Generate a Python program that has the following intention:

The intention of the program is to implement the binary search algorithm. The binary search algorithm is a search algorithm that works by dividing the search space in half at each step and recursively searching the two halves until the target value is found or until one of the halves is determined to be empty.

In the program, the p1 function is the outer function that calls the p2 function, which is the recursive function that implements the binary search algorithm. The p2 function takes two arguments: start and end, which represent the current left and right boundaries of the search space, respectively.

The p2 function first calculates the midpoint of the search space using the mid variable and then checks if the target value x is equal to the value at the midpoint. If the target value is found, the function returns the index of the midpoint. If the target value is not found, the function checks if the left half of the search space is empty by comparing x with the value at the midpoint. If the left half is empty, the function returns -1 to indicate that the target value is not present in the search space. If the left half is not empty, the function recursively calls p2 on the left half of the search space.

If the left half is not empty, the function checks if the right half is empty by comparing x with the value at the midpoint. If the right half is empty, the function recursively calls p2 on the right half of the search space. If the right half is not empty, the function recursively calls p2 on the right half of the search space.

The p1 function calls the p2 function with arr as the input array and x as the target value. The p2 function then recursively searches the search space until the target value is found or until one of the halves is determined to be empty. The p1 function then returns the index of the target value if it is found, or -1 if it is not found.

Overall, the program implements a binary search algorithm that can be used to search for a target value in an input array in O(log n) time, where n is the length of the input array.