def pig_latin(w):
    """Return the Pig Latin equivalent of English word w."""
    if starts_with_a_vowel(w):
        return w + 'ay'
    return pig_latin(w[1:] + w[0])

def starts_with_a_vowel(w):
    """Return whether w begins with a vowel."""
    return w[0].lower() in 'aeiou'

def fact_iter(n):
    """Return n! for n > 0."""
    total, k = 1, 1
    while k <= n:
        total, k = total*k, k+1
    return total

def fact(n):
    """Return n! for n > 0."""
    if n == 1:
        return 1
    return n * fact(n-1)

def reverse(s):
    """Return the reverse of string s.

    >>> reverse('sub')
    'bus'
    """
    if s == '':
        return s
    return reverse(s[1:]) + s[0]

def reverse_iter(s):
    """Return the reverse of string s.

    >>> reverse_iter('sub')
    'bus'
    """
    r, i = '', 0
    while i < len(s):
        r, i = s[i] + r, i + 1
    return r

def reverse2(s):
    """Return the reverse of string s.

    >>> reverse2('sub')
    'bus'
    """
    def reverse_s(r, i):
        if not i < len(s):
            return r
        return reverse_s(s[i] + r, i + 1)
    return reverse_s('', 0)

def approx_eq(x, y, tolerance=1e-12):
    return abs(x-y) <= tolerance

def sqrt(x):
    """Compute the square root of x by mutually recursive Newton's method.

    >>> sqrt(9)
    3.0
    >>> sqrt(2)
    1.414213562373095
    """
    def guess_sqrt(guess):
        if approx_eq(guess*guess, x):
            return guess
        return guess_again(guess)
    def guess_again(guess):
        return guess_sqrt(0.5 * (guess + x/guess))
    return guess_sqrt(1)