bpo-35431: Implemented math.comb

[FR4NKESTI3N]

Implemented the function math.comb to calculate binomial coefficient. Simple O(k) algorithm is used for now. ValueError if condition 0 <= k <= n is not satisfied and TypeError if non-integer arguments passed to function.

https://bugs.python.org/issue35431

[pablogsal]

You can use Jebelean’s exact division algorithm to speed up the division as if you factor out the coefficients as the product of(n-k+1)* PROD(i,2,k,(n-k+i)/i), i will always divide the already calculated product.

[FR4NKESTI3N]

@pablogsal I haven't started working on optimization yet since Raymond Hettinger suggested to wait till everyone has agreed on the behaviour : https://bugs.python.org/issue35431#msg332838
Can I consider that current behaviour is acceptable and write optimizations?

I was also considering using prime factorization for smaller k values : https://dl.acm.org/citation.cfm?id=26272 but seems like an overkill now

[KellerFuchs]

@FR4NKESTI3N Thanks a lot for starting on this. I merged our PRs in #11018, as the testsuite I had started there was more complete (and used as the basis for discussing the desired behaviour, in the bugs.python.org issue).

[KellerFuchs]

PS: I added you as a collaborator on my repository, so if you accept the invite you can push directly to that PR's branch.

[bedevere-bot]

A Python core developer has requested some changes be made to your pull request before we can consider merging it. If you could please address their requests along with any other requests in other reviews from core developers that would be appreciated.

Once you have made the requested changes, please leave a comment on this pull request containing the phrase I have made the requested changes; please review again. I will then notify any core developers who have left a review that you're ready for them to take another look at this pull request.

[FR4NKESTI3N]

I have made the requested changes; please review again
@rhettinger I have used long_long instead of long in the boilerplate.
Using object allowed evaluation of expressions like comb(10**400, 200) and overflow was caused if min(k, n-k) was larger than LLONG_MAX. I feel using object conversion gave the function more flexibility.

[bedevere-bot]

Thanks for making the requested changes!

@rhettinger: please review the changes made to this pull request.

[serhiy-storchaka]

1. Use PyNumber_Index() for converting arguments to Python integers. See gcd() for example.
2. Do not convert n to C long long. comb(2**1000, 5) should work.
3. Do not use PyNumber_Long().

It is convenient to implement perm(n, k) first. comb(n, k) can be easily implemented as perm(n, k) // factorial(k) (or perm(n, n-k) // factorial(n-k) if k > n-k).

[FR4NKESTI3N]

@serhiy-storchaka So boilerplate should be changed back to have n and k as objects right?
@KellerFuchs I will be busy for 2 weeks, can you get in on this.