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Are you defining a numeric interface that should work with more than just ints and floats? Are you annotating that interface for documentation and type-checking? Were you excited by PEP 3141’s glitz and gloss promising a clean, straightforward number type definition mechanism, only to learn the hard way—after many hours of searching, tweaking, hacking, and testing ever more convoluted code, again and again—that you could’t actually make it work with Python’s type-checking system? Do you now wonder whether numbers were something new to computing in general because nothing else would explain such a gaping hole in a programming language so popular with the STEM crowd that has been around since the early 1990s? Does the number 3186 haunt you in your dreams? Do you find yourself shouting to no one in particular, “There has to be a better way?”

Well I’m here to tell you there isn’t. But until there is, there’s …

numerary—Now with Protocol Power™

That’s right!

For a hopefully limited time, you too can benefit from someone else’s deranged work-arounds for the enormous chasms in Python that lie between the esoteric fields of computation that are “typing” and “numbers” instead of having to roll your own out of sheer desperation from first principles! If you still have no idea what I’m talking about, this may help illustrate.

numerary is a pure-Python codified rant for signaling that your interface is usable with non-native numeric primitives¹ without breaking type-checking. More simply, numerary aspires to a world where numbers and types can work together.

If you’re thinking that you shouldn’t need a 🤬ing library for that, you’re right.

This madness should enjoy no audience. It should not exist. Yet here we are. Its author gauges its success by how quickly it can be forgotten, relegated to the annals of superfluous folly.

numerary is licensed under the MIT License. See the accompanying LICENSE file for details. It should be considered experimental for now, but should settle down quickly. See the release notes for a summary of version-to-version changes. Source code is available on GitHub.

If you find it lacking in any way, please don’t hesitate to bring it to my attention.

You had me at, “numbers and types can work together”

numerary strives to define composable, efficient protocols that one can use to construct numeric requirements. If all you deal with are integrals and reals, and what you want is broad arithmetic operator compatibility, this will probably get you where you likely want to go:

>>> from numerary import IntegralLike, RealLike

>>> def deeper_thot(arg: RealLike) -> IntegralLike:
...   assert arg != 0 and arg ** 0 == 1
...   return arg // arg + 42

Beyond default compositions for common use cases, numerary expands on the Supports pattern used in the standard library. For example, numerary.types.SupportsIntegralOps is a @typing.runtime_checkable protocol that approximates the unary and binary operators introduced by numbers.Integral.

>>> from numerary.types import SupportsIntegralOps

>>> def shift_right_one(arg: SupportsIntegralOps) -> SupportsIntegralOps:
...   assert isinstance(arg, SupportsIntegralOps)
...   return arg >> 1

>>> shift_right_one(2)
1

>>> from sympy import sympify
>>> two = sympify("2") ; type(two)
<class 'sympy.core.numbers.Integer'>
>>> res = shift_right_one(two) ; res
1
>>> type(res)
<class 'sympy.core.numbers.One'>

>>> from fractions import Fraction
>>> shift_right_one(Fraction(1, 2))  # type: ignore [arg-type]  # properly caught by Mypy
Traceback (most recent call last):
  ...
AssertionError

!!! note

Until 1.9, ``sympy.Integer`` [lacked the requisite bitwise operators](https://github.com/sympy/sympy/issues/19311).
``numerary`` catches that!
The above properly results in both a type-checking error as well as a runtime failure for [SymPy](https://www.sympy.org/) versions prior to 1.9.

numerary’s Supports protocols can be composed to refine requirements. For example, let’s say one wanted to ensure type compatibility with primitives that support both __abs__ and __divmod__.

