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Chapter 6 Fruitful functions
Many of the Python functions we have used, such as the math
functions, produce return values. But the functions we’ve written
are all void: they have an effect, like printing a value
or moving a turtle, but they don’t have a return value. In
this chapter you will learn to write fruitful functions.
6.1 Return values
Calling the function generates a return
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)
The functions we have written so far are void. Speaking casually,
they have no return value; more precisely,
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):
a = math.pi * radius**2
return a
We have seen the return statement before, but in a fruitful
function the return statement includes
an expression. This statement means: “Return immediately from
this function and use the following expression as a return value.”
The expression 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 a can 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 runs. As soon as a return statement runs, 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
if x > 0:
return x
This function is incorrect 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
By the way, Python provides a built-in function called
abs that computes absolute values.
As an exercise, write a compare function
takes two values, x and y, and returns 1 if x > y,
0 if x == y, and -1 if x < y.
6.2 Incremental development
As you write larger functions, you might 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:
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 inputs are two points, which you can represent
using four numbers. The return value is the distance represented by
a floating-point value. Immediately 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: - Start with a working program and make small incremental changes.
At any point, if there is an error, you should have a good idea
where it is.
- Use variables to hold intermediate values so you can
display and check them.
- Once the program is working, you might want to remove some of
the scaffolding or consolidate multiple statements into compound
expressions, but only if it does not make the program difficult to
read.
As an exercise, use incremental development to write a function
called hypotenuse that returns the length of the hypotenuse of a
right triangle given the lengths of the other two legs as arguments.
Record each stage of the development process as you go.
6.3 Composition
As you should expect by now, you can call one function from within
another. 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. We just wrote 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;
we just wrote that, too: result = area(radius)
Encapsulating these steps 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))
6.4 Boolean functions
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. As an exercise, write a function is_between(x, y, z) that
returns True if x ≤ y ≤ z or False otherwise.
6.5 More recursion
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 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.
For a more complete (and accurate) discussion of the Turing Thesis,
I recommend Michael Sipser’s book Introduction to the
Theory of Computation. 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: - vorpal:
- An adjective used to describe something that is vorpal.
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:
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
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 takes an integer: 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 equals 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.
Figure 6.1 shows what the stack diagram looks like for
this sequence of function calls.
| Figure 6.1: Stack diagram. |
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 does not run.
6.6 Leap of faith
Following the flow of execution is one way to read programs, but
it can quickly become overwhelming. 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—by examining the code and testing—we can use the function
without looking at the body 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 (returns the correct result) and then ask
yourself, “Assuming that I can find the factorial of n−1, can I
compute the factorial of n?” 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!
6.7 One more example
After factorial, the most common example of a recursively
defined mathematical function is fibonacci, which has the
following definition (see
http://en.wikipedia.org/wiki/Fibonacci_number):
| | | fibonacci(0) = 0 |
| | | fibonacci(1) = 1 |
| | | fibonacci(n) = fibonacci(n−1) + fibonacci(n−2)
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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.
6.8 Checking types
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. How can that be? The function
has 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
(more negative), 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 not defined for negative integers.')
return None
elif n == 0:
return 1
else:
return n * factorial(n-1)
The first base case handles nonintegers; the
second handles negative integers. In both cases, the program prints
an error message and returns None to indicate that something
went wrong: >>> print(factorial('fred'))
Factorial is only defined for integers.
None
>>> print(factorial(-2))
Factorial is not defined for negative integers.
None
If we get past both checks, we know that n is positive or
zero, so 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. In Section 11.4 we will see a more flexible alternative to printing
an error message: raising an exception.
6.9 Debugging
Breaking a large program into smaller functions creates natural
checkpoints for debugging. If a function is not
working, there are three possibilities to consider:
- There is something wrong with the arguments the function
is getting; a precondition is violated.
- There is something wrong with the function; a postcondition
is violated.
- There is something wrong with the return value or the
way it is being used.
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). Or you can write code
that checks the preconditions explicitly.
If the parameters look good, add a print statement before each
return statement and display the return value. If
possible, check the result by hand. Consider calling the
function with values that make it easy to check the result
(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!).
Adding print statements at the beginning and end of a function
can help make the flow of execution more visible.
For example, here is a version of factorial with
print statements: 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(4) : factorial 4
factorial 3
factorial 2
factorial 1
factorial 0
returning 1
returning 1
returning 2
returning 6
returning 24
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 a little bit of scaffolding can save a lot of debugging.
6.10 Glossary
- temporary variable:
- A variable used to store an intermediate value in
a complex calculation.
- dead code:
- Part of a program that can never run, often because
it appears after a return statement.
- incremental development:
- A program development plan intended to
avoid debugging by adding and testing only
a small amount of code at a time.
- scaffolding:
- Code that is used during program development but is
not part of the final version.
- guardian:
- A programming pattern that uses a conditional
statement to check for and handle circumstances that
might cause an error.
6.11 Exercises
Exercise 1 Draw a stack diagram for the following program. What does the program print?
def b(z):
prod = a(z, z)
print(z, prod)
return prod
def a(x, y):
x = x + 1
return x * y
def c(x, y, z):
total = x + y + z
square = b(total)**2
return square
x = 1
y = x + 1
print(c(x, y+3, x+y))
Exercise 3
A palindrome is a word that is spelled the same backward and
forward, like “noon” and “redivider”. Recursively, a word
is a palindrome if the first and last letters are the same
and the middle is a palindrome.
The following are functions that take a string argument and
return the first, last, and middle letters: def first(word):
return word[0]
def last(word):
return word[-1]
def middle(word):
return word[1:-1]
We’ll see how they work in Chapter 8. - Type these functions into a file named palindrome.py
and test them out. What happens if you call middle with
a string with two letters? One letter? What about the empty
string, which is written
'' and contains no letters? - Write a function called
is_palindrome that takes
a string argument and returns True if it is a palindrome
and False otherwise. Remember that you can use the
built-in function len to check the length of a string.
Solution: http://thinkpython2.com/code/palindrome_soln.py. Exercise 4 A number, a, is a power of b if it is divisible by b
and a/b is a power of b. Write a function called
is_power that takes parameters a and b
and returns True if a is a power of b.
Note: you will have to think about the base case. Exercise 5
The greatest common divisor (GCD) of a and b is the largest number
that divides both of them with no remainder. One way to find the GCD of two numbers is based on the observation
that if r is the remainder when a is divided by b, then gcd(a,
b) = gcd(b, r). As a base case, we can use gcd(a, 0) = a. Write a function called
gcd that takes parameters a and b
and returns their greatest common divisor. Credit: This exercise is based on an example from Abelson and
Sussman’s Structure and Interpretation of Computer Programs.
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