What is wrong with mod(t, 0,5)?

In the specification of problem 1, students are asked to print their results every 0.5s. So, the skeleton of someone’s implementation looks like:

t = 0;
dt = 0.02;
tf = 15;
tol = 1e-6;

while t <= tf
    % your code
    if mod(t, 0.5) < tol
        fprintf("%.1f\n", t);
    end
    t = t + dt;
end

The code looks good and is logically correct. As a result, we should expect a sequence of numbers from 0.0 to 15.0 with increment 0.5 to be printed on the screen. However, there is always a gap between dream and reality. It turns out we got:

>>0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0

Why it is not what we thought? Let’s investigate in detail what the hell is happening in our code.
First off, our while loop eventually ternminated which means our t was updated properly and finally went beyond tf. We may prove our assumption by adding another line to print out the value of t after the while loop terminated:

t = 0;
dt = 0.02;
tf = 15;
tol = 1e-6;

while t <= tf
    % your code
    if mod(t, 0.5) < tol
        fprintf("%.1f\n", t);
    end
    t = t + dt;
end
fprintf("final t = %.02f\n", t);

The output is final t = 15.02 which is what we expected. So, the while loop makes sense and then the if statement looks very suspicious. It seems our code doesn’t print out the value of t after it is greater than or equal to 5.0. As 0 to 15 is a little bit too long, let’s make our tf to be 5 in order to better observe what exacly happend there.

t = 0;
dt = 0.02;
tf = 5; % we changed it!!!
tol = 1e-6;

while t <= tf
    % your code
    fprintf("t=%.2f\tmod=%.2f\n", t, mod(t, 0.5));
    t = t + dt;
end
fprintf("final t = %.02f\n", t);

Here is the output:

