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Efficient pure-Python implementation of the integer square root function.

Project description

Efficient pure-Python implementation of the integer square root function.

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Given an arbitrarily large non-negative integer n, the integer square root function finds the largest integer r such that r**2 <= n and (r + 1)**2 > n.

The built-in math.isqrt function was introduced in Python 3.8 and should normally be used instead of the function defined in this library. To provide the best performance possible while retaining backwards-compatible behavior for this library, the implementation in this library invokes math.isqrt when it is available. If math.isqrt is not available, this library attempts to use math.sqrt and then (if math.sqrt does not produce the correct result or the input is outside the range supported by math.sqrt) defaults to a pure-Python implementation in which the number of executed Python arithmetic operations is linear in the bit length of the input integer.

Installation and Usage

This library is available as a package on PyPI:

python -m pip install isqrt

The library can be imported in the usual way:

from isqrt import isqrt


The exported function isqrt provides a pure-Python implementation of the integer square root algorithm:

>>> isqrt(4)
>>> isqrt(16)
>>> list(map(isqrt, range(16, 26)))
[4, 4, 4, 4, 4, 4, 4, 4, 4, 5]
>>> isqrt(2**30000) == 2**15000


All installation and development dependencies are fully specified in pyproject.toml. The project.optional-dependencies object is used to specify optional requirements for various development tasks. This makes it possible to specify additional options (such as test, lint, and so on) when performing installation using pip:

python -m pip install .[test,lint]

Testing and Conventions

All unit tests are executed and their coverage is measured when using pytest (see the pyproject.toml file for configuration details):

python -m pip install .[test]
python -m pytest

Alternatively, all unit tests are included in the module itself and can be executed using doctest:

python src/isqrt/ -v

Style conventions are enforced using Pylint:

python -m pip install .[lint]
python -m pylint src/isqrt


The initial version of this function was posted on Stack Overflow. A more efficient version was implemented by Alexander Gosselin. The implementation in this package is adapted directly from these previous implementations.


In order to contribute to the source code, open an issue or submit a pull request on the GitHub page for this library.


Beginning with version 0.10.0, the version number format for this library and the changes to the library associated with version number increments conform with Semantic Versioning 2.0.0.


This library can be published as a package on PyPI by a package maintainer. First, install the dependencies required for packaging and publishing:

python -m pip install .[publish]

Ensure that the correct version number appears in the pyproject.toml file. Create and push a tag for this version (replacing ?.?.? with the version number):

git tag ?.?.?
git push origin ?.?.?

Remove any old build/distribution files. Then, package the source into a distribution archive:

rm -rf build dist src/*.egg-info
python -m build --sdist --wheel .

Finally, upload the package distribution archive to PyPI:

python -m twine upload dist/*

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