Source code for sympy.core.compatibility

Reimplementations of constructs introduced in later versions of Python than
we support. Also some functions that are needed SymPy-wide and are located
here for easy import.
from __future__ import print_function, division

import operator
from collections import defaultdict
from sympy.external import import_module

Python 2 and Python 3 compatible imports

String and Unicode compatible changes:
    * `unicode()` removed in Python 3, import `unicode` for Python 2/3
      compatible function
    * `unichr()` removed in Python 3, import `unichr` for Python 2/3 compatible
    * Use `u()` for escaped unicode sequences (e.g. u'\u2020' -> u('\u2020'))
    * Use `u_decode()` to decode utf-8 formatted unicode strings
    * `string_types` gives str in Python 3, unicode and str in Python 2,
      equivalent to basestring

Integer related changes:
    * `long()` removed in Python 3, import `long` for Python 2/3 compatible
    * `integer_types` gives int in Python 3, int and long in Python 2

Types related changes:
    * `class_types` gives type in Python 3, type and ClassType in Python 2

Renamed function attributes:
    * Python 2 `.func_code`, Python 3 `.__func__`, access with
    * Python 2 `.func_globals`, Python 3 `.__globals__`, access with
    * Python 2 `.func_name`, Python 3 `.__name__`, access with

Moved modules:
    * `reduce()`
    * `StringIO()`
    * `cStringIO()` (same as `StingIO()` in Python 3)
    * Python 2 `__builtins__`, access with Python 3 name, `builtins`

Iterator/list changes:
    * `xrange` removed in Python 3, import `xrange` for Python 2/3 compatible
      iterator version of range

    * Use `exec_()`, with parameters `exec_(code, globs=None, locs=None)`

    * Use `with_metaclass()`, examples below
        * Define class `Foo` with metaclass `Meta`, and no parent:
            class Foo(with_metaclass(Meta)):
        * Define class `Foo` with metaclass `Meta` and parent class `Bar`:
            class Foo(with_metaclass(Meta, Bar)):

import sys
PY3 = sys.version_info[0] > 2

if PY3:
    class_types = type,
    integer_types = (int,)
    string_types = (str,)
    long = int

    # String / unicode compatibility
    unicode = str
    unichr = chr
    def u(x):
        return x
    def u_decode(x):
        return x

    Iterator = object

    # Moved definitions
    get_function_code = operator.attrgetter("__code__")
    get_function_globals = operator.attrgetter("__globals__")
    get_function_name = operator.attrgetter("__name__")

    import builtins
    from functools import reduce
    from io import StringIO
    cStringIO = StringIO

    exec_=getattr(builtins, "exec")

    import codecs
    import types

    class_types = (type, types.ClassType)
    integer_types = (int, long)
    string_types = (str, unicode)
    long = long

    # String / unicode compatibility
    unicode = unicode
    unichr = unichr
    def u(x):
        return codecs.unicode_escape_decode(x)[0]
    def u_decode(x):
        return x.decode('utf-8')

    class Iterator(object):
        def next(self):
            return type(self).__next__(self)

    # Moved definitions
    get_function_code = operator.attrgetter("func_code")
    get_function_globals = operator.attrgetter("func_globals")
    get_function_name = operator.attrgetter("func_name")

    import __builtin__ as builtins
    reduce = reduce
    from StringIO import StringIO
    from cStringIO import StringIO as cStringIO

    def exec_(_code_, _globs_=None, _locs_=None):
        """Execute code in a namespace."""
        if _globs_ is None:
            frame = sys._getframe(1)
            _globs_ = frame.f_globals
            if _locs_ is None:
                _locs_ = frame.f_locals
            del frame
        elif _locs_ is None:
            _locs_ = _globs_
        exec("exec _code_ in _globs_, _locs_")

def with_metaclass(meta, *bases):
    Create a base class with a metaclass.

    For example, if you have the metaclass

    >>> class Meta(type):
    ...     pass

    Use this as the metaclass by doing

    >>> from sympy.core.compatibility import with_metaclass
    >>> class MyClass(with_metaclass(Meta, object)):
    ...     pass

    This is equivalent to the Python 2::

        class MyClass(object):
            __metaclass__ = Meta

    or Python 3::

        class MyClass(object, metaclass=Meta):

    That is, the first argument is the metaclass, and the remaining arguments
    are the base classes. Note that if the base class is just ``object``, you
    may omit it.

