I am done

rl/Lib/site-packages/sympy/external/__init__.py

@@ -0,0 +1,20 @@
+"""
+Unified place for determining if external dependencies are installed or not.
+
+You should import all external modules using the import_module() function.
+
+For example
+
+>>> from sympy.external import import_module
+>>> numpy = import_module('numpy')
+
+If the resulting library is not installed, or if the installed version
+is less than a given minimum version, the function will return None.
+Otherwise, it will return the library. See the docstring of
+import_module() for more information.
+
+"""
+
+from sympy.external.importtools import import_module
+
+__all__ = ['import_module']

rl/Lib/site-packages/sympy/external/gmpy.py

@@ -0,0 +1,341 @@
+import os
+from ctypes import c_long, sizeof
+from functools import reduce
+from typing import Tuple as tTuple, Type
+from warnings import warn
+
+from sympy.external import import_module
+
+from .pythonmpq import PythonMPQ
+
+from .ntheory import (
+    bit_scan1 as python_bit_scan1,
+    bit_scan0 as python_bit_scan0,
+    remove as python_remove,
+    factorial as python_factorial,
+    sqrt as python_sqrt,
+    sqrtrem as python_sqrtrem,
+    gcd as python_gcd,
+    lcm as python_lcm,
+    gcdext as python_gcdext,
+    is_square as python_is_square,
+    invert as python_invert,
+    legendre as python_legendre,
+    jacobi as python_jacobi,
+    kronecker as python_kronecker,
+    iroot as python_iroot,
+    is_fermat_prp as python_is_fermat_prp,
+    is_euler_prp as python_is_euler_prp,
+    is_strong_prp as python_is_strong_prp,
+    is_fibonacci_prp as python_is_fibonacci_prp,
+    is_lucas_prp as python_is_lucas_prp,
+    is_selfridge_prp as python_is_selfridge_prp,
+    is_strong_lucas_prp as python_is_strong_lucas_prp,
+    is_strong_selfridge_prp as python_is_strong_selfridge_prp,
+    is_bpsw_prp as python_is_bpsw_prp,
+    is_strong_bpsw_prp as python_is_strong_bpsw_prp,
+)
+
+
+__all__ = [
+    # GROUND_TYPES is either 'gmpy' or 'python' depending on which is used. If
+    # gmpy is installed then it will be used unless the environment variable
+    # SYMPY_GROUND_TYPES is set to something other than 'auto', 'gmpy', or
+    # 'gmpy2'.
+    'GROUND_TYPES',
+
+    # If HAS_GMPY is 0, no supported version of gmpy is available. Otherwise,
+    # HAS_GMPY will be 2 for gmpy2 if GROUND_TYPES is 'gmpy'. It used to be
+    # possible for HAS_GMPY to be 1 for gmpy but gmpy is no longer supported.
+    'HAS_GMPY',
+
+    # SYMPY_INTS is a tuple containing the base types for valid integer types.
+    # This is either (int,) or (int, type(mpz(0))) depending on GROUND_TYPES.
+    'SYMPY_INTS',
+
+    # MPQ is either gmpy.mpq or the Python equivalent from
+    # sympy.external.pythonmpq
+    'MPQ',
+
+    # MPZ is either gmpy.mpz or int.
+    'MPZ',
+
+    'bit_scan1',
+    'bit_scan0',
+    'remove',
+    'factorial',
+    'sqrt',
+    'is_square',
+    'sqrtrem',
+    'gcd',
+    'lcm',
+    'gcdext',
+    'invert',
+    'legendre',
+    'jacobi',
+    'kronecker',
+    'iroot',
+    'is_fermat_prp',
+    'is_euler_prp',
+    'is_strong_prp',
+    'is_fibonacci_prp',
+    'is_lucas_prp',
+    'is_selfridge_prp',
+    'is_strong_lucas_prp',
+    'is_strong_selfridge_prp',
+    'is_bpsw_prp',
+    'is_strong_bpsw_prp',
+]
+
+
+#
+# Tested python-flint version. Future versions might work but we will only use
+# them if explicitly requested by SYMPY_GROUND_TYPES=flint.
+#
+_PYTHON_FLINT_VERSION_NEEDED = ["0.6", "0.7", "0.8", "0.9"]
+
+
+def _flint_version_okay(flint_version):
+    major, minor = flint_version.split('.')[:2]
+    flint_ver = f'{major}.{minor}'
+    return flint_ver in _PYTHON_FLINT_VERSION_NEEDED
+
+#
+# We will only use gmpy2 >= 2.0.0
+#
+_GMPY2_MIN_VERSION = '2.0.0'
+
+
+def _get_flint(sympy_ground_types):
+    if sympy_ground_types not in ('auto', 'flint'):
+        return None
+
+    try:
+        import flint
+        # Earlier versions of python-flint may not have __version__.
+        from flint import __version__ as _flint_version
+    except ImportError:
+        if sympy_ground_types == 'flint':
+            warn("SYMPY_GROUND_TYPES was set to flint but python-flint is not "
+                 "installed. Falling back to other ground types.")
+        return None
+
+    if _flint_version_okay(_flint_version):
+        return flint
+    elif sympy_ground_types == 'auto':
+        return None
+    else:
+        warn(f"Using python-flint {_flint_version} because SYMPY_GROUND_TYPES "
+             f"is set to flint but this version of SymPy is only tested "
+             f"with python-flint versions {_PYTHON_FLINT_VERSION_NEEDED}.")