>>> from typing import TypeVar
>>> T_co = TypeVar("T_co", covariant=True)
>>> from numerary.types import (
...   CachingProtocolMeta, Protocol, runtime_checkable,
...   SupportsAbs, SupportsDivmod,
... )

>>> @runtime_checkable
... class MyType(
...   SupportsAbs[T_co], SupportsDivmod[T_co],
...   Protocol, metaclass=CachingProtocolMeta,
... ):
...   pass

>>> my_type: MyType

>>> my_type = 3.5
>>> isinstance(my_type, MyType)
True
>>> abs(my_type)
3.5
>>> divmod(my_type, 2)
(1.0, 1.5)

>>> from fractions import Fraction
>>> my_type = Fraction(22, 7)
>>> isinstance(my_type, MyType)
True
>>> abs(my_type)
Fraction(22, 7)
>>> divmod(my_type, 2)
(1, Fraction(8, 7))

>>> from decimal import Decimal
>>> my_type = Decimal("5.2")
>>> isinstance(my_type, MyType)
True
>>> abs(my_type)
Decimal('5.2')
>>> divmod(my_type, 2)
(Decimal('2'), Decimal('1.2'))

>>> my_type = "nope"  # type: ignore [assignment]  # properly caught by Mypy
>>> isinstance(my_type, MyType)
False

Remember that scandal where complex defined exception-throwing comparators it wasn’t supposed to have, which confused runtime protocol checking, and then its type definitions lied about it to cover it up? Yeah, that shit ends here.

>>> from numerary.types import SupportsRealOps
>>> isinstance(1.0, SupportsRealOps)  # all good
True
>>> has_real_ops: SupportsRealOps = complex(1)  # type: ignore [assignment]  # properly caught by Mypy
>>> isinstance(complex(1), SupportsRealOps)  # you're not fooling anyone, buddy
False

numerary not only caches runtime protocol evaluations, but allows overriding those evaluations when the default machinery gets it wrong.

>>> from abc import abstractmethod
>>> from numerary.types import CachingProtocolMeta, Protocol, runtime_checkable

>>> @runtime_checkable
... class MySupportsOne(Protocol, metaclass=CachingProtocolMeta):
...   @abstractmethod
...   def one(self) -> int:
...     pass

>>> class Imposter:
...   def one(self) -> str:
...     return "one"

>>> imp: MySupportsOne = Imposter()  # type: ignore [assignment]  # properly caught by Mypy
>>> isinstance(imp, MySupportsOne)  # fool me once, shame on you ...
True

>>> MySupportsOne.excludes(Imposter)
>>> isinstance(imp, MySupportsOne)  # ... can't get fooled again
False

numerary has default overrides to correct for known oddities with native types (like our old friend, complex) and with popular libraries like numpy² and sympy. Others will be added as they are identified. If I’ve missed any, or if you would like numerary to support additional number implementations out of the box, please let me know.

Performance Enhanced Protocols—A different kind of “PEP” for your step

By default, protocols frustrate runtime type-checking performance.

A lot.

numerary applies two distinct, layered optimization strategies:

1. Cached __instancecheck__ results for numerary-defined protocols; and
2. Optional(-ish) short-circuit type enumerations.

Cached __instancecheck__ results

To understand why numerary protocols are faster for runtime checks, it helps to understand why non-numerary protocols are so slow. At runtime (i.e., via isinstance), the default Protocol implementation delegates to type(Protocol).__instancecheck__ to perform a crude comparison of an instance’s callable attributes against the protocol’s. More attributes means more comparisons. Further, it performs these comparisons … Every. Single. 🤬ing. Time.

Protocols provided by numerary use instead CachingProtocolMeta as their meta class. CachingProtocolMeta derives from type(beartype.typing.Protocol) which caches results based on instance type. numerary’s version allows for runtime check overrides of those results.

Conceptually:

>>> isinstance(1, SupportsIntegralOps)  # first check for an int is delegated to type(Protocol).__instancecheck__
True
>>> isinstance(2, SupportsIntegralOps)  # cached result
True
>>> isinstance(1.0, SupportsIntegralOps)  # the first check for a float is delegated to type(Protocol).__instancecheck__
False
>>> isinstance(2.0, SupportsIntegralOps)  # cached result
False

These offer significant performance improvements, especially where protocols define many methods.

--8<-- "docs/perf_supports_complex.txt"

Source: perf_supports_complex.ipy

--8<-- "docs/perf_supports_complex.ipy"

Unions for inclusion

Sometimes we might want types that don’t comply with protocol definitions to validate anyway (e.g., because we know they will work at runtime). For example, floats in Python versions prior to 3.9 officially lacked __floor__ and __ceil__ methods, but were registered with the numeric tower and worked just fine with math.floor and math.ceil.

