[Four fours][1] is a famous puzzle (made trivial with logarithms). For this puzzle, we take inspiration from [Glen O's challenge][2] from a few years back. The rules here are slightly different, but the goal is the same. Your goal is to approximate $\pi$ to the highest accuracy (per operation) as possible, using only four fours and the following operations:

 - The classic arithmetic operations: $+,-,\times,\div$ 
 - Exponentiation and log-base as binary operations: $\log_a b$ and $a^b$. You may only take logs of positive real numbers.
 - Unary operations: $(\cdot)!$ for integer arguments only (so you can't use $\frac12!$ to get $\sqrt{\pi}/2$), unary negation $-$, the square root $\sqrt{\cdot}$, and floor/ceiling for rounding down/up: $\lfloor\cdot\rfloor$ and $\lceil \cdot\rceil$.

Any other operations are not allowed, including [double factorials][3] and decimal points. In addition, you can do the following with no penalty:

 - Parentheses (for grouping purposes only, no binomial coefficients etc.)
 - Concatenation of 4's. That is, you can use 44 as a single number without costing an operation. You cannot concatenate things that are not fours, e.g. you can't concatenate $\sqrt{4}$ and $4!$ to get 224.
 - You do not have to use all four fours (e.g. [msh210's answer][4] of $\lfloor 4\rfloor$ is allowed).

Your score is equal to the number of digits of accuracy *per operation* used. That is, if you got the approximation $A$ by using $n$ operations, your score is $$
\frac{-\log_{10}|\pi - A|}{n}
$$
To avoid division by 0, you must use at least one operation. 

---

As an example, if you submit $\sqrt{(44 - 4!)/4} = \sqrt{5}\approx 2.24$, that has 4 operations, so your score would be:$$
\frac{-\log_{10} |\pi - \sqrt{5}|}4 \approx 0.01438...
$$


  [1]: https://en.wikipedia.org/wiki/Four_fours
  [2]: https://puzzling.stackexchange.com/questions/12325/making-%CF%80-from-1-2-3-4-5-6-7-8-9/91156#91156
  [3]: https://en.wikipedia.org/wiki/Double_factorial
  [4]: https://puzzling.stackexchange.com/questions/97895/four-fours-to-get-pi/97896#97896