    >>> MyClass.__mro__
    (<class 'MyClass'>, <... 'object'>)
    >>> type(MyClass)
    <class 'Meta'>

    # This requires a bit of explanation: the basic idea is to make a dummy
    # metaclass for one level of class instantiation that replaces itself with
    # the actual metaclass.
    # Code copied from the 'six' library.
    class metaclass(meta):
        def __new__(cls, name, this_bases, d):
            return meta(name, bases, d)
    return type.__new__(metaclass, "NewBase", (), {})

# These are in here because telling if something is an iterable just by calling
# hasattr(obj, "__iter__") behaves differently in Python 2 and Python 3.  In
# particular, hasattr(str, "__iter__") is False in Python 2 and True in Python 3.
# I think putting them here also makes it easier to use them in the core.

class NotIterable:
    Use this as mixin when creating a class which is not supposed to return
    true when iterable() is called on its instances. I.e. avoid infinite loop
    when calling e.g. list() on the instance

[docs]def iterable(i, exclude=(string_types, dict, NotIterable)): """ Return a boolean indicating whether ``i`` is SymPy iterable. True also indicates that the iterator is finite, i.e. you e.g. call list(...) on the instance. When SymPy is working with iterables, it is almost always assuming that the iterable is not a string or a mapping, so those are excluded by default. If you want a pure Python definition, make exclude=None. To exclude multiple items, pass them as a tuple. You can also set the _iterable attribute to True or False on your class, which will override the checks here, including the exclude test. As a rule of thumb, some SymPy functions use this to check if they should recursively map over an object. If an object is technically iterable in the Python sense but does not desire this behavior (e.g., because its iteration is not finite, or because iteration might induce an unwanted computation), it should disable it by setting the _iterable attribute to False. See also: is_sequence Examples ======== >>> from sympy.utilities.iterables import iterable >>> from sympy import Tuple >>> things = [[1], (1,), set([1]), Tuple(1), (j for j in [1, 2]), {1:2}, '1', 1] >>> for i in things: ... print('%s %s' % (iterable(i), type(i))) True <... 'list'> True <... 'tuple'> True <... 'set'> True <class 'sympy.core.containers.Tuple'> True <... 'generator'> False <... 'dict'> False <... 'str'> False <... 'int'> >>> iterable({}, exclude=None) True >>> iterable({}, exclude=str) True >>> iterable("no", exclude=str) False """ if hasattr(i, '_iterable'): return i._iterable try: iter(i) except TypeError: return False if exclude: return not isinstance(i, exclude) return True
[docs]def is_sequence(i, include=None): """ Return a boolean indicating whether ``i`` is a sequence in the SymPy sense. If anything that fails the test below should be included as being a sequence for your application, set 'include' to that object's type; multiple types should be passed as a tuple of types. Note: although generators can generate a sequence, they often need special handling to make sure their elements are captured before the generator is exhausted, so these are not included by default in the definition of a sequence. See also: iterable Examples ======== >>> from sympy.utilities.iterables import is_sequence >>> from types import GeneratorType >>> is_sequence([]) True >>> is_sequence(set()) False >>> is_sequence('abc') False >>> is_sequence('abc', include=str) True >>> generator = (c for c in 'abc') >>> is_sequence(generator) False >>> is_sequence(generator, include=(str, GeneratorType)) True """ return (hasattr(i, '__getitem__') and iterable(i) or bool(include) and isinstance(i, include))
