#### 8.3Examples, Testing, and Program Checking

Back in Documenting Functions with Examples, we began to develop your habit of writing concrete examples of functions. In Task Plans, we showed you how to develop examples of intermediate values to help you plan the code for you to write. As these examples show, there are many ways to write down examples. We could write them on a board, on paper, or even as comments in a computer document. These are all reasonable and indeed, often, the best way to begin working on a problem. However, if we can write our examples in a precise form that a computer can understand, we achieve two things:
• When we’re done writing our purported solution, we can have the computer check whether we got it right.

• In the process of writing down our expectation, we often find it hard to express with the precision that a computer expects. Sometimes this is because we’re still formulating the details and haven’t yet pinned them down, but at other times it’s because we don’t yet understand the problem. In such situations, the force of precision actually does us good, because it helps us understand the weakness of our understanding.

##### 8.3.1From Examples to Tests

Until now, we have written examples in where: blocks for two purposes: to help us figure out what a function needs to do, and to provide guidance to someone reading our code as to what behavior they can expect when using our function. For the smaller programs that we have written until now, where-based examples have been sufficient. As our programs get more complicated, however, a small set of related illustrative examples won’t suffice. We need to think about being much more thorough in the sets of inputs that we consider.

Consider for example a function count-uses that counts how many times a specific string appears in a list (this could be used to tally votes, to compute the frequency of using a discount code, and so on). What input scenarios might we need to check before using our function to run an actual election or a business?

• The result for a string that is in the list once

• The result for a string that is in the list multiple times

• The result for a string that is at the end of a longer list (to make sure we are checking all of the elements)

• The result for a string that isn’t in the list

• The result for a string that is in the list but with different capitalization

• The result for a string that is a typo-away from a word in the list

Notice that here we are considering many more situations, including fairly nuanced ones that affect how robust our code would be under realistic situations. Once we start considering situations like these, we are shifting from examples to illustrate our code to tests to thoroughly test our code.

In Pyret, we use where blocks inside function definitions for examples. We use a check block outside the function definition for tests. For example:

fun count-uses(of-string :: String, in-list :: List<String>) -> Number:
...
where:
count-uses("pepper", [list:]) is 0
count-uses("pepper", [list: "onion"]) is 0
count-uses("pepper", [list: "pepper", "onion"]) is 1
count-uses("pepper", [list: "pepper", "pepper", "onion"]) is 2
end

check:
count-uses("ppper", [list: "pepper"]) is 0
count-uses("ONION", [list: "pepper", "onion"]) is 1
count-uses("tomato",
[list: "pepper", "onion", "onion", "pepper", "tomato",
"tomato", "onion", "tomato"])
is 3
...
end

As a guiding rule, we put illustrative cases that would help someone else reading our code into the where block, while we put the nitty-gritty checks that our code handles the wider range of usage scenarios (including error cases) into the check. Sometimes, the line between these two isn’t clear: for example, one could easily argue that the second test (the function handles different capitalization) belongs in where instead. The third test about using a really long list would remain in check, however, as longer inputs are generally not instructive to a reader of your code.

Putting tests in a block that lives outside the function has another advantage at the level of professional programming: it allows your tests to live in a separate file from your code. This has two key benefits. First, it makes it easier for someone to read the essential parts of your code (if they are building on your work). Second, it makes it easier to control when tests are run. When your check blocks are in the same file as your code, all the tests will be checked when you run your code. When they are in a different file, an organization can choose when to run the tests. During development, tests are run frequently to make sure no errors have been introduced. Once code is tested and ready to be deployed or used, tests are not run along with the program (unless there has been a modification or someone has discovered an error with the code). This is standard practice in software projects.

It is also worth noting that the collection of tests grows throughout the development process, moreso than do the collection of examples. As you are developing code, every time you find a bug in your code, add a test for it in your check block so you don’t accidentally introduce that same error again later. Whereas we develop examples up front as we figure out what we want our program to do, we augment our tests as we discover what our program actually does (and perhaps should not do). In practice, developers write an initial set of checks on the scenarios they thought of before and while writing the code, then expand those tests as they try out more scenarios and gain users who report scenarios where the code does not work.

Nearly all programming languages come with some constructs or packages in which you can write tests in separate files. Pyret is unique in supporting the distinction between examples and tests (both for learning and for readability of code by others). Many programming tools that support professionals expect you to put all tests in separate folders and files (offering no support for examples). In this book, we emphasize the difference between these two uses of input-output pairs in programming because we find them extremely useful both professionally and pedagogically.

##### 8.3.2More Refined Comparisons

Sometimes, a direct comparison via is isn’t enough for testing. We have already seen this in the case of raises tests (Computing Genetic Parents from an Ancestry Table). As another example, when doing some computations, especially involving math with approximations, the exact match of is isn’t feasible. For example, consider these tests for distance-to-origin:

check:
distance-to-origin(point(1, 1)) is ???
end

What can we check here? Typing this into the REPL, we can find that the answer prints as 1.4142135623730951. That’s an approximation of the real answer, which Pyret cannot represent exactly. But it’s hard to know that this precise answer, to this decimal place, and no more, is the one we should expect up front, and thinking through the answers is supposed to be the first thing we do!

