I have always had this question , and yet I do not have a good solid answer.
Is it the number of accurate decimal places?
Here : https://mathworld.wolfram.com/AperysConstantApproximations.html
$$\zeta(3) \approx \gamma^{\frac{-1}{3}}+\pi^{-4} \left(1+2\gamma-\frac{2}{130+\pi^{2}}\right)^{-3} \tag{1}$$
is given but is only accurate to 4 decimal places.
while I may provide for example :
$$\Gamma \left(\frac{-1}{4}\right) \approx -2\,(2\pi)^{\frac{3}{4}}\sqrt{\frac{4+\ln(2)}{2\sqrt{2}\,\pi+\frac{4(\pi-1)(6\pi-5)}{35\pi^2+8\pi-22}+\sqrt{2}\,\pi\ln(2)}} \tag{2} $$
which is accurate up to 10 decimal places .. and yet I consider the approximation given in $(1)$ to be more beautiful due to its simplicity.
Q = So what is the criteria that must be taken into account when deciding if an approximation to a constant is good/bad or worth mentioning or not?
Best Answer
In both cases, it's preferable a formula which shows - or allows to deduce - a reasonable bound for the accuracy that can be attained, which is lacking in both formulas you presented.
Keeping firm the above, even more preferable is a formula that allows you to extend the computation till the accuracy you need, e.g. a series, a continued fraction, a recursion ...