Re: Unexpected warning with ArcTan

• To: mathgroup at smc.vnet.net
• Subject: [mg71809] Re: Unexpected warning with ArcTan
• From: "Andrew Moylan" <andrew.j.moylan at gmail.com>
• Date: Thu, 30 Nov 2006 06:04:59 -0500 (EST)
• References: <ekh7ab\$sar\$1@smc.vnet.net><ekjep8\$cb1\$1@smc.vnet.net>

```Yup, the warning is only issued when the exact expression contains E to
a sufficiently high power. It all points to machine precision
arithmetic being undertaken somewhere.

On Nov 29, 7:01 pm, "dimitris" <dimmec... at yahoo.com> wrote:
> ArcTan[Simplify[1/(1/2 + (1/2)*(-1 + 1/E^63))]]
> ArcTan[2*E^10]
>
> Strangely,
>
> In[42]:=
> ArcTan[1/(1/2 + 1/2*(-1 + 1/E^10))]
> ArcTan[1/(1/2 + 1/2*(-1 + 1/E^20))]
> ArcTan[1/(1/2 + 1/2*(-1 + 1/E^30))]
> ArcTan[1/(1/2 + 1/2*(-1 + 1/E^40))]
>
> Out[42]=
> ArcTan[1/(1/2 + 1/2*(-1 + 1/E^10))]
>
> Out[43]=
> ArcTan[1/(1/2 + 1/2*(-1 + 1/E^20))]
>
> Out[44]=
> ArcTan[1/(1/2 + 1/2*(-1 + 1/E^30))]
>
> Out[45]=
> Power::"infy" : "Infinite expression \!\(1\/0.`\) encountered."
> ArcTan[1/(1/2 + 1/2*(-1 + 1/E^40))]
>
> Regards
> Dimitris
>
> Ï/Ç Andrew Moylan Ýãñáøå:
>
> > Hi all.
>
> > 1/(1/2 + (1/2)*(-1 + 1/E^63)) is an exact expression that's equal to
> > 2*E^63:
>
> >   1/(1/2 + (1/2)*(-1 + 1/E^63)) // Simplify
> > gives
> >   2*E^63
>
> > Now, evaluating
> >   ArcTan[2*E^63]
> > gives
> >   ArcTan[2*E^63]
>
> > Evaluating
> >   ArcTan[1/(1/2 + (1/2)*(-1 + 1/E^63))]
> > similarly gives
> >   ArcTan[1/(1/2 + (1/2)*(-1 + 1/E^63))]
> > but first it issues the following warnings:
>
> > Power::"infy" : "Infinite expression \!\(1\/0.`\) encountered."
> > Power::"infy" : "Infinite expression \!\(1\/0.`\^1.`\) encountered."
>
> > The warnings suggest finite- or machine-precision arithmetic is being
> > used somewhere (1/0.); but this is an exact expression.
>
> > If ArcTan is replaced by e.g. ArcSin the same warnings are generated.
> > If it is replaced by e.g. Sin, no warnings are generated. Either way,
> > Mathematica returns the exact expression unchanged.
>
> > Can anyone explain the origin of these warnings?
>
> > Cheers,
> > Andrew

```

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