True division of integers could be more accurate

[mdickinson]

BPO	1811
Nosy	@tim-one, @terryjreedy, @mdickinson, @tiran
Files	int_truediv.py: More accurate truediv for integers: example code long_division.patch: patch for Objects/longobject.c long_division2.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 = 'https://github.com/mdickinson'
closed_at = <Date 2009-12-27.15:10:11.018>
created_at = <Date 2008-01-12.05:20:28.608>
labels = ['interpreter-core', 'type-bug']
title = 'True division of integers could be more accurate'
updated_at = <Date 2009-12-27.15:10:11.016>
user = 'https://github.com/mdickinson'

bugs.python.org fields:

activity = <Date 2009-12-27.15:10:11.016>
actor = 'mark.dickinson'
assignee = 'mark.dickinson'
closed = True
closed_date = <Date 2009-12-27.15:10:11.018>
closer = 'mark.dickinson'
components = ['Interpreter Core']
creation = <Date 2008-01-12.05:20:28.608>
creator = 'mark.dickinson'
dependencies = []
files = ['9137', '10363', '15667']
hgrepos = []
issue_num = 1811
keywords = ['patch']
message_count = 12.0
messages = ['59798', '59824', '59830', '59838', '59839', '59988', '64034', '67031', '91020', '96834', '96840', '96910']
nosy_count = 4.0
nosy_names = ['tim.peters', 'terry.reedy', 'mark.dickinson', 'christian.heimes']
pr_nums = []
priority = 'normal'
resolution = 'fixed'
stage = 'resolved'
status = 'closed'
superseder = None
type = 'behavior'
url = 'https://bugs.python.org/issue1811'
versions = ['Python 2.7', 'Python 3.2']

[mdickinson]

Division of two longs can produce results that are needlessly
inaccurate:

>>> from __future__ import division
>>> 10**40/10**39
10.000000000000002

The correct result is, of course, 10.0, which is exactly representable
as a float.

The attached snippet of Python code shows that things don't have to be
this way.

[tiran]

How fast is your algorithm compared to the current algorithm and where
starts the break even zone? Most users use only small integers so it
might be a good idea to use a simpler algorithm for longs with Py_SIZE()
== 1. This is important for py3k.

[mdickinson]

It would be easy and safe to just use a/b = float(a)/float(b) if both a and b are less
than 2**53 (assuming IEEE doubles). Then there wouldn't be any loss of speed for
small integers.

For large integers the algorithm I posted should run in time linear in the number of
digits of max(a, b), at least in the worst case (though with appropriate optimizations
it could be made much faster for 'random' inputs). The current algorithm has
essentially O(1) runtime.

To get proper timings I'd have to code this up properly. I'll do this, unless there's
a consensus that it would be a waste of time :)

[tiran]

Mark Dickinson wrote:

To get proper timings I'd have to code this up properly. I'll do this, unless there's
a consensus that it would be a waste of time :)

Why don't you ask Tim? He is the right person for the discussion. I'm
only an interested amateur mathematician.

Christian

[mdickinson]

Tim: is this worth fixing?

[mdickinson]

A related problem is that float(n) isn't always correctly rounded for an integer
n. A contrived example:

>>> n = 2**68 + 2**16 - 1
>>> float(n)
2.9514790517935283e+20

Here the difference between float(n) and the true value of n is around 0.99998
ulps; a correctly rounded float() would have error at most 0.5 ulps.

I don't regard this as terribly serious: from looking at the code, I *think*
it's always true that the error is strictly less than 1 ulp, which is just
enough to guarantee that float(n) == n whenever n is exactly representable as a
float.

In contrast, the division of two integers can produce results that are up to 3.5
ulps out from the true value. This is, in my opinion, a worryingly large error
for a simple calculation.

[terryjreedy]

To my mind, the inaccurate result is a bug that should be fixed.

Note: (3.0a3)
>>> 10e40/10e39
10.0

The rationale for the division change is that (as far as reasonably
possible) arithmetic operations with same values should give same result
regardless of types.

I have not looked at either algorithm, but if long/long started by
finding divmod(), but added fractional value when remainer is non-zero
instead of tossing it, exact quotients would easily be exact (unless too
large).

[mdickinson]

Here's a patch that fixes the rounding of integer division. It includes a
fast path for the case where both integers are small (less than about
3.5e12).

[mdickinson]

An example of a case that's almost 3.5 ulps out (Python 2.6):

Python 2.6.2 (r262:71600, Jul  8 2009, 09:56:31) 
[GCC 4.0.1 (Apple Inc. build 5490)] on darwin
Type "help", "copyright", "credits" or "license" for more information.
>>> from __future__ import division
>>> m = 295147931372582273023
>>> n = 295147932265116303360
>>> m/n
0.99999999697597697

The correctly rounded result would be the float given by
0.9999999969759773.

[mdickinson]

Stealing this from Tim, with the intention of acting on it in the next
couple of weeks.

[mdickinson]

Here's an updated patch, against py3k. On my machine, a/b is a touch faster with this patch when abs(a),
abs(b) are smaller than 1e15 or so; it's (inevitably) slower than the existing implementation for larger a
and b. For 'random' a and b, average running time is proportional to the size of b, and is independent of
the size of a; worst-case running time (which occurs when a has many trailing zero bits) grows as
max(size(a), size(b)).

Changing versions to 2.7 and 3.2, but I'm mostly aiming for 3.2. It may not be worth backporting to 2.7,
given the extra effort required to deal correctly with ints as well as with longs.

[mdickinson]

Fixed in r77062 (trunk), r77063 (py3k).