It depends how many numbers you add. If you compute A+B it will be slower because the Carrie's still have to be handled after the addition. But if you compute A+B+C+D+E you do 4 fast addition and only deal with carry bits once. TFA allows up to about 8000 numbers to be summed for a single pass of carry propagation.
I can't tell from your comments whether you realized yet, but the trick is only useful if you add several 256 bit numbers. You can add up to 2^13 numbers (each one 256 bits) before you actually have to do the four carry shift-outs once. In effect, you do reduce the number of carry operations (to four, albeit slightly more involved), and that is what ends up saving time.
There is no performance gain if you only add two numbers, even though that's the setup the article discusses for 90% of its length. It seems that this is leading to some confusion.
Yeah, I re-read the whole thing again, and I see the mistake I made. I took a wrong turn when op went into base 37 numbers, which made me consider every bit carry not just the carry on word boundary.