+        return flint
+
+
+def _get_gmpy2(sympy_ground_types):
+    if sympy_ground_types not in ('auto', 'gmpy', 'gmpy2'):
+        return None
+
+    gmpy = import_module('gmpy2', min_module_version=_GMPY2_MIN_VERSION,
+            module_version_attr='version', module_version_attr_call_args=())
+
+    if sympy_ground_types != 'auto' and gmpy is None:
+        warn("gmpy2 library is not installed, switching to 'python' ground types")
+
+    return gmpy
+
+
+#
+# SYMPY_GROUND_TYPES can be flint, gmpy, gmpy2, python or auto (default)
+#
+_SYMPY_GROUND_TYPES = os.environ.get('SYMPY_GROUND_TYPES', 'auto').lower()
+_flint = None
+_gmpy = None
+
+#
+# First handle auto-detection of flint/gmpy2. We will prefer flint if available
+# or otherwise gmpy2 if available and then lastly the python types.
+#
+if _SYMPY_GROUND_TYPES in ('auto', 'flint'):
+    _flint = _get_flint(_SYMPY_GROUND_TYPES)
+    if _flint is not None:
+        _SYMPY_GROUND_TYPES = 'flint'
+    else:
+        _SYMPY_GROUND_TYPES = 'auto'
+
+if _SYMPY_GROUND_TYPES in ('auto', 'gmpy', 'gmpy2'):
+    _gmpy = _get_gmpy2(_SYMPY_GROUND_TYPES)
+    if _gmpy is not None:
+        _SYMPY_GROUND_TYPES = 'gmpy'
+    else:
+        _SYMPY_GROUND_TYPES = 'python'
+
+if _SYMPY_GROUND_TYPES not in ('flint', 'gmpy', 'python'):
+    warn("SYMPY_GROUND_TYPES environment variable unrecognised. "
+         "Should be 'auto', 'flint', 'gmpy', 'gmpy2' or 'python'.")
+    _SYMPY_GROUND_TYPES = 'python'
+
+#
+# At this point _SYMPY_GROUND_TYPES is either flint, gmpy or python. The blocks
+# below define the values exported by this module in each case.
+#
+
+#
+# In gmpy2 and flint, there are functions that take a long (or unsigned long)
+# argument. That is, it is not possible to input a value larger than that.
+#
+LONG_MAX = (1 << (8*sizeof(c_long) - 1)) - 1
+
+#
+# Type checkers are confused by what SYMPY_INTS is. There may be a better type
+# hint for this like Type[Integral] or something.
+#
+SYMPY_INTS: tTuple[Type, ...]
+
+if _SYMPY_GROUND_TYPES == 'gmpy':
+
+    assert _gmpy is not None
+
+    flint = None
+    gmpy = _gmpy
+
+    HAS_GMPY = 2
+    GROUND_TYPES = 'gmpy'
+    SYMPY_INTS = (int, type(gmpy.mpz(0)))
+    MPZ = gmpy.mpz
+    MPQ = gmpy.mpq
+
+    bit_scan1 = gmpy.bit_scan1
+    bit_scan0 = gmpy.bit_scan0
+    remove = gmpy.remove
+    factorial = gmpy.fac
+    sqrt = gmpy.isqrt
+    is_square = gmpy.is_square
+    sqrtrem = gmpy.isqrt_rem
+    gcd = gmpy.gcd
+    lcm = gmpy.lcm
+    gcdext = gmpy.gcdext
+    invert = gmpy.invert
+    legendre = gmpy.legendre
+    jacobi = gmpy.jacobi
+    kronecker = gmpy.kronecker
+
+    def iroot(x, n):
+        # In the latest gmpy2, the threshold for n is ULONG_MAX,
+        # but adjust to the older one.
+        if n <= LONG_MAX:
+            return gmpy.iroot(x, n)
+        return python_iroot(x, n)
+
+    is_fermat_prp = gmpy.is_fermat_prp
+    is_euler_prp = gmpy.is_euler_prp
+    is_strong_prp = gmpy.is_strong_prp
+    is_fibonacci_prp = gmpy.is_fibonacci_prp
+    is_lucas_prp = gmpy.is_lucas_prp
+    is_selfridge_prp = gmpy.is_selfridge_prp
+    is_strong_lucas_prp = gmpy.is_strong_lucas_prp
+    is_strong_selfridge_prp = gmpy.is_strong_selfridge_prp
+    is_bpsw_prp = gmpy.is_bpsw_prp
+    is_strong_bpsw_prp = gmpy.is_strong_bpsw_prp
+
+elif _SYMPY_GROUND_TYPES == 'flint':
+
+    assert _flint is not None
+
+    flint = _flint
+    gmpy = None
+
+    HAS_GMPY = 0
+    GROUND_TYPES = 'flint'
+    SYMPY_INTS = (int, flint.fmpz) # type: ignore
+    MPZ = flint.fmpz # type: ignore
+    MPQ = flint.fmpq # type: ignore
+
+    bit_scan1 = python_bit_scan1
+    bit_scan0 = python_bit_scan0
+    remove = python_remove
+    factorial = python_factorial
+
+    def sqrt(x):
+        return flint.fmpz(x).isqrt()
+
+    def is_square(x):
+        if x < 0:
+            return False
+        return flint.fmpz(x).sqrtrem()[1] == 0
+
+    def sqrtrem(x):
+        return flint.fmpz(x).sqrtrem()
+
+    def gcd(*args):
+        return reduce(flint.fmpz.gcd, args, flint.fmpz(0))
+
+    def lcm(*args):
+        return reduce(flint.fmpz.lcm, args, flint.fmpz(1))
+
+    gcdext = python_gcdext
+    invert = python_invert
+    legendre = python_legendre
+
+    def jacobi(x, y):
+        if y <= 0 or not y % 2:
+            raise ValueError("y should be an odd positive integer")
+        return flint.fmpz(x).jacobi(y)
+
+    kronecker = python_kronecker
+
+    def iroot(x, n):
+        if n <= LONG_MAX:
+            y = flint.fmpz(x).root(n)
+            return y, y**n == x
+        return python_iroot(x, n)
+
+    is_fermat_prp = python_is_fermat_prp
+    is_euler_prp = python_is_euler_prp
+    is_strong_prp = python_is_strong_prp
+    is_fibonacci_prp = python_is_fibonacci_prp
+    is_lucas_prp = python_is_lucas_prp
+    is_selfridge_prp = python_is_selfridge_prp
+    is_strong_lucas_prp = python_is_strong_lucas_prp