How does numerary’s SupportsFloorCeil deal with this situation? Not very well, unfortunately, at least not on its own.

>>> import math, sys
>>> from numerary.types import SupportsFloorCeil

>>> def my_dumb_floor_func(arg: SupportsFloorCeil) -> int:
...   assert isinstance(arg, SupportsFloorCeil)  # will work, even for floats, thanks to default overrides
...   return math.floor(arg)  # type: ignore [arg-type]  # doesn't understand SupportsFloorCeil

>>> float_val: float = 1.6180339887
>>> # For illustration only until <https://github.com/python/mypy/issues/5940> is fixed
>>> if sys.version_info < (3, 9):
...   my_dumb_floor_func(float_val)  # type: ignore [arg-type]  # still results in a Mypy error for Python version <3.9
... else:
...   my_dumb_floor_func(float_val)  # validates
1

Unions allow a work-around.

>>> from typing import Union
>>> from numerary.types import SupportsFloorCeil, __floor__
>>> SupportsFloorCeilU = Union[float, SupportsFloorCeil]

>>> import sys
>>> def my_floor_func(arg: SupportsFloorCeilU) -> int:
...   assert isinstance(arg, SupportsFloorCeil)
...   return __floor__(arg)

>>> my_floor_func(float(1.2))  # works in 3.9+
1

This is largely a contrived example, since math.floor and math.ceil happily accept SupportsFloat, but it is useful for illustration.

Limitations

There are some downsides, though. (Aren’t there always?)

Sometimes protocols are too trusting

Protocols trust numeric tower registrations. TODO(@posita): Is this really true? But sometimes, out there in the real world, implementations lie.

Consider:

>>> from numbers import Integral
>>> hasattr(Integral, "real") and hasattr(Integral, "imag")
True
>>> import sympy
>>> pants_on_fire = sympy.Integer(1)
>>> isinstance(pants_on_fire, Integral)
True
>>> hasattr(pants_on_fire, "real") or hasattr(pants_on_fire, "imag")  # somebody's tellin' stories
False
>>> from numerary.types import SupportsRealImag
>>> real_imag: SupportsRealImag = pants_on_fire  # fails to detect the lie
>>> real_imag.real
Traceback (most recent call last):
  ...
AttributeError: 'One' object has no attribute 'real'

In this particular case, numerary provides us with a defensive mechanism.

>>> from numerary.types import SupportsRealImagMixedU, real, imag
>>> real_imag_defense: SupportsRealImagMixedU = pants_on_fire
>>> real(real_imag_defense)
1
>>> imag(real_imag)
0

Protocols loses fidelity during runtime checking

At runtime, protocols match names, not signatures. For example, SupportsNumeratorDenominator’s numerator and denominator properties will match sage.rings.integer.Integer’s similarly named functions. In other words, isinstance(sage_integer, SupportsNumeratorDenominator) will return True. Further, if the short-circuiting approach is used, because sage.rings.integer.Integer registers itself with the numeric tower, this may³ not be caught by Mypy.

>>> class SageLikeRational:
...   def __init__(self, numerator: int, denominator: int = 1):
...     self._numerator = numerator
...     self._denominator = denominator
...   def numerator(self) -> int:
...     return self._numerator
...   def denominator(self) -> int:
...     return self._denominator

>>> from numerary.types import SupportsNumeratorDenominator
>>> frac: SupportsNumeratorDenominator = Fraction(29, 3)  # no typing error
>>> sage_rational1: SupportsNumeratorDenominator = SageLikeRational(29, 3)  # type: ignore [assignment]  # Mypy catches this
>>> isinstance(sage_rational1, SupportsNumeratorDenominator)  # isinstance does not
True
>>> sage_rational1.numerator
<...method...numerator...>
>>> frac.numerator
29

Known warts could be cured by cache overriding as discussed above. However, to combat this particular situation, numerary provides an alternative: the SupportsNumeratorDenominatorMethods protocol and the numerator and denominator helper functions. These accommodate rational implementations like Sage’s that are mostly compliant with the exception of their respective numerator and denominator implementations.

