If you're coming from a language like C, Haskell can take some getting used to. It's typical for a new language to feel a little different, but in Haskell the differences are more dramatic, and more fundamental. In particular...

`for`

loops?In most imperative languages, `for`

loops are all
over the place, and are used for a wide variety of *different
things*. Whether you're squaring every value of an array or
finding its sum, you're probably using a `for`

loop.

In Haskell, control structures are more expressive. Sure,
there's a counterpart to C's `for`

(Haskell's
`forM_`

). But that's a discussion for another time.
Today, we'll see some `for`

loops in C that can be
written very differently in Haskell, and why that's a good
thing.

Consider the simple example of computing the norm of a vector.
For a vector **x**∈ℝ^{n}, this is just

Conceptually, there are three stages to this computation:

: Square each element`mapSq`

: Compute the sum`sum`

: Compute the square root`sqrt`

We can think of the first step as building (at least abstractly)
a new array whose `i`

th element is ```
y[i] = x[i] *
x[i]
```

. In functional programming, we refer to this as
*mapping* the square function over the array.

Putting everything in terms of functions, we can write this (in Haskell-like pseudocode) as

`norm(x) = sqrt(sum(mapSq(x))) ,`

To clean up the syntax a bit, we can instead use the notation for function composition and write

`norm(x) = (sqrt ○ sum ○ mapSq)(x) ,`

or just

`norm = sqrt ○ sum ○ mapSq .`

In C, this modular approach leads to something like

```
void mapSq(double *x, double *y, int n) {
int i;
for(i=0; i<n; i++) {
y[i] = x[i] * x[i];
}
}
double sum(double *x, int n) {
double result = 0;
int i;
for(i=0; i<n; i++) {
result += x[i];
}
return result;
}
double norm1(double *x, int n) {
double *y = malloc(n * sizeof(double));
mapSq(x, y, n);
theSum = sum(y, n);
free(y);
return sqrt(theSum);
}
```

We'd probably never do it this way, for a few reasons:

- For such a simple computation, it's way too verbose.
- It uses two separate
`for`

loops, where only one is needed. - There's an unnecessary
`malloc`

that's unlikely to be removed by the compiler.

If it weren't for these issues, we probably would write code like the above. It has the advantage of being more modular and of faithfully representing the concepts. But the problems are too big to ignore, so we typically make some code transformations in our head, and instead write

```
double norm2(double *x, int n) {
double theSum = 0.0;
for (int i = 0; i < n; i++) {
theSum += x[i]*x[i]; }
return sqrt(theSum);
}
```

This code is very clear, and performs well, but we've entirely lost modularity. In this case the code is very short, so we don't give the compromise a second thought. But at a larger scale, this kind of manual optimization reduces code reuse and makes components more complex. Code becomes harder to understand, and harder to test for correctness.

In Haskell, we could write `norm`

as

```
import Prelude hiding (sum)
import Data.List (foldl')
-- show
mapSq x = map sq x
where
sq xi = xi * xi
sum x = foldl' (+) 0.0 x
norm x = (sqrt . sum . mapSq) x
main = print (norm [1,2,3,4,5])
```

While the C example is in terms of arrays, this Haskell example
instead uses *lazy linked lists*. We'll change this in a
bit.

Let's step through this code. The first function,
`mapSq`

, is defined in terms of the `map`

function, and produces a new list by squaring every element of the
original list. The squaring function `sq`

can also be
written as `(\x -> x*x)`

, allowing us to write the
function as

mapSq x = map (\x -> x*x) x

or simply

mapSq = map (\x -> x*x)

Next, the `sum`

function is defined in terms of
`foldl'`

. Before you worry too much about the name, you
should know that a *fold* is just a function that traverses
some data structure, using a given function to update some
*accumulator* as it goes. In the current case, the accumulator
is the counterpart of "`theSum`

" in the C version.
Accumulation is via addition, and starts at 0.0.

Lists can be folded from either side. In our case, we're using
`foldl'`

. The "`l`

" in this function name is
for "left", the side we're starting on, and the "`'`

"
indicates that this fold is *strict*. Haskell is lazy by
default, but for numeric there's no point in delaying evaluation of
the accumulator (and for large lists it can lead to a stack
overflow), so in this case we prefer to evaluate it at every
step.

As it turns out, our `sum`

function is common enough
that it's included in Haskell's Prelude; we can use it exactly as
described with no need to define it.^{1} Our
`norm`

function is now simple:

