A Modular-Positional Computation Technique for Multiple-Precision Floating-Point Arithmetic

Floating-point machine precision is often not sufficient to correctly solve large scientific and engineering problems. Moreover, computation time is a critical parameter here. Therefore, any research aimed at developing high-speed methods for multiple-precision arithmetic is of great immediate interest. This paper deals with a new technique of multiple-precision computations, based on the use of modular-positional floating-point format for representation of numbers. In this format, the significands are represented in residue number system RNS, thus enabling high-speed processing of the significands with possible parallelization by RNS modules. Number exponents and signs are represented in the binary number system. The interval-positional characteristic method is used to increase the speed of executing complex non-modular operations in RNS. Algorithms for rounding off and aligning the exponents of numbers in modular-positional format are presented. The structure and features of a new multiple-precision library are given. Some results of an experimental study on the efficiency of this library are also presented.

[1]  Gian Carlo Cardarilli,et al.  Programmable power-of-two RNS scaler and its application to a QRNS polyphase filter , 2005, 2005 IEEE International Symposium on Circuits and Systems.

[2]  A. Omondi,et al.  Residue Number Systems: Theory and Implementation , 2007 .

[3]  Eisuke Kinoshita,et al.  A Residue Arithmetic Extension for Reliable Scientific Computation , 1997, IEEE Trans. Computers.

[4]  Josef Stoer,et al.  Numerische Mathematik 1 , 1989 .

[5]  Behrooz Parhami,et al.  Computer arithmetic - algorithms and hardware designs , 1999 .

[6]  Jonathan M. Borwein,et al.  High-precision computation: Mathematical physics and dynamics , 2010, Appl. Math. Comput..

[7]  David Defour,et al.  Reproducible and Accurate Matrix Multiplication for High-Performance Computing , 2014 .

[8]  Jean-Michel Muller,et al.  Handbook of Floating-Point Arithmetic (2nd Ed.) , 2018 .

[9]  C. H. Huang.  A Fully Parallel Mixed-Radix Conversion Algorithm for Residue Number Applications , 1983, IEEE Transactions on Computers.

[10]  Hideo Kosako,et al.  Floating-Point Arithmetic Algorithms in the Symmetric Residue Number System , 1974, IEEE Transactions on Computers.

[11]  Pepe Siy,et al.  A new Mixed Radix Conversion algorithm MRC-II , 2007, J. Syst. Archit..

[12]  H. Garner.  The residue number system , 1959, IRE-AIEE-ACM '59 (Western).

[13]  A Sasaki.  The basis for implementation of additive operations in the residue number system , 1986 .

[14]  T. J. Dekker,et al.  A floating-point technique for extending the available precision , 1971 .

[15]  Xiaoye S. Li,et al.  ARPREC: An arbitrary precision computation package , 2002 .

[16]  Sorin Cotofana,et al.  An O(n) Residue Number System to Mixed Radix Conversion technique , 2009, 2009 IEEE International Symposium on Circuits and Systems.

[17]  Vincent Lefèvre,et al.  MPFR: A multiple-precision binary floating-point library with correct rounding , 2007, TOMS.

[18]  Ulrich W. Kulisch,et al.  Computer Arithmetic and Validity - Theory, Implementation, and Applications , 2008, de Gruyter studies in mathematics.

[19]  Siegfried M. Rump,et al.  Algorithms for verified inclusions—theory and practice , 1988 .

[20]  Marc Tajchman.  Programming Experiences Using the X10 Language , 2010, Computing in Science & Engineering.

[21]  Vincent Lefèvre,et al.  Why and How to Use Arbitrary Precision , 2010, Comput. Sci. Eng..

[22]  Akio Sasaki.  The Basis for Implementation of Ad idive Operations in the Residue Number System , 1968, IEEE Transactions on Computers.

[23]  Mi Lu,et al.  Floating-point numbers in residue number systems , 1991 .