>>> from numerary.types import numerator
>>> numerator(sage_rational1)
29
>>> numerator(frac)
29

>>> from numerary.types import SupportsNumeratorDenominatorMethods, numerator
>>> sage_rational2: SupportsNumeratorDenominatorMethods = SageLikeRational(3, 29)  # no type error
>>> numerator(sage_rational2)
3

numerary also defines:

SupportsNumeratorDenominatorMixedU = Union[
    SupportsNumeratorDenominator,
    SupportsNumeratorDenominatorMethods,
]
SupportsNumeratorDenominatorMixedT = (
    SupportsNumeratorDenominator,
    SupportsNumeratorDenominatorMethods,
)

>>> from numerary.types import SupportsNumeratorDenominatorMixedU, numerator
>>> chimera_rational: SupportsNumeratorDenominatorMixedU
>>> chimera_rational = Fraction(29, 3)  # no type error
>>> numerator(chimera_rational)
29
>>> chimera_rational = SageLikeRational(3, 29)  # still no type error
>>> numerator(chimera_rational)
3

The SupportsNumeratorDenominator* primitives provide the basis for analogous numerary.types.RationalLike* primitives, which should provide sufficient (if idiosyncratic) coverage for dealing with (seemingly mis-appropriately named) rationals.

Pass-through caching with composition is pretty sketchy

This is really getting into where the sausage is made, but full transparency is important, because CachingProtocolMeta does change how protocols are validated at runtime.

Let’s say we register an errant implementation as non-compliant using the CachingProtocolMeta.excludesmethod.

>>> from numerary.types import SupportsFloat

>>> class FloatImposter:
...   def __float__(self) -> float:
...     raise NotImplementedError("Haha! JK! @#$% you!")
...   def __int__(self) -> int:
...     return 42

>>> float_imp = FloatImposter()
>>> isinstance(float_imp, SupportsFloat)
True
>>> SupportsFloat.excludes(FloatImposter)
>>> isinstance(float_imp, SupportsFloat)
False

For registration to be ergonomic, it should be indelible, survive composition, and afford preference to subsequent overrides by inheritors.

>>> from numerary.types import (
...   CachingProtocolMeta, Protocol, runtime_checkable,
...   SupportsInt,
... )

>>> @runtime_checkable
... class MySupportsFloatInt(
...   SupportsFloat, SupportsInt,
...   Protocol,
... ):
...   pass

>>> isinstance(float_imp, MySupportsFloatInt)  # composition picks up override from base
False

>>> SupportsFloat.reset_for(FloatImposter)  # base resets override
>>> isinstance(float_imp, SupportsFloat)
True
>>> isinstance(float_imp, MySupportsFloatInt)  # picks up base’s changes
True

>>> MySupportsFloatInt.excludes(FloatImposter)  # composition overrides
>>> isinstance(float_imp, MySupportsFloatInt)
False
>>> SupportsFloat.includes(FloatImposter)  # base changes
>>> isinstance(float_imp, FloatImposter)
True
>>> isinstance(float_imp, MySupportsFloatInt)  # composition remains unchanged
False

>>> MySupportsFloatInt.reset_for(FloatImposter)  # removes override in composition
>>> isinstance(float_imp, MySupportsFloatInt)  # base is visible again
True
>>> SupportsFloat.excludes(FloatImposter)
>>> isinstance(float_imp, MySupportsFloatInt)  # base’s changes are visible to composition again
False

For this to work under the current implementation, we cannot rely exclusively on the standard library’s implementation of __instancecheck__, since it flattens and inspects all properties (with some proprietary exceptions) of all classes in the inheritance tree (in order of the MRO). In practical terms, this means one can’t easily delegate to an ancestor’s __instancecheck__ method and a protocol’s cache is effectively hidden from its progeny. In other words, leaning on the default behavior would require one to register exceptions with every inheritor. That would suck, so let’s not do that.

