The eight divisors of $24$ are $1, 2, 3, 4, 6, 8, 12$ and $24$. The ten numbers not exceeding $100$ having exactly eight divisors are $24, 30, 40, 42, 54, 56, 66, 70, 78$ and $88$. Let $f(n)$ be the count of numbers not exceeding $n$ with exactly eight divisors.
You are given $f(100) = 10$, $f(1000) = 180$ and $f(10^6) = 224427$.
Find $f(10^{12})$.