The formal verification of a pipelined double-precision IEEE floating-point multiplier

Floating-point circuits are notoriously difficult to design and verify. For verification, simulation barely offers adequate coverage, conventional model-checking techniques are infeasible, and theorem-proving based verification is not sufficiently mature. In this paper we present the formal verification of a radix-eight, pipelined, IEEE double-precision floating-point multiplier. The verification was carried out using a mixture of model-checking and theorem-proving techniques in the Voss hardware verification system. By combining model-checking and theorem-proving we were able to build on the strengths of both areas and achieve significant results with a reasonable amount of effort.

Formal Verification of an Iterative Low-Power x86 Floating-Point Multiplier with Redundant Feedback

We present the formal verification of a low-power x86 floating-point multiplier. The multiplier operates iteratively and feeds back intermediate results in redundant representation. It supports x87 and SSE instructions in various precisions and can block the issuing of new instructions. The design has been optimized for low-power operation and has not been constrained by the formal verification effort. Additional improvements for the implementation were identified through formal verification. The formal verification of the design also incorporates the implementation of clock-gating and control logic. The core of the verification effort was based on ACL2 theorem proving. Additionally, model checking has been used to verify some properties of the floating-point scheduler that are relevant for the correct operation of the unit.Comment: In Proceedings ACL2 2011, arXiv:1110.447

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

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

Optimisations arithmétiques et synthèse de haut niveau

High-level synthesis (HLS) tools offer increased productivity regarding FPGA programming.However, due to their relatively young nature, they still lack many arithmetic optimizations.This thesis proposes safe arithmetic optimizations that should always be applied.These optimizations are simple operator specializations, following the C semantic.Other require to a lift the semantic embedded in high-level input program languages, which are inherited from software programming, for an improved accuracy/cost/performance ratio.To demonstrate this claim, the sum-of-product of floating-point numbers is used as a case study. The sum is performed on a fixed-point format, which is tailored to the application, according to the context in which the operator is instantiated.In some cases, there is not enough information about the input data to tailor the fixed-point accumulator.The fall-back strategy used in this thesis is to generate an accumulator covering the entire floating-point range.This thesis explores different strategies for implementing such a large accumulator, including new ones.The use of a 2's complement representation instead of a sign+magnitude is demonstrated to save resources and to reduce the accumulation loop delay.Based on a tapered precision scheme and an exact accumulator, the posit number systems claims to be a candidate to replace the IEEE floating-point format.A throughout analysis of posit operators is performed, using the same level of hardware optimization as state-of-the-art floating-point operators.Their cost remains much higher that their floating-point counterparts in terms of resource usage and performance. Finally, this thesis presents a compatibility layer for HLS tools that allows one code to be deployed on multiple tools.This library implements a strongly typed custom size integer type along side a set of optimized custom operators.À cause de la nature relativement jeune des outils de synthèse de haut-niveau (HLS), de nombreuses optimisations arithmétiques n'y sont pas encore implémentées. Cette thèse propose des optimisations arithmétiques se servant du contexte spécifique dans lequel les opérateurs sont instanciés.Certaines optimisations sont de simples spécialisations d'opérateurs, respectant la sémantique du C.D'autres nécéssitent de s'éloigner de cette sémantique pour améliorer le compromis précision/coût/performance.Cette proposition est démontré sur des sommes de produits de nombres flottants.La somme est réalisée dans un format en virgule-fixe défini par son contexte.Quand trop peu d’informations sont disponibles pour définir ce format en virgule-fixe, une stratégie est de générer un accumulateur couvrant l'intégralité du format flottant.Cette thèse explore plusieurs implémentations d'un tel accumulateur.L'utilisation d'une représentation en complément à deux permet de réduire le chemin critique de la boucle d'accumulation, ainsi que la quantité de ressources utilisées. Un format alternatif aux nombres flottants, appelé posit, propose d'utiliser un encodage à précision variable.De plus, ce format est augmenté par un accumulateur exact.Pour évaluer précisément le coût matériel de ce format, cette thèse présente des architectures d'opérateurs posits, implémentés avec le même degré d'optimisation que celui de l'état de l'art des opérateurs flottants.Une analyse détaillée montre que le coût des opérateurs posits est malgré tout bien plus élevé que celui de leurs équivalents flottants.Enfin, cette thèse présente une couche de compatibilité entre outils de HLS, permettant de viser plusieurs outils avec un seul code. Cette bibliothèque implémente un type d'entiers de taille variable, avec de plus une sémantique strictement typée, ainsi qu'un ensemble d'opérateurs ad-hoc optimisés

