Add math.gcd()

[scoder]

BPO	22486
Nosy	@mdickinson, @abalkin, @pitrou, @scoder, @vstinner, @serhiy-storchaka, @wm75
PRs	bpo-39350: Remove deprecated fractions.gcd() #18021
Files	lehmer_gcd_4.patch lehmer_gcd_5.patch lehmer_gcd_6.patch lehmer_gcd_7.patch euclidean_gcd.patch lehmer_gcd_8.patch

^{Note: these values reflect the state of the issue at the time it was migrated and might not reflect the current state.}

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GitHub fields:

assignee = None
closed_at = <Date 2015-05-12.23:26:09.896>
created_at = <Date 2014-09-24.20:40:35.148>
labels = ['library', 'performance']
title = 'Add math.gcd()'
updated_at = <Date 2020-01-16.10:02:58.267>
user = 'https://github.com/scoder'

bugs.python.org fields:

activity = <Date 2020-01-16.10:02:58.267>
actor = 'vstinner'
assignee = 'none'
closed = True
closed_date = <Date 2015-05-12.23:26:09.896>
closer = 'benjamin.peterson'
components = ['Library (Lib)']
creation = <Date 2014-09-24.20:40:35.148>
creator = 'scoder'
dependencies = []
files = ['36722', '36723', '36726', '36741', '36742', '37422']
hgrepos = []
issue_num = 22486
keywords = ['patch']
message_count = 48.0
messages = ['227487', '227507', '227522', '227524', '227525', '227527', '227528', '227529', '227531', '227532', '227533', '227534', '227535', '227536', '227537', '227538', '227554', '227555', '227570', '227619', '227620', '227623', '227629', '227630', '227637', '227638', '227640', '227641', '227663', '227664', '227670', '227672', '227711', '228544', '228545', '228546', '228547', '228548', '228549', '228550', '228551', '228552', '228556', '232521', '232523', '242048', '243015', '360112']
nosy_count = 9.0
nosy_names = ['mark.dickinson', 'belopolsky', 'pitrou', 'scoder', 'vstinner', 'python-dev', 'serhiy.storchaka', 'wolma', 'gladman']
pr_nums = ['18021']
priority = 'normal'
resolution = 'fixed'
stage = 'resolved'
status = 'closed'
superseder = None
type = 'performance'
url = 'https://bugs.python.org/issue22486'
versions = ['Python 3.5']

[scoder]

fractions.gcd() is required for normalising numerator and denominator of the Fraction data type. Some speed improvements were applied to Fraction in bpo-22464, now the gcd() function takes up about half of the instantiation time in the benchmark in bpo-22458, which makes it quite a heavy part of the overall Fraction computation time.

The current implementation is

def gcd(a, b):
    while b:
        a, b = b, a%b
    return a

Reimplementing it in C would provide for much faster calculations. Here is a Cython version that simply drops the calculation loop into C as soon as the numbers are small enough to fit into a C long long int:

def _gcd(a, b):
    # Try doing all computation in C space.  If the numbers are too large
    # at the beginning, retry until they are small enough.
    cdef long long ai, bi
    while b:
        try:
            ai, bi = a, b
        except OverflowError:
            pass
        else:
            # switch to C loop
            while bi:
                ai, bi = bi, ai%bi
            return ai
        a, b = b, a%b
    return a

It's substantially faster already because the values will either be small enough right from the start or quickly become so after a few iterations with Python objects.

Further improvements should be possible with a dedicated PyLong implementation based on Lehmer's GCD algorithm:

https://en.wikipedia.org/wiki/Lehmer_GCD_algorithm

[mdickinson]

In case it's useful, see issue bpo-1682 for my earlier Lehmer gcd implementation. At the time, that approach was dropped as being premature optimisation.

[pitrou]

If Python grows an optimized implementation, how about exposing it in the math module?

[mdickinson]

If Python grows an optimized implementation, how about exposing it in the math module?

+1.

[scoder]

That's what the patch does anyway. +1

[pitrou]

Hmm... which patch?

[scoder]

http://bugs.python.org/file9486/lehmer_gcd.patch

(see bpo-1682)

[serhiy-storchaka]

Here is updated Mark's patch from bpo-1682. It is ported to 3.5, slightly simplified and optimized (I did not touched the main algorithm still), utilized in the fractions module, added tests and documentation.

It speeds up Stefan's fractions benchmark about 20%.

[scoder]

The problem is that this changes the behaviour of fractions.gcd() w.r.t. negative numbers. It's a public function, so we should keep it for backwards compatibility reasons, *especially* when adding a new function in the math module.