THE DESIGN OF AN IC HALF PRECISION FLOATING POINT ARITHMETIC LOGIC UNIT

A 16 bit floating point (FP) Arithmetic Logic Unit (ALU) was designed and implemented in 0.35µm CMOS technology. Typical uses of the 16 bit FP ALU include graphics processors and embedded multimedia applications. The ALU of the modern microprocessors use a fused multiply add (FMA) design technique. An advantage of the FMA is to remove the need for a comparator which is required for a normal FP adder. The FMA consists of a multiplier, shifters, adders and rounding circuit. A fast multiplier based on the Wallace tree configuration was designed. The number of partial products was greatly reduced by the use of the modified booth encoder. The Wallace tree was chosen to reduce the number of reduction layers of partial products. The multiplier also involved the design of a pass transistor based 4:2 compressor. The average delay of the pass transistor based compressor was 55ps and was found to be 7 times faster than the full adder based 4:2 compressor. The shifters consist of separate left and right shifters using multiplexers. The shift amount is calculated using the exponents of the three operands. The addition operation is implemented using a carry skip adder (CSK). The average delay of the CSK was 1.05ns and was slower than the carry look ahead adder by about 400ps. The advantages of the CSK are reduced power, gate count and area when compared to the similar sized carry look ahead adder. The adder computes the addition of the multiplier result and the shifted value of the addend. In most modern computers, division is performed using software thereby eliminating the need for a separate hardware unit. FMA hardware unit was utilized to perform FP division. The FP divider uses the Newton Raphson algorithm to solve division by iteration. The initial approximated value with five bit accuracy was assumed to be pre-stored in cache memory and a separate clock cycle for cache read was assumed before the start of the FP division operation. In order to significantly reduce the area of the design, only one multiplier was used. Rounding to nearest technique was implemented using an 11 bit variable CSK adder. This is the best rounding technique when compared to other rounding techniques. In both the FMA and division, rounding was performed after the computation of the final result during the last clock cycle of operation. Testability analysis is performed for the multiplier which is the most complex and critical part of the FP ALU. The specific aim of testability was to ensure the correct operation of the multiplier and thus guarantee the correctness of the FMA circuit at the layout stage. The multiplier\u27s output was tested by identifying the minimal number of input vectors which toggle the inputs of the 4:2 compressors of the multiplier. The test vectors were identified in a semi automated manner using Perl scripting language. The multiplier was tested with a test set of thirty one vectors. The fault coverage of the multiplier was found to be 90.09%. The layout was implemented using IC station of Mentor Graphics CAD tool and resulted in a chip area of 1.96mm2. The specifications for basic arithmetic operations were met successfully. FP Division operation was completed within six clock cycles. The other arithmetic operations like FMA, FP addition, FP subtraction and FP multiplication were completed within three clock cycles

Single-Precision and Double-Precision Merged Floating-Point Multiplication and Addition Units on FPGA

Floating-point (FP) operations defined in IEEE 754-2008 Standard for Floating-Point Arithmetic can provide wider dynamic range and higher precision than fixed-point operations. Many scientific computations and multimedia applications adopt FP operations. Among all the FP operations, addition and multiplication are the most frequent operations. In this thesis, the single-precision (SP) and double-precision (DP) merged FP multiplier and FP adder architectures are proposed. The proposed efficient iterative FP multiplier is designed based on the Karatsuba algorithm and implemented with the pipelined architecture. It can accomplish two parallel SP multiplication operations in one iteration with a latency of 6 clock cycles or one DP multiplication operation in two iterations with a latency of 9 clock cycles. Implemented on Xilinx Virtex-5 (xc5vlx155ff1760-3) FPGA device, the proposed multiplier runs at 348 MHz using 6 DSP48E blocks, 1117 LUTs, and 1370 FFs. Compared to previous FPGA based multiple-precision FP multiplier, the proposed designs runs at 4% faster clock frequency with reduction of 33% of DSP blocks, 17% latency for SP multiplication, and 28% latency for DP multiplication. The proposed high performance FP adder is designed based one the two-path FP addition algorithm. With fully pipelined architecture, the proposed adder can accomplish one DP or two parallel SP addition/subtraction operations in 6 clock cycles. The proposed adder architecture is implemented on both Altera and Xilinx 65nm process FPGA devices. The proposed adder can run up to 336 MHz with 1694 FFs, 1420 LUTs on Xilinx Virtex-5 (xc5vlx155ff1760-3) FPGA device. Compared to the combination of one DP and two SP architecture built with Xilinx FP operator, the proposed adder has 11.3% faster clock frequency. On Altera Stratix-III (EP3SL340F1760C2) FPGA device, the maximum clock frequency of the proposed adder can reach 358 MHz and 1686 ALUTs and 1556 registers are occupied. The proposed adder is 11.6% faster than the combination of one DP and two SP architecture built with Altera FP megafunction. For the reference of other researchers, the implementation results of the proposed FP multiplier and FP adder on the latest Xilinx Virtex-7 device and Altera Arria 10 device are also provided

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

Comparison of logarithmic and floating-point number systems implemented on Xilinx Virtex-II field-programmable gate arrays

The aim of this thesis is to compare the implementation of parameterisable LNS (logarithmic number system) and floating-point high dynamic range number systems on FPGA. The Virtex/Virtex-II range of FPGAs from Xilinx, which are the most popular FPGA technology, are used to implement the designs. The study focuses on using the low level primitives of the technology in an efficient way and so initially the design issues in implementing fixed-point operators are considered. The four basic operations of addition, multiplication, division and square root are considered. Carry- free adders, ripple-carry adders, parallel multipliers and digit recurrence division and square root are discussed. The floating-point operators use the word format and exceptions as described by the IEEE std-754. A dual-path adder implementation is described in detail, as are floating-point multiplier, divider and square root components. Results and comparisons with other works are given. The efficient implementation of function evaluation methods is considered next. An overview of current FPGA methods is given and a new piecewise polynomial implementation using the Taylor series is presented and compared with other designs in the literature. In the next section the LNS word format, accuracy and exceptions are described and two new LNS addition/subtraction function approximations are described. The algorithms for performing multiplication, division and powering in the LNS domain are also described and are compared with other designs in the open literature. Parameterisable conversion algorithms to convert to/from the fixed-point domain from/to the LNS and floating-point domain are described and implementation results given. In the next chapter MATLAB bit-true software models are given that have the exact functionality as the hardware models. The interfaces of the models are given and a serial communication system to perform low speed system tests is described. A comparison of the LNS and floating-point number systems in terms of area and delay is given. Different functions implemented in LNS and floating-point arithmetic are also compared and conclusions are drawn. The results show that when the LNS is implemented with a 6-bit or less characteristic it is superior to floating-point. However, for larger characteristic lengths the floating-point system is more efficient due to the delay and exponential area increase of the LNS addition operator. The LNS is beneficial for larger characteristics than 6-bits only for specialist applications that require a high portion of division, multiplication, square root, powering operations and few additions