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Here's my first attempt at a Riley! Good luck!

The thought of my prefix brings many a smile;

Sit down, relax, or sleep for a while.

My infix is, mathematically;

One point six times five point eight, then less three.

My suffix is a speech, mad and long;

Which definitely is not a song.

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  • $\begingroup$ Nice try! Very interesting! $\endgroup$ – Omega Krypton Oct 31 '18 at 11:34
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    $\begingroup$ -1 because I thought it would be about Pokemon. Was hoping to see something like myWho's this Pokemon?. Kidding! :) $\endgroup$ – gsamaras Oct 31 '18 at 13:27
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Restaurant

Prefix

Rest is a synonym for relax, etc. and people look forward to it wistfully, with smiles on their faces.

Infix

Tau (twice pi) is approximately $6.28$, which is $1.6 * 5.8 - 3$. Clued in by @Omega Krypton.

Suffix

Rant, which is an unhappy speech

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  • $\begingroup$ FGITW, by two seconds! $\endgroup$ – Chowzen Oct 31 '18 at 11:49
  • $\begingroup$ +1 Obviously, I did not see your answer when I posted. $\endgroup$ – hexomino Oct 31 '18 at 11:51
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    $\begingroup$ @hexomino If you had that would have been incredible speed on the typing. $\endgroup$ – WAF Oct 31 '18 at 11:53
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I think the answer is

Restaurant

The thought of my prefix brings many a smile;
Sit down, relax, or sleep for a while.

Rest - which means to relax and is pleasing for many people.

My infix is, mathematically;
One point six times five point eight, then less three.

Tau, as pointed out by WAF's comment on Omega Krypton's answer $$ (1.6 \times 5.8) - 3 = 6.28 \approx 2\pi \equiv \tau $$

My suffix is a speech, mad and long;
Which definitely is not a song.

Rant - A lengthy, angry impassioned speech.

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Partial Answer:

Infix 1.6*5.8-3=6.28, which is 2*pi, so infix="pipi"

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