However, overriding the behavior is problematic, because the standard library uses non-public interfaces to perform its attribute enumeration. We certainly don’t want to re-implement protocol runtime checking from scratch. (At least not yet.)

beartype.typing.Protocol’s meta class tries to work around this by sneakily limiting its evaluation to directly defined attributes, and then delegating isinstance evaluation to its __base__ classes. In doing so, it picks up its bases’ then-cached values, but at the cost of re-implementing the attribute check as well as taking a dependency on various implementation details of the standard library, which creates a fragility. Further, for post-inheritance updates, CachingProtocolMeta extends beartype’s version to implement a simplistic publish/subscribe mechanism that dirties non-overridden caches in inheritors when member protocols caches are updated. That’s completely off the beaten path and there are probably some gremlins hiding out there.

One subtlety is that the implementation deviates from performing checks in MRO order (and may perform redundant checks). This is probably fine as long as runtime comparisons remain limited to crude checks whether attributes merely exist. It would likely fail if runtime checking becomes more sophisticated, at which time, this implementation will need to be revisited. Hopefully by then, we can just delete numerary as the aspirationally unnecessary hack it is and move on with our lives.

(See beartype.typing and numerary’s extension for details.)

License

numerary is licensed under the MIT License. See the included LICENSE file for details. Source code is available on GitHub.

Installation

Installation can be performed via PyPI.

% pip install numerary
...

Alternately, you can download the source and install manually.

% git clone https://github.com/posita/numerary.git
...
% cd numerary
% python -m pip install .  # -or- python -c 'from setuptools import setup ; setup()' install .
...

Requirements

numerary requires a relatively modern version of Python:

• CPython (3.9+)
• PyPy (CPython 3.9+ compatible) See beartype/beartype#324

It has the following runtime dependencies:

• beartype for caching protocols

numerary will not use beartype internally unless the NUMERARY_BEARTYPE environment variable is set to a truthy⁴ value before numerary is loaded.

See the hacking quick-start for additional development and testing dependencies.

Customers

• dyce - a pure-Python library for modeling arbitrarily complex dice mechanics and mother birthing code base of numerary!
• 👻 phantom-types - predicates and other type constraints without runtime overhead
• The next one could be you! 👋

Do you have a project that suffers problems made slightly less annoying by numerary? Let me know, and I’ll promote it here!

And don’t forget to do your part in perpetuating gratuitous badge-ification!

<!-- Markdown -->
As of version 0.4.1, ``dyce`` is
[![numerary-encumbered](https://raw.githubusercontent.com/posita/numerary/latest/docs/numerary-encumbered.svg)][numerary-encumbered]!
[numerary-encumbered]: https://posita.github.io/numerary/ "numerary-encumbered"

..
    reStructuredText - see https://docutils.sourceforge.io/docs/ref/rst/directives.html#image

As of version 0.4.1, ``dyce`` is |numerary-encumbered|!

.. |numerary-encumbered| image:: https://raw.githubusercontent.com/posita/numerary/latest/docs/numerary-encumbered.svg
   :align: top
   :target: https://posita.github.io/numerary/
   :alt: numerary-encumbered

<!-- HTML -->
As of version 0.4.1, <code>dyce</code> is <a href="https://posita.github.io/numerary/"><img
  src="https://raw.githubusercontent.com/posita/numerary/latest/docs/numerary-encumbered.svg"
  alt="numerary-encumbered"
  style="vertical-align: middle;"></a>!

Footnotes

1. You know, super weird, off-the-wall shit, like members of the numeric tower, or standard library primitives that remain non-members for some 🤬ed up reason, or legitimate non-members because they predate PEP 3141 and conforming would amount to breaking changes, or—I don’t know—oodles of libraries and applications that have been around for literally decades that bring huge value to vast scientific and mathematic audiences, but whose number primitives break type-checking if one abides by the ubiquitous bum steer, “I don’t have any experience trying to do what you’re doing, but just use float, bro.”

   Because, hey, 🤬 numbers! Am I right? ↩

2. !!! bug

   `numpy` no longer validates on assignment as it once did.
   See [posita/numerary#16](https://github.com/posita/numerary/issues/16) for details.
   

   ↩

3. I say may because I don’t really understand how Sage’s number registrations work. ↩

4. I.E., one of: 1, on, t, true, and yes. ↩