simplefractions 1.2.1

Find the simplest fraction for a given float or interval

The simplefractions package finds the simplest fraction that converts to a given float, or more generally, the simplest fraction that lies within a given interval.

Definition. Given fractions x = a/b and y = c/d (written in lowest terms with positive denominators), say that x is simpler than y if abs(a) <= abs(c), b <= d, and at least one of these two inequalities is strict.

For example, 22/7 is simpler than 23/8, but neither of 3/8 and 4/7 is simpler than the other.

Then it's a theorem that given any subinterval I of the real line that contains at least one fraction, that interval contains a unique simplest fraction. That is, there's a fraction a/b in I such that a/b is simpler (in the above sense) than all other fractions in I. As a consequence, for any given finite Python float x, there's a unique simplest fraction that rounds to that float.

The simplefractions package provides two functions:

• simplest_from_float returns, for a given float x, the unique simplest fraction with the property that float(simplest_from_float(x)) == x.
• simplest_in_interval returns the unique simplest fraction in a given (open or closed, bounded or unbounded) nonempty interval.

Example usage

Start by importing the functions from the module:

>>> from simplefractions import *

The simplest_from_float function takes a single finite float x and produces a Fraction object that recovers that float:

>>> simplest_from_float(0.25)
Fraction(1, 4)
>>> simplest_from_float(0.33)
Fraction(33, 100)

No matter what x is given, the invariant float(simplest_from_float(x)) == x will always be true.

>>> x = 0.7429667872099244
>>> simplest_from_float(x)
Fraction(88650459, 119319545)
>>> float(simplest_from_float(x))
0.7429667872099244
>>> float(simplest_from_float(x)) == x
True

If the float x was constructed by dividing two small integers, then more often than not, simplest_from_float will recover those integers:

>>> x = 231 / 199
>>> x
1.1608040201005025
>>> simplest_from_float(x)
Fraction(231, 199)

More precisely, if x was constructed by dividing two relatively prime integers smaller than or equal to 67114657 in absolute value, simplest_from_float will recover those integers.

>>> simplest_from_float(64841043 / 66055498)
Fraction(64841043, 66055498)

But 67114657 is the best we can do here:

>>> simplest_from_float(67114658 / 67114657)
Fraction(67114657, 67114656)

In larger cases, simplest_from_float might discover a simpler fraction that gives the same float:

>>> x = 818421477165 / 1580973145504
>>> simplest_from_float(x)
Fraction(5171, 9989)
>>> 818421477165 / 1580973145504 == 5171 / 9989
True

Note that simplest_from_float does not magically fix floating-point inaccuracies. For example:

>>> x = 1.1 + 2.2
>>> simplest_from_float(x)
Fraction(675539944105597, 204709073971393)

You might have expected Fraction(33, 10) here, but when converted to float, that gives a value very close to, but not exactly equal to, x. In contrast, the return value of simplest_from_float(x) will always produce exactly x when converted to float.

To fix this, you might want to ask for the simplest float that lies within some small error bound of x - for example, within 5 ulps (units in the last place) in either direction. simplest_from_float can't do that, but simplest_in_interval can! For example

>>> from math import ulp
>>> x = 1.1 + 2.2
>>> simplest_in_interval(x - 5*ulp(x), x + 5*ulp(x))
Fraction(33, 10)

Alternatively, you might ask for the simplest fraction approximating x with a relative error of at most 0.000001:

>>> relerr = 1e-6
>>> simplest_in_interval(x - relerr*x, x + relerr*x)
Fraction(33, 10)

Here are some more examples of simplest_in_interval at work. The inputs to simplest_in_interval can be floats, integers, or Fraction objects.

>>> simplest_in_interval(3.14, 3.15)
Fraction(22, 7)

By default, simplest_in_interval assumes that you're specifying an open interval:

>>> simplest_in_interval(3, 4)
Fraction(7, 2)

Keyword arguments include_left and include_right allow you to specify that one or both endpoints should be included in the interval:

>>> simplest_in_interval(3, 4, include_left=True, include_right=True)
Fraction(3, 1)

The left and right endpoints of the interval are also both optional, alowing a semi-infinite or infinite interval to be specified:

>>> simplest_in_interval(right=4)  # simplest in (-inf, 4)
Fraction(0, 1)
>>> simplest_in_interval(left=4, include_left=True)  # simplest in [4, inf)
Fraction(4, 1)
>>> simplest_in_interval()  # simplest in (-inf, inf)
Fraction(0, 1)