Some of the built-in functions we have used, such as the math functions, produce results. Calling the function generates a new value, which we usually assign to a variable or use as part of an expression.
e = math.exp(1.0) height = radius * math.sin(radians)
All of the functions we have written so far are void; they print something or move turtles around, but their return value is None.
In this chapter, we are (finally) going to write fruitful functions. The first example is area, which returns the area of a circle with the given radius:
def area(radius):
temp = math.pi * radius**2
return temp
We have seen the return statement before, but in a fruitful function the return statement includes a return value. This statement means: “Return immediately from this function and use the following expression as a return value.” The expression provided can be arbitrarily complicated, so we could have written this function more concisely:
def area(radius):
return math.pi * radius**2
On the other hand, temporary variables like temp often make debugging easier.
Sometimes it is useful to have multiple return statements, one in each branch of a conditional:
def absolute_value(x):
if x < 0:
return -x
else:
return x
Since these return statements are in an alternative conditional, only one will be executed.
As soon as a return statement executes, the function terminates without executing any subsequent statements. Code that appears after a return statement, or any other place the flow of execution can never reach, is called dead code.
In a fruitful function, it is a good idea to ensure that every possible path through the program hits a return statement. For example:
def absolute_value(x):
if x < 0:
return -x
elif x > 0:
return x
This program is not correct because if x happens to be 0, neither condition is true, and the function ends without hitting a return statement. If the flow of execution gets to the end of a function, the return value is None, which is not the absolute value of 0.
>>> print absolute_value(0) None
As you write larger functions, you might start find yourself spending more time debugging.
To deal with increasingly complex programs, you might want to try a process called incremental development. The goal of incremental development is to avoid long debugging sessions by adding and testing only a small amount of code at a time.
As an example, suppose you want to find the distance between two points, given by the coordinates (x1, y1) and (x2, y2). By the Pythagorean theorem, the distance is:
| distance = | √ |
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The first step is to consider what a distance function should look like in Python. In other words, what are the inputs (parameters) and what is the output (return value)?
In this case, the two points are the inputs, which you can represent using four parameters. The return value is the distance, which is a floating-point value.
Already you can write an outline of the function:
def distance(x1, y1, x2, y2):
return 0.0
Obviously, this version doesn't compute distances; it always returns zero. But it is syntactically correct, and it runs, which means that you can test it before you make it more complicated.
To test the new function, call it with sample arguments:
>>> distance(1, 2, 4, 6) 0.0
I chose these values so that the horizontal distance is 3 and the vertical distance is 4; that way, the result is 5 (the hypotenuse of a 3-4-5 triangle). When testing a function, it is useful to know the right answer.
At this point we have confirmed that the function is syntactically correct, and we can start adding code to the body. A reasonable next step is to find the differences x2 − x1 and y2 − y1. The next version stores those values in temporary variables and prints them.
def distance(x1, y1, x2, y2):
dx = x2 - x1
dy = y2 - y1
print 'dx is', dx
print 'dy is', dy
return 0.0
If the function is working, it should display 'dx is 3' and 'dy is 4'. If so, we know that the function is getting the right arguments and performing the first computation correctly. If not, there are only a few lines to check.
Next we compute the sum of squares of dx and dy:
def distance(x1, y1, x2, y2):
dx = x2 - x1
dy = y2 - y1
dsquared = dx**2 + dy**2
print 'dsquared is: ', dsquared
return 0.0
Again, you would run the program at this stage and check the output (which should be 25).
Finally, you can use math.sqrt to compute and return the result:
def distance(x1, y1, x2, y2):
dx = x2 - x1
dy = y2 - y1
dsquared = dx**2 + dy**2
result = math.sqrt(dsquared)
return result
If that works correctly, you are done. Otherwise, you might want to print the value of result before the return statement.
The final version of the function doesn't display anything when it runs; it only returns a value. The print statements we wrote are useful for debugging, but once you get the function working, you should remove them. Code like that is called scaffolding because it is helpful for building the program but is not part of the final product.
When you start out, you should add only a line or two of code at a time. As you gain more experience, you might find yourself writing and debugging bigger chunks. Either way, incremental development can save you a lot of debugging time.
The key aspects of the process are:
As you should expect by now, you can call one function from within another. This ability is called composition.
As an example, we'll write a function that takes two points, the center of the circle and a point on the perimeter, and computes the area of the circle.