+    is_strong_selfridge_prp = python_is_strong_selfridge_prp
+    is_bpsw_prp = python_is_bpsw_prp
+    is_strong_bpsw_prp = python_is_strong_bpsw_prp
+
+elif _SYMPY_GROUND_TYPES == 'python':
+
+    flint = None
+    gmpy = None
+
+    HAS_GMPY = 0
+    GROUND_TYPES = 'python'
+    SYMPY_INTS = (int,)
+    MPZ = int
+    MPQ = PythonMPQ
+
+    bit_scan1 = python_bit_scan1
+    bit_scan0 = python_bit_scan0
+    remove = python_remove
+    factorial = python_factorial
+    sqrt = python_sqrt
+    is_square = python_is_square
+    sqrtrem = python_sqrtrem
+    gcd = python_gcd
+    lcm = python_lcm
+    gcdext = python_gcdext
+    invert = python_invert
+    legendre = python_legendre
+    jacobi = python_jacobi
+    kronecker = python_kronecker
+    iroot = python_iroot
+    is_fermat_prp = python_is_fermat_prp
+    is_euler_prp = python_is_euler_prp
+    is_strong_prp = python_is_strong_prp
+    is_fibonacci_prp = python_is_fibonacci_prp
+    is_lucas_prp = python_is_lucas_prp
+    is_selfridge_prp = python_is_selfridge_prp
+    is_strong_lucas_prp = python_is_strong_lucas_prp
+    is_strong_selfridge_prp = python_is_strong_selfridge_prp
+    is_bpsw_prp = python_is_bpsw_prp
+    is_strong_bpsw_prp = python_is_strong_bpsw_prp
+
+else:
+    assert False

rl/Lib/site-packages/sympy/external/importtools.py

@@ -0,0 +1,187 @@
+"""Tools to assist importing optional external modules."""
+
+import sys
+import re
+
+# Override these in the module to change the default warning behavior.
+# For example, you might set both to False before running the tests so that
+# warnings are not printed to the console, or set both to True for debugging.
+
+WARN_NOT_INSTALLED = None  # Default is False
+WARN_OLD_VERSION = None  # Default is True
+
+
+def __sympy_debug():
+    # helper function from sympy/__init__.py
+    # We don't just import SYMPY_DEBUG from that file because we don't want to
+    # import all of SymPy just to use this module.
+    import os
+    debug_str = os.getenv('SYMPY_DEBUG', 'False')
+    if debug_str in ('True', 'False'):
+        return eval(debug_str)
+    else:
+        raise RuntimeError("unrecognized value for SYMPY_DEBUG: %s" %
+                           debug_str)
+
+if __sympy_debug():
+    WARN_OLD_VERSION = True
+    WARN_NOT_INSTALLED = True
+
+
+_component_re = re.compile(r'(\d+ | [a-z]+ | \.)', re.VERBOSE)
+
+def version_tuple(vstring):
+    # Parse a version string to a tuple e.g. '1.2' -> (1, 2)
+    # Simplified from distutils.version.LooseVersion which was deprecated in
+    # Python 3.10.
+    components = []
+    for x in _component_re.split(vstring):
+        if x and x != '.':
+            try:
+                x = int(x)
+            except ValueError:
+                pass
+            components.append(x)
+    return tuple(components)
+
+
+def import_module(module, min_module_version=None, min_python_version=None,
+        warn_not_installed=None, warn_old_version=None,
+        module_version_attr='__version__', module_version_attr_call_args=None,
+        import_kwargs={}, catch=()):
+    """
+    Import and return a module if it is installed.
+
+    If the module is not installed, it returns None.
+
+    A minimum version for the module can be given as the keyword argument
+    min_module_version.  This should be comparable against the module version.
+    By default, module.__version__ is used to get the module version.  To
+    override this, set the module_version_attr keyword argument.  If the
+    attribute of the module to get the version should be called (e.g.,
+    module.version()), then set module_version_attr_call_args to the args such
+    that module.module_version_attr(*module_version_attr_call_args) returns the
+    module's version.
+
+    If the module version is less than min_module_version using the Python <
+    comparison, None will be returned, even if the module is installed. You can
+    use this to keep from importing an incompatible older version of a module.
+
+    You can also specify a minimum Python version by using the
+    min_python_version keyword argument.  This should be comparable against
+    sys.version_info.
+
+    If the keyword argument warn_not_installed is set to True, the function will
+    emit a UserWarning when the module is not installed.
+
+    If the keyword argument warn_old_version is set to True, the function will
+    emit a UserWarning when the library is installed, but cannot be imported
+    because of the min_module_version or min_python_version options.
+
+    Note that because of the way warnings are handled, a warning will be
+    emitted for each module only once.  You can change the default warning
+    behavior by overriding the values of WARN_NOT_INSTALLED and WARN_OLD_VERSION
+    in sympy.external.importtools.  By default, WARN_NOT_INSTALLED is False and
+    WARN_OLD_VERSION is True.
