What is the most "efficient" closed-form expression for approximating a transcendental number?

Question

$\frac{9}{5}+\sqrt{\frac{9}{5}}=3.1416...$ is a closed-form approximation of $\pi$ with a precision of 3 digits. After reading the wiki page on approximations of pi, I read about many clever methods for finding closed-form approximations of $\pi$. I was curious how much "information" you can pack in a closed-form expression to approximate some transcendental number. I wouldn't know how to measure the "efficiency" of a closed-form expression. For instance, the number of integers in the expression divided by the the number of digits of precision. That way, you get information divided by precision. Is there a limit to how much you can pack in an expression?

Answer 1

As pointed out in the comments here, the expression:

$$ \pi = 20 \arctan \frac{1}{7} + 8 \arctan \frac{3}{79} $$

has a "precision of over $12$ digits" only in the sense that it is an exact equation. The transcendental number $\pi$ is expressed as a sum of two more transcendental numbers.

That is an expression due to Euler among a class of exact representations commonly called Machin-like formulas after John Machin, who gave (1706):

$$ \pi = 16 \arctan \frac{1}{5} - 4 \arctan \frac{1}{239} $$

and used it to compute $\pi$ to a thousand decimal places. The "simplest" of these exact expressions was also found by Euler:

$$ \pi = 4 \arctan \frac{1}{2} + 4 \arctan \frac{1}{3} $$

but it is less practical for computation because of the relatively slow convergence at these arguments of the Taylor series of $\arctan x$.

But since these are not "approximations" of a transcendental number by algebraic numbers, the remainder of this post will point out references pertinent to that general subject (and not solely to approximating $\pi$).

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Most authors begin their treatment of this subject by revisiting the approximation of real numbers by rational numbers, This classic "Diophantine" topic is fully developed by Knuth in AOCP Vol. II, Semi-numerical Algorithms, and has been previously been discussed here and here at Math.SE in terms of continued fraction convergents and semi-convergents.

The basic result, going back to Dirichlet, is that for any irrational real number $r$, there exist an infinite number of rational approximations $p/q$ such that:

$$ \left| r - \frac{p}{q} \right| \lt \frac{1}{q^2} $$

where $p,q$ are coprime and $q\gt 0$, and one way to obtain such approximations is from the convergents of the simple continued fraction of $r$. By allowing the denominator $q$ to grow large, we obtain rational approximations as close to $r$ as we wish. But if $q$ is bounded, then there are only finitely many such approximations, and the best precision is therefore limited.

To generalize this to approximation of transcendental real numbers by algebraic numbers, it was conjectured by Eduard Wirsing (1960) that for any real $\xi$ not algebraic of degree $d$ or less over the integers, there are an infinite number of algebraic numbers $\alpha$ of degree $d$ or less over the integers such that:

$$ \left| \xi - \alpha \right| < c H(\alpha)^{-(d+1)} $$

where $c$ is a positive constant (depending on $\xi$) and "height" $H(\alpha)$ is the maximum absolute value of coefficients in the minimal integer polynomial of $\alpha$.

This conjecture remains an open problem, but was proven for almost all real numbers by V. Sprindžuk (1965). For $d=2$ it was proven without exceptions by Harold Davenport and Wolfgang M. Schmidt (1967).

There is a book-length treatment by Yann Bugeaud (Cambridge Univ. Press, 2004) titled Approximation by Algebraic Numbers. See also the survey by Vladislav Frank (2007), "Approximation to real numbers by algebraic numbers of bounded degree" at the level of a Master's Thesis.