1

March 2012
IEEE Transactions on Computers: Volume 61 Issue 3, March 2012

**Publisher:** IEEE Computer Society

This paper presents a study of some basic blocks needed in the design of floating-point summation algorithms. In particular, in radix-2 floating-point arithmetic, we show that among the set of the algorithms with no comparisons performing only floating-point additions/subtractions, the 2Sum algorithm introduced by Knuth is minimal, both in terms ...

**Keywords**:
Floating-point arithmetic, summation algorithms, correct rounding, 2Sum and Fast2Sum algorithms.

2

July 2011
ARITH '11: Proceedings of the 2011 IEEE 20th Symposium on Computer Arithmetic

**Publisher:** IEEE Computer Society

Define an "augmented precision" algorithm as an algorithm that returns, in precision-p floating-point arithmetic, its result as the unevaluated sum of two floating-point numbers, with a relative error of the order of 2^(-2p). Assuming an FMA instruction is available, we perform a tight error analysis of an augmented precision algorithm ...

**Keywords**:
Floating-point arithmetic, compensated algorithms, square-root, Correct rounding, 2D-norms, accurate computations

3

November 2010

Fundamental arithmetic operations support virtually all of the engineering, scientific, and financial computations required for practical applications, from cryptography, to financial planning, to rocket science. This comprehensive reference provides researchers with the thorough understanding of number representations that is a necessary foundation for designing efficient arithmetic algorithms. Using the elementary ...

4

January 2010
ACM Transactions on Mathematical Software (TOMS): Volume 37 Issue 1, January 2010

**Publisher:** ACM

**Bibliometrics**:

Citation Count: 1

Downloads (6 Weeks): 1, Downloads (12 Months): 11, Downloads (Overall): 291

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We introduce several algorithms for accurately evaluating powers to a positive integer in floating-point arithmetic, assuming a fused multiply-add (fma) instruction is available. For bounded, yet very large values of the exponent, we aim at obtaining correctly rounded results in round-to-nearest mode, that is, our algorithms return the floating-point number ...

**Keywords**:
Correct rounding, floating-point arithmetic, integer power function

5

February 2009
IEEE Transactions on Computers: Volume 58 Issue 2, February 2009

**Publisher:** IEEE Computer Society

The eight articles in this special section focus on computer arithmetic. They are grouped onto four categories: decimal arithmetic, number systems, multiplication and elementary functions, and formal proofs.

6

May 2006
IEEE Transactions on Computers: Volume 55 Issue 5, May 2006

**Publisher:** IEEE Computer Society

Redundant number representations are generally used to allow constant time additions, based on the fact that only bounded carry-ripples take place. But, carries may ripple out into positions which may not be needed to represent the final value of the result and, thus, a certain amount of leading guard digits ...

**Keywords**:
Redundant representations, leading guard digits, multioperand additions, pseudo overflows., Redundant representations, pseudo overflows., leading guard digits, multioperand additions

7

February 2006
Theoretical Computer Science - Real numbers and computers: Volume 351 Issue 1, 14 February 2006

**Publisher:** Elsevier Science Publishers Ltd.

We aim at finding the best possible seed values when computing a 1/ p using the Newton-Raphson iteration in a given interval. A natural choice of the seed value would be the one that best approximates the expected result. It turns out that in most cases, the best seed value ...

**Keywords**:
root extraction, square-root reciprocal, Newton-Raphson iteration, computer arithmetic, division, square-root

8

May 2005
Journal of VLSI Signal Processing Systems: Volume 40 Issue 1, May 2005

**Publisher:** Kluwer Academic Publishers

Recently there has been quite a number of papers discussing the use of redundant 4-to-2 adders for the accumulation of partial products in multipliers, claiming one type to be superior to other types. This paper analyses a recent proposal of various 3- and 4-element redundant digit sets for radix 2, ...

**Keywords**:
digit encodings, digit sets, multiplier trees, redundant adders

9

March 2005
IEEE Transactions on Computers: Volume 54 Issue 3, March 2005

**Publisher:** IEEE Computer Society

Range-reduction is a key point for getting accurate elementary function routines. We introduce a new algorithm that is fast for input arguments belonging to the most common domains, yet accurate over the full double-precision range.

