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    Re: Rounding decimal fractions
    From: UNK
    Date: 2013 Jul 19, 10:02 -0700

    Stan,

    It's exactly the opposite of what you state regarding standard rounding versus banker's . The flaw in your argument that "[...] 'standard' rounding has five values that round down and five that round up, i.e. half down and half up." is that you do not take into consideration the sizes of the individual rounding errors. Simply put, when you "round" 0, you are not really rounding, so don't count it in!

    More correctly argued:

    Consider integers 0 to 9, evenly distributed. The following table shows the signed rounding error when rounding to the nearest 10. (The reason why we choose the NEAREST 10 is that we want to minimize absolute rounding error in each case.)

    0 ... 0
    1 ... -1
    2 ... -2
    3 ... -3
    4 ... -4
    5 ... ?
    6 ... +4
    7 ... +3
    8 ... +2
    9 ... +1

    The arithmetic sum of all cases in the table is zero. Any unbiased rounding algorithms must therefore use a procedure that ensures that the average rounding error for 5 comes out to zero. 'Rounding to even' is the cheapest way of implementing this with reproducibility (= pseudo-randomly).

    Herbert Prinz

    P.S.

    The real problem with the v and d correction table in the N.A. is not at all a rounding problem. This table has a systematic error which I will outline in a separate post. The increments for sun and planets, on the other hand, (and, in all likelihood, also that for Aries and Moon) does have a rounding problem. It uses standard rounding instead of the seemingly more appropriate 'round to even' method. I never inquired why this choice was made.


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