Using timeit

add_digits.py

def add_digits_optimized(num):
    """
    Recursively sum al digits of a number until result is one digit.
    Optimized version using simple integer logic.

    :param int num: Number to sum all it's digits.
    :rtype: int
    :return: The recusive sum of all digits.
    """
    result = 0
    while True:
        while num:
            result += num % 10
            num = num // 10
        if result < 10:
            break
        num = result
        result = 0
    return result


def add_digits(num):
    """
    Ugly unoptimized version.
    """
    list = [int(i) for i in str(num)]
    while len(list) != 1:
        sum = list[0] + list[1]
        if len(str(sum)) == 1:
            list[1] = sum
            list.remove(list[0])
        else:
            list[0] = int(str(sum)[0])
            list[1] = int(str(sum)[1])
    return list[0]


if __name__ == '__main__':

    # Assert that for the same input value the function
    # remains functionally the same
    for num in range(100):
        original = add_digits(num)
        optimized = add_digits_optimized(num)
        print(
            'add_digits({0}) -> {1:d}\t'
            'add_digits_optimized({0}) -> {2:d}'.format(
                num, original, optimized
            )
        )
        assert original == optimized

    # Calculate performance improvement
    print('Running benchmark...')

    def stress(func, num):
        def binded():
            for idx in range(num):
                func(idx)
        return binded

    from timeit import repeat

    pre_opt = min(
        repeat(stress(add_digits, 1000), repeat=10, number=100)
    )
    post_opt = min(
        repeat(stress(add_digits_optimized, 1000), repeat=10, number=100)
    )
    gain = ((pre_opt - post_opt) / pre_opt) * 100.0

    print('Best execution time:')
    print('Non optimized version: {}'.format(pre_opt))
    print('Optimized version: {}'.format(post_opt))
    print('Performance gain: -{:.2f}%'.format(gain))

    exit(0)