An approach for the formal verification of DSP designs using Theorem proving

This paper proposes a framework for the incorporation of formal methods in the design flow of digital signal processing (DSP) systems in a rigorous way. In the proposed approach, DSP descriptions were modeled and verified at different abstraction levels using higher order logic based on the higher order logic (HOL) theorem prover. This framework enables the formal verification of DSP designs that in the past could only be done partially using conventional simulation techniques. To this end, a shallow embedding of DSP descriptions in HOL at the floating-point (FP), fixed-point (FXP), behavioral, register transfer level (RTL), and netlist gate levels is provided. The paper made use of existing formalization of FP theory in HOL and a parallel one developed for FXP arithmetic. The high ability of abstraction in HOL allows a seamless hierarchical verification encompassing the whole DSP design path, starting from top-level FP and FXP algorithmic descriptions down to RTL, and gate level implementations. The paper illustrates the new verification framework on the fast Fourier transform (FFT) algorithm as a case study

Table Driven Adaptive Effectively Heterogeneous Multi-core Architectures

Exploiting flexibilities and scope of multi-core architectures for performance enhancement is one of the highly used approaches used by many researchers. However, with increasing dynamic nature of the workloads of everyday computing, even general multi-core architectures seem to just touch an upper limit on the deliverable performance. This has paved way for meticulous consideration of heterogeneous multi-core architectures. Such architectures can be further enhanced, by making the heterogeneity of the cores dynamic in nature. s work proposes techniques, which change configurations of these cores dynamically with workload. Thus, depending on requirements and pre-programmed preferences, each core can arrange itself to be power optimized or speed optimized. In addition, the project has been designed using RTL (Verilog) to provide completely realistic grip on the silicon investment. The project can be simulated using SPEC2000 and SPEC2006 benchmarks, and is completely synthesizable using IBM_LPE library for 65nm (IBM65LPE).School of Electrical & Computer Engineerin

Automated Dynamic Error Analysis Methods for Optimization of Computer Arithmetic Systems

Computer arithmetic is one of the more important topics within computer science and engineering. The earliest implementations of computer systems were designed to perform arithmetic operations and cost if not all digital systems will be required to perform some sort of arithmetic as part of their normal operations. This reliance on the arithmetic operations of computers means the accurate representation of real numbers within digital systems is vital, and an understanding of how these systems are implemented and their possible drawbacks is essential in order to design and implement modern high performance systems. At present the most widely implemented system for computer arithmetic is the IEEE754 Floating Point system, while this system is deemed to the be the best available implementation it has several features that can result in serious errors of computation if not implemented correctly. Lack of understanding of these errors and their effects has led to real world disasters in the past on several occasions. Systems for the detection of these errors are highly important and fast, efficient and easy to use implementations of these detection systems is a high priority. Detection of floating point rounding errors normally requires run-time analysis in order to be effective. Several systems have been proposed for the analysis of floating point arithmetic including Interval Arithmetic, Affine Arithmetic and Monte Carlo Arithmetic. While these systems have been well studied using theoretical and software based approaches, implementation of systems that can be applied to real world situations has been limited due to issues with implementation, performance and scalability. The majority of implementations have been software based and have not taken advantage of the performance gains associated with hardware accelerated computer arithmetic systems. This is especially problematic when it is considered that systems requiring high accuracy will often require high performance. The aim of this thesis and associated research is to increase understanding of error and error analysis methods through the development of easy to use and easy to understand implementations of these techniques

Efficient arithmetic for high speed DSP implementation on FPGAs

The author was sponsored by EnTegra Ltd, a company who develop hardware and software products and services for the real time implementation of DSP and RF systems. The field programmable gate array (FPGA) is being used increasingly in the field of DSP. This is due to the fact that the parallel computing power of such devices is ideal for today’s truly demanding DSP algorithms. Algorithms such as the QR-RLS update are computationally intensive and must be carried out at extremely high speeds (MHz). This means that the DSP processor is simply not an option. ASICs can be used but the expense of developing custom logic is prohibitive. The increased use of the FPGA in DSP means that there is a significant requirement for efficient arithmetic cores that utilises the resources on such devices. This thesis presents the research and development effort that was carried out to produce fixed point division and square root cores for use in a new Electronic Design Automation (EDA) tool for EnTegra, which is targeted at FPGA implementation of DSP systems. Further to this, a new technique for predicting the accuracy of CORDIC systems computing vector magnitudes and cosines/sines is presented. This work allows the most efficient CORDIC design for a specified level of accuracy to be found quickly and easily without the need to run lengthy simulations, as was the case before. The CORDIC algorithm is a technique using mainly shifts and additions to compute many arithmetic functions and is thus ideal for FPGA implementation