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For unums, there is good evidence (see figure 5) that accuracy is better than IEEE floats. (Note: I use the term "unum" broadly to refer to any of the various iterations and revisions to the format.)

However, generally we regard "stability as more of a constraint than accuracy". Are there numerical algorithms which can be made stable using some unum format that is unstable using IEEE floating point?

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  • $\begingroup$ In your experience, can unums be largely described as (variable length) multi-precision intervals, so that algorithms, correctly implemented, return the "best possible" approximation and a correct (possibly too pessimistic) error bound? $\endgroup$ Apr 28, 2022 at 5:41
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    $\begingroup$ Implicit in the question is "given that we're put so much effort into hardware implementations of IEEE floating point, would there be any real benefit to doing a hardware implementation of unums?" $\endgroup$
    – user14717
    Apr 28, 2022 at 17:07
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    $\begingroup$ Maybe start by asking the reverse question: What numerical algorithms currently suffer because they are using FP numbers that provide insufficient accuracy? (I can't think of much in this regard, but it's also not exactly my field.) $\endgroup$ Apr 28, 2022 at 18:27
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    $\begingroup$ How about how Gaussian elimination is not guaranteed to be stable? (Most matrices are stable under Gaussian elimination in floating point, but there exist matrices which diverge.) $\endgroup$
    – user14717
    Apr 28, 2022 at 18:56
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    $\begingroup$ @WolfgangBangerth: "Yet it was realized in around 1960 by Wilkinson and others that for certain exceptional matrices, Gaussian elimination is still unstable, even with pivoting" people.maths.ox.ac.uk/trefethen/NAessay.pdf $\endgroup$
    – user14717
    Apr 29, 2022 at 18:40

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