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    Rounding decimal fractions
    From: Paul Hirose
    Date: 2013 Jul 18, 12:28 -0700

    Stan K wrote:
     > A couple of years ago a friend of mine and I were discussing how to
    implement Increments and Corrections in my Celestial Tools program and
    his Navigation Calculator Workbook spreadsheet.  Looking at the output
    of his spreadsheet, we noticed that the Almanac values apparently did
    not use what we call "standard" rounding, in this case, for instance,
    0.00 through 0.04 would round down to 0.0 and 0.05 through 0.09 would
    round up to 0.1.
    I thought the convention, when a decimal can be rounded up or down with
    equal accuracy, is "round to even." E.g., 1.05 rounded to the nearest
    tenth is 1.0, but 1.15 is 1.2. But I see that my HP 49G calculator
    rounds up in these borderline cases. With 2 decimal point precision
    selected, 1/8 = .13, 3/8 = .38, 5/8 = .63, 7/8 = .88.
    The composite formatting feature in Microsoft's .NET Framework does the
    same thing as the calculator. But I discovered the .NET Math.Round class
    gives the programmer a "round to even" option. Results from a test program:
    0.125 0.13 0.12
    0.375 0.38 0.38
    0.625 0.63 0.62
    0.875 0.88 0.88
    The first column is the fraction to full accuracy. In the second and
    third columns I used composite formatting with precision of two decimal
    places. The difference is that in the third column the value was first
    rounded with Math.Round.
    The sum of the second column is a little high (1/8 + 3/8 + 5/8 + 7/8 = 2
    exactly) because rounding all borderline cases up introduces a small
    systematic error.
    I have to admit I never pay attention to these fine points in my own
    software. Since Tinyac and Lunar3 both have selectable precision, I
    assume the user will use, say, .01 precision if "tenths" are critical.
    I filter out messages with attachments or HTML.

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