A barrier to wide-spread use of Field Programmable Gate Arrays (FPGAs) has been the complexity of programming, but recent advances in High-Level Synthesis (HLS) have made it possible for non-experts to easily create floating-point numerical accelerators from C-like code. However, HLS users are limited to the set of numerical primitives provided by HLS vendors and designers of floating-point IP cores, and cannot easily implement new fast or accurate numerical primitives. This paper presents a method for automatically creating high-performance pipelined floating-point function approximations, which can be integrated as IP cores into numerical accelerators, whether derived from HLS or traditional design methods. Both input and output are floating-point, but internally the function approximator uses fixed-point polynomial segments, guaranteeing a faithfully rounded output. A robust and automated non-uniform segmentation scheme is used to segment any twice-differentiable input function and produce platform-independent VHDL. The approach is demonstrated across ten functions, which are automatically generated then placed and routed in Xilinx devices. The method provides a 1.1x-3x improvement in area over composite numerical approximations, while providing similar performance and significantly better relative error.
{"title":"A General-Purpose Method for Faithfully Rounded Floating-Point Function Approximation in FPGAs","authors":"David B. Thomas","doi":"10.1109/ARITH.2015.27","DOIUrl":"https://doi.org/10.1109/ARITH.2015.27","url":null,"abstract":"A barrier to wide-spread use of Field Programmable Gate Arrays (FPGAs) has been the complexity of programming, but recent advances in High-Level Synthesis (HLS) have made it possible for non-experts to easily create floating-point numerical accelerators from C-like code. However, HLS users are limited to the set of numerical primitives provided by HLS vendors and designers of floating-point IP cores, and cannot easily implement new fast or accurate numerical primitives. This paper presents a method for automatically creating high-performance pipelined floating-point function approximations, which can be integrated as IP cores into numerical accelerators, whether derived from HLS or traditional design methods. Both input and output are floating-point, but internally the function approximator uses fixed-point polynomial segments, guaranteeing a faithfully rounded output. A robust and automated non-uniform segmentation scheme is used to segment any twice-differentiable input function and produce platform-independent VHDL. The approach is demonstrated across ten functions, which are automatically generated then placed and routed in Xilinx devices. The method provides a 1.1x-3x improvement in area over composite numerical approximations, while providing similar performance and significantly better relative error.","PeriodicalId":6526,"journal":{"name":"2015 IEEE 22nd Symposium on Computer Arithmetic","volume":"13 1","pages":"42-49"},"PeriodicalIF":0.0,"publicationDate":"2015-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88001808","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we show how to speed up the computation of fast Fourier transforms over complex numbers for "medium" precisions, typically in the range from 100 until 400 bits. On the one hand, such precisions are usually not supported by hardware. On the other hand, asymptotically fast algorithms for multiple precision arithmetic do not pay off yet. The main idea behind our algorithms is to develop efficient vectorial multiple precision fixed point arithmetic, capable of exploiting SIMD instructions in modern processors.
{"title":"Faster FFTs in Medium Precision","authors":"J. Hoeven, Grégoire Lecerf","doi":"10.1109/ARITH.2015.10","DOIUrl":"https://doi.org/10.1109/ARITH.2015.10","url":null,"abstract":"In this paper, we show how to speed up the computation of fast Fourier transforms over complex numbers for \"medium\" precisions, typically in the range from 100 until 400 bits. On the one hand, such precisions are usually not supported by hardware. On the other hand, asymptotically fast algorithms for multiple precision arithmetic do not pay off yet. The main idea behind our algorithms is to develop efficient vectorial multiple precision fixed point arithmetic, capable of exploiting SIMD instructions in modern processors.","PeriodicalId":6526,"journal":{"name":"2015 IEEE 22nd Symposium on Computer Arithmetic","volume":"55 1","pages":"75-82"},"PeriodicalIF":0.0,"publicationDate":"2015-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91381649","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The atan2 function computes the polar angle arctan(y/x) of a point given by its cartesian coordinates. It is widely used in digital signal processing to recover the phase of a signal. This article studies for this context the implementation of atan2 with fixed-point inputs and outputs. It compares the prevalent CORDIC shift-and-add algorithm to two multiplier-based techniques. The first one computes the bivariate atan2 function as the composition of two univariate functions: the reciprocal, and the arctangent, each evaluated using bipartite or polynomial approximation methods. The second technique directly uses piecewise bivariate polynomial approximations of degree 1 or 2. Each of these approaches requires a relevant argument reduction, which is also discussed. All the algorithms are last-bit accurate, and implemented with similar care in the open-source FloPoCo framework. Based on synthesis results on FPGAs, their relevance domains are discussed.
