We have drastically improved our earlier result. We have recently proved that for an N-bit constant, the upper-bound in cascaded additions/subtractions using the radix-2r arithmetic is equal to log[(N+1)/r]/log(2)+r/2-1, where r=2.W[((N+1).log(2))^0.5]/log(2) and W is the Lambert function. This is a near-optimal bound, knowing that the lower-bound is log[(N+1)/2]/log(2)

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