Start this problem set by downloading starter code, which contains a helpful components for both exercises. Complete Exercise 15.1.3, but skip count-proper-descendants.

2 Generative Recursion

1. Using the starter code's guess-with-linear-search function as a model, along with the binary search lecture notes, define the function:

   ; guess-with-binary-search : (integer -> answer) integer integer -> guess-result

   ; Guesses a secret using binary search.

   (define (guess-with-binary-search secret low high) ---)

   Note that guess-with-linear-search returns both the number of guesses used and the secret, in a guess-result structure, just as your function should. To count the guesses, you can use the increment-guesses helper function in essentially the same way as in guess-with-linear-search. Also note that the symbols (e.g. 'toosmall) in the starter code don't exactly match the posted lecture notes, and if you build off of the lecture notes code you'll need to adjust it accordingly.
   Lastly, to help you test your function, here is a check-expect that tests whether it indeed gets a secret of 50 correct on the first try. You will want to create at least two additional tests:
   (check-expect (guess-with-binary-search (lambda (guess) (check-guess guess 50)) 1 100) (make-guess-result 50 1))

2. Using the supplied run-game function in the starter code, run your binary search function with make-uniform-secret between 1 and 100, a total of 10000 times. (For help on calling run-game, follow the examples in the starter code). How many guesses are required on average? How does this compare with linear search?
3. Now try 10000 runs in run-game with your binary search and make-special-secret between 1 and 100. How many guesses are required on average? Try making a small alteration to your binary search function to achieve better performance for special secrets, and report on any improvements you observe.