unswitch while loop optimization in c - c

Question

I'm having a hard time optimizing the following while loop by means of Loop Unswitching. I have tried applying the example from wiki, however I am having a hard time applying it to a while loop. I have the following code:
int n = 5,
m = 5,
i = 0,
val = 0;
while (i < n ) {
j = 0;
while (j < m ) {
if (i < j ) {
val = val + i ;
}
else if ( j == i ) {
val = val - 1;
}
else {
val = val + j ;
}
j = j + 1;
}
i = i + 1;
}
And have tried unswitching it the following way:
while (i < n ) {
j = 0;
if (i < j ) {
while (j < m ) {
val = val + i;
j = j + 1;
}
}
if ( j == i ) {
while (j < m) {
val = val - 1;
j = j + 1;
}
}
if (i > j) {
while (j < m) {
val = val + j;
j = j + 1;
}
}
i = i + 1;
}
What could I be doing wrong.

Answer 1

Such loops are best unrolled with the help of pencil and paper. You want the sum of the following grid:
0 1 2 3 4 | 5 n
0 -1 0 0 0 0 | 0 0
1 0 -1 1 1 1 | 1 1
2 0 1 -1 2 2 | 2 2
3 0 1 2 -1 3 | 3 3
m 0 1 2 3 -1 | 4 4
The grid can be subdivided into three parts: the diagonal, the upper and lower triangles next to the diagonal in the square part and the rectangular block when n and m differ.
Let's represent the dimension of the grid by means of the square part, k² and the rectangular part, k·r:
k = min(n, m)
r = max(m, n) - k
Now you can see which sums the three parts contribute:
val = 2·∑(k - i - 1)·i # two triangles
+ r·∑(i) # rectangle
- k # diagonal
(All sums run from i = 0; i < n; i++.) This sum can be rearranged to:
val = 2·(k - 1)·∑(i) - 2*∑(i²) + r·(i) - k
= (2·k + r - 2)·∑(i) - 2*∑(i²) - k
This reduces your two nested loops two two independent loops to do the sums of the natural numbers and of their squares. Fortunately, these sums can be expressed by simple relations:
∑(i) = (n - 1)·n / 2
∑(i²) = (2·n - 1)·(n - 1)·n / 6
You now have a constant-time formula for your resulting sum:
int val(int n, int m)
{
int k = (n < m) ? n : m;
int r = ((n > m) ? n : m) - k;
return (2*k + r - 2) * (k - 1) * k / 2
- (2*k - 1) * k * (k - 1) / 3 - k;
}
All this doesn't have anything to do with loop unrolling, of course.

Answer 2

Loop unswitching according to wikipedia is a compiler optimization so I'm a little confused about why you'd need to do this yourself, but I think this is as good as it gets in terms of breaking apart the for..ifs
for (i = 0; i < n; ++i) {
// j < i
for (j = 0; j < i; ++j) {
val = val + j;
}
// j == i
val = val - 1;
// j > i
for (j = i + 1; j < m; ++j) {
val = val + i;
}
}
This is not a loop you can traditionally unswitch because the conditional variable here is the loop variable.