{"title":"Hardware Implementations of Fixed-Point Atan2","authors":"F. D. Dinechin, Matei Iştoan","doi":"10.1109/ARITH.2015.23","DOIUrl":"https://doi.org/10.1109/ARITH.2015.23","url":null,"abstract":"The atan2 function computes the polar angle arctan(y/x) of a point given by its cartesian coordinates. It is widely used in digital signal processing to recover the phase of a signal. This article studies for this context the implementation of atan2 with fixed-point inputs and outputs. It compares the prevalent CORDIC shift-and-add algorithm to two multiplier-based techniques. The first one computes the bivariate atan2 function as the composition of two univariate functions: the reciprocal, and the arctangent, each evaluated using bipartite or polynomial approximation methods. The second technique directly uses piecewise bivariate polynomial approximations of degree 1 or 2. Each of these approaches requires a relevant argument reduction, which is also discussed. All the algorithms are last-bit accurate, and implemented with similar care in the open-source FloPoCo framework. Based on synthesis results on FPGAs, their relevance domains are discussed.","PeriodicalId":6526,"journal":{"name":"2015 IEEE 22nd Symposium on Computer Arithmetic","volume":"17 1","pages":"34-41"},"PeriodicalIF":0.0,"publicationDate":"2015-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75720041","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Summary form only given, as follows. The full paper was not made available as part of this conference proceedings. Long time integrations of the planetary motion in the Solar System has been a challenging work in the past decades. The progress have followed the improvements of computer technology, but also the improvements in the integration algorithms. This quest has led to the development of high order dedicated symplectic integrators that have a stable behavior over long time scales. As important in the increase of the computing performances is the use of parallel algorithms that have divided the computing times by an order of magnitude. A specific aspect of these long term computation is also a careful monitoring of the accumulation of the roundoff error in the numerical algorithms, where all bias should be avoided. It should also be noted that for these computations, not only compensated summation is required, but also 80 bits extended precision floating point arithmetics. Integrating the equation of motion is only a part of the work. One needs also to determine precise initial conditions in order to ensure that the long time integration represent actually the motion of the real Solar System. Once these steps are fulfilled, the main limitation in the obtention of a precise solution of the planetary motion will be given by the chaotic nature of the Solar system that will strictly limit the possibility of precise prediction for the motion of the planets to about 60 Myr.
