Normalizing or not normalizing? An open question for floating-point arithmetic in embedded systems

Emerging embedded applications lack of a specific standard when they require floating-point arithmetic. In this situation they use the IEEE-754 standard or ad hoc variations of it. However, this standard was not designed for this purpose. This paper aims to open a debate to define a new extension of the standard to cover embedded applications. In this work, we only focus on the impact of not performing normalization. We show how eliminating the condition of normalized numbers, implementation costs can be dramatically reduced, at the expense of a moderate loss of accuracy. Several architectures to implement addition and multiplication for non-normalized numbers are proposed and analyzed. We show that a combined architecture (adder-multiplier) can halve the area and power consumption of its counterpart IEEE-754 architecture. This saving comes at the cost of reducing an average of about 10 dBs the Signal-to-Noise Ratio for the tested algorithms. We think these results should encourage researchers to perform further investigation in this issue.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

Fast HUB Floating-point Adder for FPGA

Several previous publications have shown the area and delay reduction when implementing real number computation using HUB formats for both floating-point and fixed-point. In this paper, we present a HUB floating-point adder for FPGA which greatly improves the speed of previous proposed HUB designs for these devices. Our architecture is based on the double path technique which reduces the execution time since each path works in parallel. We also deal with the implementation of unbiased rounding in the proposed adder. Experimental results are presented showing the goodness of the new HUB adder for FPGA.TIN2016- 80920-R, JA2012 P12-TIC-1692, JA2012 P12-TIC-147

New Results on Non-normalized Floating-point Formats

Compulsory normalization of the represented numbers is a key requirement of the floating-point standard. This requirement contributes to fundamental characteristics of the standard, such as taking the most of the precision available, reproducibility and facilitation of comparison and other operations. However, it also imposes a high restriction in effectiveness of basic arithmetic operation implementation. In many embedded applications may be worth to sacrifice the benefits of normalization for gaining in implementation metrics. This paper analyzes and measures the effect of removing the normalization requirement in terms of precision and implementation savings for embedded applications. We propose several adder and multiplier architectures to deal with non-normalized floating-point numbers, and quantify the accuracy loss and the improvements in hardware implementation. Our experiments show that it is possible to reduce the area and power consumption up to 78% in ASIC and 50% in FPGA implementations with a reasonable accuracy loss.TIN2016-80920-R, JA2012 P12-TIC-1692, JA2012 P12-TIC-147

Floating Point Arithmetic for Transport Triggered Architectures

Laskentajärjestelmiin kohdistuu usein suorituskyky- ja virrankulutusvaatimuksia, joita ei pystytä saavuttamaan yleiskäyttöisellä prosessorilla. Toistaalta laitteistokiihdyttimien suunnittelu voi vaatia kohtuuttoman paljon työaikaa. Ongelmaa voidaan lähestyä käyttämällä sovellusta varten räätälöityä sovelluskohtaista käskykantaprosessoria (Application-Specific Instruction set Processor, ASIP), joka on kuitenkin ohjelmoitava. Prosessorin räätälöinnin täytyy olla pitkälle automatisoitua säästääkseen kustannuksia. TTA-based Codesign Environment (TCE) on siirtoliipaistuun prosessoriarkkitehtuuriin (Transport Triggered Architecture, TTA) perustuva ASIP-kehitysympäristö. TTA on arkkitehtuurina helposti räätälöitävä ja joustaa pienistä ytimistä suuritehoisiin pitkän käskysanan suorittimiin. Useat tieteellisen laskennan ja signaalinkäsittelyn sovellukset, joissa TTA:n skaalautuvuudesta ja käskytason rinnakkaisuudesta olisi erityistä hyötyä, vaativat tuen laitteistokiihdytetylle liukulukulaskennalle. Tässä diplomityössä suunniteltiin ja toteutettiin TCE-projektia varten sarja liukulukuyksiköitä. Yksiköiden suunnittelussa pyrittiin alustariippumattomuuteen sekä korkeaan suorituskykyyn Field Programmable Gate Array alustoilla (FPGA) jopa tinkimällä tuetusta liukulukustandardista. Yksiköt sisältävät työkalut puolen tarkkuuden liukulukulaskentaan. Lisäksi työssä esitetään erikoiskäskyihin perustuvat nopeat algoritmit liukulukujakolaskun ja -neliöjuuren laskentaan. Yksiköiden toiminta varmistettiin automaattisella rekisterisiirtotason (Register Transfer Level, RTL) testipenkillä. Vertailussa Altera Stratix-II-FPGA:lla yksiköt pääsivät lähelle Alteran omien liukulukuyksiköiden suorituskykyä. Uudemmalla Xilinx Virtex-6-FPGA:lla korkein mahdollinen suorituskyky vaatisi tiheämpää liukuhihnoitusta

