Addition of long integers suffers from the problem that carries have to be propagated from right to left and the final result is not known until this process has been completed.
The principle of the Kochanski algorithm is one of making guesses as to whether or not r should be subtracted, based on the most significant few bits of the carry-save value in the accumulator.
Such a guess will be wrong some of the time, since there is no way of knowing whether latent carries in the less significant digits (which have not been examined) might not invalidate the result of the comparison.
At the end of a complete modular multiplication, the true binary result of the operation has to be evaluated and it is possible that an additional addition or subtraction of r will be needed as a result of the carries that are then discovered; but the cost of that extra step is small when amortized over the hundreds of shift-and-add steps that dominate the overall cost of the multiplication.
Brickell[3] has published a similar algorithm that requires greater complexity in the electronics for each digit of the accumulator.