Calculating Factorials using Stirling Approximation

We are aware of calculating factorials using loops or recursion, but if we are asked to calculate factorial without using any loop or recursion. Yes, this is possible through a well-known approximation algorithm known as Stirling approximation. 

Examples:

Input : n = 6
Output : 720

Input : n = 2
Output : 2

Stirling approximation: is an approximation for calculating factorials. It is also useful for approximating the log of a factorial. 
n! ~ sqrt(2*pi*n) * pow((n/e), n) 
Note: This formula will not give the exact value of the factorial because it is just the approximation of the factorial.

C++

// CPP program for calculating factorial
// of a number using Stirling
// Approximation
#include <bits/stdc++.h>
using namespace std;

// function for calculating factorial
long int stirlingFactorial(int n)
{
    if (n == 1)
        return 1;
    long int z;
    float e = 2.71; // value of natural e

    // evaluating factorial using
    // stirling approximation
    z = sqrt(2 * 3.14 * n) * pow((n / e), n);
    return z;
}

// driver program
int main()
{
    cout << stirlingFactorial(1) << endl;
    cout << stirlingFactorial(2) << endl;
    cout << stirlingFactorial(3) << endl;
    cout << stirlingFactorial(4) << endl;
    cout << stirlingFactorial(5) << endl;
    cout << stirlingFactorial(6) << endl;
    cout << stirlingFactorial(7) << endl;
    return 0;
}

Java

// Java program for calculating
// factorial of a number using
// Stirling Approximation
class GFG
{

// function for
// calculating factorial
public static int stirlingFactorial(double n)
{
    if (n == 1)
        return 1;
    double z;
    double e = 2.71; // value of natural e

    // evaluating factorial using
    // stirling approximation
    z = Math.sqrt(2 * 3.14 * n) *
        Math.pow((n / e), n);
    return (int)(z);
}

// Driver Code
public static void main(String[] args)
{
    System.out.println(stirlingFactorial(1));
    System.out.println(stirlingFactorial(2));
    System.out.println(stirlingFactorial(3));
    System.out.println(stirlingFactorial(4));
    System.out.println(stirlingFactorial(5));
    System.out.println(stirlingFactorial(6));
    System.out.println(stirlingFactorial(7));
}
}

// This code is contributed by mits.

Python3

# Python3 program for calculating
# factorial of a number using
# Stirling Approximation
import math

# Function for calculating factorial
def stirlingFactorial(n):
    if (n == 1):
        return 1

    # value of natural e
    e = 2.71

    # evaluating factorial using
    # stirling approximation
    z = (math.sqrt(2 * 3.14 * n) * math.pow((n / e), n))
    return math.floor(z)

# Driver Code
print(stirlingFactorial(1))
print(stirlingFactorial(2))
print(stirlingFactorial(3))
print(stirlingFactorial(4))
print(stirlingFactorial(5))
print(stirlingFactorial(6))
print(stirlingFactorial(7))

# This code is contributed by mits

C#

// C# program for calculating
// factorial of a number using
// Stirling Approximation

class GFG
{

// function for
// calculating factorial
public static int stirlingFactorial(double n)
{
    if (n == 1)
        return 1;
    double z;
    double e = 2.71; // value of natural e

    // evaluating factorial using
    // stirling approximation
    z = System.Math.Sqrt(2 * 3.14 * n) *
        System.Math.Pow((n / e), n);
    return (int)(z);
}

// Driver Code
public static void Main()
{
    System.Console.WriteLine(stirlingFactorial(1));
    System.Console.WriteLine(stirlingFactorial(2));
    System.Console.WriteLine(stirlingFactorial(3));
    System.Console.WriteLine(stirlingFactorial(4));
    System.Console.WriteLine(stirlingFactorial(5));
    System.Console.WriteLine(stirlingFactorial(6));
    System.Console.WriteLine(stirlingFactorial(7));
}
}

// This code is contributed by mits.

PHP

<?php
// PHP program for calculating factorial
// of a number using Stirling
// Approximation

// Function for calculating factorial
function stirlingFactorial($n)
{
    if ($n == 1)
        return 1;
    $z;

    // value of natural e
    $e = 2.71;

    // evaluating factorial using
    // stirling approximation
    $z = sqrt(2 * 3.14 * $n) *
         pow(($n / $e), $n);
    return floor($z);
}

    // Driver Code
    echo stirlingFactorial(1),"\n";
    echo stirlingFactorial(2) ,"\n";
    echo stirlingFactorial(3) ,"\n";
    echo stirlingFactorial(4), "\n" ;
    echo stirlingFactorial(5) ,"\n";
    echo stirlingFactorial(6) ," \n";
    echo stirlingFactorial(7) ," \n";

// This code is contributed by anuj_67.
?>

Javascript

<script>
// Javascript program for calculating factorial
// of a number using Stirling
// Approximation

// Function for calculating factorial
function stirlingFactorial(n)
{
    if (n == 1)
        return 1;
    let z;

    // value of natural e
    let e = 2.71;

    // evaluating factorial using
    // stirling approximation
    z = Math.sqrt(2 * 3.14 * n) *
         Math.pow((n / e), n);
    return Math.floor(z);
}

    // Driver Code
    document.write( stirlingFactorial(1) + "<br>");
    document.write( stirlingFactorial(2) + "<br>");
    document.write( stirlingFactorial(3) + "<br>");
    document.write( stirlingFactorial(4) + "<br>");
    document.write( stirlingFactorial(5) + "<br>");
    document.write( stirlingFactorial(6) + "<br>");
    document.write( stirlingFactorial(7) + "<br>");

// This code is contributed by _saurabh_jaiswal.
</script>

Time complexity: O(logn)
Auxiliary space: O(1)

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