t=0.00	mod=0.00  % <-- looks good
t=0.02	mod=0.02
t=0.04	mod=0.04
t=0.06	mod=0.06
t=0.08	mod=0.08
t=0.10	mod=0.10
t=0.12	mod=0.12
t=0.14	mod=0.14
t=0.16	mod=0.16
t=0.18	mod=0.18
t=0.20	mod=0.20
t=0.22	mod=0.22
t=0.24	mod=0.24
t=0.26	mod=0.26
t=0.28	mod=0.28
t=0.30	mod=0.30
t=0.32	mod=0.32
t=0.34	mod=0.34
t=0.36	mod=0.36
t=0.38	mod=0.38
t=0.40	mod=0.40
t=0.42	mod=0.42
t=0.44	mod=0.44
t=0.46	mod=0.46
t=0.48	mod=0.48
t=0.50	mod=0.00  % <-- looks good
t=0.52	mod=0.02
t=0.54	mod=0.04
t=0.56	mod=0.06
t=0.58	mod=0.08
t=0.60	mod=0.10
t=0.62	mod=0.12
t=0.64	mod=0.14
t=0.66	mod=0.16
t=0.68	mod=0.18
t=0.70	mod=0.20
t=0.72	mod=0.22
t=0.74	mod=0.24
t=0.76	mod=0.26
t=0.78	mod=0.28
t=0.80	mod=0.30
t=0.82	mod=0.32
t=0.84	mod=0.34
t=0.86	mod=0.36
t=0.88	mod=0.38
t=0.90	mod=0.40
t=0.92	mod=0.42
t=0.94	mod=0.44
t=0.96	mod=0.46
t=0.98	mod=0.48
t=1.00	mod=0.00  % <-- looks good
t=1.02	mod=0.02
t=1.04	mod=0.04
t=1.06	mod=0.06
t=1.08	mod=0.08
t=1.10	mod=0.10
t=1.12	mod=0.12
t=1.14	mod=0.14
t=1.16	mod=0.16
t=1.18	mod=0.18
t=1.20	mod=0.20
t=1.22	mod=0.22
t=1.24	mod=0.24
t=1.26	mod=0.26
t=1.28	mod=0.28
t=1.30	mod=0.30
t=1.32	mod=0.32
t=1.34	mod=0.34
t=1.36	mod=0.36
t=1.38	mod=0.38
t=1.40	mod=0.40
t=1.42	mod=0.42
t=1.44	mod=0.44
t=1.46	mod=0.46
t=1.48	mod=0.48
t=1.50	mod=0.00  % <-- looks good
t=1.52	mod=0.02
t=1.54	mod=0.04
t=1.56	mod=0.06
t=1.58	mod=0.08
t=1.60	mod=0.10
t=1.62	mod=0.12
t=1.64	mod=0.14
t=1.66	mod=0.16
t=1.68	mod=0.18
t=1.70	mod=0.20
t=1.72	mod=0.22
t=1.74	mod=0.24
t=1.76	mod=0.26
t=1.78	mod=0.28
t=1.80	mod=0.30
t=1.82	mod=0.32
t=1.84	mod=0.34
t=1.86	mod=0.36
t=1.88	mod=0.38
t=1.90	mod=0.40
t=1.92	mod=0.42
t=1.94	mod=0.44
t=1.96	mod=0.46
t=1.98	mod=0.48
t=2.00	mod=0.00  % <-- looks good
t=2.02	mod=0.02
t=2.04	mod=0.04
t=2.06	mod=0.06
t=2.08	mod=0.08
t=2.10	mod=0.10
t=2.12	mod=0.12
t=2.14	mod=0.14
t=2.16	mod=0.16
t=2.18	mod=0.18
t=2.20	mod=0.20
t=2.22	mod=0.22
t=2.24	mod=0.24
t=2.26	mod=0.26
t=2.28	mod=0.28
t=2.30	mod=0.30
t=2.32	mod=0.32
t=2.34	mod=0.34
t=2.36	mod=0.36
t=2.38	mod=0.38
t=2.40	mod=0.40
t=2.42	mod=0.42
t=2.44	mod=0.44
t=2.46	mod=0.46
t=2.48	mod=0.48
t=2.50	mod=0.00  % <-- looks good
t=2.52	mod=0.02
t=2.54	mod=0.04
t=2.56	mod=0.06
t=2.58	mod=0.08
t=2.60	mod=0.10
t=2.62	mod=0.12
t=2.64	mod=0.14
t=2.66	mod=0.16
t=2.68	mod=0.18
t=2.70	mod=0.20
t=2.72	mod=0.22
t=2.74	mod=0.24
t=2.76	mod=0.26
t=2.78	mod=0.28
t=2.80	mod=0.30
t=2.82	mod=0.32
t=2.84	mod=0.34
t=2.86	mod=0.36
t=2.88	mod=0.38
t=2.90	mod=0.40
t=2.92	mod=0.42
t=2.94	mod=0.44
t=2.96	mod=0.46
t=2.98	mod=0.48
t=3.00	mod=0.00  % <-- looks good
t=3.02	mod=0.02
t=3.04	mod=0.04
t=3.06	mod=0.06
t=3.08	mod=0.08
t=3.10	mod=0.10
t=3.12	mod=0.12
t=3.14	mod=0.14
t=3.16	mod=0.16
t=3.18	mod=0.18
t=3.20	mod=0.20
t=3.22	mod=0.22
t=3.24	mod=0.24
t=3.26	mod=0.26
t=3.28	mod=0.28
t=3.30	mod=0.30
t=3.32	mod=0.32
t=3.34	mod=0.34
t=3.36	mod=0.36
t=3.38	mod=0.38
t=3.40	mod=0.40
t=3.42	mod=0.42
t=3.44	mod=0.44
t=3.46	mod=0.46
t=3.48	mod=0.48
t=3.50	mod=0.00  % <-- looks good
t=3.52	mod=0.02
t=3.54	mod=0.04
t=3.56	mod=0.06
t=3.58	mod=0.08
t=3.60	mod=0.10
t=3.62	mod=0.12
t=3.64	mod=0.14
t=3.66	mod=0.16
t=3.68	mod=0.18
t=3.70	mod=0.20
t=3.72	mod=0.22
t=3.74	mod=0.24
t=3.76	mod=0.26
t=3.78	mod=0.28
t=3.80	mod=0.30
t=3.82	mod=0.32
t=3.84	mod=0.34
t=3.86	mod=0.36
t=3.88	mod=0.38
t=3.90	mod=0.40
t=3.92	mod=0.42
t=3.94	mod=0.44
t=3.96	mod=0.46
t=3.98	mod=0.48
t=4.00	mod=0.00  % <-- looks good
t=4.02	mod=0.02
t=4.04	mod=0.04
t=4.06	mod=0.06
t=4.08	mod=0.08
t=4.10	mod=0.10
t=4.12	mod=0.12
t=4.14	mod=0.14
t=4.16	mod=0.16
t=4.18	mod=0.18
t=4.20	mod=0.20
t=4.22	mod=0.22
t=4.24	mod=0.24
t=4.26	mod=0.26
t=4.28	mod=0.28
t=4.30	mod=0.30
t=4.32	mod=0.32
t=4.34	mod=0.34
t=4.36	mod=0.36
t=4.38	mod=0.38
t=4.40	mod=0.40
t=4.42	mod=0.42
t=4.44	mod=0.44
t=4.46	mod=0.46
t=4.48	mod=0.48
t=4.50	mod=0.50  % HERE IT IS !!!!!!!!!!!!!!!!!!!!!! I FOUND IT!!!!!!!!!!!!!!!!!!!!!!
t=4.52	mod=0.02
t=4.54	mod=0.04
t=4.56	mod=0.06
t=4.58	mod=0.08
t=4.60	mod=0.10
t=4.62	mod=0.12
t=4.64	mod=0.14
t=4.66	mod=0.16
t=4.68	mod=0.18
t=4.70	mod=0.20
t=4.72	mod=0.22
t=4.74	mod=0.24
t=4.76	mod=0.26
t=4.78	mod=0.28
t=4.80	mod=0.30
t=4.82	mod=0.32
t=4.84	mod=0.34
t=4.86	mod=0.36
t=4.88	mod=0.38
t=4.90	mod=0.40
t=4.92	mod=0.42
t=4.94	mod=0.44
t=4.96	mod=0.46
t=4.98	mod=0.48
t=5.00	mod=0.50
final t = 5.02

