Given a non-negative number x, we want to approximate its square root in C++ without using the sqrt function from the ma

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Given a non-negative number x, we want to approximate its square
root in C++ without using the sqrt function from the math library
(we will use it only to compare its answer with our approximation).
For simplicity we will assume that x ≤ 1 and thus x ≤ √ x ≤ 1.
How to approximate the square root?
Approach 1: Exhaustive search
We know that the square root is in the interval [x, 1]
Divide the interval [x,1] into say 1000000 equally spaced
points
Sequentially loop over the points to find the point p such that
p*p is closest to x
Very slow!
Approach 2:
The bisection method
Compute the midpoint of the interval [x, 1]: mid = x+1 2
Compare mid ∗ mid with x: if mid ∗ mid < x, narrow down the
search to the interval [mid, 1], else, narrow it down to [x,
mid]
Repeat the above process until abs(mid∗mid−x) ≤ epsilon, where
epsilon is the approximation error margin that we will allow (e.g.,
you can take epsilon = 0.00001)
Write a C++ program that reads the value of x from the user and
prints an error message if the input is not in the valid range
(should be between 0 and 1). If the input is valid, compute and
print the approximate square root of x using the bisection method
described above. To verify the accuracy of your approximation, also
print the square root using the sqrt function from the math
library.
Hint: define two variables: low and high to keep track of the
boundaries of the search interval. Initially, low = x and high = 1.
As the algorithm proceeds, update these limits as needed. Submit
your solution in the source file bisection.cpp.