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[mathcad] Re: Rounding up/down error
| [mathcad] Re: Rounding up/down error |
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Author: Steve Ellison
Posted: Thu, 17 Jul 2003 10:41:37 +0100
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Chris & co;
The numerical precision isn't causing this behaviour. Mathcad is apparently implementing ISO/IEEE round-to-even rules exactly correctly. Chris is correct as to the reason for that rule; on large data sets, always rounding 5 up causes a small positive bias (because only 9 (ie 1-9) of the ten available digits are ever rounded - noone rounds zero. To have rounding average out to zero, half the digits must round up and half down. Half of 9 is 4.5 - which means one needs to be rounded up half the time).
The 'unexpectedness' arises partly because the round-toeven rule is relatively recent and rarely taght at school. Also, though, any rounding rules do odd things if you progressively round; 1.50 rounds to 2 under most rounding rules, but 1.49 rounds correctly to 1.5 to two sig figs, and correctly to 1.0 if rounded to one sig fig. Only if you incorrectly round first from 1.49 to 1.5 and then round 1.5 does is get inconsistent. Moral 1 - never round progressively!
Also, don't round if it matters! Remember we only round for reporting convenience (or sometimes paper calculation); in either case, the general rule is never to throw away significant information. So no argument over rounding behaviour should, almost by definition, ever be important. :)
Steve E.
>>> 15/07/2003 09:04:29 >>>
I suspect the reason is this: numbers are represented internally by a
normalised mantissa (52 bits) times a power of 2. Most decimal fractions
cannot be represented exactly, so some rounding takes place. Then when you
display the number, it is converted back to decimal, and it gets rounded
again. The first rounding could take it up or down, by a very tiny amount.
The actual IEEE floating point format specifies that if the number ends in
exactly 0.5, it is rounded to the nearest even integer. So 1.5 and 2.5 are
both rounded to 2. This is so that, statistically, 0.5 is rounded up or
down with equal probablility.
Chris
At 09:12 15/07/2003 +0200, you wrote:
>I have been using a calculation with a number of 0.04425 N/mm^2. However,
>I noticed a strange behavior when I modified the properties of a result to
>show me 4 digits. Normally, it would then show me 0.0443 as a 5 always
>rounds up. However, as you can see in the attached mathcad-sheet, MathCad
>decides that it does not want to show 0.0443, but 0.0442. I have been
>playing with a few other numbers, but this is the only one I have found.
>Can someone tell me, whether this is a bug in MathCad, or that there is an
>option included in MathCad that makes sure that some results are rounded
>to a different value? (Or perhaps someone else has encountered the same
>rounding error somewhere else...)
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