```
norm = sqrt . sum . map (\x -> x*x)
main = print (norm [1,2,3,4,5])
```

This looks remarkably like the mathematical specification; some
have affectionately refered to Haskell as an *executable
specification language*.

Let's compare performance with C. In order to measure easily across languages, we'll make the vector much bigger - lets say a billion elements. We'll start with a standard list implementation in Haskell and compare this to an implementation in C using arrays. Then we'll make a couple of minor changes in Haskell that allow us to use unboxed vectors.

We'll test using an AMD A6-3670 APU running Ubuntu 13.04. For
compilation, we'll use GHC 7.6.3 (with `-O2`

) and GCC
4.7.3 (with `-O3`

).

Our Haskell version is certainly elegant, but using linked lists
in this way is not the best approach for performance. For a billion
elements, the running time is **over 2 minutes**.
What's going on here?

Haskell lists are *lazy* (only those elements required by
later calculations are computed) and *polymorphic* (you can
have a list of elements of any type, even if elements of that type
don't take up a fixed number of bytes). In order to implement this,
a list in Haskell is really a *list of pointers to elements*.
In cases like this, all that pointer chasing adds up.

We'll soon see an easy way to improve this and get Haskell running really fast.

Before we get Haskell zipping along, let's look at how we would
do this in C. We'll use the faster version of our norm in C
(`norm2`

above). Here's the plan:

`malloc`

an array of a billion elements- Fill the array so that
`x[i] = i`

- Find the norm of the vector
- Free the array and print the result

We'll use the OS to measure the time of the whole process, and insert some code to measure step (3) alone. So all together we have this:

```
#include <time.h>
#include <math.h>
#include <stdio.h>
#include <stdlib.h>
#define x_size 1000000000
double norm(double *x, int n) {
double theSum = 0.0;
int i;
for (i=0; i<n; i++) {
theSum += x[i]*x[i];
}
return sqrt(theSum);
}
int main() {
double *x = malloc(x_size * sizeof(double));
int i;
for(i=0; i<x_size; i++) {
x[i] = (double) i;
}
clock_t start = clock();
double result = norm(x, x_size);
clock_t end = clock();
printf("%3.2f\n", result);
printf("%3.2f sec\n", (end-start)/(double)CLOCKS_PER_SEC);
free(x);
return 0;
}
```

The whole thing takes about **7 seconds** to run,
with the `norm`

function itself taking **1.9
seconds**.

To close in on C, let's make some minor changes. Instead of
using lists, we'll use unboxed *vector*s. The "unboxed"
qualifier just means that the elements are available directly,
without the need to follow a pointer.

From a data structure standpoint, Haskell's vectors are a lot
like C's arrays. There is an important difference, though; many
operations on vectors (even boxed ones) are subject to
*fusion*. A `map`

followed by a `sum`

will be represented as a tight loop, but the code can remain
modular. Remember the malloc in our original C code? Through
fusion, it's eliminated entirely.

Updating the original code to operate on vectors is
straightforward. And To stay entirely outside the realm of lists,
we can generate our vector using `iterateN`

, which plays
the role of our data-filling loop in C.

[EDIT: Because School of Haskell compilation does not use
`-O2`

, there is no fusion, so we'll need to keep the
values a bit smaller. In this case, I'll use an unusually small
value for `oneBillion`

, though for benchmarking I still
use 10^{9}]

```
import Data.Vector.Unboxed as V
norm :: Vector Double -> Double
norm = sqrt . V.sum . V.map (\x -> x*x)
oneBillion = 100
main = print $ norm $ V.iterateN oneBillion (+1) 0.0
```

This code runs in about **2.5 seconds**. It's just
a bit longer than C's inner loop alone, but much less than the 7
seconds if the malloc is included.

EDIT: As Pedro Vasconcelos points out, tight numerical loops
like this can often benefit from using the LLVM back-end, via
`-fllvm`

. This brings the time for the entire Haskell
run down to **1.9 seconds**, the same as the inner
loop alone in C!

Haskell's control structures express that a reduction (a
*fold* in Haskell) is very different than a
*map*.^{2} Libraries like **vector**
implement powerful fusion techniques to combine loops and eliminate
intermediate data structures. And *unboxing* elminates
excessive pointer chasing required by built-in lists.

All in all, this lets us write *high-level code without
sacrificing high performance*. The need for any compromise
between the two is becoming more rare every day.

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