The Conversion of a High Order Programming Language from Floating-Point Arithmetic to Range Arithmetic

<p>Rounded interval arithmetic is very easy to implement by means of directed rounding arithmetic operators. Such operators are available in the IEEE floating point arithmetic of the transputer. When a few small pieces of assembly language code are used to access the directed rounding operators, the four basic rounded interval arithmetic operators can easily be expressed in the programming language Occam.</p><p>The performance of this implementation is assessed and it is shown that the time consuming part of the calculation are not the directed rounding floating point operations as one might have expected. Most of the time is spent with transport of operands to and from the on-chip floating point unit and the procedure call/parameter passing overhead. Based on this experience the implementation is improved. This implementation runs with 0.15 MIOPS (Million Interval Operations Per Second) or 0.30 MFLOPS on an example interval calculation proposed by Moore. Furthermore, it is demonstrated that an advanced interval language compiler may provide a performance of 0.30 MIOPS or 0.59 MFLOPS on this example calculation.</p>

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Deductive Verification of Floating-Point Java Programs in KeY

Tools and Algorithms for the Construction and Analysis of Systems - Lecture Notes in Computer Science ◽

10.1007/978-3-030-72013-1_13 ◽

2021 ◽

pp. 242-261

Author(s):

Rosa Abbasi ◽

Jonas Schiffl ◽

Eva Darulova ◽

Mattias Ulbrich ◽

Wolfgang Ahrendt

Keyword(s):

Programming Language ◽

Floating Point ◽

Deductive Verification ◽

Java Programming ◽

Special Values ◽

Floating Point Arithmetic ◽

Smt Solvers ◽

Java Programs ◽

Transcendental Functions ◽

Point Arithmetic

AbstractDeductive verification has been successful in verifying interesting properties of real-world programs. One notable gap is the limited support for floating-point reasoning. This is unfortunate, as floating-point arithmetic is particularly unintuitive to reason about due to rounding as well as the presence of the special values infinity and ‘Not a Number’ (NaN). In this paper, we present the first floating-point support in a deductive verification tool for the Java programming language. Our support in the KeY verifier handles arithmetic via floating-point decision procedures inside SMT solvers and transcendental functions via axiomatization. We evaluate this integration on new benchmarks, and show that this approach is powerful enough to prove the absence of floating-point special values—often a prerequisite for further reasoning about numerical computations—as well as certain functional properties for realistic benchmarks.

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Real Floating-Point Arithmetic—The Heart of a Scientific Programming Language

Pascal-Sc ◽

10.1016/b978-0-12-111155-7.50006-1 ◽

1987 ◽

pp. 42-57

Author(s):

Gerd Bohlender ◽

Christian Ullrich ◽

Jürgen Wolff von Gudenberg ◽

Louis B. Rall

Keyword(s):

Programming Language ◽

Floating Point ◽

Floating Point Arithmetic ◽

Point Arithmetic

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On the numerical solution of chaotic dynamical systems using extend precision floating point arithmetic and very high order numerical methods

Nonlinear Analysis Modelling and Control ◽

10.15388/na.16.3.14097 ◽

2011 ◽

Vol 16 (3) ◽

pp. 340-352 ◽

Cited By ~ 10

Author(s):

Scott A. Sarra ◽

Clyde Meador

Keyword(s):

Dynamical Systems ◽

Numerical Solution ◽

Numerical Method ◽

High Order ◽

Floating Point ◽

Time Step ◽

Floating Point Arithmetic ◽

Chaotic Dynamical Systems ◽

Point Arithmetic ◽

Very High

Multiple results in the literature exist that indicate that all computed solutions to chaotic dynamical systems are time-step dependent. That is, solutions with small but different time steps will decouple from each other after a certain (small) finite amount of simulation time. When using double precision floating point arithmetic time step independent solutions have been impossible to compute, no matter how accurate the numerical method. Taking the well-known Lorenz equations as an example, we examine the numerical solution of chaotic dynamical systems using very high order methods as well as extended precision floating point number systems. Time step independent solutions are obtained over a finite period of time. However even with a sixteenth order numerical method and with quad-double floating point numbers, there is a limit to this approach.

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Specification for binary floating point arithmetic for microprocessor systems

10.3403/00218955u ◽

2015 ◽

Keyword(s):

Floating Point ◽

Floating Point Arithmetic ◽

Point Arithmetic

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Numerical algorithms for high-performance computational science

Philosophical Transactions of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rsta.2019.0066 ◽

2020 ◽

Vol 378 (2166) ◽

pp. 20190066 ◽

Cited By ~ 2

Author(s):

Jack Dongarra ◽

Laura Grigori ◽

Nicholas J. Higham

Keyword(s):

High Performance ◽

Numerical Algorithms ◽

Computational Science ◽

Floating Point ◽

Important Criterion ◽

Data Movement ◽

Floating Point Arithmetic ◽

High Performance Computers ◽

Point Arithmetic ◽

Speed And Accuracy

A number of features of today’s high-performance computers make it challenging to exploit these machines fully for computational science. These include increasing core counts but stagnant clock frequencies; the high cost of data movement; use of accelerators (GPUs, FPGAs, coprocessors), making architectures increasingly heterogeneous; and multi- ple precisions of floating-point arithmetic, including half-precision. Moreover, as well as maximizing speed and accuracy, minimizing energy consumption is an important criterion. New generations of algorithms are needed to tackle these challenges. We discuss some approaches that we can take to develop numerical algorithms for high-performance computational science, with a view to exploiting the next generation of supercomputers. This article is part of a discussion meeting issue ‘Numerical algorithms for high-performance computational science’.

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