Re: floor problems

• To: mathgroup at smc.vnet.net
• Subject: [mg8176] Re: floor problems
• From: murray at math.umass.edu (Murray Eisenberg)
• Date: Mon, 18 Aug 1997 03:06:59 -0400
• Organization: University of Massachusetts, Amherst
• Sender: owner-wri-mathgroup at wolfram.com

```Tom (toad at planet.eon.net) wrote:
: Hello Mathematica users.

: I had a very frustrating time trying to debug some work I was doing in
: Mathematica.  I finally pared it down to a problem with the Floor
: function, and I have illustrated my concern below.  I am using M 3.0 and
: a Mac computer.  Does anyone have any advice or could you tell me if I
: am doing something wrong?

: Floor[(.7 67 10)]

: 469

: Floor[(67 10 .7)]

: 468

Mathematica is NOT making a mistake, even though if you simply ask for
the products .7 67 10 and 67 10 .7 you get the same output 469.  This
is an inherent problem with approximate machine arithmetic -- it is
NOT associative! And .7 does not have an exact binary representation
(no matter how many bits of precision you might want).

To see what's happening, you can look at the actual binary
representation of the two products.  Do

\$NumberBits[.7 67 10]
\$NumberBits[67 10 .7]

and you will get different results.  Evidently, when Mathematica takes
the floor (of the binary representation), it therefore gets different
results.

Welcome to the world of numerical analysis where, as one wit put it,
no number that anyone really cares about has ever been calculated exactly.

--
Murray Eisenberg                       Internet:  murray at math.umass.edu
Mathematics & Statistics Dept.            Voice:  413-545-2859 (W)
University of Massachusetts                       413-549-1020 (H)
Amherst, MA 01003                           Fax:  413-545-1801

```

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