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Introduction

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Handbook of Floating-Point Arithmetic

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Abstract

Representing and manipulating real numbers efficiently is required in many fields of science, engineering, finance, and more. Since the early years of electronic computing, many different ways of approximating real numbers on computers have been introduced. One can cite (this list is far from being exhaustive): fixed-point arithmetic, logarithmic [337, 585] and semi-logarithmic [444] number systems, intervals [428], continued fractions [349, 622], rational numbers [348] and possibly infinite strings of rational numbers [418], level-index number systems [100, 475], fixed-slash and floating-slash number systems [412], tapered floating-point arithmetic [432, 22], 2-adic numbers [623], and most recently unums and posits [228, 229].

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Notes

  1. 1.

    For legal reasons, financial calculations frequently require special rounding rules that are very tricky to implement if the underlying arithmetic is binary: this is illustrated in [320, Section 2].

  2. 2.

    If p and β were real numbers, the value of β that would minimize β × p while letting β p be constant would be e = 2. 7182818⋯.

  3. 3.

    Even if sometimes you need to dive into the compiler documentation to find the right options; see Chapter 6

  4. 4.

    http://www.cs.berkeley.edu/~wkahan/.

  5. 5.

    That conjecture asserts that there are infinitely many pairs of prime numbers that differ by 2.

  6. 6.

    A detailed analysis of this bug can be found at http://www.lomont.org/Math/Papers/2007/Excel2007/Excel2007Bug.pdf.

  7. 7.

    See http://www.science20.com/news_articles/what_happens_bridge_when_one_side_ uses_mediterranean_sea_level_and_another_north_sea-121600.

  8. 8.

    Interestingly enough, the decimal value a 0 = 1. 71828182845904523536028747135 given in Program 1.2 is less than e − 1, but when rounded to the nearest binary64/double precision floating-point number, it becomes larger than e − 1.

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Author information

Authors and Affiliations

  1. CNRS - LIP, Lyon, France

    Jean-Michel Muller

  2. Kalray, Grenoble, France

    Nicolas Brunie

  3. INSA-Lyon - CITI, Villeurbanne, France

    Florent de Dinechin

  4. Inria - LIP, Lyon, France

    Claude-Pierre Jeannerod, Vincent Lefèvre & Nathalie Revol

  5. CNRS - LAAS, Toulouse, France

    Mioara Joldes

  6. Inria - LRI, Orsay, France

    Guillaume Melquiond

  7. ENS-Lyon - LIP, Lyon, France

    Serge Torres

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Muller, JM. et al. (2018). Introduction. In: Handbook of Floating-Point Arithmetic. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-76526-6_1

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