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    Re: Bowditch 2002
    From: John Kabel
    Date: 2002 Oct 5, 15:39 -0400

    John Kabel replies:
    As a long-time HP user, I quote from the HP 48G Series User's Guide:
    "The display mode controls how the HP 48 displays numbers. (Note that
    numbers can be displayed differently than they are stored.  Regardless
    of the display mode, numbers are always stored as signed, 12-digit
    mantissas with signed, 3-digit exponents.)
    The HP 48 has four display modes . . . "
    So, internal precision is governed by where the decimal place is in two
    numbers on an addition, and the lowest number of significant digits on
    a mutilplication, etc..  The calculator internally behaves as a 12-
    digit machine no matter how many decimal places you enter.  You have to
    control the rounding of answers in keeping with the quality of the
    numbers used to start the calculation.
    John Kabel
    London, Ontario
    "Where Lili came to roost with a wallop last night."
    > MBurrill@LEESBURGVA.ORG said:
    > > I told my HP48 to leave default mode, gave it 7 decimal places, asked
    > > it for 5, and it disagreed with Bowditch. How many  places should I
    > > give it ?
    > That question piqued my interest and the answer is 9, at least for sines.
    > First I downoaded the pdf file for Table 2 and extracted its contents.
    > I managed to reconstruct all the significant data except the difference
    > columns.
    > Then I made use of a calculation program on Linux/Unix systems (bc)
    > that does it's calculations all in decimal and permits selection of
    > the number of decimal places it keeps. It doesn't round, it truncates
    > calculations to the specified number of places.
    > I programmed bc to calculated sines using the classic series
    >         x - x^3/3! + x^5/5! - ...
    > calculation terminated when the term became 0 at the specified scale.
    > I compared these results against the data from Bowditch. With all the
    > calculations done with 5 decimal places there were, as you might expect,
    > many disagreements in the data sets. About half the values disagreed
    > in the last decimal place, but the differences were all no more than
    > +/-0.00002.
    > Carried to 6 decimal places, the calculated values differed from Bowditch
    > by at least 0.000005 in 619 values (out of 5401) but never more than
    > +/-0.000007.
    > At 7 digits there were 38 discrepancies of  0.0000050 or more. The greatest
    > being 0.0000052.
    > At 8 digits, there were five of precisely 0.00000500
    > dd:mm Bowditch   bc
    > ===== ======== ========
    > 21:37 0.36839 .36839500
    > 23:40 0.40141 .40141500
    > 28:8  0.47153 .47152500
    > 35:19 0.57810 .57809500
    > 56:47 0.83660 .83660500
    > At 9 digits, all differences were less than 0.000005000.
    > --
    >         Mike Wescott
    >         Wescott_Mike@EMC.COM

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