Is my solution to a mathematics puzzle I created the most efficient solution there is to it?

Question

Here's a puzzle of mine that I created around 2 hours ago:

Let $f(x):=x^2$ and $g(x):=x-4$. Starting with x=0, what is the least amount of times you need to apply the functions $f$ and $g$ so that at some point, the output is $1024$?

Restrictions

Negative numbers less than $-8$ are not allowed.

I wanted to try my hand at getting the most efficient solution to this, so here is my most efficient attempt so far (my second attempt, first attempt used up a total of 60 functions):

Total functions: 14$$f(g(g(g(g(g(g(g(g(f(g(g(f(g(0))))))))))))))$$

1. Apply $g(x)$ and then $f(x)$ to get $16$ (total of $2$ functions right now)
2. Apply $g(x)$ twice to get $8$ and then apply $f(x)$ to get $64$ (total of $5$ functions right now)
3. Apply $g(x)$ a total of $8$ times to get $32$ and then apply $f(x)$ to get $1024$ (total of $14$ functions)

However, my question is: Is this truly the most efficient solution to my puzzle, or can YOU find a solution that is more efficient?

_{unsure if I need more tags or not or if tags need to be changed/removed}

_{Of course I know the most efficient solution, I'm just seeing if you can find it :)}

Answer 1

Let's see what an optimal path would look like. For starters,

Our number will always be a multiple of $4$. If the result of the last $f$ is greater than $1024$, it's at least $36^2=1296$, and we'll save many, many applications by reducing the argument of the last $f$ to $32$ instead of reducing its output to $1024$. (It can't be $-32$, because that's forbidden.)
The result of the f prior to that has to be the square of another multiple of $4$. $16$ is too small, so the same argument tells us that the second-last $f$ must output $64$. Your path used $f(8)$, but $f(-8)$ is allowed, and this is where the reduction lies.
$f(g^8) fgg(0) = 1024$ in $12$ applications.