{"title":"Numerical challenges in long term integrations of the solar system","authors":"J. Laskar","doi":"10.1109/ARITH.2015.35","DOIUrl":"https://doi.org/10.1109/ARITH.2015.35","url":null,"abstract":"Summary form only given, as follows. The full paper was not made available as part of this conference proceedings. Long time integrations of the planetary motion in the Solar System has been a challenging work in the past decades. The progress have followed the improvements of computer technology, but also the improvements in the integration algorithms. This quest has led to the development of high order dedicated symplectic integrators that have a stable behavior over long time scales. As important in the increase of the computing performances is the use of parallel algorithms that have divided the computing times by an order of magnitude. A specific aspect of these long term computation is also a careful monitoring of the accumulation of the roundoff error in the numerical algorithms, where all bias should be avoided. It should also be noted that for these computations, not only compensated summation is required, but also 80 bits extended precision floating point arithmetics. Integrating the equation of motion is only a part of the work. One needs also to determine precise initial conditions in order to ensure that the long time integration represent actually the motion of the real Solar System. Once these steps are fulfilled, the main limitation in the obtention of a precise solution of the planetary motion will be given by the chaotic nature of the Solar system that will strictly limit the possibility of precise prediction for the motion of the planets to about 60 Myr.","PeriodicalId":6526,"journal":{"name":"2015 IEEE 22nd Symposium on Computer Arithmetic","volume":"63 1","pages":"104"},"PeriodicalIF":0.0,"publicationDate":"2015-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87061054","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Carry tree sparseness is used in high-performance binary adders to achieve better energy-delay trade-off. To determine the energy optimal degree of sparseness, a detailed analysis is performed in this work. An analytical expression for the upper bound of sparseness is derived. The effect of increased sparseness on partial sum block and total energy is explored on 32-, 64-, 128-, and 256-bit adders. Higher degrees of sparseness in the carry generation block is achieved by employing parallel adders in the sum block instead of serial ripple carry adders. 64-bit adders with various sparseness degrees using leading addition algorithms are synthesized and optimized with a standard cell library in 45nm CMOS technology. Post layout simulations revealed that the optimal sparse carry tree adders provide up to 50% and 22% improvement in energy at same performance over full carry tree Kogge-Stone and Ladner-Fischer adder designs, respectively.
{"title":"Minimizing Energy by Achieving Optimal Sparseness in Parallel Adders","authors":"M. Aktan, D. Baran, V. Oklobdzija","doi":"10.1109/ARITH.2015.13","DOIUrl":"https://doi.org/10.1109/ARITH.2015.13","url":null,"abstract":"Carry tree sparseness is used in high-performance binary adders to achieve better energy-delay trade-off. To determine the energy optimal degree of sparseness, a detailed analysis is performed in this work. An analytical expression for the upper bound of sparseness is derived. The effect of increased sparseness on partial sum block and total energy is explored on 32-, 64-, 128-, and 256-bit adders. Higher degrees of sparseness in the carry generation block is achieved by employing parallel adders in the sum block instead of serial ripple carry adders. 64-bit adders with various sparseness degrees using leading addition algorithms are synthesized and optimized with a standard cell library in 45nm CMOS technology. Post layout simulations revealed that the optimal sparse carry tree adders provide up to 50% and 22% improvement in energy at same performance over full carry tree Kogge-Stone and Ladner-Fischer adder designs, respectively.","PeriodicalId":6526,"journal":{"name":"2015 IEEE 22nd Symposium on Computer Arithmetic","volume":"9 1","pages":"10-17"},"PeriodicalIF":0.0,"publicationDate":"2015-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81951137","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We describe a new implementation of the elementary transcendental functions exp, sin, cos, log and atan for variable precision up to approximately 4096 bits. Compared to the MPFR library, we achieve a maximum speedup ranging from a factor 3 for cos to 30 for atan. Our implementation uses table-based argument reduction together with rectangular splitting to evaluate Taylor series. We collect denominators to reduce the number of divisions in the Taylor series, and avoid overhead by doing all multiprecision arithmetic using the mpn layer of the GMP library. Our implementation provides rigorous error bounds.
