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Re: Numerical accuracy/precision - this is a bug or a feature?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg120274] Re: Numerical accuracy/precision - this is a bug or a feature?
  • From: Bill Rowe <readnews at sbcglobal.net>
  • Date: Sat, 16 Jul 2011 05:41:27 -0400 (EDT)

On 7/14/11 at 5:20 AM, fateman at cs.berkeley.edu (Richard Fateman)
wrote:

>On 7/13/2011 12:13 AM, Bill Rowe wrote:
>>On 7/12/11 at 6:59 AM, slawek at host.pl (slawek) wrote:

>>The documentation for Equal (==) specifically states:

>>Approximate numbers with machine precision or higher are considered
>>equal if they differ in at most their last seven binary digits
>>(roughly their last two decimal digits).

>You mean "considered Equal[] by Mathematica.

No. I meant exactly what I wrote. The excerpt was copy/paste
from the documentation for Equal. Now if you want to interpret
the documentation fine. But this is what is in the
documentation. And if your comment is to the effect something
other than Mathematica might treat 1.4 as unequal to 14/10, so
what? The existence of other ways to treating this has no impact
on what is in the Mathematica documentation or how Mathematica works.

>>Ultimately, since the developers are unlikely to change such a
>>fundamental aspect of Mathematica, the only sensible thing is to
>>understand how Mathematica does things if you want to effectively
>>use Mathematica. The alternative would be to find a system that
>>operates more to your liking.

>It might be fun to test to see if any of your code broke if you did
>this:

>Unprotect[Equal] Equal[a_Real,b_]:=
>Equal[Rationalize[SetAccuracy[a,Infinity]],b] Equal[a_,b_Real]:=
>Equal[a,Rationalize[SetAccuracy[b,Infinity]]]

Fun?? Modifying built-in Mathematica commands is risky. To much
of Mathematica uses other Mathematica functions. I could do then
this spend lots of time testing and getting results that lead me
to believe there is no impact only to get burned much later and
likely have a very difficult time determining why something
didn't work as expected. This isn't my idea of fun. I have far
better things to do with my time.

>For example, 0.1d0 is exactly p= 3602879701896397/36028797018963968

>so my new and improved Equal thinks that 0.1d0 and 1/10 are NOT
>equal,  (indeed, they differ by 1/180143985094819840)

>but

>0.1d0 and p   ARE equal.

>So the question is:  would any of YOUR code break if you used this
>patch on Equal?  Really?

I don't know the answer. And since I assume you don't have
access to Mathematica's source code, you cannot know the answer
either. I have no intention of attempting to determine the
answer for the reasons stated above. So, a discussion of whether
this patch would or would not break some code is pointless to me.



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