How Can We Optimize Factorial Calculations Using Fast Mathematical Operations and Efficient Algorithms?

The provided text thoroughly explains how to optimize factorial calculations by utilizing fast mathematical operations such as additions, subtractions, and bit shifts. It also delves into efficient algorithms like Karatsuba multiplication and discusses the complexities involved in optimizing such calculations. While the text provides a detailed analysis of the code in the question, it does not contain code that implements the T2 term. To provide the missing code specifically, here's a Python implementation based on the provided analysis:

def T2(x):
  if x == 0: return 1
  t = [1] * (4 * x + 1)
  for p in primes:
    if p > 4 * x: break
    while x % p == 0:
      x /= p
      for j in range(p-1, 4 * x, p):
        t[j] *= p
  return prod(t)

def fact(x):
  return prod([(2 * y)! for y in range(x // 2 + 1)] + [T2(x)])

This function follows the strategy outlined in the text:

1. Initializes a list t with size 4 * x 1 and all elements set to 1.
2. Iterates over the primes less than or equal to 4 * x.
3. For each prime p, repeatedly divides x by p as long as it is divisible.
4. For each multiple of p in the range [p-1, 4 * x], multiplies the corresponding element in the list t by p.
5. The product of all the elements in t is stored as T2(x).
6. Finally, fact(x) is calculated as the product of the factorials of all integers from 1 to x // 2, followed by T2(x).

Note that the prod function used in this code is not defined, but it can be any function that computes the product of a list of numbers efficiently.

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