Slow recursion in python

Question

I know this subject is well discussed but I've come around a case I don't really understand how the recursive method is "slower" than a method using "reduce,lambda,xrange".

def factorial2(x, rest=1):
    if x <= 1:
        return rest
    else:
        return factorial2(x-1, rest*x)


def factorial3(x):
    if x <= 1:
        return 1
    return reduce(lambda a, b: a*b, xrange(1, x+1))

I know python doesn't optimize tail recursion so the question isn't about it. To my understanding, a generator should still generate n amount of numbers using the +1 operator. So technically, fact(n) should add a number n times just like the recursive one. The lambda in the reduce will be called n times just as the recursive method... So since we don't have tail call optimization in both case, stacks will be created/destroyed and returned n times. And a if in the generator should check when to raise a StopIteration exception.

This makes me wonder why does the recursive method still slowlier than the other one since the recursive one use simple arithmetic and doesn't use generators.

In a test, I replaced rest*x by x in the recursive method and the time spent got reduced on par with the method using reduce.

Here are my timings for fact(400), 1000 times

factorial3 : 1.22370505333

factorial2 : 1.79896998405

Edit:

Making the method start from 1 to n doesn't help either instead of n to 1. So not an overhead with the -1.

Also, can we make the recursive method faster. I tried multiple things like global variables that I can change... Using a mutable context by placing variables in cells that I can modify like an array and keep the recursive method without parameters. Sending the method used for recursion as a parameter so we don't have to "dereference" it in our scope...?! But nothings makes it faster.

I'll point out that I have a version of the fact that use a forloop that is much faster than both of those 2 methods so there is clearly space for improvement but I wouldn't expect anything faster than the forloop.

Answer 1

The slowness of the recursive version comes from the need to resolve on each call the named (argument) variables. I have provided a different recursive implementation that has only one argument and it works slightly faster.

$ cat fact.py 
def factorial_recursive1(x):
    if x <= 1:
        return 1
    else:
        return factorial_recursive1(x-1)*x

def factorial_recursive2(x, rest=1):
    if x <= 1:
        return rest
    else:
        return factorial_recursive2(x-1, rest*x)

def factorial_reduce(x):
    if x <= 1:
        return 1
    return reduce(lambda a, b: a*b, xrange(1, x+1))

# Ignore the rest of the code for now, we'll get back to it later in the answer
def range_prod(a, b):
    if a + 1 < b:
        c = (a+b)//2
        return range_prod(a, c) * range_prod(c, b)
    else:
        return a
def factorial_divide_and_conquer(n):
    return 1 if n <= 1 else range_prod(1, n+1)

$ ipython -i fact.py 
In [1]: %timeit factorial_recursive1(400)
10000 loops, best of 3: 79.3 µs per loop
In [2]: %timeit factorial_recursive2(400)
10000 loops, best of 3: 90.9 µs per loop
In [3]: %timeit factorial_reduce(400)
10000 loops, best of 3: 61 µs per loop

Since in your example very large numbers are involved, initially I suspected that the performance difference might be due to the order of multiplication. Multiplying on every iteration a large partial product by the next number is proportional to the number of digits/bits in the product, so the time complexity of such a method is O(n²), where n is the number of bits in the final product. Instead it is better to use a divide and conquer technique, where the final result is obtained as a product of two approximately equally long values each of which is computed recursively in the same manner. So I implemented that version too (see factorial_divide_and_conquer(n) in the above code) . As you can see below it still loses to the reduce()-based version for small arguments (due to the same problem with named parameters) but outperforms it for large arguments.

In [4]: %timeit factorial_divide_and_conquer(400)
10000 loops, best of 3: 90.5 µs per loop
In [5]: %timeit factorial_divide_and_conquer(4000)
1000 loops, best of 3: 1.46 ms per loop
In [6]: %timeit factorial_reduce(4000)
100 loops, best of 3: 3.09 ms per loop

UPDATE

Trying to run the factorial_recursive?() versions with x=4000 hits the default recursion limit, so the limit must be increased:

In [7]: sys.setrecursionlimit(4100)
In [8]: %timeit factorial_recursive1(4000)
100 loops, best of 3: 3.36 ms per loop
In [9]: %timeit factorial_recursive2(4000)
100 loops, best of 3: 7.02 ms per loop