based implementations that dominated until now. (2) Identifying the special cases of operands that require software assistance due to possible overflow, underflow, or loss of precision of intermediate results. This study was initiated in an attempt to prove the IEEE correctness for a class of divide and square root algorithms based on the Newton-Rapshson iterative methods. As more insight into the inner workings of these algorithms was gained, it became obvious that a formal study and proof were necessary in order to achieve the desired objectives. The result is a complete and rigorous proof of IEEE correctness for floating-point divide and square root algorithms based on the Newton-Raphson iterative method. Even more, the method used in proving the IEEE correctness of the square root algorithm is applicable in principle to any iterative algorithm, not only based on the Newton-Raphson method. Conditions requiring Software Assistance (SWA) were also determined, and were used to identify cases when alternate algorithms are needed to generate correct results. Overall, this is one important step toward flawless implementation of these floating-point operations based on software implementations

Formal methods and digital systems validation for airborne systems

This report has been prepared to supplement a forthcoming chapter on formal methods in the FAA Digital Systems Validation Handbook. Its purpose is as follows: to outline the technical basis for formal methods in computer science; to explain the use of formal methods in the specification and verification of software and hardware requirements, designs, and implementations; to identify the benefits, weaknesses, and difficulties in applying these methods to digital systems used on board aircraft; and to suggest factors for consideration when formal methods are offered in support of certification. These latter factors assume the context for software development and assurance described in RTCA document DO-178B, 'Software Considerations in Airborne Systems and Equipment Certification,' Dec. 1992