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Re: Odd behavior?

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  • Subject: [mg101859] Re: [mg101850] Odd behavior?
  • From: Andrzej Kozlowski <akozlowski at>
  • Date: Mon, 20 Jul 2009 05:59:39 -0400 (EDT)
  • References: <> <>

Just one more comment: in my below I carelessly used MachinePrecision  
and $MachinePrecision as interchangeable, which they are not.  
$MachinePrecision is a global constant whose value depends on your  
processor; MachinePrecision is a symbol used to refer to the precision  
of floating point computations performed by your processor.

Andrzej Kozlowski

On 19 Jul 2009, at 22:10, Andrzej Kozlowski wrote:

> First and above all you have to understand that the results you are  
> getting have (essentially) nothing to do with Mathematica. You are  
> doing computations with MachinePrecision which means that they area  
> all performed entirely by your processor and Mathematica is only  
> reporting the results. So if you want to blame someone for these  
> "poor solutions" your choice is:
> 1. Processor makers for making binary (rather then decimal)  
> processors which cannot represent these decimal fractions with  
> perfect accuracy
> 2. The laws of numerical analysis, which cause the order of  
> computations to matter in doing this sort of inexact arithmetic
> If you are not looking for someone to blame you could follow the  
> common advice in such situations and read the manual (as well as  
> looking into to an introductory text on basic numerical analysis,  
> particularly the section on binary representation of numbers with  
> finite precision).
> As for Mathematica: there are only a few bits of information  
> relevant to this. One is that it is Precision rather than Accuracy  
> that gives you relevant information, and your case the Precision is  
> always the same: $MachinePrecision. If the precision is  
> $MachinePrecision then Accuracy is defined purely formally as  
> $MachinePrecision-Log[10,Abs[x]] and carries no useful information  
> beside what is already included in the value Abs[x]. (In other  
> words, reporting the value of your computation and its Accuracy you  
> were simply reporting the same thing twice).
> The second useful piece of information is that:
> Chop[5.1 - 4.5 - 0.6]
> 0
> Andrzej Kozlowski
> On 19 Jul 2009, at 04:15, elliot wrote:
>> The expression 5.1 - 4.5 - 0.6 gives different answers depending on  
>> the version of Mathematica. Version 6 says 0. (which is obviously  
>> what we expect) with an accuracy of about 307. Version 7 says  
>> -3.33067x10^-16 with an accuracy of about 31. Also, in version 7,  
>> if the expression is changed to 5.1 - (4.5 + 0.6) the answer it  
>> gives is 0. with an accuracy of about 307. Both versions were run  
>> on three different computers.
>> Anybody know what changed or what I can do to prevent such poor  
>> solutions (besides forcing a specific order of evaluation)? Thanks.

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