## CS 1711 Lab 4 - October 13 and 15, 1998 Functions and Real Numbers

### Watching Program Variables

It is often useful to set a ``Watch'' on program variables, to see if their values change during a program run. Watching a variable can be a productive way of uncovering bugs in a program.

1. Move the cursor on top of any occurrence of the variable you want to watch.
2. Choose ``Add Watch'' from the Debug menu. You should be given a dialog box with the variable name already in the box.
3. Press Enter to set up a watch on the variable. (you can repeat this for other variables.)
4. Move the Watch window to the right.
5. Now step through the program, keeping an eye on the Watch window. You should see the values of of the watched variable change during a run of the program.

### Euclid's Algorithm

In the file GCD.CPP (in the usual directory) is the program GCD, which will output the greatest common divisor (GCD) of the two positive integers entered by the user. The GCD of two integers is the largest integer that divides both of them.

The program is based on Euclid's algorithm for finding the GCD of two positive integers m and n, i.e. GCD(m,n). This algorithm says that:

[a.] The GCD is n if n is the smaller number and n divides m.
[b.] If m is the smaller number, then the GCD determination should be performed with the arguments transposed (switched).
[c.] If n does not divide m, the answer is obtained by finding the GCD of n and the remainder of m divided by n.

For example, to find GCD(28,48):

 apply to get: b GCD(48,28) c GCD(28, remainder of 48 divided by 28) = GCD(28,20) c GCD(20, remainder of 28 divided by 20) = GCD(20,8) c GCD(8, remainder of 20 divided by 8) = GCD(8,4) a 4

The program has a bug in it. Although you may be able to identify the bug without the use of your debugging tools, you should still follow the steps listed here, so that you can practice using the tools. The true value of the tools is discovered when you have a large program, thousands of lines long, that is hiding a small bug.

1. Obtain the program from the file GCD.CPP.
2. Test the program. What does it output as the GCD of 42 and 18 (enter 42 for the first number and 18 for the second number)? Is that correct?
3. Test the program again. What does it output as the GCD of 18 and 42? Is that correct?
4. Use the debugging tools to set up a Watch of the variables m and n. You want to determine each value that m becomes during an execution of the program. Run the program again with the input values of 48 and 28. Show the sequence of values for m and n:

#### n:

5. Compare your trace of the values of m and n, to the values shown in the table on the previous page. Study the program and determine the error. What line of the program needs to be changed?
6. Fix the bug in the program and save the program. What is the GCD of 16380 and 1911? Before leaving the lab, you will be asked to load this program and point out the bug that was fixed.
7. Use your debugger to answer the following questions about the corrected program: [a.] Write down the sequence of values for variable m if the input is 5913 and 7592.
[b.] How many times is the then part of the if-then-else statement executed if the input is 5913 and 7592?

### Accuracy

The section is really an experiment in which you are given the basic programs you need, and you are asked specific questions about the results. The objective of the experiment is to establish how many decimal places you can view of a real number on your system.

1. Obtain the file TESTREAL.CPP. Look it over. You will see a new loop construct known as a `do-while' loop, it is described in Chapter 7 of the text if you are interested but essentially it works the same as a `while' loop.
2. This program will output the value 2/3 repeatedly, each time using a higher value for the number of decimal places to display. The sleep(1) function causes the program to pause for 1 second before proceeding so you can view the output.
3. Run the program and determine the maximum number of decimal places that your system reports to you.
4. How many decimal places can you use?
5. Turbo C++provides provides a few alternative types for representing real numbers. Now change the type `float' to be one of the following: double, and long double. How many decimal places can you use with each of these types?
6. A theorem states: the limit as x approaches zero of sinx divided by x equals 1.
7. The program in file TESTSINE.CPP investigates the computer's evaluation and reporting of this formula. Given an initial number of radians, the program reports sinx divided by x for that number, then for that number divided by 2, then divided by 4, then 8, and so on. The program reports the value in both a narrow format and a broad format. Obtain the file and study it. Edit the program and set the value of the constant numdecs on line 9 to whatever answer you got for question number 5 above (with the double type). This way, when you run the program, it will report as much information as possible on your system.
[a.] What is the purpose of the if statement on line 55?
[b.] On what line is the radian number cut in half?
[c.] On what line is the formula evaluated?
9. Run the program. Use an initial value for the radians of 99.
[a.] At what count does the computer show in the narrow format that the value of sinx/x is 1?
[b.] At what count does the computer show in the broad format that the value of sinx/x is 1?
[c.] What is the value of the radians at that point, i.e., when the value of sinx/x is 1 in the broad format.
[d.] At what count does the computer show that the value of the Boolean expression sinx/x = 1.0 is true?
[e.] At what count does the computer report the value of the radians as zero?
10. Now open the file SAME.CPP. Notice that a is set to 27 and b is set to 9. The variable x then is set to a2 and y is set to b3; clearly a2 and b3 are the same value. Run the program to see what output you get.
11. Now note that if a were 1/27 and b were 1/9 then the relation that a2 equals b3 is still correct. Change the values of a and b in the program then run it again. What happens?

### Purpose of Above Excercise

When dealing with real numbers (float or double):
1. You can't always trust what you see. Just because it looks like a value of 1 doesn't mean it actually is.
2. Even with all digits of accuracy switched on, 1 doesn't even equal 1! This problem will arise in a later assignment; it is very difficult to check two real numbers for equality.
3. Keep in mind that the answer you see on the screen for sinx divided by x should have never been 1.
Question: Its obvious that if you want more accuracy, you would use double or long double. Why would you ever want to use the type float?