try: from functools import cmp_to_key except ImportError: # <= Python 2.6 def cmp_to_key(mycmp): """ Convert a cmp= function into a key= function """ class K(object): def __init__(self, obj, *args): self.obj = obj def __lt__(self, other): return mycmp(self.obj, other.obj) < 0 def __gt__(self, other): return mycmp(self.obj, other.obj) > 0 def __eq__(self, other): return mycmp(self.obj, other.obj) == 0 def __le__(self, other): return mycmp(self.obj, other.obj) <= 0 def __ge__(self, other): return mycmp(self.obj, other.obj) >= 0 def __ne__(self, other): return mycmp(self.obj, other.obj) != 0 return K try: from itertools import zip_longest except ImportError: # <= Python 2.7 from itertools import izip_longest as zip_longest try: from string import maketrans except ImportError: maketrans = str.maketrans try: from itertools import combinations_with_replacement except ImportError: # <= Python 2.6 def combinations_with_replacement(iterable, r): """Return r length subsequences of elements from the input iterable allowing individual elements to be repeated more than once. Combinations are emitted in lexicographic sort order. So, if the input iterable is sorted, the combination tuples will be produced in sorted order. Elements are treated as unique based on their position, not on their value. So if the input elements are unique, the generated combinations will also be unique. See also: combinations Examples ======== >>> from sympy.core.compatibility import combinations_with_replacement >>> list(combinations_with_replacement('AB', 2)) [('A', 'A'), ('A', 'B'), ('B', 'B')] """ pool = tuple(iterable) n = len(pool) if not n and r: return indices = [0] * r yield tuple(pool[i] for i in indices) while True: for i in reversed(range(r)): if indices[i] != n - 1: break else: return indices[i:] = [indices[i] + 1] * (r - i) yield tuple(pool[i] for i in indices)
[docs]def as_int(n): """ Convert the argument to a builtin integer. The return value is guaranteed to be equal to the input. ValueError is raised if the input has a non-integral value. Examples ======== >>> from sympy.core.compatibility import as_int >>> from sympy import sqrt >>> 3.0 3.0 >>> as_int(3.0) # convert to int and test for equality 3 >>> int(sqrt(10)) 3 >>> as_int(sqrt(10)) Traceback (most recent call last): ... ValueError: ... is not an integer """ try: result = int(n) if result != n: raise TypeError except TypeError: raise ValueError('%s is not an integer' % n) return result
def default_sort_key(item, order=None): """Return a key that can be used for sorting. The key has the structure: (class_key, (len(args), args), exponent.sort_key(), coefficient) This key is supplied by the sort_key routine of Basic objects when ``item`` is a Basic object or an object (other than a string) that sympifies to a Basic object. Otherwise, this function produces the key. The ``order`` argument is passed along to the sort_key routine and is used to determine how the terms *within* an expression are ordered. (See examples below) ``order`` options are: 'lex', 'grlex', 'grevlex', and reversed values of the same (e.g. 'rev-lex'). The default order value is None (which translates to 'lex'). Examples ======== >>> from sympy import S, I, default_sort_key, sin, cos, sqrt >>> from sympy.core.function import UndefinedFunction >>> from sympy.abc import x The following are equivalent ways of getting the key for an object: >>> x.sort_key() == default_sort_key(x) True Here are some examples of the key that is produced: >>> default_sort_key(UndefinedFunction('f')) ((0, 0, 'UndefinedFunction'), (1, ('f',)), ((1, 0, 'Number'), (0, ()), (), 1), 1) >>> default_sort_key('1') ((0, 0, 'str'), (1, ('1',)), ((1, 0, 'Number'), (0, ()), (), 1), 1) >>> default_sort_key(S.One) ((1, 0, 'Number'), (0, ()), (), 1) >>> default_sort_key(2) ((1, 0, 'Number'), (0, ()), (), 2) While sort_key is a method only defined for SymPy objects, default_sort_key will accept anything as an argument so it is more robust as a sorting key. For the following, using key= lambda i: i.sort_key() would fail because 2 doesn't have a sort_key method; that's why default_sort_key is used. Note, that it also handles sympification of non-string items likes ints: >>> a = [2, I, -I] >>> sorted(a, key=default_sort_key) [2, -I, I] The returned key can be used anywhere that a key can be specified for a function, e.g. sort, min, max, etc...: >>> a.sort(key=default_sort_key); a[0] 2 >>> min(a, key=default_sort_key) 2 Note ---- The key returned is useful for getting items into a canonical order that will be the same across platforms. It is not directly useful for sorting lists of expressions: >>> a, b = x, 1/x Since ``a`` has only 1 term, its value of sort_key is unaffected by ``order``: >>> a.sort_key() == a.sort_key('rev-lex') True If ``a`` and ``b`` are combined then the key will differ because there are terms that can be ordered: >>> eq = a + b >>> eq.sort_key() == eq.sort_key('rev-lex') False >>> eq.as_ordered_terms() [x, 1/x] >>> eq.as_ordered_terms('rev-lex') [1/x, x] But since the keys for each of