Introduction to Functional Programming – Lab 1: “Power Function” | TDA555 / DIT440, LP1 2016 |

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Introduction to Functional Programming – Lab 1: “Power Function” | TDA555 / DIT440, LP1 2016 |

Home | Schedule | Labs | Lectures | Exercises | Exam | About | FAQ | Fire | Forum | TimeEdit | YouTube | Links |

(When you are done, please submit your solution using the Fire system)

In this lab assignment, you will implement the well-known `power`

function in two different new ways. The power function takes two arguments *n* and *k* and computes *n*^{k}. Your implementation only has to work for non-negative *k*.

We have already seen one implementation of this function in the lecture:

```
power :: Integer -> Integer -> Integer
power n k | k < 0 = error "power: negative argument"
power n 0 = 1
power n k = n * power n (k-1)
```

You will implement two more ways in this lab assignment.

*Note:* Please make sure you follow the submission guidelines when you write your code.

In order to calculate `power n k`

, for a given `n`

and `k`

, how many computing “steps” are being used?

*Hint:* `power n 0`

takes 1 step.

`power n 1`

takes 1 step, and then uses `power n 0`

.

`power n 2`

takes 1 step, and then uses `power n 1`

.

`power n 3`

takes 1 step, and then uses `power n 2`

.

And so forth.

Write your answer *as a comment* in the Haskell file you submit.

A different way of computing the power function is to use the standard Haskell function `product`

, which calculates the product (multiplication) of all elements in a list.

(You may not yet have learned about lists, but you can read about them here.)

To calculate `power n k`

, first construct a list with `k`

elements, all being `n`

, and then use `product`

.

Implement this idea as a Haskell function `power1`

.

*Hint:* You have to come up with a way of producing a list with `k`

elements, all being equal to `n`

. Use a list comprehension, or use the standard Haskell function `replicate`

. If you use `replicate`

, you might want to use the function `fromInteger`

too! Use Hoogle to find out more about standard functions.

A different approach to calculating the power function uses fewer computing steps.

We use the fact that, if *k* is even, we can calculate *n*^{k} as follows:

n^{k}= (n^{2})^{k/2}(kis even)

In other words:

n^{k}= (n⋅n)^{k/2}(kis even)

So, instead of recursively using the case for *k* − 1, we use the (much smaller) case for *k*/2.

If *k* is not even, we simply go one step down in order to arrive at an even *k*, just as in the original definition of power:

n^{k}=n⋅ (n^{k − 1}) (kis odd)

To sum up, to calculate `power n k`

:

- If
`k`

is even, we use (*n*⋅*n*)^{k/2} - If
`k`

is odd, we use*n*⋅ (*n*^{k − 1})

Implement this idea as a Haskell function `power2`

.

*Hints:*

- Do not forget to add a base case (what do you do when
*k*= 0?) - You need to find out when numbers are even or odd. Use the standard Haskell functions
`even`

and/or`odd`

. - To divide integer numbers, use the function
`div`

(and not the function`/`

, which is used to divide floating point and rational numbers).

We would like the three functions `power`

, `power1`

, and `power2`

to calculate the same thing. It is probably a good idea to test this!

**A.** Come up with a number of test cases (inputs you will test your functions on). Argue why you have chosen these test cases. (Think about for what inputs the functions are defined, and for what inputs the functions are not defined.)

**B.** Implement two functions: One function `comparePower1`

(which given `n`

and `k`

, compares the result of the `power`

function with your `power1`

function), and a function `comparePower2`

(which does the same for `power`

and `power2`

). Your tests should give the result `True`

when the test succeeds, and `False`

if it runs without an error but fails otherwise. These two functions will make your tests easier to write!

**C.** Write all your test cases as one or two Haskell functions that perform all tests. It is probably a good idea to use the functions `comparePower1`

and `comparePower2`

here.

*Hint:* You can use a list comprehension to combine all possible cases you would like to test for `n`

and `k`

. Use the standard Haskell function `and`

to combine the results. If you are not familiar with list comprehensions, you do not have to use these.

Just for fun, measure and compare the speed of the different power implementations. You can do this by downloading the MeasureTime module to the same directory as the file for lab 1. Then add the following line at the top of the file for lab 1:

`import MeasureTime`

Now you can time the power implementations in the following way in the terminal:

```
*MeasureTime> measureTime2 power 10 100000
Start evaluation
Done after 2.9s
```

(This is an extra assignment. You do not have to do this assignment to pass the course, but you will learn more when you do!)

Write a Haskell function `table`

that takes `x`

and a maximum `n`

as an argument, and then generates a table in the following way:

```
*Main> table 2 10
n power power1 power2
0 1 1 1
1 2 2 2
2 4 4 4
3 8 8 8
4 16 16 16
5 32 32 32
6 64 64 64
7 128 128 128
8 256 256 256
9 512 512 512
10 1024 1024 1024
```

Exact details on how many spaces to insert where, how to line up the numbers, etc. are freely choosable for you.

*Hint:* Use `show`

to convert a number to a string, and use the functions `length`

and `replicate`

to calculate the right number of spaces between the strings. Use `unlines`

to convert a list of lines into one big text. Use `putStr`

to display the generated string on the screen.

Use Hoogle to find out more information about these functions and others.

Use the XHtml library to render the above table as an HTML file that you can view in the browser.

You can explore the library yourself, but here is a simple example to get you started:

```
import Text.XHtml hiding (table)
-- Hiding `table` because it clashes with the function in assignment 5
tab :: Html
tab = simpleTable [] []
[ [toHtml "1", toHtml "2"]
, [toHtml "3", toHtml "4"]
]
main = writeFile "table.html" (renderHtml tab)
```

Running `main`

in GHCi produces the file `table.html`

which, when loaded in a web browser shows the following table:

1 2 3 4

This lab has two deadlines:

- First deadline (Wednesday, Sept. 7 at 23:59 2016): You need to have made a serious effort on completing the complete lab assignment.
- Final deadline (Friday, Sept. 16 at 23:59 2016): You need to have gotten a pass on the lab.

Write your answers in one file, called **Lab1.hs**. For each assignment, use Haskell comments to indicate what part of the file contains the answer to the assignment. For answers in natural language, you can use Swedish or English (English is preferred as some of the graders do not yet speak Swedish); write your answers also in Haskell comments. Remove irrelevant things from the file.

Before you submit your code, Clean It Up! Remember, submitting clean code is Really Important, and simply the polite thing to do. After you feel you are done, spend some time on cleaning your code; make it simpler, remove unneccessary things, etc. We will reject your solution if it is not clean. Clean code:

- Does not have long lines (< 78 characters)
- Has a consistent layout (do not use TAB characters in your code)
- Has type signatures for all top-level functions
- Has good comments where needed
- Has no junk (junk is unused code, commented code, unneccessary comments)
- Has no overly complicated function definitions
- Does not contain any repetitive code (copy-and-paste programming)

Once you have submitted, please arrange to present your solution to one of the course assistants according to the instructions on the lab overview page.

Good luck!