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  • Round to Nearest at Binade

    From MitchAlsup1@21:1/5 to All on Tue Apr 22 00:50:39 2025
    In a message over on Unum NG, there was a statement which got
    me thinking about whether Round to Nearest is properly defined
    in IEEE 754 (all the way back to 1985) in a theoretic sense
    not in a practical sense.

    Consider a calculation which produces in a result prior to
    rounding with a fraction:: 1.1111111...11111 and 1 as the
    ½ ULP bit--properly ½ way between nearest representable
    points except for the binade switching points. (ignore sticky}

    Since rounding up increments the exponent, the closest number
    above the result is 2 units of ½ ULP above while the closest
    number below is only 1 unit of ½ ULP below. A span of 3 units
    of ½ ULP. The midpoint of a span of 3×½ is 3/4 not 1/2.

    Would not the Round to Nearest rule want the rounding point to
    be 3/4 instead of 1/2 ?? as this is the point closest to both
    representable IEEE 754 numbers ??

    I know this ship sailed <gosh> 40 years ago, and I am not trying
    to alter it, I am trying to wrap my head around the numerics of
    it only.

    I also know that the gain in accuracy is triflingly small
    0.5 ULP - 1/2^53 better, that it might not be noticed in
    practice.

    Comments anyone ??

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  • From Anton Ertl@21:1/5 to [email protected] on Tue Apr 22 15:09:09 2025
    [email protected] (MitchAlsup1) writes:
    In a message over on Unum NG, there was a statement which got
    me thinking about whether Round to Nearest is properly defined
    in IEEE 754 (all the way back to 1985) in a theoretic sense
    not in a practical sense.

    Consider a calculation which produces in a result prior to
    rounding with a fraction:: 1.1111111...11111 and 1 as the
    ½ ULP bit--properly ½ way between nearest representable
    points except for the binade switching points. (ignore sticky}

    Since rounding up increments the exponent, the closest number
    above the result is 2 units of ½ ULP above

    No.

    Let's consider a (before rounding) 4-bit mantissa (plus hidden bit),
    and a 3-bit mantissa after rounding, and the full-length unrounded
    number in binary is

    x = 1.1111 e0

    The closest number above is (before normalization)

    y = 10.0000 e0

    and after normalization and with the correct number of mantissa bits:

    y = 1.000 e1

    Nothing is lost in this normalization. The closest representable
    number below x is

    z = 1.111 e0

    Both y and z are representable in FP with a 3 bit mantissa (plus
    hidden bit), and both are 0.0001 e0 away from x. So when you round
    from 1.1111 e0 to a 3-bit mantissa with round-to-nearest-or-even, the
    effect you claim does not exist. And thanks to the or-even rule, the
    rounded number is y.

    However, there is one case where the effect probably exists. Consider

    1.1111 eM

    where M is the maximum finite exponent for our FP numbers. Do you
    round up towards infinity, or down to the maximum finite FP number
    with 3-bit mantissa? I have not tried, but I expect rounding to
    infinity in this case.

    If you rounded to the maximum finite FP number, at what intermediate
    result would you stop doing that and round to infinity?

    while the closest
    number below is only 1 unit of ½ ULP below. A span of 3 units
    of ½ ULP. The midpoint of a span of 3×½ is 3/4 not 1/2.

    Unums may have this problem, but IEEE FP numbers don't.

    I know this ship sailed <gosh> 40 years ago, and I am not trying
    to alter it, I am trying to wrap my head around the numerics of
    it only.

    Maybe hanging around in a Unum NG is not good for you:-)

    - anton
    --
    'Anyone trying for "industrial quality" ISA should avoid undefined behavior.'
    Mitch Alsup, <[email protected]>

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