Assume that the center point is stored in the variables xc and yc, and the perimeter point is in xp and yp. The first step is to find the radius of the circle, which is the distance between the two points. Fortunately, there is a function, distance, that does that:
radius = distance(xc, yc, xp, yp)
The next step is to find the area of a circle with that radius:
result = area(radius)
Wrapping that up in a function, we get:
def circle_area(xc, yc, xp, yp):
radius = distance(xc, yc, xp, yp)
result = area(radius)
return result
The temporary variables radius and result are useful for development and debugging, but once the program is working, we can make it more concise by composing the function calls:
def circle_area(xc, yc, xp, yp):
return area(distance(xc, yc, xp, yp))
Functions can return booleans, which is often convenient for hiding complicated tests inside functions. For example:
def is_divisible(x, y):
if x % y == 0:
return True
else:
return False
It is common to give boolean functions names that sound like yes/no questions; is_divisible returns either True or False to indicate whether x is divisible by y.
Here is an example:
>>> is_divisible(6, 4) False >>> is_divisible(6, 3) True
The result of the == operator is a boolean, so we can write the function more concisely by returning it directly:
def is_divisible(x, y):
return x % y == 0
Boolean functions are often used in conditional statements:
if is_divisible(x, y):
print 'x is divisible by y'
It might be tempting to write something like:
if is_divisible(x, y) == True:
print 'x is divisible by y'
But the extra comparison is unnecessary.
We have only covered a small subset of Python, but you might be interested to know that this subset is a complete programming language, which means that anything that can be computed can be expressed in this language. Any program ever written could be rewritten using only the language features you have learned so far (actually, you would need a few commands to control devices like the keyboard, mouse, disks, etc., but that's all).
Proving that claim is a nontrivial exercise first accomplished by Alan Turing, one of the first computer scientists (some would argue that he was a mathematician, but a lot of early computer scientists started as mathematicians). Accordingly, it is known as the Turing Thesis. If you take a course on the Theory of Computation, you will have a chance to see the proof.
To give you an idea of what you can do with the tools you have learned so far, we'll evaluate a few recursively defined mathematical functions. A recursive definition is similar to a circular definition, in the sense that the definition contains a reference to the thing being defined. A truly circular definition is not very useful:
If you saw that definition in the dictionary, you might be annoyed. On the other hand, if you looked up the definition of the factorial function, denoted with the symbol !, you might get something like this:
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This definition says that the factorial of 0 is 1, and the factorial of any other value, n, is n multiplied by the factorial of n−1.
So 3! is 3 times 2!, which is 2 times 1!, which is 1 times 0!. Putting it all together, 3! equals 3 times 2 times 1 times 1, which is 6.
If you can write a recursive definition of something, you can usually write a Python program to evaluate it. The first step is to decide what the parameters should be. In this case it should be clear that factorial has a single parameter:
def factorial(n):
If the argument happens to be 0, all we have to do is return 1:
def factorial(n):
if n == 0:
return 1
Otherwise, and this is the interesting part, we have to make a recursive call to find the factorial of n−1 and then multiply it by n:
def factorial(n):
if n == 0:
return 1
else:
recurse = factorial(n-1)
result = n * recurse
return result
The flow of execution for this program is similar to the flow of countdown in Section 5.8. If we call factorial with the value 3:
Since 3 is not 0, we take the second branch and calculate the factorial of n-1...
Since 2 is not 0, we take the second branch and calculate the factorial of n-1...Since 1 is not 0, we take the second branch and calculate the factorial of n-1...Since 0 is 0, we take the first branch and return 1 without making any more recursive calls.The return value (1) is multiplied by n, which is 1, and the result is returned.
The return value (1) is multiplied by n, which is 2, and the result is returned.
The return value (2) is multiplied by n, which is 3, and the result, 6, becomes the return value of the function call that started the whole process.
Here is what the stack diagram looks like for this sequence of function calls:
The return values are shown being passed back up the stack. In each frame, the return value is the value of result, which is the product of n and recurse.
In the last frame, the local variables recurse and result do not exist, because the branch that creates them did not execute.
Following the flow of execution is one way to read programs, but it can quickly become labyrinthine. An alternative is what I call the “leap of faith.” When you come to a function call, instead of following the flow of execution, you assume that the function works correctly and returns the right result.
In fact, you are already practicing this leap of faith when you use built-in functions. When you call math.cos or math.exp, you don't examine the bodies of those functions. You just assume that they work because the people who wrote the built-in functions were good programmers.
The same is true when you call one of your own functions. For example, in Section 6.4, we wrote a function called is_divisible that determines whether one number is divisible by another. Once we have convinced ourselves that this function is correct—examining the code and testing—we can use the function without looking at the code again.
