 Solution

Evaluate the mixed radix representation from its most significant end, using Horner's method. There is no need to compute the positional weights separately.The digit conditions $0\le xs(i)<bs(i)$ are additional preconditions. The loop invariant at the loop test isAn empty sum is zero and an empty product is one, so initialization at $i=N$ is correct. For the next digit, writing $i^{\prime }=i-1$ giveswhich preserves the loop invariant. The loop variant is $i$. At termination $i=0$, givingThe code executes exactly $N$ multiplications, meeting the bound, and uses $O(1)$ auxiliary space. When $N=0$ it returns zero. For the time representation in the source, it computes $s+60(m+60h)$.