Implementation: Division

Just like for multiplication, the integer (floor) division may be treated as repeated subtractions. The quotient \( \lfloor A/B \rfloor \) tells us how often \(B\) can be subtracted from \(A\) before it becomes negative.

The naïve floor division can thus be implemented as:

def naive_div(a, b):
    r = -1
    while a >= 0:
        a -= b
        r += 1
    return r

Just like naïve multiplication, this division algorithm scales linearly with the size of dividend, if the effect of divisor is ignored.

Can we do better?

In school, we have learned to do long divisions. This can also be done using binary numbers. We at first left-align the divisor b with the dividend a and compare the sizes of the overlapping part. If the divisor is smaller, it goes once into the dividend. Therefore, the quotient at that bit becomes 1 and the dividend is subtracted from the part of divisor. Otherwise, this quotient bit will be 0 and no subtraction takes place. Afterwards, the dividend is right-shifted and the whole process repeated. This algorithm is illustrated in the following table for the example 11 // 2:

An example implementation is given in the following listing:

def div(a, b):
    n = a.bit_length()
    tmp = b << n
    r = 0
    for _ in range(0, n + 1):
        r <<= 1
        if tmp <= a:
            a -= tmp
            r += 1
        tmp >>= 1
    return r

In this implementation, rather than setting the result bitwise like described in the table above, it is initialized to 0 and appended with 0 or 1. Also, the divisor is shifted by the bit-length of a instead of the difference between a and b. This may increase the number of loops, but prevents negative shifts, when bit-length of a is smaller than that of b.

This algorithm is linear in the bit-length of the dividend and thus a \(\mathcal{\log(n)}\) algorithm. Again, we want to quantify the performance of both algorithms by timing them.

Since the size of the divisor does not have a simple relation with the execution time, we shall fix its size. Here we choose nb = 10. An example function for timing is shown in the following listing:

    from random import randrange
    from time import perf_counter
    
    def time_division(func, na, nb, cycles=1):
        total_time = 0.0
        for i in range(0, cycles):
            a = randrange(0, na)
            b = randrange(1, nb)
    
            t_start = perf_counter()
            func(a, b)
            t_stop = perf_counter()
    
            total_time += (t_stop - t_start)
    
        return total_time / float(cycles)

Since we cannot divide by zero, the second argument, the divisor in this case, is chosen between 1 and n instead of 0 and n.

The execution time per execution for different sizes are listed below:

Again, although the naïve method is faster for smaller numbers, its scaling prevents it from being used for larger numbers.