+
+    This function uses __import__() to import the module.  To pass additional
+    options to __import__(), use the import_kwargs keyword argument.  For
+    example, to import a submodule A.B, you must pass a nonempty fromlist option
+    to __import__.  See the docstring of __import__().
+
+    This catches ImportError to determine if the module is not installed.  To
+    catch additional errors, pass them as a tuple to the catch keyword
+    argument.
+
+    Examples
+    ========
+
+    >>> from sympy.external import import_module
+
+    >>> numpy = import_module('numpy')
+
+    >>> numpy = import_module('numpy', min_python_version=(2, 7),
+    ... warn_old_version=False)
+
+    >>> numpy = import_module('numpy', min_module_version='1.5',
+    ... warn_old_version=False) # numpy.__version__ is a string
+
+    >>> # gmpy does not have __version__, but it does have gmpy.version()
+
+    >>> gmpy = import_module('gmpy', min_module_version='1.14',
+    ... module_version_attr='version', module_version_attr_call_args=(),
+    ... warn_old_version=False)
+
+    >>> # To import a submodule, you must pass a nonempty fromlist to
+    >>> # __import__().  The values do not matter.
+    >>> p3 = import_module('mpl_toolkits.mplot3d',
+    ... import_kwargs={'fromlist':['something']})
+
+    >>> # matplotlib.pyplot can raise RuntimeError when the display cannot be opened
+    >>> matplotlib = import_module('matplotlib',
+    ... import_kwargs={'fromlist':['pyplot']}, catch=(RuntimeError,))
+
+    """
+    # keyword argument overrides default, and global variable overrides
+    # keyword argument.
+    warn_old_version = (WARN_OLD_VERSION if WARN_OLD_VERSION is not None
+        else warn_old_version or True)
+    warn_not_installed = (WARN_NOT_INSTALLED if WARN_NOT_INSTALLED is not None
+        else warn_not_installed or False)
+
+    import warnings
+
+    # Check Python first so we don't waste time importing a module we can't use
+    if min_python_version:
+        if sys.version_info < min_python_version:
+            if warn_old_version:
+                warnings.warn("Python version is too old to use %s "
+                    "(%s or newer required)" % (
+                        module, '.'.join(map(str, min_python_version))),
+                    UserWarning, stacklevel=2)
+            return
+
+    try:
+        mod = __import__(module, **import_kwargs)
+
+        ## there's something funny about imports with matplotlib and py3k. doing
+        ##    from matplotlib import collections
+        ## gives python's stdlib collections module. explicitly re-importing
+        ## the module fixes this.
+        from_list = import_kwargs.get('fromlist', ())
+        for submod in from_list:
+            if submod == 'collections' and mod.__name__ == 'matplotlib':
+                __import__(module + '.' + submod)
+    except ImportError:
+        if warn_not_installed:
+            warnings.warn("%s module is not installed" % module, UserWarning,
+                    stacklevel=2)
+        return
+    except catch as e:
+        if warn_not_installed:
+            warnings.warn(
+                "%s module could not be used (%s)" % (module, repr(e)),
+                stacklevel=2)
+        return
+
+    if min_module_version:
+        modversion = getattr(mod, module_version_attr)
+        if module_version_attr_call_args is not None:
+            modversion = modversion(*module_version_attr_call_args)
+        if version_tuple(modversion) < version_tuple(min_module_version):
+            if warn_old_version:
+                # Attempt to create a pretty string version of the version
+                if isinstance(min_module_version, str):
+                    verstr = min_module_version
+                elif isinstance(min_module_version, (tuple, list)):
+                    verstr = '.'.join(map(str, min_module_version))
+                else:
+                    # Either don't know what this is.  Hopefully
+                    # it's something that has a nice str version, like an int.
+                    verstr = str(min_module_version)
+                warnings.warn("%s version is too old to use "
+                    "(%s or newer required)" % (module, verstr),
+                    UserWarning, stacklevel=2)
+            return
+
+    return mod

rl/Lib/site-packages/sympy/external/ntheory.py

@@ -0,0 +1,637 @@
+# sympy.external.ntheory
+#
+# This module provides pure Python implementations of some number theory
+# functions that are alternately used from gmpy2 if it is installed.
+
+import sys
+import math
+
+import mpmath.libmp as mlib
+
+
+_small_trailing = [0] * 256
+for j in range(1, 8):
+    _small_trailing[1 << j :: 1 << (j + 1)] = [j] * (1 << (7 - j))
+
+
+def bit_scan1(x, n=0):
+    if not x:
+        return
+    x = abs(x >> n)
+    low_byte = x & 0xFF
+    if low_byte:
+        return _small_trailing[low_byte] + n
+
+    t = 8 + n
+    x >>= 8
+    # 2**m is quick for z up through 2**30
+    z = x.bit_length() - 1
+    if x == 1 << z:
+        return z + t
+
+    if z < 300:
+        # fixed 8-byte reduction
+        while not x & 0xFF:
+            x >>= 8
+            t += 8
+    else:
+        # binary reduction important when there might be a large
+        # number of trailing 0s
+        p = z >> 1
+        while not x & 0xFF:
+            while x & ((1 << p) - 1):
+                p >>= 1
+            x >>= p
+            t += p
+    return t + _small_trailing[x & 0xFF]
+
+
+def bit_scan0(x, n=0):
+    return bit_scan1(x + (1 << n), n)
+
+
+def remove(x, f):
+    if f < 2:
+        raise ValueError("factor must be > 1")
+    if x == 0:
+        return 0, 0
+    if f == 2:
+        b = bit_scan1(x)
+        return x >> b, b
+    m = 0
+    y, rem = divmod(x, f)
+    while not rem:
+        x = y
+        m += 1
+        if m > 5:
+            pow_list = [f**2]
+            while pow_list:
+                _f = pow_list[-1]
+                y, rem = divmod(x, _f)
+                if not rem:
+                    m += 1 << len(pow_list)
+                    x = y
+                    pow_list.append(_f**2)
+                else:
+                    pow_list.pop()
+        y, rem = divmod(x, f)
+    return x, m
+
+
+def factorial(x):
+    """Return x!."""