run.log

$ python add_digits.py 
add_digits(0) -> 0	add_digits_optimized(0) -> 0
add_digits(1) -> 1	add_digits_optimized(1) -> 1
add_digits(2) -> 2	add_digits_optimized(2) -> 2
add_digits(3) -> 3	add_digits_optimized(3) -> 3
add_digits(4) -> 4	add_digits_optimized(4) -> 4
add_digits(5) -> 5	add_digits_optimized(5) -> 5
add_digits(6) -> 6	add_digits_optimized(6) -> 6
add_digits(7) -> 7	add_digits_optimized(7) -> 7
add_digits(8) -> 8	add_digits_optimized(8) -> 8
add_digits(9) -> 9	add_digits_optimized(9) -> 9
add_digits(10) -> 1	add_digits_optimized(10) -> 1
add_digits(11) -> 2	add_digits_optimized(11) -> 2
add_digits(12) -> 3	add_digits_optimized(12) -> 3
add_digits(13) -> 4	add_digits_optimized(13) -> 4
add_digits(14) -> 5	add_digits_optimized(14) -> 5
add_digits(15) -> 6	add_digits_optimized(15) -> 6
add_digits(16) -> 7	add_digits_optimized(16) -> 7
add_digits(17) -> 8	add_digits_optimized(17) -> 8
add_digits(18) -> 9	add_digits_optimized(18) -> 9
add_digits(19) -> 1	add_digits_optimized(19) -> 1
add_digits(20) -> 2	add_digits_optimized(20) -> 2
add_digits(21) -> 3	add_digits_optimized(21) -> 3
add_digits(22) -> 4	add_digits_optimized(22) -> 4
add_digits(23) -> 5	add_digits_optimized(23) -> 5
add_digits(24) -> 6	add_digits_optimized(24) -> 6
add_digits(25) -> 7	add_digits_optimized(25) -> 7
add_digits(26) -> 8	add_digits_optimized(26) -> 8
add_digits(27) -> 9	add_digits_optimized(27) -> 9
add_digits(28) -> 1	add_digits_optimized(28) -> 1
add_digits(29) -> 2	add_digits_optimized(29) -> 2
add_digits(30) -> 3	add_digits_optimized(30) -> 3
add_digits(31) -> 4	add_digits_optimized(31) -> 4
add_digits(32) -> 5	add_digits_optimized(32) -> 5
add_digits(33) -> 6	add_digits_optimized(33) -> 6
add_digits(34) -> 7	add_digits_optimized(34) -> 7
add_digits(35) -> 8	add_digits_optimized(35) -> 8
add_digits(36) -> 9	add_digits_optimized(36) -> 9
add_digits(37) -> 1	add_digits_optimized(37) -> 1
add_digits(38) -> 2	add_digits_optimized(38) -> 2
add_digits(39) -> 3	add_digits_optimized(39) -> 3
add_digits(40) -> 4	add_digits_optimized(40) -> 4
add_digits(41) -> 5	add_digits_optimized(41) -> 5
add_digits(42) -> 6	add_digits_optimized(42) -> 6
add_digits(43) -> 7	add_digits_optimized(43) -> 7
add_digits(44) -> 8	add_digits_optimized(44) -> 8
add_digits(45) -> 9	add_digits_optimized(45) -> 9
add_digits(46) -> 1	add_digits_optimized(46) -> 1
add_digits(47) -> 2	add_digits_optimized(47) -> 2
add_digits(48) -> 3	add_digits_optimized(48) -> 3
add_digits(49) -> 4	add_digits_optimized(49) -> 4
add_digits(50) -> 5	add_digits_optimized(50) -> 5
add_digits(51) -> 6	add_digits_optimized(51) -> 6
add_digits(52) -> 7	add_digits_optimized(52) -> 7
add_digits(53) -> 8	add_digits_optimized(53) -> 8
add_digits(54) -> 9	add_digits_optimized(54) -> 9
add_digits(55) -> 1	add_digits_optimized(55) -> 1
add_digits(56) -> 2	add_digits_optimized(56) -> 2
add_digits(57) -> 3	add_digits_optimized(57) -> 3
add_digits(58) -> 4	add_digits_optimized(58) -> 4
add_digits(59) -> 5	add_digits_optimized(59) -> 5
add_digits(60) -> 6	add_digits_optimized(60) -> 6
add_digits(61) -> 7	add_digits_optimized(61) -> 7
add_digits(62) -> 8	add_digits_optimized(62) -> 8
add_digits(63) -> 9	add_digits_optimized(63) -> 9
add_digits(64) -> 1	add_digits_optimized(64) -> 1
add_digits(65) -> 2	add_digits_optimized(65) -> 2
add_digits(66) -> 3	add_digits_optimized(66) -> 3
add_digits(67) -> 4	add_digits_optimized(67) -> 4
add_digits(68) -> 5	add_digits_optimized(68) -> 5
add_digits(69) -> 6	add_digits_optimized(69) -> 6
add_digits(70) -> 7	add_digits_optimized(70) -> 7
add_digits(71) -> 8	add_digits_optimized(71) -> 8
add_digits(72) -> 9	add_digits_optimized(72) -> 9
add_digits(73) -> 1	add_digits_optimized(73) -> 1
add_digits(74) -> 2	add_digits_optimized(74) -> 2
add_digits(75) -> 3	add_digits_optimized(75) -> 3
add_digits(76) -> 4	add_digits_optimized(76) -> 4
add_digits(77) -> 5	add_digits_optimized(77) -> 5
add_digits(78) -> 6	add_digits_optimized(78) -> 6
add_digits(79) -> 7	add_digits_optimized(79) -> 7
add_digits(80) -> 8	add_digits_optimized(80) -> 8
add_digits(81) -> 9	add_digits_optimized(81) -> 9
add_digits(82) -> 1	add_digits_optimized(82) -> 1
add_digits(83) -> 2	add_digits_optimized(83) -> 2
add_digits(84) -> 3	add_digits_optimized(84) -> 3
add_digits(85) -> 4	add_digits_optimized(85) -> 4
add_digits(86) -> 5	add_digits_optimized(86) -> 5
add_digits(87) -> 6	add_digits_optimized(87) -> 6
add_digits(88) -> 7	add_digits_optimized(88) -> 7
add_digits(89) -> 8	add_digits_optimized(89) -> 8
add_digits(90) -> 9	add_digits_optimized(90) -> 9
add_digits(91) -> 1	add_digits_optimized(91) -> 1
add_digits(92) -> 2	add_digits_optimized(92) -> 2
add_digits(93) -> 3	add_digits_optimized(93) -> 3
add_digits(94) -> 4	add_digits_optimized(94) -> 4
add_digits(95) -> 5	add_digits_optimized(95) -> 5
add_digits(96) -> 6	add_digits_optimized(96) -> 6
add_digits(97) -> 7	add_digits_optimized(97) -> 7
add_digits(98) -> 8	add_digits_optimized(98) -> 8
add_digits(99) -> 9	add_digits_optimized(99) -> 9
Running benchmark...
Best execution time:
Non optimized version: 0.372886896133
Optimized version: 0.0686509609222
Performance gain: -81.59%