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Subject: Re: Bowditch 2002
From: Michael Wescott (wescott_mike@XXX.XXX)
Date: Sat Oct 05 2002 - 12:03:43 EDT
MBurrill@XXX.XXX 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@XXX.XXX
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