+    return int(mlib.ifac(int(x)))
+
+
+def sqrt(x):
+    """Integer square root of x."""
+    return int(mlib.isqrt(int(x)))
+
+
+def sqrtrem(x):
+    """Integer square root of x and remainder."""
+    s, r = mlib.sqrtrem(int(x))
+    return (int(s), int(r))
+
+
+if sys.version_info[:2] >= (3, 9):
+    # As of Python 3.9 these can take multiple arguments
+    gcd = math.gcd
+    lcm = math.lcm
+
+else:
+    # Until python 3.8 is no longer supported
+    from functools import reduce
+
+
+    def gcd(*args):
+        """gcd of multiple integers."""
+        return reduce(math.gcd, args, 0)
+
+
+    def lcm(*args):
+        """lcm of multiple integers."""
+        if 0 in args:
+            return 0
+        return reduce(lambda x, y: x*y//math.gcd(x, y), args, 1)
+
+
+def _sign(n):
+    if n < 0:
+        return -1, -n
+    return 1, n
+
+
+def gcdext(a, b):
+    if not a or not b:
+        g = abs(a) or abs(b)
+        if not g:
+            return (0, 0, 0)
+        return (g, a // g, b // g)
+
+    x_sign, a = _sign(a)
+    y_sign, b = _sign(b)
+    x, r = 1, 0
+    y, s = 0, 1
+
+    while b:
+        q, c = divmod(a, b)
+        a, b = b, c
+        x, r = r, x - q*r
+        y, s = s, y - q*s
+
+    return (a, x * x_sign, y * y_sign)
+
+
+def is_square(x):
+    """Return True if x is a square number."""
+    if x < 0:
+        return False
+
+    # Note that the possible values of y**2 % n for a given n are limited.
+    # For example, when n=4, y**2 % n can only take 0 or 1.
+    # In other words, if x % 4 is 2 or 3, then x is not a square number.
+    # Mathematically, it determines if it belongs to the set {y**2 % n},
+    # but implementationally, it can be realized as a logical conjunction
+    # with an n-bit integer.
+    # see https://mersenneforum.org/showpost.php?p=110896
+    # def magic(n):
+    #     s = {y**2 % n for y in range(n)}
+    #     s = set(range(n)) - s
+    #     return sum(1 << bit for bit in s)
+    # >>> print(hex(magic(128)))
+    # 0xfdfdfdedfdfdfdecfdfdfdedfdfcfdec
+    # >>> print(hex(magic(99)))
+    # 0x5f6f9ffb6fb7ddfcb75befdec
+    # >>> print(hex(magic(91)))
+    # 0x6fd1bfcfed5f3679d3ebdec
+    # >>> print(hex(magic(85)))
+    # 0xdef9ae771ffe3b9d67dec
+    if 0xfdfdfdedfdfdfdecfdfdfdedfdfcfdec & (1 << (x & 127)):
+        return False  # e.g. 2, 3
+    m = x % 765765 # 765765 = 99 * 91 * 85
+    if 0x5f6f9ffb6fb7ddfcb75befdec & (1 << (m % 99)):
+        return False  # e.g. 17, 68
+    if 0x6fd1bfcfed5f3679d3ebdec & (1 << (m % 91)):
+        return False  # e.g. 97, 388
+    if 0xdef9ae771ffe3b9d67dec & (1 << (m % 85)):
+        return False  # e.g. 793, 1408
+    return mlib.sqrtrem(int(x))[1] == 0
+
+
+def invert(x, m):
+    """Modular inverse of x modulo m.
+
+    Returns y such that x*y == 1 mod m.
+
+    Uses ``math.pow`` but reproduces the behaviour of ``gmpy2.invert``
+    which raises ZeroDivisionError if no inverse exists.
+    """
+    try:
+        return pow(x, -1, m)
+    except ValueError:
+        raise ZeroDivisionError("invert() no inverse exists")
+
+
+def legendre(x, y):
+    """Legendre symbol (x / y).
+
+    Following the implementation of gmpy2,
+    the error is raised only when y is an even number.
+    """
+    if y <= 0 or not y % 2:
+        raise ValueError("y should be an odd prime")
+    x %= y
+    if not x:
+        return 0
+    if pow(x, (y - 1) // 2, y) == 1:
+        return 1
+    return -1
+
+
+def jacobi(x, y):
+    """Jacobi symbol (x / y)."""
+    if y <= 0 or not y % 2:
+        raise ValueError("y should be an odd positive integer")
+    x %= y
+    if not x:
+        return int(y == 1)
+    if y == 1 or x == 1:
+        return 1
+    if gcd(x, y) != 1:
+        return 0
+    j = 1
+    while x != 0:
+        while x % 2 == 0 and x > 0:
+            x >>= 1
+            if y % 8 in [3, 5]:
+                j = -j
+        x, y = y, x
+        if x % 4 == y % 4 == 3:
+            j = -j
+        x %= y
+    return j
+
+
+def kronecker(x, y):
+    """Kronecker symbol (x / y)."""
