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Find square root of number upto given precision using binary search
  • Difficulty Level : Medium
  • Last Updated : 24 Jun, 2021

Given a positive number n and precision p, find the square root of number upto p decimal places using binary search. 
Note : Prerequisite : Binary search 
Examples: 
 

Input : number = 50, precision = 3
Output : 7.071

Input : number = 10, precision = 4
Output : 3.1622

 

We have discussed how to compute the integral value of square root in Square Root using Binary Search
Approach : 
1) As the square root of number lies in range 0 <= squareRoot <= number, therefore, initialize start and end as : start = 0, end = number.
2) Compare the square of the mid integer with the given number. If it is equal to the number,  the square root is found. Else look for the same in the left or right side depending upon the scenario. 
3) Once we are done with finding an integral part, start computing the fractional part. 
4) Initialize the increment variable by 0.1 and iteratively compute the fractional part up to P places. For each iteration, the increment changes to 1/10th of its previous value. 
5) Finally return the answer computed.
Below is the implementation of above approach : 
 

C++




// C++ implementation to find
// square root of given number
// upto given precision using
// binary search.
#include <bits/stdc++.h>
using namespace std;
 
// Function to find square root
// of given number upto given
// precision
float squareRoot(int number, int precision)
{
    int start = 0, end = number;
    int mid;
 
    // variable to store the answer
    float ans;
 
    // for computing integral part
    // of square root of number
    while (start <= end) {
        mid = (start + end) / 2;
        if (mid * mid == number) {
            ans = mid;
            break;
        }
 
        // incrementing start if integral
        // part lies on right side of the mid
        if (mid * mid < number) {
            start = mid + 1;
            ans = mid;
        }
 
        // decrementing end if integral part
        // lies on the left side of the mid
        else {
            end = mid - 1;
        }
    }
 
    // For computing the fractional part
    // of square root upto given precision
    float increment = 0.1;
    for (int i = 0; i < precision; i++) {
        while (ans * ans <= number) {
            ans += increment;
        }
 
        // loop terminates when ans * ans > number
        ans = ans - increment;
        increment = increment / 10;
    }
    return ans;
}
 
// Driver code
int main()
{
    // Function calling
    cout << squareRoot(50, 3) << endl;
 
    // Function calling
    cout << squareRoot(10, 4) << endl;
 
    return 0;
}

Java




// Java implementation to find
// square root of given number
// upto given precision using
// binary search.
import java.io.*;
 
class GFG {
 
    // Function to find square root
    // of given number upto given
    // precision
    static float squareRoot(int number, int precision)
    {
        int start = 0, end = number;
        int mid;
 
        // variable to store the answer
        double ans = 0.0;
 
        // for computing integral part
        // of square root of number
        while (start <= end) {
            mid = (start + end) / 2;
 
            if (mid * mid == number) {
                ans = mid;
                break;
            }
 
            // incrementing start if integral
            // part lies on right side of the mid
            if (mid * mid < number) {
                start = mid + 1;
                ans = mid;
            }
 
            // decrementing end if integral part
            // lies on the left side of the mid
            else {
                end = mid - 1;
            }
        }
 
        // For computing the fractional part
        // of square root upto given precision
        double increment = 0.1;
        for (int i = 0; i < precision; i++) {
            while (ans * ans <= number) {
                ans += increment;
            }
 
            // loop terminates when ans * ans > number
            ans = ans - increment;
            increment = increment / 10;
        }
        return (float)ans;
    }
 
    // Driver code
    public static void main(String[] args)
    {
        // Function calling
        System.out.println(squareRoot(50, 3));
 
        // Function calling
        System.out.println(squareRoot(10, 4));
    }
}
 
// This code is contributed by vt_m.

Python3




# Python3 implementation to find
# square root of given number
# upto given precision using
# binary search.
 
# Function to find square root of
# given number upto given precision
 
 
def squareRoot(number, precision):
 
    start = 0
    end, ans = number, 1
 
    # For computing integral part
    # of square root of number
    while (start <= end):
        mid = int((start + end) / 2)
 
        if (mid * mid == number):
            ans = mid
            break
 
        # incrementing start if integral
        # part lies on right side of the mid
        if (mid * mid < number):
            start = mid + 1
            ans = mid
 
        # decrementing end if integral part
        # lies on the left side of the mid
        else:
            end = mid - 1
 
    # For computing the fractional part
    # of square root upto given precision
    increment = 0.1
    for i in range(0, precision):
        while (ans * ans <= number):
            ans += increment
 
