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Handbook of Floating-Point Arithmetic pp 513–552Cite as

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Extending the Precision

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Abstract

Though satisfactory in most situations, the fixed-precision floating-point formats that are available in hardware or software in our computers may sometimes prove insufficient. There are reasonably rare cases when the binary64/decimal64 or binary128/decimal128 floating-point numbers of the IEEE 754 standard are too crude as approximations of the real numbers. Also, at the time of writing these lines, the binary128 and decimal128 formats are very seldom implemented in hardware.

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Notes

  1. 1.

    To our knowledge, the only commercially significant platform that has supported binary128 in hardware for the last decade has been the IBM z Systems [387].

  2. 2.

    Such as 2Sum (Algorithm 4.4, page 108), Fast2Sum (Algorithm 4.3, page 104), Dekker product (Algorithm 4.10, page 116), and 2MultFMA (Algorithm 4.8, page 112).

  3. 3.

    Of course, when we write x h + x ℓ , the addition symbol corresponds to the exact, mathematical addition.

  4. 4.

    BLAS is an acronym for Basic Linear Algebra Subroutines.

  5. 5.

    QD is available at http://crd-legacy.lbl.gov/~dhbailey/mpdist/.

  6. 6.

    CAMPARY is available at http://homepages.laas.fr/mmjoldes/campary/.

  7. 7.

    By the way, “correct rounding” is not clearly defined for double-words: Does this mean that we get the double-word closest to the exact value, or that we get the 2p-digit number closest to the exact value (if the underlying arithmetic is of precision p)? This can be quite different. For instance, in radix-2, precision-p arithmetic, the double-word closest to a = 2p + 2−p + 2−p−1 is a itself, whereas the precision-2p number closest to a is 2p + 2−p+1.

  8. 8.

    Note that even if the general idea remains the same, the notion of overlap slightly changes depending on the authors!

  9. 9.

    CRlibm was developed by the Arénaire/AriC team of CNRS, INRIA, UCBL and ENS Lyon, France. It is available at https://gforge.inria.fr/scm/browser.php?group_id=5929&extra=crlibm.

  10. 10.

    INTLAB is the Matlab toolbox for self-validating algorithms; it is available at http://www.ti3.tu-harburg.de/rump/intlab/.

  11. 11.

    Magma is available at http://magma.maths.usyd.edu.au/magma/.

  12. 12.

    NTL is a library for performing number theory, available at http://www.shoup.net/ntl/.

  13. 13.

    ARPREC is available at http://crd-legacy.lbl.gov/~dhbailey/mpdist/.

  14. 14.

    GNU MP is available at https://gmplib.org/.

  15. 15.

    MPFR is available at http://www.mpfr.org/.

  16. 16.

    The minimum precision was initially chosen as 2 because the ties-to-even rule was not defined for precision 1 in IEEE 754. But since this case can occur in conversions to character strings, a complete definition has been proposed for the 2018 revision (see Section 3.1.2.1), and the minimum precision could be changed to 1 in MPFR, even though the context is different (there are no formats with 1-bit precision in IEEE 754).

  17. 17.

    Application Binary Interface, which specifies the sizes of the scalar types, in particular.

  18. 18.

    Fused multiply-subtract.

  19. 19.

    Some linkers will include only unresolved symbols seen so far in the command line, and the requisite GMP symbols are known only after -lmpfr has been taken into account.

  20. 20.

    This is how functions with a variable number of arguments are usually designed in C.

  21. 21.

    This can easily be detected dynamically, and one can choose to stop the iterations when this is no longer true, as the enclosing interval gets too wide to provide any interesting information.

  22. 22.

    The results are actually correctly rounded, so that even more precision may be needed, but the issue mainly comes from the accuracy requirement, and the future support for faithful arithmetic will not avoid it.

  23. 23.

    Arb is available at http://arblib.org/.

  24. 24.

    GNU MPC is available at http://www.multiprecision.org/mpc/.

  25. 25.

    MPFI is available at https://gforge.inria.fr/projects/mpfi/.

  26. 26.

    iRRAM is available at http://irram.uni-trier.de/.

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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). Extending the Precision. In: Handbook of Floating-Point Arithmetic. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-76526-6_14

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