Four fours is a famous puzzle (made trivial with logarithms). For this puzzle, we take inspiration from Glen O's challenge 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.
• The root function as a binary operator $\sqrt[b]{a}$.
• 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 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 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:$$ \frac{-\log_{10} |\pi - \sqrt{5}|}4 \approx 0.01438... $$