        # loop terminates when ans * ans > number
        ans = ans - increment
        increment = increment / 10
 
    return ans
 
 
# Driver code
print(round(squareRoot(50, 3), 4))
print(round(squareRoot(10, 4), 4))
 
# This code is contributed by Smitha Dinesh Semwal.

C#




// C# implementation to find
// square root of given number
// upto given precision using
// binary search.
using System;
class GFG {
 
    // Function to find square root
    // of given number upto given
    // precision
    static float squareRoot(int number, int precision)
    {
        int start = 0, end = number;
        int mid;
 
        // variable to store the answer
        double ans = 0.0;
 
        // for computing integral part
        // of square root of number
        while (start <= end) {
            mid = (start + end) / 2;
 
            if (mid * mid == number) {
                ans = mid;
                break;
            }
 
            // incrementing start if integral
            // part lies on right side of the mid
            if (mid * mid < number) {
                start = mid + 1;
                ans = mid;
            }
 
            // decrementing end if integral part
            // lies on the left side of the mid
            else {
                end = mid - 1;
            }
        }
 
        // For computing the fractional part
        // of square root upto given precision
        double increment = 0.1;
        for (int i = 0; i < precision; i++) {
            while (ans * ans <= number) {
                ans += increment;
            }
 
            // loop terminates when ans * ans > number
            ans = ans - increment;
            increment = increment / 10;
        }
        return (float)ans;
    }
 
    // Driver code
    public static void Main()
    {
        // Function calling
        Console.WriteLine(squareRoot(50, 3));
 
        // Function calling
        Console.WriteLine(squareRoot(10, 4));
    }
}
 
// This code is contributed by Sheharaz Sheikh

PHP




<?php
// PHP implementation to find
// square root of given number
// upto given precision using
// binary search.
 
// Function to find square root
// of given number upto given
// precision
function squareRoot($number, $precision)
{
    $start=0;
    $end=$number;
    $mid;
 
    // variable to store
    // the answer
    $ans;
 
    // for computing integral part
    // of square root of number
    while ($start <= $end)
    {
        $mid = ($start + $end) / 2;
        if ($mid * $mid == $number)
        {
            $ans = $mid;
            break;
        }
 
        // incrementing start if integral
        // part lies on right side of the mid
        if ($mid * $mid < $number)
        {
            $start = $mid + 1;
            $ans = $mid;
        }
 
        // decrementing end if integral part
        // lies on the left side of the mid
        else
        {
            $end = $mid - 1;
        }
    }
 
    // For computing the fractional part
    // of square root upto given precision
    $increment = 0.1;
    for ($i = 0; $i < $precision; $i++)
    {
        while ($ans * $ans <= $number)
        {
            $ans += $increment;
        }
 
        // loop terminates when
        // ans * ans > number
        $ans = $ans - $increment;
        $increment = $increment / 10;
    }
    return $ans;
}
 
    // Driver code
    // Function calling
    echo squareRoot(50, 3),"\n";
 
    // Function calling
    echo squareRoot(10, 4),"\n";
 
// This code is contributed by ajit.
?>

Javascript




<script>
 
// JavaScript program implementation to find
// square root of given number
// upto given precision using
// binary search.
 
    // Function to find square root
    // of given number upto given
    // precision
    function squareRoot(number, precision)
    {
        let start = 0, end = number;
        let mid;
   
        // variable to store the answer
        let ans = 0.0;
   
        // for computing integral part
        // of square root of number
        while (start <= end)
        {
            mid = (start + end) / 2;
               
            if (mid * mid == number)
            {
                ans = mid;
                break;
            }
   
            // incrementing start if integral
            // part lies on right side of the mid
            if (mid * mid < number) {
                start = mid + 1;
                ans = mid;
            }
   
            // decrementing end if integral part
            // lies on the left side of the mid
            else {
                end = mid - 1;
            }
        }
   
        // For computing the fractional part
        // of square root upto given precision
        let increment = 0.1;
        for (let i = 0; i < precision; i++) {
            while (ans * ans <= number) {
                ans += increment;
            }
   
            // loop terminates when ans * ans > number
            ans = ans - increment;
            increment = increment / 10;
        }
        return ans;
    }
 
// Driver code
 
         // Function calling
        document.write(squareRoot(50, 3) + "<br/>");
   
        // Function calling
        document.write(squareRoot(10, 4) + "<br/>");
                                
</script>

Output: 

7.071
3.1622

Time Complexity : The time required to compute the integral part is O(log(number)) and constant i.e, = precision for computing the fractional part. Therefore, overall time complexity is O(log(number) + precision) which is approximately equal to O(log(number)).
 

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