Algorithm to find largest prime number smaller than x [closed]

Answer 1

Some good info here on the function pi(x). Apparently,

pi(x) = the number of primes less than x

and you can approximate pi(x) with

x/(log x - 1)

while

the n-th prime of that list of primes is equal to approximately n(log n)

Answer 2

How fast you need the program runs? And how frequently you calculate this problem?

If you need a fast achievement and don't care about the memory. You can generate an increasing prime table by sieve method then hold it during the program lifetime, and then when you what to find 'the largest prime number smaller than value x', just look up the table and in O(log N) time you can find an exact answer.