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From: David Bailey on 2 Feb 2010 03:23
Daniel Lichtblau wrote:

> I had in mind the spoiler answer Richard Fateman provided in his first
> post mentioning this particular tangent, err, example.
>
> http://forums.wolfram.com/mathgroup/archive/2010/Jan/msg00638.html
>
> At the bottom we find:
> ---
> I would especially avoid .nb objects, and most especially on topics of
> numerical analysis, where the design flaws are, in my opinion, so
> fundamental. Example (mathematica 7.0):
> {x >= 1, x > 1, x > 0, x}
> evaluates to
> {True, False, False, 0.}
>
> can you construct x?
>
> RJF
>
> One possible answer, below....
>
> x=0``-.5
> ---
>
> The point is that with Mathematica's version of significance arithmetic,
> equality, I believe, is effectively treated as having a nontrivial an
> intersection (of the implicit intervals defining two numbers). If
> neither has any fuzz (i.e. both are exact), then Equal allows for no
> fuzz, so this is only a subtlety if at least one of the values is
> approximate.
>
> One implication is that a "zero" of sufficiently low (as in bad)
> accuracy can be regarded as 1, or -1, or Pi, if those values happen to
> fall within the accuracy (which I refer to as fuzz).
>
> The other inequalities follow from the preservation of trichotomy. For
> explicitly real values we regard that as important. mathematica makes no
> pretense that Equal is transitive and I do not see any way to do that
> and also have useful approximate arithmetic.
>
> There has been some amount of communication off-line on this topic,
> which is why some of us (well, me, at least) sometimes forget the
> examples are not universally obvious to those who have not memorized the
> enitre thread.
>
> Daniel
>

Maybe an notebook option to flag numbers of extremely low precision with
a colour might be useful. I guess this might be useful more generally in
numerical analysis.

David Bailey
http://www.dbaileyconsultancy.co.uk

From: Richard Fateman on 3 Feb 2010 06:13
David Bailey wrote:
....
>>
>
> Maybe an notebook option to flag numbers of extremely low precision with
> a colour might be useful. I guess this might be useful more generally in
> numerical analysis.
>

This (previously posted) program will do it.

$LowPrecisionWarningLimit=1
PrintShowFuzz[x_] :=
x /. (r_Real /; (Precision[r] < $LowPrecisionWarningLimit) ->
Style[InputForm[r], Red])
$PrePrint=PrintShowFuzz

RJF

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