我们描述了一个新的实现初等超越函数exp, sin, cos, log和atan的可变精度高达约4096位。与MPFR库相比，我们实现了从cos的3倍到atan的30倍的最大加速。我们的实现使用基于表的参数约简和矩形分裂来计算泰勒级数。我们收集分母以减少泰勒级数中的除法次数，并通过使用GMP库的mpn层进行所有多精度算术来避免开销。我们的实现提供了严格的错误界限。
{"title":"Efficient Implementation of Elementary Functions in the Medium-Precision Range","authors":"Fredrik Johansson","doi":"10.1109/ARITH.2015.16","DOIUrl":"https://doi.org/10.1109/ARITH.2015.16","url":null,"abstract":"We describe a new implementation of the elementary transcendental functions exp, sin, cos, log and atan for variable precision up to approximately 4096 bits. Compared to the MPFR library, we achieve a maximum speedup ranging from a factor 3 for cos to 30 for atan. Our implementation uses table-based argument reduction together with rectangular splitting to evaluate Taylor series. We collect denominators to reduce the number of divisions in the Taylor series, and avoid overhead by doing all multiprecision arithmetic using the mpn layer of the GMP library. Our implementation provides rigorous error bounds.","PeriodicalId":6526,"journal":{"name":"2015 IEEE 22nd Symposium on Computer Arithmetic","volume":"28 1","pages":"83-89"},"PeriodicalIF":0.0,"publicationDate":"2014-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74894592","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Summary form only given, as follows. High-precision floating-point arithmetic software, ranging from "double-double" or "quad" precision to arbitrarily high-precision (hundreds or thousands of digits), has been available for years. Such facilities are standard features of Mathematica and Maple, and software packages such as MPFR, QD and ARPREC are available on the Internet. Some of these packages include high-level language interface modules that make conversion of standard-precision programs a relatively simple task. However, until recently such facilities were widely considered as novelty items - why would anyone need such exalted levels of numeric precision in "practical" research or engineering? In fact, during the past decade or two, numerous applications have arisen for high-precision floatingpoint arithmetic. This presentation will briefly describe some of these applications, which mostly arise in mathematical physics, applied physics and mathematics. Many heretofore unknown identities and relationships have been discovered, and features have been identified in computed data that were not "visible" with ordinary 64-bit precision. Applications of double-double (31 digits) or quad-double precision (62 digits) are particularly common, but there are also some interesting applications for as high as 50,000 digits. The speaker will also outline what is needed in improved facilities for high-precision computation to address challenges that lie ahead.
{"title":"High-precision computation: Applications and challenges [Keynote I]","authors":"D. Bailey","doi":"10.1109/ARITH.2013.39","DOIUrl":"https://doi.org/10.1109/ARITH.2013.39","url":null,"abstract":"Summary form only given, as follows. High-precision floating-point arithmetic software, ranging from \"double-double\" or \"quad\" precision to arbitrarily high-precision (hundreds or thousands of digits), has been available for years. Such facilities are standard features of Mathematica and Maple, and software packages such as MPFR, QD and ARPREC are available on the Internet. Some of these packages include high-level language interface modules that make conversion of standard-precision programs a relatively simple task. However, until recently such facilities were widely considered as novelty items - why would anyone need such exalted levels of numeric precision in \"practical\" research or engineering? In fact, during the past decade or two, numerous applications have arisen for high-precision floatingpoint arithmetic. This presentation will briefly describe some of these applications, which mostly arise in mathematical physics, applied physics and mathematics. Many heretofore unknown identities and relationships have been discovered, and features have been identified in computed data that were not \"visible\" with ordinary 64-bit precision. Applications of double-double (31 digits) or quad-double precision (62 digits) are particularly common, but there are also some interesting applications for as high as 50,000 digits. The speaker will also outline what is needed in improved facilities for high-precision computation to address challenges that lie ahead.","PeriodicalId":6526,"journal":{"name":"2015 IEEE 22nd Symposium on Computer Arithmetic","volume":"526 1","pages":"3"},"PeriodicalIF":0.0,"publicationDate":"2013-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77692565","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Summary form only given, as follows. Doing arithmetic has probably been necessary since civilization began. We now know that the ancient Greeks were able to make mechanical devices capable of calculation. The Antikythera Mechanism is an extraordinary device containing over thirty gear wheels dating from the 1st century BC, and is an order of magnitude more complicated than any surviving mechanism from the following millennium. It is clear from its structure and inscriptions that its purpose was astronomical, including eclipse prediction. In this illustrated talk, I will show the results from our international research team, which has used modern imaging methods to probe its functions and details. The Mechanism's design is very sophisticated. I will outline how its technology may have almost disappeared from sight for over a thousand years and then been extended to more general mechanical clocks, calculators and computers from around 1200 AD through to the 19th century.