The same is true of recursive programs. When you get to the recursive call, instead of following the flow of execution, you should assume that the recursive call works (yields the correct result) and then ask yourself, “Assuming that I can find the factorial of n−1, can I compute the factorial of n?” In this case, it is clear that you can, by multiplying by n.
Of course, it's a bit strange to assume that the function works correctly when you haven't finished writing it, but that's why it's called a leap of faith!
After factorial, the most common example of a recursively defined mathematical function is fibonacci, which has the following definition:
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Translated into Python, it looks like this:
def fibonacci (n):
if n == 0:
return 0
elif n == 1:
return 1
else:
return fibonacci(n-1) + fibonacci(n-2)
If you try to follow the flow of execution here, even for fairly small values of n, your head explodes. But according to the leap of faith, if you assume that the two recursive calls work correctly, then it is clear that you get the right result by adding them together.
What happens if we call factorial and give it 1.5 as an argument?
>>> factorial(1.5) RuntimeError: Maximum recursion depth exceeded
It looks like an infinite recursion. But how can that be? There is a base case—when n == 0. But if n is not an integer, we can miss the base case and recurse forever.
In the first recursive call, the value of n is 0.5. In the next, it is -0.5. From there, it gets smaller and smaller, but it will never be 0.
We have two choices. We can try to generalize the factorial function to work with floating-point numbers, or we can make factorial check the type of its argument. The first option is called the gamma function and it's a little beyond the scope of this book. So we'll go for the second.
We can use the built-in function isinstance to verify the type of the argument. While we're at it, we can also make sure the argument is positive:
def factorial (n):
if not isinstance(n, int):
print 'Factorial is only defined for integers.'
return None
elif n < 0:
print 'Factorial is only defined for positive integers.'
return None
elif n == 0:
return 1
else:
return n * factorial(n-1)
Now we have three base cases. The first catches nonintegers and the second catches negative integers. In both cases, the program prints an error message and returns None to indicate that something went wrong:
>>> factorial('fred')
Factorial is only defined for integers.
None
>>> factorial(-2)
Factorial is only defined for positive integers.
None
If we get past both checks, then we know that n is a positive integer, and we can prove that the recursion terminates.
This program demonstrates a pattern sometimes called a guardian. The first two conditionals act as guardians, protecting the code that follows from values that might cause an error. The guardians make it possible to prove the correctness of the code.
As you start writing bigger programs, you might find yourself spending more time debugging. More code means more chances to make an error and more place for bugs to hide.
One way to cut your debugging time is “debugging by bisection.” For example, if there are 100 lines in your program and you check them one at a time, it would take 100 steps.
Instead, try to break the problem in half. Look at the middle of the program, or near it, for an intermediate value you can check. Add a print statement (or something else that has a verifiable effect) and run the program.
If the mid-point check is incorrect, the problem must be in the first half of the program. If it is correct, the problem is in the second half.
Every time you perform a check like this, you halve the number of lines you have to search. After six steps (which is much less than 100), you would be down to one or two lines of code.
At least in theory. In practice it is not always clear what the “middle of the program” is and not always possible to check it.
That's where functions come in. By breaking a large program into smaller, well-defined functions, you create natural checkpoints for debugging. If you suspect there is something wrong with a particular function, there are three possibilities to consider:
To rule out the first possibility, you can add a print statement at the beginning of the function and display the values of the parameters (and maybe their types).
If the parameters look good, add a print statement before each return statement that displays the return value. If possible, check the result by hand. If necessary, call the function with special values where you know what the result should be (as in Section 6.2).
If the function seems to be working, look at the function call to make sure the return value is being used correctly (or used at all!).
For example, here is a version of factorial that demonstrates the flow of execution:
def factorial(n):
space = ' ' * (4 * n)
print space, 'factorial', n
if n == 0:
print space, 'returning 1'
return 1
else:
recurse = factorial(n-1)
result = n * recurse
print space, 'returning', result
return result
space is a string of space characters that controls the indentation of the output. Here is the result of factorial(5) :
factorial 5
factorial 4
factorial 3
factorial 2
factorial 1
factorial 0
returning 1
returning 1
returning 2
returning 6
returning 24
returning 120
If you are confused about the flow of execution, this kind of output can be helpful. It takes some time to develop effective scaffolding, but according to the Fifth Theorem of Debugging:
A little bit of scaffolding can save a lot of debugging.