**Keywords**:
Index Terms- Range-reduction, elementary function evaluation, floating-point arithmetic.

10

March 2005
IEEE Transactions on Computers: Volume 54 Issue 3, March 2005

**Publisher:** IEEE Computer Society

The quotient digit selection in the SRT division algorithm is based on a few most significant bits of the remainder and divisor, where the remainder is usually represented in a redundant representation. The number of leading bits needed depends on the quotient radix and digit set, and is usually found ...

**Keywords**:
Index Terms- Digit selection, division, square root.

11

June 2003
ARITH '03: Proceedings of the 16th IEEE Symposium on Computer Arithmetic (ARITH-16'03)

**Publisher:** IEEE Computer Society

The quotient digit selection in the SRT division algorithm is based on a few most significant bits of the remainder and divisor, where the remainder is usually represented in a redundant representation. The number of leading bits needed depends on the quotient radix and digit set, and is usually found ...

12

January 2003
Theoretical Computer Science - Real numbers and computers: Volume 291 Issue 2, 5 January 2003

**Publisher:** Elsevier Science Publishers Ltd.

13

July 2002
ASAP '02: Proceedings of the IEEE International Conference on Application-Specific Systems, Architectures, and Processors

**Publisher:** IEEE Computer Society

Recently there has been quite a number of papers discussing the use of redundant 4-to-2 adders for the accumulation of partial products in multipliers, claiming one type to be superior to other types. This paper analyses the use of various 3- and 4-element redundant digit sets for radix 2, and ...

14

June 2001
ARITH '01: Proceedings of the 15th IEEE Symposium on Computer Arithmetic

**Publisher:** IEEE Computer Society

Abstract: We present a new RNS modular multiplication for very large operands. The algorithm is based on Montgomery's method adapted to residue arithmetic. By choosing the moduli of the RNS system reasonably large, an effect corresponding to a redundant high-radix implementation is achieved, due to the carry-free nature of residue ...

15

July 2000
IEEE Transactions on Computers - Special issue on computer arithmetic: Volume 49 Issue 7, July 2000

**Publisher:** IEEE Computer Society

First Page of the Article

16

May 2000
Journal of VLSI Signal Processing Systems: Volume 25 Issue 1, May 2000

**Publisher:** Kluwer Academic Publishers

Moments of images are widely used in pattern recognition, because in suitable form they can be made invariant to variations in translation, rotation and size. However the computation of discrete moments by their definition requires many multiplications which limits the speed of computation. In this paper we express the moments ...

17

May 2000
Journal of VLSI Signal Processing Systems: Volume 25 Issue 1, May 2000

**Publisher:** Kluwer Academic Publishers

Moments of images are widely used in pattern recognition, because in suitable form they can be made invariant to variations in translation, rotation and size. However the computation of discrete moments by their definition requires many multiplications which limits the speed of computation. In this paper we express the moments ...

18

November 1999
IEEE Transactions on Computers: Volume 48 Issue 11, November 1999

**Publisher:** IEEE Computer Society

This paper presents an analysis of radix representations of elements from general rings; in particular, we study the questions of redundancy and completeness in such representations. Mappings into radix representations, as well as conversions between such, are discussed, in particular where the target system is redundant. Results are shown valid ...

**Keywords**:
Radix representation of rings, integer and computer radix number systems, redundancy, number system conversion, computer arithmetic.

19

April 1999
ARITH '99: Proceedings of the 14th IEEE Symposium on Computer Arithmetic

**Publisher:** IEEE Computer Society

This note presents necessary and sufficient conditions for parallel and constant time conversions from one digit-set into another, and thus also for constant time addition. In the integer domain it is generally believed that such conversion and addition is possible if the target digit-set is redundant and complete. This is ...

20

December 1998
Journal of VLSI Signal Processing Systems: Volume 20 Issue 3, Dec. 1998

**Publisher:** Kluwer Academic Publishers

This paper presents a new fast Discrete Fourier Transform (DFT) algorithm. By rewriting the DFT, a new algorithm is obtained that uses 2n�2(3n�13)+4n�2 real multiplications and 2n�2(7n�29)+6n+2 real additions for a real data N=2n point DFT, comparable to the number of operations in the Split-Radix method, but with slightly fewer ...