I might not have the optimum yet, but I think I can at least do a lot better than 4.4 e11...
I can do it with at maximum
96 pieces (--> 65)
As a starting point:
There is a very nice way to convert a rectangle into a square by dissection with only
two cuts. (or 3 pieces)
If you do the following:
I wish I could claim having found that solution myself, but it is actually from
New Mexico State University; Department of Mathematical Sciences
Now, this solution has a problem:
It is only valid for rectangles where the two sides fulfil: a < b < 2a
(So a is the shorter side.)
Our "starting point" is a rectangle of size a x 2015a.
But we can make it to fulfil the condition! All we need is to cut stripes until b<2a.
Now we have
a striped rectangle which matches the condition and used up 32 pieces(31 cuts). By applying the cutting in the first image, we will at maximum create 3 pieces hence we will need no more than 3 x 32 = 96 pieces to accomplish the task.
Note that X is not splitting the longer side of the rectangle exactly at 1/2, because the longer side is not exactly 63 but 62.96875. The solution is still valid by virtue of the general statement at the beginning of the proof.
Looking at my own answer above, I think there is a very straight-forward way to optimize the solution!
Starting with the solution above:
It can be seen that
The tiles of the upper half (except the very tip) can of course be merged! This reduces 62 tiles to 31 brings the solution down to 96-31 = 65 tiles.