Looking for any idea for reduction from $SAT \leq 3-SAT$ where $SAT$ is known to have $d$ variables at most in each clause. I am looking for a reduction in which the resulting formula will not depend on the value of $d$. The classical reduction adds $O(d)$ new variables per clause, so this is not good to my case.
Do such reductions even exist?
Edit: Even if it's impossible to find such reductions, maybe there are other interesting reductions which are not the classical one, they can also be interesting?