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Calculating Factorials using Stirling Approximation

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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)



Last Updated : 15 Sep, 2023
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