A new floating-point adder FPGA-based implementation using RN-coding of numbers

A well-known problem in the computer science area is related to numerical data representation, which directly affects adder circuits’ design and a reason to have different formats: IEEE Std. 754, Half-Unit-Biased (HUB), and Round-to-Nearest (RN). RN has an advantage that rounding to nearest is equivalent to a word truncation. It avoids double rounding errors and intermediate rounding steps with an exact conversion between formats, making it applicable to general problems. However, there is a lack of research on the hardware implementation of the RN representation. In this work, we propose hardware architectures for binary and floating-point adders, analyzing for the latter its performance in terms of error and resource consumption in FPGAs. To accomplish this, we have developed a one-bit RN-based adder that allows modular designs, considering an efficient signal propagation to obtain new architectures for both binary and floating-point single-precision adders. The results open new perspectives for further applications

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

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

Using reconfigurable computing technology to accelerate matrix decomposition and applications

Matrix decomposition plays an increasingly significant role in many scientific and engineering applications. Among numerous techniques, Singular Value Decomposition (SVD) and Eigenvalue Decomposition (EVD) are widely used as factorization tools to perform Principal Component Analysis for dimensionality reduction and pattern recognition in image processing, text mining and wireless communications, while QR Decomposition (QRD) and sparse LU Decomposition (LUD) are employed to solve the dense or sparse linear system of equations in bioinformatics, power system and computer vision. Matrix decompositions are computationally expensive and their sequential implementations often fail to meet the requirements of many time-sensitive applications. The emergence of reconfigurable computing has provided a flexible and low-cost opportunity to pursue high-performance parallel designs, and the use of FPGAs has shown promise in accelerating this class of computation. In this research, we have proposed and implemented several highly parallel FPGA-based architectures to accelerate matrix decompositions and their applications in data mining and signal processing. Specifically, in this dissertation we describe the following contributions: • We propose an efficient FPGA-based double-precision floating-point architecture for EVD, which can efficiently analyze large-scale matrices. • We implement a floating-point Hestenes-Jacobi architecture for SVD, which is capable of analyzing arbitrary sized matrices. • We introduce a novel deeply pipelined reconfigurable architecture for QRD, which can be dynamically configured to perform either Householder transformation or Givens rotation in a manner that takes advantage of the strengths of each. • We design a configurable architecture for sparse LUD that supports both symmetric and asymmetric sparse matrices with arbitrary sparsity patterns. • By further extending the proposed hardware solution for SVD, we parallelize a popular text mining tool-Latent Semantic Indexing with an FPGA-based architecture. • We present a configurable architecture to accelerate Homotopy l1-minimization, in which the modification of the proposed FPGA architecture for sparse LUD is used at its core to parallelize both Cholesky decomposition and rank-1 update. Our experimental results using an FPGA-based acceleration system indicate the efficiency of our proposed novel architectures, with application and dimension-dependent speedups over an optimized software implementation that range from 1.5ÃÂ to 43.6ÃÂ in terms of computation time

Design and implementation of a Multimedia Extension for a RISC Processor

Design and implementation of a Multimedia Extension for a RISC Processor in a FPG