these terms are independent of ``order``'s value, they don't sort differently when they appear separately in a list: >>> sorted(eq.args, key=default_sort_key) [1/x, x] >>> sorted(eq.args, key=lambda i: default_sort_key(i, order='rev-lex')) [1/x, x] The order of terms obtained when using these keys is the order that would be obtained if those terms were *factors* in a product. Although it is useful for quickly putting expressions in canonical order, it does not sort expressions based on their complexity defined by the number of operations, power of variables and others: >>> sorted([sin(x)*cos(x), sin(x)], key=default_sort_key) [sin(x)*cos(x), sin(x)] >>> sorted([x, x**2, sqrt(x), x**3], key=default_sort_key) [sqrt(x), x, x**2, x**3] See Also ======== ordered, sympy.core.expr.as_ordered_factors, sympy.core.expr.as_ordered_terms """ from .singleton import S from .basic import Basic from .sympify import sympify, SympifyError from .compatibility import iterable if isinstance(item, Basic): return item.sort_key(order=order) if iterable(item, exclude=string_types): if isinstance(item, dict): args = item.items() unordered = True elif isinstance(item, set): args = item unordered = True else: # e.g. tuple, list args = list(item) unordered = False args = [default_sort_key(arg, order=order) for arg in args] if unordered: # e.g. dict, set args = sorted(args) cls_index, args = 10, (len(args), tuple(args)) else: if not isinstance(item, string_types): try: item = sympify(item) except SympifyError: # e.g. lambda x: x pass else: if isinstance(item, Basic): # e.g int -> Integer return default_sort_key(item) # e.g. UndefinedFunction # e.g. str cls_index, args = 0, (1, (str(item),)) return (cls_index, 0, item.__class__.__name__ ), args, S.One.sort_key(), S.One def _nodes(e): """ A helper for ordered() which returns the node count of ``e`` which for Basic objects is the number of Basic nodes in the expression tree but for other objects is 1 (unless the object is an iterable or dict for which the sum of nodes is returned). """ from .basic import Basic if isinstance(e, Basic): return e.count(Basic) elif iterable(e): return 1 + sum(_nodes(ei) for ei in e) elif isinstance(e, dict): return 1 + sum(_nodes(k) + _nodes(v) for k, v in e.items()) else: return 1 def ordered(seq, keys=None, default=True, warn=False): """Return an iterator of the seq where keys are used to break ties in a conservative fashion: if, after applying a key, there are no ties then no other keys will be computed. Two default keys will be applied if 1) keys are not provided or 2) the given keys don't resolve all ties (but only if `default` is True). The two keys are `_nodes` (which places smaller expressions before large) and `default_sort_key` which (if the `sort_key` for an object is defined properly) should resolve any ties. If ``warn`` is True then an error will be raised if there were no keys remaining to break ties. This can be used if it was expected that there should be no ties between items that are not identical. Examples ======== >>> from sympy.utilities.iterables import ordered >>> from sympy import count_ops >>> from sympy.abc import x, y The count_ops is not sufficient to break ties in this list and the first two items appear in their original order (i.e. the sorting is stable): >>> list(ordered([y + 2, x + 2, x**2 + y + 3], ... count_ops, default=False, warn=False)) ... [y + 2, x + 2, x**2 + y + 3] The default_sort_key allows the tie to be broken: >>> list(ordered([y + 2, x + 2, x**2 + y + 3])) ... [x + 2, y + 2, x**2 + y + 3] Here, sequences are sorted by length, then sum: >>> seq, keys = [[[1, 2, 1], [0, 3, 1], [1, 1, 3], [2], [1]], [ ... lambda x: len(x), ... lambda x: sum(x)]] ... >>> list(ordered(seq, keys, default=False, warn=False)) [[1], [2], [1, 2, 1], [0, 3, 1], [1, 1, 3]] If ``warn`` is True, an error will be raised if there were not enough keys to break ties: >>> list(ordered(seq, keys, default=False, warn=True)) Traceback (most recent call last): ... ValueError: not enough keys to break ties Notes ===== The decorated sort is one of the fastest ways to sort a sequence for which special item comparison is desired: the sequence is decorated, sorted on the basis of the decoration (e.g. making all letters lower case) and then undecorated. If one wants to break ties for items that have the same decorated value, a second key can be used. But if the second