Interval Arithmetic

Muller, Jean-Michel; Brunie, Nicolas; Dinechin, Florent de; Jeannerod, Claude-Pierre; Joldeş, Mioara; Lefèvre, Vincent; Melquiond, Guillaume; Revol, Nathalie; Torres, Serge

doi:10.1007/978-3-319-76526-6_12

Cited by 2 publications

(8 citation statements)

References 34 publications

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“…Now we discuss how to modify classical algorithms for addition and multiplication of floating-point numbers [16, ch. 7], [27, §4.2.1], [47, ch.…”

Section: Methodsmentioning

confidence: 99%

“…8] in order to obtain algorithms that support SR and can readily be implemented in software or hardware. Algorithms for other operations to include SR such as fused multiply-add (FMA) or division can be similarly derived by modifying the original algorithms for the IEEE 754 arithmetic operations [16,47]. To the best of our knowledge, these algorithms are new in that no general methods to round to precision p taking normalization into account had so far been proposed in the literature.…”

Section: Methodsmentioning

confidence: 99%

“…This restriction does not affect our main observations pertaining to the implementation of SR. The role that the sign of the operands plays in this algorithm is discussed, for instance, in [16, §7.3]. We make additional observations about subtraction when necessary.…”

Section: Methodsmentioning

confidence: 99%

“…If the sum of two floating-point numbers is subnormal, then it is exact and no rounding is required [16, thm. 4.2].…”

Section: Methodsmentioning

confidence: 99%

“…For RN, the bound in (6.1) can be tightened to | δ | ≤ u /(1 + u ) < u [16, thm. 2.3], [27, eqn (18)].…”

Section: Rounding Error Analysis With Srmentioning

confidence: 99%

See 4 more Smart Citations

Stochastic rounding: implementation, error analysis and applications

et al. 2022

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Stochastic rounding (SR) randomly maps a real number x to one of the two nearest values in a finite precision number system. The probability of choosing either of these two numbers is 1 minus their relative distance to x . This rounding mode was first proposed for use in computer arithmetic in the 1950s and it is currently experiencing a resurgence of interest. If used to compute the inner product of two vectors of length n in floating-point arithmetic, it yields an error bound with constant n u with high probability, where u is the unit round-off. This is not necessarily the case for round to nearest (RN), for which the worst-case error bound has constant nu . A particular attraction of SR is that, unlike RN, it is immune to the phenomenon of stagnation, whereby a sequence of tiny updates to a relatively large quantity is lost. We survey SR by discussing its mathematical properties and probabilistic error analysis, its implementation, and its use in applications, with a focus on machine learning and the numerical solution of differential equations.

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“…Now we discuss how to modify classical algorithms for addition and multiplication of floating-point numbers [16, ch. 7], [27, §4.2.1], [47, ch.…”

Section: Methodsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

“…If the sum of two floating-point numbers is subnormal, then it is exact and no rounding is required [16, thm. 4.2].…”

Section: Methodsmentioning

confidence: 99%

“…For RN, the bound in (6.1) can be tightened to | δ | ≤ u /(1 + u ) < u [16, thm. 2.3], [27, eqn (18)].…”

Section: Rounding Error Analysis With Srmentioning

confidence: 99%

See 3 more Smart Citations

Stochastic rounding: implementation, error analysis and applications

et al. 2022

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Platform-independent Specification and Verification of the Standard Mathematical Square Root Function

Shilov

Кондратьев

Anureev

et al. 2018

Model. anal. inf. sist.

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Research project "Platform-independent approach to formal specification and verification of standard mathematical functions" is aimed onto a development of an incremental combined approach to the specification and verification of the standard mathematical functions like sqrt, cos, sin, etc. Platform-independence means that we attempt to design a relatively simple axiomatization of the computer arithmetic in terms of real, rational, and integer arithmetic (i.e. the fields R and Q of real and rational numbers, the ring Z of integers) but dont specify neither base of the computer arithmetic, nor a format of numbers representation. Incrementality means that we start with the most straightforward specification of the simplest easy to verify algorithm in real numbers and finish with a realistic specification and a verification of an algorithm in computer arithmetic. We call our approach combined because we start with a manual (pen-and-paper) verification of some selected algorithm in real numbers, then use these algorithm and verification as a draft and proof-outlines for the algorithm in computer arithmetic and its manual verification, and finish with a computer-aided validation of our manual proofs with some proof-assistant system (to avoid appeals to "obviousness" that are very common in human-carried proofs). In the paper we present first steps towards a platform-independent incremental combined approach to specification and verification of the standard functions cos and sin that implement mathematical trigonometric functions cos and sin.

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Interval Arithmetic

Cited by 2 publications

References 34 publications

Stochastic rounding: implementation, error analysis and applications

Stochastic rounding: implementation, error analysis and applications

Platform-independent Specification and Verification of the Standard Mathematical Square Root Function

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