+    if gcd(x, y) != 1:
+        return 0
+    if y == 0:
+        return 1
+    sign = -1 if y < 0 and x < 0 else 1
+    y = abs(y)
+    s = bit_scan1(y)
+    y >>= s
+    if s % 2 and x % 8 in [3, 5]:
+        sign = -sign
+    return sign * jacobi(x, y)
+
+
+def iroot(y, n):
+    if y < 0:
+        raise ValueError("y must be nonnegative")
+    if n < 1:
+        raise ValueError("n must be positive")
+    if y in (0, 1):
+        return y, True
+    if n == 1:
+        return y, True
+    if n == 2:
+        x, rem = mlib.sqrtrem(y)
+        return int(x), not rem
+    if n >= y.bit_length():
+        return 1, False
+    # Get initial estimate for Newton's method. Care must be taken to
+    # avoid overflow
+    try:
+        guess = int(y**(1./n) + 0.5)
+    except OverflowError:
+        exp = math.log2(y)/n
+        if exp > 53:
+            shift = int(exp - 53)
+            guess = int(2.0**(exp - shift) + 1) << shift
+        else:
+            guess = int(2.0**exp)
+    if guess > 2**50:
+        # Newton iteration
+        xprev, x = -1, guess
+        while 1:
+            t = x**(n - 1)
+            xprev, x = x, ((n - 1)*x + y//t)//n
+            if abs(x - xprev) < 2:
+                break
+    else:
+        x = guess
+    # Compensate
+    t = x**n
+    while t < y:
+        x += 1
+        t = x**n
+    while t > y:
+        x -= 1
+        t = x**n
+    return x, t == y
+
+
+def is_fermat_prp(n, a):
+    if a < 2:
+        raise ValueError("is_fermat_prp() requires 'a' greater than or equal to 2")
+    if n < 1:
+        raise ValueError("is_fermat_prp() requires 'n' be greater than 0")
+    if n == 1:
+        return False
+    if n % 2 == 0:
+        return n == 2
+    a %= n
+    if gcd(n, a) != 1:
+        raise ValueError("is_fermat_prp() requires gcd(n,a) == 1")
+    return pow(a, n - 1, n) == 1
+
+
+def is_euler_prp(n, a):
+    if a < 2:
+        raise ValueError("is_euler_prp() requires 'a' greater than or equal to 2")
+    if n < 1:
+        raise ValueError("is_euler_prp() requires 'n' be greater than 0")
+    if n == 1:
+        return False
+    if n % 2 == 0:
+        return n == 2
+    a %= n
+    if gcd(n, a) != 1:
+        raise ValueError("is_euler_prp() requires gcd(n,a) == 1")
+    return pow(a, n >> 1, n) == jacobi(a, n) % n
+
+
+def _is_strong_prp(n, a):
+    s = bit_scan1(n - 1)
+    a = pow(a, n >> s, n)
+    if a == 1 or a == n - 1:
+        return True
+    for _ in range(s - 1):
+        a = pow(a, 2, n)
+        if a == n - 1:
+            return True
+        if a == 1:
+            return False
+    return False
+
+
+def is_strong_prp(n, a):
+    if a < 2:
+        raise ValueError("is_strong_prp() requires 'a' greater than or equal to 2")
+    if n < 1:
+        raise ValueError("is_strong_prp() requires 'n' be greater than 0")
+    if n == 1:
+        return False
+    if n % 2 == 0:
+        return n == 2
+    a %= n
+    if gcd(n, a) != 1:
+        raise ValueError("is_strong_prp() requires gcd(n,a) == 1")
+    return _is_strong_prp(n, a)
+
+
+def _lucas_sequence(n, P, Q, k):
+    r"""Return the modular Lucas sequence (U_k, V_k, Q_k).
+
+    Explanation
+    ===========
+
+    Given a Lucas sequence defined by P, Q, returns the kth values for
+    U and V, along with Q^k, all modulo n. This is intended for use with
+    possibly very large values of n and k, where the combinatorial functions
+    would be completely unusable.
+
+    .. math ::
+        U_k = \begin{cases}
+             0 & \text{if } k = 0\\
+             1 & \text{if } k = 1\\
+             PU_{k-1} - QU_{k-2} & \text{if } k > 1
+        \end{cases}\\
+        V_k = \begin{cases}
+             2 & \text{if } k = 0\\
+             P & \text{if } k = 1\\
+             PV_{k-1} - QV_{k-2} & \text{if } k > 1
+        \end{cases}
+
+    The modular Lucas sequences are used in numerous places in number theory,
+    especially in the Lucas compositeness tests and the various n + 1 proofs.
+
+    Parameters
+    ==========
+
+    n : int
+        n is an odd number greater than or equal to 3
+    P : int
+    Q : int
+        D determined by D = P**2 - 4*Q is non-zero
+    k : int
+        k is a nonnegative integer
+
+    Returns
+    =======
+
+    U, V, Qk : (int, int, int)
+        `(U_k \bmod{n}, V_k \bmod{n}, Q^k \bmod{n})`
+
+    Examples
+    ========
+
+    >>> from sympy.external.ntheory import _lucas_sequence
+    >>> N = 10**2000 + 4561
+    >>> sol = U, V, Qk = _lucas_sequence(N, 3, 1, N//2); sol
+    (0, 2, 1)
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Lucas_sequence
+
+    """
+    if k == 0:
+        return (0, 2, 1)
+    D = P**2 - 4*Q
+    U = 1
+    V = P
+    Qk = Q % n
+    if Q == 1:
+        # Optimization for extra strong tests.