The result is self-explantory. It seems after t is equal to 4.5, the result of operation mod(t, 0.5) will be 0.5 instead of 0.0! But why is that?! Why matlab cannot even finish this simple calculation?! The answer is the precision of floatiing-point numbers. You may either google the IEEE standard for floating-point numbers or go back to watch the first half hour of discussion on week 4 to know more about how floating-point numbers are implementated in matlab.
The problem happened in our code was that even though we saw the value of t is 4.5, the actual number stored in the machine was 4.49999… As a result, when matlab takes the remainder of 4.4999999… / 0.5, it realizes that 0.4999999… should be the remainder and that is the reason why our mod is 0.5 (actually it is 0.4999999).
To prove our assumption:

t = 0;
dt = 0.02;
tf = 5; % we changed it!!!
tol = 1e-6;

for i = 1: 225
    t = t + dt;
end
fprintf("t=%.2f\t==4.5?%d\t<4.5?%d\n", t, t==4.5, t<4.5);

>>
t=4.50  ==4.5?0	<4.5?1

Here we go! t looks 4.5, it is not equal to 4.5 and it is actually less than 4.5! Done!
Then how do we fix this problem?
I will recommand you to use iteration counts as the condition instead of time because integer is more reliable than floating-point numbers. So, the final code should look like:

t = 0;
dt = 0.02;
tf = 15;
tol = 1e-6;
iterCount = 0;

while t <= tf
    % your code
    if mod(iterCount, 25) < tol
        fprintf("%.1f\n", t);
    end
    t = t + dt;
    iterCount = iterCount + 1;
end

>>
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
10.5
11.0
11.5
12.0
12.5
13.0
13.5
14.0
14.5
15.0

Yeah! Cheers!