Arithmetic Algorithms for Extended Precision Using Floating-Point Expansions

Joldeş, Mioara; Marty, Olivier; Muller, Jean-Michel; Popescu, Valentina

doi:10.1109/tc.2015.2441714

Cited by 24 publications

(33 citation statements)

References 14 publications

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“…introduced arithmetic algorithms using floating points expansion, yielding better precision. The work offers improvement in performing normalization, division, and square root [13]. Similar work has also been done by Muller, et al, who focused their research on a multiplication operation algorithm using floatingpoint expansions [14].…”

Section: Related Workmentioning

confidence: 74%

Performance Analysis of BigDecimal Arithmetic Operation in Java

2018

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The Java programming language provides binary floating-point primitive data types such as float and double to represent decimal numbers. However, these data types cannot represent decimal numbers with complete accuracy, which may cause precision errors while performing calculations. To achieve better precision, Java provides the BigDecimal class. Unlike float and double, which use approximation, this class is able to represent the exact value of a decimal number. However, it comes with a drawback: BigDecimal is treated as an object and requires additional CPU and memory usage to operate with. In this paper, statistical data are presented of performance impact on using BigDecimal compared to the double data type. As test cases, common mathematical processes were used, such as calculating mean value, sorting, and multiplying matrices.

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Section: Related Workmentioning

confidence: 74%

Performance Analysis of BigDecimal Arithmetic Operation in Java

2018

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“…Unfortunately, this fails when zeros appear in the FPE, therefore we have to account for the fact that we may have intermediate zeros inside our FPEs. This case was not considered in [10], so nothing was proved when a zero is involved. We get rid of this flaw and handle possible intermediate zeros in the proofs; in particular, we want to prove the FPE has the wanted property P , even if we remove the zeros.…”

Section: Coq Definitionsmentioning

confidence: 99%

“…numbers are represented as the sum of 2 or 4 double-precision FP numbers. Campary [11,10] supports FPEs with an arbitrary number of terms, and also targets GPU implementation. Other libraries that manipulate FPEs are presented in [20,17].…”

Section: Introductionmentioning

confidence: 99%

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Formal Verification of a Floating-Point Expansion Renormalization Algorithm

Boldo

Joldeş

Muller³

et al. 2017

Interactive Theorem Proving

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Many numerical problems require a higher computing precision than the one offered by standard floating-point formats. A common way of extending the precision is to use floating-point expansions. As the problems may be critical and as the algorithms used have very complex proofs (many sub-cases), a formal guarantee of correctness is a wish that can now be fulfilled, using interactive theorem proving. In this article we give a formal proof in Coq for one of the algorithms used as a basic brick when computing with floating-point expansions, the renormalization, which is usually applied after each operation. It is a critical step needed to ensure that the resulted expansion has the same property as the input one, and is more "compressed". The formal proof uncovered several gaps in the pen-and-paper proof and gives the algorithm a very high level of guarantee.

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“…The method we use for computing the reciprocal and the square root of a FP expansion is based on an adapted Newton-Raphson iteration, where the intermediate calculations are done using "truncated" operations (additions, multiplications) involving FP expansions. We gave a thorough error analysis showing that it allows for very accurate computations (see [13]). We also introduced a new multiplication algorithm for FP expansions with arbitrary precision, up to the order of tens of FP elements in mind.…”

Section: Key Featuresmentioning

confidence: 99%

CAMPARY: Cuda Multiple Precision Arithmetic Library and Applications

Joldeş

Muller

Popescu

et al. 2016

Lecture Notes in Computer Science

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Abstract. Many scientific computing applications demand massive numerical computations on parallel architectures such as Graphics Processing Units (GPUs). Usually, either floating-point single or double precision arithmetic is used. Higher precision is generally not available in hardware, and software extended precision libraries are much slower and rarely supported on GPUs. We develop CAMPARY: a multipleprecision arithmetic library, using the CUDA programming language for the NVidia GPU platform. In our approach, the precision is extended by representing real numbers as the unevaluated sum of several standard machine precision floating-point numbers. We make use of error-free transforms algorithms, which are based only on native precision operations, but keep track of all rounding errors that occur when performing a sequence of additions and multiplications. This offers the simplicity of using hardware highly optimized floating-point operations, while also allowing for rigorously proven rounding error bounds. This also allows for easy implementation of an interval arithmetic. Currently, all basic multiple-precision arithmetic operations are supported. Our target applications are in chaotic dynamical systems or automatic control.

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Arithmetic Algorithms for Extended Precision Using Floating-Point Expansions

Cited by 24 publications

References 14 publications

Performance Analysis of BigDecimal Arithmetic Operation in Java

Performance Analysis of BigDecimal Arithmetic Operation in Java

Formal Verification of a Floating-Point Expansion Renormalization Algorithm

CAMPARY: Cuda Multiple Precision Arithmetic Library and Applications

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