+        for b in bin(k)[3:]:
+            U = (U*V) % n
+            V = (V*V - 2) % n
+            if b == "1":
+                U, V = U*P + V, V*P + U*D
+                if U & 1:
+                    U += n
+                if V & 1:
+                    V += n
+                U, V = U >> 1, V >> 1
+    elif P == 1 and Q == -1:
+        # Small optimization for 50% of Selfridge parameters.
+        for b in bin(k)[3:]:
+            U = (U*V) % n
+            if Qk == 1:
+                V = (V*V - 2) % n
+            else:
+                V = (V*V + 2) % n
+                Qk = 1
+            if b == "1":
+                # new_U = (U + V) // 2
+                # new_V = (5*U + V) // 2 = 2*U + new_U
+                U, V  = U + V, U << 1
+                if U & 1:
+                    U += n
+                U >>= 1
+                V += U
+                Qk = -1
+        Qk %= n
+    elif P == 1:
+        for b in bin(k)[3:]:
+            U = (U*V) % n
+            V = (V*V - 2*Qk) % n
+            Qk *= Qk
+            if b == "1":
+                # new_U = (U + V) // 2
+                # new_V = new_U - 2*Q*U
+                U, V  = U + V, (Q*U) << 1
+                if U & 1:
+                    U += n
+                U >>= 1
+                V = U - V
+                Qk *= Q
+            Qk %= n
+    else:
+        # The general case with any P and Q.
+        for b in bin(k)[3:]:
+            U = (U*V) % n
+            V = (V*V - 2*Qk) % n
+            Qk *= Qk
+            if b == "1":
+                U, V = U*P + V, V*P + U*D
+                if U & 1:
+                    U += n
+                if V & 1:
+                    V += n
+                U, V = U >> 1, V >> 1
+                Qk *= Q
+            Qk %= n
+    return (U % n, V % n, Qk)
+
+
+def is_fibonacci_prp(n, p, q):
+    d = p**2 - 4*q
+    if d == 0 or p <= 0 or q not in [1, -1]:
+        raise ValueError("invalid values for p,q in is_fibonacci_prp()")
+    if n < 1:
+        raise ValueError("is_fibonacci_prp() requires 'n' be greater than 0")
+    if n == 1:
+        return False
+    if n % 2 == 0:
+        return n == 2
+    return _lucas_sequence(n, p, q, n)[1] == p % n
+
+
+def is_lucas_prp(n, p, q):
+    d = p**2 - 4*q
+    if d == 0:
+        raise ValueError("invalid values for p,q in is_lucas_prp()")
+    if n < 1:
+        raise ValueError("is_lucas_prp() requires 'n' be greater than 0")
+    if n == 1:
+        return False
+    if n % 2 == 0:
+        return n == 2
+    if gcd(n, q*d) not in [1, n]:
+        raise ValueError("is_lucas_prp() requires gcd(n,2*q*D) == 1")
+    return _lucas_sequence(n, p, q, n - jacobi(d, n))[0] == 0
+
+
+def _is_selfridge_prp(n):
+    """Lucas compositeness test with the Selfridge parameters for n.
+
+    Explanation
+    ===========
+
+    The Lucas compositeness test checks whether n is a prime number.
+    The test can be run with arbitrary parameters ``P`` and ``Q``, which also change the performance of the test.
+    So, which parameters are most effective for running the Lucas compositeness test?
+    As an algorithm for determining ``P`` and ``Q``, Selfridge proposed method A [1]_ page 1401
+    (Since two methods were proposed, referred to simply as A and B in the paper,
+    we will refer to one of them as "method A").
+
+    method A fixes ``P = 1``. Then, ``D`` defined by ``D = P**2 - 4Q`` is varied from 5, -7, 9, -11, 13, and so on,
+    with the first ``D`` being ``jacobi(D, n) == -1``. Once ``D`` is determined,
+    ``Q`` is determined to be ``(P**2 - D)//4``.
+
+    References
+    ==========
+
+    .. [1] Robert Baillie, Samuel S. Wagstaff, Lucas Pseudoprimes,
+           Math. Comp. Vol 35, Number 152 (1980), pp. 1391-1417,
+           https://doi.org/10.1090%2FS0025-5718-1980-0583518-6
+           http://mpqs.free.fr/LucasPseudoprimes.pdf
+
+    """
+    for D in range(5, 1_000_000, 2):
+        if D & 2: # if D % 4 == 3
+            D = -D
+        j = jacobi(D, n)
+        if j == -1:
+            return _lucas_sequence(n, 1, (1-D) // 4, n + 1)[0] == 0
+        if j == 0 and D % n:
+            return False
+        # When j == -1 is hard to find, suspect a square number
+        if D == 13 and is_square(n):
+            return False
+    raise ValueError("appropriate value for D cannot be found in is_selfridge_prp()")
+
+
+def is_selfridge_prp(n):
+    if n < 1:
+        raise ValueError("is_selfridge_prp() requires 'n' be greater than 0")
+    if n == 1:
+        return False
+    if n % 2 == 0:
+        return n == 2
+    return _is_selfridge_prp(n)
+
+
+def is_strong_lucas_prp(n, p, q):
+    D = p**2 - 4*q
+    if D == 0:
+        raise ValueError("invalid values for p,q in is_strong_lucas_prp()")
+    if n < 1:
+        raise ValueError("is_selfridge_prp() requires 'n' be greater than 0")
+    if n == 1:
+        return False
+    if n % 2 == 0:
+        return n == 2
+    if gcd(n, q*D) not in [1, n]:
+        raise ValueError("is_strong_lucas_prp() requires gcd(n,2*q*D) == 1")
+    j = jacobi(D, n)
+    s = bit_scan1(n - j)
+    U, V, Qk = _lucas_sequence(n, p, q, (n - j) >> s)
+    if U == 0 or V == 0:
+        return True
+    for _ in range(s - 1):
+        V = (V*V - 2*Qk) % n
+        if V == 0:
+            return True
