Consider the following problem:
You have a list of floating point numbers. No nasty tricks - these aren’t NaN or Infinity, just normal “simple” floating point numbers.
Now: Calculate the mean (average). Can you do it?
It turns out this is a hard problem. It’s hard to get it even close to right. Lets see why.
Consider the following test case using Hypothesis:
from hypothesis import given from hypothesis.strategies import lists, floats @given(lists(floats(allow_nan=False, allow_infinity=False)), min_size=1) def test_mean_is_within_reasonable_bounds(ls): assert min(ls) <= mean(ls) <= max(ls)
This isn’t testing much about correctness, only that the value of the mean is within reasonable bounds for the list: There are a lot of functions that would satisfy this without being the mean. min and max both satisfy this, as does the median, etc.
However, almost nobody’s implementation of the mean satisfies this.
To see why, lets write our own mean:
def mean(ls): return sum(ls) / len(ls)
This seems reasonable enough - it’s just the definition of the mean - but it’s wrong:
assert inf <= 8.98846567431158e+307 + where inf = mean([8.988465674311579e+307, 8.98846567431158e+307]) + and 8.98846567431158e+307 = max([8.988465674311579e+307, 8.98846567431158e+307]) Falsifying example: test_mean_is_within_reasonable_bounds( ls=[8.988465674311579e+307, 8.98846567431158e+307] )
The problem is that finite floating point numbers may be large enough that their sum overflows to infinity. When you then divide infinity by a finite number you still get infinity, which is out of the range.
So to prevent that overflow, lets try to bound the size of our numbers by the length first:
def mean(ls): return sum(l / len(ls) for l in ls)
assert min(ls) <= mean(ls) <= max(ls) assert 1.390671161567e-309 <= 1.390671161566996e-309 where 1.390671161567e-309 = min([1.390671161567e-309, 1.390671161567e-309, 1.390671161567e-309]) and 1.390671161566996e-309 = mean([1.390671161567e-309, 1.390671161567e-309, 1.390671161567e-309]) Falsifying example: test_mean_is_within_reasonable_bounds( ls=[1.390671161567e-309, 1.390671161567e-309, 1.390671161567e-309] )
In this case the problem you run into is not overflow, but the lack of precision of floating point numbers: Floating point numbers are only exact up to powers of two times an integer, so dividing by three will cause rounding errors. In this case we have the problem that (x / 3) * 3 may not be equal to x in general.
So now we’ve got a sense of why this might be hard. Lets see how existing implementations do at satisfying this test.
First let’s try numpy:
import numpy as np def mean(ls): return np.array(ls).mean()
This runs into the problem we had in our first implementation:
assert min(ls) <= mean(ls) <= max(ls) assert inf <= 8.98846567431158e+307 where inf = mean([8.988465674311579e+307, 8.98846567431158e+307]) and 8.98846567431158e+307 = max([8.988465674311579e+307, 8.98846567431158e+307]) Falsifying example: test_mean_is_within_reasonable_bounds( ls=[8.988465674311579e+307, 8.98846567431158e+307] )
There’s also the new statistics module from Python 3.4. Unfortunately, this is broken too (this is fixed in 3.5.2):
OverflowError: integer division result too large for a float Falsifying example: test_mean_is_within_reasonable_bounds( ls=[8.988465674311579e+307, 8.98846567431158e+307] )
In the case where we previously overflowed to infinity this instead raises an error. The reason for this is that internally the statistics module is converting everything to the Fraction type, which is an arbitrary precision rational type. Because of the details of where and when they were converting back to floats, this produced a rational that couldn’t be readily converted back to a float.
It’s relatively easy to write an implementation which passes this test by simply cheating and not actually calculating the mean:
def clamp(lo, v, hi): return min(hi, max(lo, v)) def mean(ls): return clamp(min(ls), sum(ls) / len(ls), max(ls))
i.e. just restricting the value to lie in the desired range.
However getting an actually correct implementation of the mean (which would pass this test) is quite hard:
To see just how hard, here’s a 30 page paper on calculating the mean of two numbers.
I wouldn’t feel obliged to read that paper if I were you. I have read it and I don’t remember many of the details.
This test is a nice instance of a general one: Once you’ve got the this code doesn’t crash, tests working, you can start to layer on additional constraints on the result value. As this example shows, even when the constraints you impose are very lax it can often catch interesting bugs.
It also demonstrates a problem: Floating point mathematics is very hard, and this makes it somewhat unsuitable for testing with Hypothesis.
This isn’t because Hypothesis is bad at testing floating point code, it’s because it’s good at showing you how hard programming actually is, and floating point code is much harder than people like to admit.
As a result, you probably don’t care about the bugs it will find: Generally speaking most peoples’ attitude to floating point errors is “Eh, those are weird numbers, we don’t really care about that. It’s probably good enough”. Very few people are actually prepared to do the required work of a numerical sensitivity analysis that is needed if you want your floating point code to be correct.
I used to use this example a lot for demonstrating Hypothesis to people, but because of these problems I tend not to any more: Telling people about bugs they’re not going to want to fix will get you neither bug fixes nor friends.
But it’s worth knowing that this is a problem: Programming is really hard, and ignoring the problems won’t make it less hard. You can ignore the correctness issues until they actually bite you, but it’s best not to be surprised when they do.
And it’s also worth remembering the general technique here, because this isn’t just useful for floating point numbers: Most code can benefit from this, and most of the time the bugs it tells you won’t be nearly this unpleasant.