Since we know we’re getting an approximation, we can really only check that the answer is roughly correct, not exactly correct. If we can check that the answer to distance-to-origin(point(1, 1)) is around, say, 1.41, and can do the same for some similar cases, that’s probably good enough for many applications, and for our purposes here. If we were calculating orbital dynamics, we might demand higher precision, but note that we’d still need to pick a cutoff! Testing for inexact results is a necessary task.

Let’s first define what we mean by “around” with one of the most precise ways we can, a function:

fun around(actual :: Number, expected :: Number) -> Boolean:
doc: "Return whether actual is within 0.01 of expected"
num-abs(actual - expected) < 0.01
where:
around(5, 5.01) is true
around(5.01, 5) is true
around(5.02, 5) is false
around(num-sqrt(2), 1.41) is true
end

The is form now helps us out. There is special syntax for supplying a user-defined function to use to compare the two values, instead of just checking if they are equal:

check:
5 is%(around) 5.01
num-sqrt(2) is%(around) 1.41
distance-to-origin(point(1, 1)) is%(around) 1.41
end

Adding %(something) after is changes the behavior of is. Normally, it would compare the left and right values for equality. If something is provided with %, however, it instead passes the left and right values to the provided function (in this example around). If the provided function produces true, the test passes, if it produces false, the test fails. This gives us the control we need to test functions with predictable approximate results.

Exercise

Extend the definition of distance-to-origin to include polar points.

Exercise

This might save you a Google search: polar conversions. Use the design recipe to write x-component and y-component, which return the x and y Cartesian parts of the point (which you would need, for example, if you were plotting them on a graph). Read about num-sin and other functions you’ll need at the Pyret number documentation.

Exercise

Write a data definition called Pay for pay types that includes both hourly employees, whose pay type includes an hourly rate, and salaried employees, whose pay type includes a total salary for the year. Use the design recipe to write a function called expected-weekly-wages that takes a Pay, and returns the expected weekly salary: the expected weekly salary for an hourly employee assumes they work 40 hours, and the expected weekly salary for a salaried employee is 1/52 of their salary.

##### 8.3.3When Tests Fail

Suppose we’ve written the function sqrt, which computes the square root of a given number. We’ve written some tests for this function. We run the program, and find that a test fails. There are two obvious reasons why this can happen.

Do Now!

What are the two obvious reasons?

The two reasons are, of course, the two “sides” of the test: the problem could be with the values we’ve written or with the function we’ve written. For instance, if we’ve written

sqrt(4) is 1.75

then the fault clearly lies with the values (because $$1.75^2$$ is clearly not $$4$$). On the other hand, if it fails the test

sqrt(4) is 2

then the odds are that we’ve made an error in the definition of sqrt instead, and that’s what we need to fix.

Note that there is no way for the computer to tell what went wrong. When it reports a test failure, all it’s saying is that there is an inconsistency between the program and the tests. The computer is not passing judgment on which one is “correct”, because it can’t do that. That is a matter for human judgment.For this reason, we’ve been doing research on peer review of tests, so students can help one another review their tests before they begin writing programs.

Actually...not so fast. There’s one more possibility we didn’t consider: the third, not-so-obvious reason why a test might fail. Return to this test:

sqrt(4) is 2

Clearly the inputs and outputs are correct, but it could be that the definition of sqrt is also correct, and yet the test fails.

Do Now!

Do you see why?

Depending on how we’ve programmed sqrt, it might return the root -2 instead of 2. Now -2 is a perfectly good answer, too. That is, neither the function nor the particular set of test values we specified is inherently wrong; it’s just that the function happens to be a relation, i.e., it maps one input to multiple outputs (that is, $$\sqrt{4} = \pm 2$$). The question now is how to write the test properly.

##### 8.3.4Oracles for Testing

In other words, sometimes what we want to express is not a concrete input-output pair, but rather check that the output has the right relationship to the input. Concretely, what might this be in the case of sqrt? We hinted at this earlier when we said that 1.75 clearly can’t be right, because squaring it does not yield 4. That gives us the general insight: that a number is a valid root (note the use of “a” instead of “the”) if squaring it yields the original number. That is, we might write a function like this:

fun is-sqrt(n):
n-root = sqrt(n)
n == (n-root * n-root)
end

and then our test looks like

check:
is-sqrt(4) is true
end

Unfortunately, this has an awkward failure case. If sqrt does not produce a number that is in fact a root, we aren’t told what the actual value is; instead, is-sqrt returns false, and the test failure just says that false (what is-sqrt returns) is not true (what the test expects)—which is both absolutely true and utterly useless.

Fortunately, Pyret has a better way of expressing the same check. Instead of is, we can write satisfies, and then the value on the left must satisfy the predicate on the right. Concretely, this looks like:

fun check-sqrt(n):
lam(n-root):
n == (n-root * n-root)
end
end

which lets us write:

check:
sqrt(4) satisfies check-sqrt(4)
end

Now, if there’s a failure, we learn of the actual value produced by sqrt(4) that failed to satisfy the predicate.