{"title":"The Antikythera Mechanism and the early history of mechanical computing","authors":"M. Edmunds","doi":"10.1109/ARITH.2013.40","DOIUrl":"https://doi.org/10.1109/ARITH.2013.40","url":null,"abstract":"Summary form only given, as follows. Doing arithmetic has probably been necessary since civilization began. We now know that the ancient Greeks were able to make mechanical devices capable of calculation. The Antikythera Mechanism is an extraordinary device containing over thirty gear wheels dating from the 1st century BC, and is an order of magnitude more complicated than any surviving mechanism from the following millennium. It is clear from its structure and inscriptions that its purpose was astronomical, including eclipse prediction. In this illustrated talk, I will show the results from our international research team, which has used modern imaging methods to probe its functions and details. The Mechanism's design is very sophisticated. I will outline how its technology may have almost disappeared from sight for over a thousand years and then been extended to more general mechanical clocks, calculators and computers from around 1200 AD through to the 19th century.","PeriodicalId":6526,"journal":{"name":"2015 IEEE 22nd Symposium on Computer Arithmetic","volume":"520 1","pages":"79"},"PeriodicalIF":0.0,"publicationDate":"2013-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77027236","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we provide new methods to generate a class of algorithms computing modular multiplication and division. All these algorithms rely on sequences derived from the Euclidean algorithm for a well chosen input. We then use these sequences as number scales of the Ostrowski number system to construct the result of either the modular multiplication or division.
{"title":"Modular Multiplication and Division Algorithms Based on Continued Fraction Expansion","authors":"Mourad Gouicem","doi":"10.1109/ARITH.2015.21","DOIUrl":"https://doi.org/10.1109/ARITH.2015.21","url":null,"abstract":"In this paper, we provide new methods to generate a class of algorithms computing modular multiplication and division. All these algorithms rely on sequences derived from the Euclidean algorithm for a well chosen input. We then use these sequences as number scales of the Ostrowski number system to construct the result of either the modular multiplication or division.","PeriodicalId":6526,"journal":{"name":"2015 IEEE 22nd Symposium on Computer Arithmetic","volume":"1 1","pages":"137-143"},"PeriodicalIF":0.0,"publicationDate":"2013-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74716289","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
David G. Hough, Bill Hay, J. Kidder, E. J. Riedy, G. Steele, James J. Thomas
The entire process of creating and executing applications that solve interesting problems with acceptable cost and accuracy involves a complex interaction among hardware, system software, programming environments, mathematical software libraries, and applications software, all mediated by standards for arithmetic, operating systems, and programming environments. This panel will discuss various issues arising among these various contending points of view, sometimes from the point of view of issues raised during the current IEEE 754R standards revision effort.
{"title":"Arithmetic Interactions: From Hardware to Applications","authors":"David G. Hough, Bill Hay, J. Kidder, E. J. Riedy, G. Steele, James J. Thomas","doi":"10.1109/ARITH.2005.10","DOIUrl":"https://doi.org/10.1109/ARITH.2005.10","url":null,"abstract":"The entire process of creating and executing applications that solve interesting problems with acceptable cost and accuracy involves a complex interaction among hardware, system software, programming environments, mathematical software libraries, and applications software, all mediated by standards for arithmetic, operating systems, and programming environments. This panel will discuss various issues arising among these various contending points of view, sometimes from the point of view of issues raised during the current IEEE 754R standards revision effort.","PeriodicalId":6526,"journal":{"name":"2015 IEEE 22nd Symposium on Computer Arithmetic","volume":"161 1","pages":"87"},"PeriodicalIF":0.0,"publicationDate":"2005-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83162099","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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