key is expensive to compute then it is inefficient to decorate all items with both keys: only those items having identical first key values need to be decorated. This function applies keys successively only when needed to break ties. By yielding an iterator, use of the tie-breaker is delayed as long as possible. This function is best used in cases when use of the first key is expected to be a good hashing function; if there are no unique hashes from application of a key then that key should not have been used. The exception, however, is that even if there are many collisions, if the first group is small and one does not need to process all items in the list then time will not be wasted sorting what one was not interested in. For example, if one were looking for the minimum in a list and there were several criteria used to define the sort order, then this function would be good at returning that quickly if the first group of candidates is small relative to the number of items being processed. """ d = defaultdict(list) if keys: if not isinstance(keys, (list, tuple)): keys = [keys] keys = list(keys) f = keys.pop(0) for a in seq: d[f(a)].append(a) else: if not default: raise ValueError('if default=False then keys must be provided') d[None].extend(seq) for k in sorted(d.keys()): if len(d[k]) > 1: if keys: d[k] = ordered(d[k], keys, default, warn) elif default: d[k] = ordered(d[k], (_nodes, default_sort_key,), default=False, warn=warn) elif warn: from sympy.utilities.iterables import uniq u = list(uniq(d[k])) if len(u) > 1: raise ValueError( 'not enough keys to break ties: %s' % u) for v in d[k]: yield v d.pop(k) # If HAS_GMPY is 0, no supported version of gmpy is available. Otherwise, # HAS_GMPY contains the major version number of gmpy; i.e. 1 for gmpy, and # 2 for gmpy2. # Versions of gmpy prior to 1.03 do not work correctly with int(largempz) # For example, int(gmpy.mpz(2**256)) would raise OverflowError. # See issue 4980. # Minimum version of gmpy changed to 1.13 to allow a single code base to also # work with gmpy2. def _getenv(key, default=None): from os import getenv return getenv(key, default) GROUND_TYPES = _getenv('SYMPY_GROUND_TYPES', 'auto').lower() HAS_GMPY = 0 if GROUND_TYPES != 'python': # Don't try to import gmpy2 if ground types is set to gmpy1. This is # primarily intended for testing. if GROUND_TYPES != 'gmpy1': gmpy = import_module('gmpy2', min_module_version='2.0.0', module_version_attr='version', module_version_attr_call_args=()) if gmpy: HAS_GMPY = 2 else: GROUND_TYPES = 'gmpy' if not HAS_GMPY: gmpy = import_module('gmpy', min_module_version='1.13', module_version_attr='version', module_version_attr_call_args=()) if gmpy: HAS_GMPY = 1 if GROUND_TYPES == 'auto': if HAS_GMPY: GROUND_TYPES = 'gmpy' else: GROUND_TYPES = 'python' if GROUND_TYPES == 'gmpy' and not HAS_GMPY: from warnings import warn warn("gmpy library is not installed, switching to 'python' ground types") GROUND_TYPES = 'python' # SYMPY_INTS is a tuple containing the base types for valid integer types. SYMPY_INTS = integer_types if GROUND_TYPES == 'gmpy': SYMPY_INTS += (type(gmpy.mpz(0)),) # check_output() is new in Python 2.7 import os try: try: from subprocess import check_output except ImportError: # <= Python 2.6 from subprocess import CalledProcessError, check_call def check_output(*args, **kwargs): with open(os.devnull, 'w') as fh: kwargs['stdout'] = fh try: return check_call(*args, **kwargs) except CalledProcessError as e: e.output = ("program output is not available for Python 2.6.x") raise e except ImportError: # running on platform like App Engine, no subprocess at all pass # lru_cache compatible with py2.6->py3.2 copied directly from # http://code.activestate.com/ # recipes/578078-py26-and-py30-backport-of-python-33s-lru-cache/ from collections import namedtuple from functools import update_wrapper from threading import RLock _CacheInfo = namedtuple("CacheInfo", ["hits", "misses", "maxsize", "currsize"]) class _HashedSeq(list): __slots__ = 'hashvalue' def __init__(self, tup, hash=hash): self[:] = tup self.hashvalue = hash(tup) def __hash__(self): return self.hashvalue def _make_key(args, kwds, typed, kwd_mark = (object(),), fasttypes = set((int, str, frozenset, type(None))), sorted=sorted, tuple=tuple, type=type, len=len): 'Make a cache key from optionally typed positional and keyword arguments' key = args if kwds: sorted_items = sorted(kwds.items()) key += kwd_mark for item in sorted_items: key += item if typed: key += tuple(type(v) for v in args) if kwds: key += tuple(type(v) for k, v in sorted_items) elif len(key) == 1 and type(key[0]) in fasttypes: return key[0] return _HashedSeq(key) def lru_cache(maxsize=100, typed=False): """Least-recently-used cache decorator. If *maxsize* is set to None, the LRU features are disabled and the cache can grow without bound. If *typed* is True, arguments of different types will be cached separately. For example, f(3.0) and f(3) will be treated as distinct calls with distinct results. Arguments to the cached function must be hashable. View the cache statistics named tuple (hits, misses, maxsize, currsize) with f.cache_info(). Clear the cache and statistics with f.cache_clear(). Access the underlying function with f.__wrapped__. See: http://en.wikipedia.org/wiki/Cache_algorithms#Least_Recently_Used """ # Users should only access the lru_cache through its public API: # cache_info, cache_clear, and f.__wrapped__ # The internals of the lru_cache are encapsulated for thread safety and # to allow the implementation to change (including a possible C version). def decorating_function(user_function): cache = dict() stats = [0, 0] # make statistics updateable non-locally HITS, MISSES = 0, 1 # names for the stats fields make_key = _make_key cache_get = cache.get # bound method to lookup key or return None _len = len # localize the global len() function lock = RLock() # because linkedlist updates aren't threadsafe root = [] # root of the circular doubly linked list root[:] = [root, root, None, None] # initialize by pointing to self nonlocal_root = [root] # make updateable non-locally PREV, NEXT, KEY, RESULT = 0, 1, 2, 3 # names for the link fields if maxsize == 0: def wrapper(*args, **kwds): # no caching, just do a statistics update after a successful call result = user_function(*args, **kwds) stats[MISSES] += 1 return result elif maxsize is None: def wrapper(*args, **kwds): # simple caching without ordering or size limit key = make_key(args, kwds, typed) result = cache_get(key, root) # root used here as a unique not-found sentinel if result is not root: stats[HITS] += 1 return result result = user_function(*args, **kwds) cache[key] = result stats[MISSES] += 1 return result else: def wrapper(*args, **kwds): # size limited caching that tracks accesses by recency try: key = make_key(args, kwds, typed) if kwds or typed else args except TypeError: stats[MISSES] += 1 return user_function(*args, **kwds) with lock: link = cache_get(key) if link is not None: # record recent use of the key by moving it to the front of the list root, = nonlocal_root link_prev, link_next, key, result = link link_prev[NEXT] = link_next link_next[PREV] = link_prev last = root[PREV] last[NEXT] = root[PREV] = link link[PREV] = last link[NEXT] = root stats[HITS] += 1 return result result = user_function(*args, **kwds) with lock: root, = nonlocal_root if key in cache: # getting here means that this same key was added to the # cache while the lock was released. since the link # update is already done, we need only return the # computed result and update the count of misses. pass elif _len(cache) >= maxsize: # use the old root to store the new key and result oldroot = root oldroot[KEY] = key oldroot[RESULT] = result # empty the oldest link and make it the new root root = nonlocal_root[0] = oldroot[NEXT] oldkey = root[KEY] oldvalue = root[RESULT] root[KEY] = root[RESULT] = None # now update the cache dictionary for the new links del cache[oldkey] cache[key] = oldroot else: # put result in a new link at the front of the list last = root[PREV] link = [last, root, key, result] last[NEXT] = root[PREV] = cache[key] = link stats[MISSES] += 1 return result def cache_info(): """Report cache statistics""" with lock: return _CacheInfo(stats[HITS], stats[MISSES], maxsize, len(cache)) def cache_clear(): """Clear the cache and cache statistics""" with lock: cache.clear() root = nonlocal_root[0] root[:] = [root, root, None, None] stats[:] = [0, 0] wrapper.__wrapped__ = user_function wrapper.cache_info = cache_info wrapper.cache_clear = cache_clear return update_wrapper(wrapper, user_function) return decorating_function ### End of backported lru_cache if sys.version_info[:2] >= (3, 3): # 3.2 has an lru_cache with an incompatible API from functools import lru_cache