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[mathcad] Re: Rounding up/down error
| [mathcad] Re: Rounding up/down error |
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Author: Pergande, Albert N
Posted: Thu, 17 Jul 2003 08:29:22 -0400
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Round to even dates back to steam engine and slide rule days - by
rounding to even, subsequent calculations where very slightly easier,
since numbers that end in an even digit as easier to divide by 2.
It's sort of like the roman carriage wheel width standard driving modern
railroad gages.
-----Original Message-----
From: Steve Ellison />
Sent: Thursday, July 17, 2003 5:42 AM
To: Mathcad Discussion List
Subject: [mathcad] Re: Rounding up/down error
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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