How Can We Efficiently Extract the T2 Term from T1 = T2 * N! Using Prime Exponent Analysis?

In this article, the author discusses a fast and accurate method to calculate the factorial of a large number using fixed-point bignumber library. A recurring question about the implementation was the extraction of the T2 term from the product T1 = T2 * N!, where T1 and N! are already known. To find the T2 term, the author conducted an analysis on prime exponents and proposed a formula to calculate it:

T2(4N) = multiplication(i=all primes<=4N) of [i^sum(j=1,2,3,4,5,...4N/(i^j)) of [(4N/(i^j))%2]]

Subterms of T2 have exponent e for prime i inside the T2(N) term that can be computed like this:

for (e=0,j=N4;j;e+=j&1,j/=p);

where e is exponent, p is prime and N4 is 4*N

The computed T2 term is then used to optimize the computation of factorials, and the resulting algorithm exhibits a computational complexity near ~ O(log(n)).

Rough time measurements are provided for the first 128 factorials. The author acknowledges that this implementation cannot be simplified further and is already highly optimized.

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