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Unexpected warning with ArcTan

• To: mathgroup at smc.vnet.net
• Subject: [mg71763] Unexpected warning with ArcTan
• From: "Andrew Moylan" <andrew.j.moylan at gmail.com>
• Date: Tue, 28 Nov 2006 06:04:17 -0500 (EST)

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