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Yesterday night, I found that $e^{e^{2}} \sim 1000\phi$, where $\phi$ is the golden ratio.

I believe that it is correct to four decimal places.

Would it be considered a relatively good approximation?

P.S. I know that an approximation is not needed, as it has a closed form, but, yeah...

$$\begin{align} e^{e^{2}} &\approx 1{,}618.17799191266 \\ 1000\phi &\approx 1{,}618.033988749895 \end{align}$$

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    $\begingroup$ There is really no framework in which to answer: "is this a very good approximation?" $\endgroup$ Jul 30, 2015 at 11:12
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    $\begingroup$ Usually, an approximation is supposed to be something that is easier to calculate. Are you going to use the golden ratio to approximate $e^{e^2}$ or opposite? Anyway, 1618 is a much easier approximation for both numbers. $\endgroup$
    – A.Γ.
    Jul 30, 2015 at 11:20
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    $\begingroup$ I wonder what you did yesterday night that led you to this... $\endgroup$
    – lhf
    Jul 30, 2015 at 11:33
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    $\begingroup$ funcfact: this the approximation you get with the RIES tool (lowest complexity) mrob.com/pub/ries/ries.php?target=1618.0339887&rst= $\endgroup$
    – Bort
    Jul 30, 2015 at 11:35
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    $\begingroup$ @Taylor Similarly, yesterday I found that $(\pi+1) / e\approx \pi ^{1/e}$ :) The relative error is about $4.2\cdot 10^{-5}$ $\endgroup$
    – grizzly
    Jul 30, 2015 at 11:47

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The relative error is about $8.9 \cdot 10^{-5}$.

It is surprising that the relative error is somewhat small because $e$ and $\phi$ are apparently unrelated (*). It is probably a mathematical coincidence.

Whether this error implies a relatively good approximation depends on why you want this approximation or where you're going to use it.

(*) More precisely, $e$ and $\phi$ are algebraically independent because $e$ is transcendental and $\phi$ is algebraic.

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One can consider that an approximation formula is good if is achieves some information compression, i.e. it requires less bits of information than the decimal expansion of comparable accuracy.

Take the case of the well-known fractional approximations of $\pi$:

$$\frac{22}7=\color{green}{3.14}2857\cdots$$ $$\frac{335}{113}=\color{green}{3.141592}92\cdots$$

In my opinion they are virtually worthless as they don't require less digits, not counting the information contained in the shape of the formula, a fraction.

The case of $\dfrac{e^{e^2}}{1000}$ isn't so easy to evaluate, but it doesn't clearly improve on the four digits of accuracy it gives.

$\dfrac{\sqrt5+1}2$ isn't much more costly, is easier to evaluate, and isn't a bad approximation :)


A four digits approximation of any constant isn't impressive. If you compute a million random simple expressions, you will quite probably end up finding several ones.

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  • $\begingroup$ On $\pi$ - I find it as beautiful as the Euler relation that $$\phi = \frac{1}{2}\csc \left( \frac{\pi}{10}\right)$$ $\endgroup$
    – Autolatry
    Jul 30, 2015 at 13:59
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    $\begingroup$ @Autolatry: indeed, the pentagon discovered $\phi$ long before us. $\endgroup$
    – user65203
    Jul 30, 2015 at 14:13
  • $\begingroup$ Ha! Pentagon indeed. $\endgroup$
    – Autolatry
    Jul 30, 2015 at 14:14
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I remember when I've found this approximation some years ago. Some other approximations of this type:

\begin{align} e^{e^{2}} &\approx 1000\phi \\ e^{e^{-2}} &\approx \ln\pi \approx \pi-2 \\ e^{e^{e^{-2}}} &\approx \pi \end{align} But these are matches just for few digits. You could find more accurate $\phi$ approximations at Golden Ratio Approximations MathWorld page. In this paper you could find relations between $e$ and $\phi$.

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