+        Qk = pow(Qk, 2, n)
+    return False
+
+
+def _is_strong_selfridge_prp(n):
+    for D in range(5, 1_000_000, 2):
+        if D & 2: # if D % 4 == 3
+            D = -D
+        j = jacobi(D, n)
+        if j == -1:
+            s = bit_scan1(n + 1)
+            U, V, Qk = _lucas_sequence(n, 1, (1-D) // 4, (n + 1) >> s)
+            if U == 0 or V == 0:
+                return True
+            for _ in range(s - 1):
+                V = (V*V - 2*Qk) % n
+                if V == 0:
+                    return True
+                Qk = pow(Qk, 2, n)
+            return False
+        if j == 0 and D % n:
+            return False
+        # When j == -1 is hard to find, suspect a square number
+        if D == 13 and is_square(n):
+            return False
+    raise ValueError("appropriate value for D cannot be found in is_strong_selfridge_prp()")
+
+
+def is_strong_selfridge_prp(n):
+    if n < 1:
+        raise ValueError("is_strong_selfridge_prp() requires 'n' be greater than 0")
+    if n == 1:
+        return False
+    if n % 2 == 0:
+        return n == 2
+    return _is_strong_selfridge_prp(n)
+
+
+def is_bpsw_prp(n):
+    if n < 1:
+        raise ValueError("is_bpsw_prp() requires 'n' be greater than 0")
+    if n == 1:
+        return False
+    if n % 2 == 0:
+        return n == 2
+    return _is_strong_prp(n, 2) and _is_selfridge_prp(n)
+
+
+def is_strong_bpsw_prp(n):
+    if n < 1:
+        raise ValueError("is_strong_bpsw_prp() requires 'n' be greater than 0")
+    if n == 1:
+        return False
+    if n % 2 == 0:
+        return n == 2
+    return _is_strong_prp(n, 2) and _is_strong_selfridge_prp(n)

rl/Lib/site-packages/sympy/external/pythonmpq.py

@@ -0,0 +1,341 @@
+"""
+PythonMPQ: Rational number type based on Python integers.
+
+This class is intended as a pure Python fallback for when gmpy2 is not
+installed. If gmpy2 is installed then its mpq type will be used instead. The
+mpq type is around 20x faster. We could just use the stdlib Fraction class
+here but that is slower:
+
+    from fractions import Fraction
+    from sympy.external.pythonmpq import PythonMPQ
+    nums = range(1000)
+    dens = range(5, 1005)
+    rats = [Fraction(n, d) for n, d in zip(nums, dens)]
+    sum(rats) # <--- 24 milliseconds
+    rats = [PythonMPQ(n, d) for n, d in zip(nums, dens)]
+    sum(rats) # <---  7 milliseconds
+
+Both mpq and Fraction have some awkward features like the behaviour of
+division with // and %:
+
+    >>> from fractions import Fraction
+    >>> Fraction(2, 3) % Fraction(1, 4)
+    1/6
+
+For the QQ domain we do not want this behaviour because there should be no
+remainder when dividing rational numbers. SymPy does not make use of this
+aspect of mpq when gmpy2 is installed. Since this class is a fallback for that
+case we do not bother implementing e.g. __mod__ so that we can be sure we
+are not using it when gmpy2 is installed either.
+"""
+
+
+import operator
+from math import gcd
+from decimal import Decimal
+from fractions import Fraction
+import sys
+from typing import Tuple as tTuple, Type
+
+
+# Used for __hash__
+_PyHASH_MODULUS = sys.hash_info.modulus
+_PyHASH_INF = sys.hash_info.inf
+
+
+class PythonMPQ:
+    """Rational number implementation that is intended to be compatible with
+    gmpy2's mpq.
+
+    Also slightly faster than fractions.Fraction.
+
+    PythonMPQ should be treated as immutable although no effort is made to
+    prevent mutation (since that might slow down calculations).
+    """
+    __slots__ = ('numerator', 'denominator')
+
+    def __new__(cls, numerator, denominator=None):
+        """Construct PythonMPQ with gcd computation and checks"""
+        if denominator is not None:
+            #
+            # PythonMPQ(n, d): require n and d to be int and d != 0
+            #
+            if isinstance(numerator, int) and isinstance(denominator, int):
+                # This is the slow part:
+                divisor = gcd(numerator, denominator)
+                numerator //= divisor
+                denominator //= divisor
+                return cls._new_check(numerator, denominator)
+        else:
+            #
+            # PythonMPQ(q)
+            #
+            # Here q can be PythonMPQ, int, Decimal, float, Fraction or str
+            #
+            if isinstance(numerator, int):
+                return cls._new(numerator, 1)
+            elif isinstance(numerator, PythonMPQ):
+                return cls._new(numerator.numerator, numerator.denominator)
+
+            # Let Fraction handle Decimal/float conversion and str parsing
+            if isinstance(numerator, (Decimal, float, str)):
+                numerator = Fraction(numerator)
+            if isinstance(numerator, Fraction):
+                return cls._new(numerator.numerator, numerator.denominator)
+        #
+        # Reject everything else. This is more strict than mpq which allows
+        # things like mpq(Fraction, Fraction) or mpq(Decimal, any). The mpq
+        # behaviour is somewhat inconsistent so we choose to accept only a
+        # more strict subset of what mpq allows.
+        #
+        raise TypeError("PythonMPQ() requires numeric or string argument")
+
+    @classmethod
+    def _new_check(cls, numerator, denominator):
+        """Construct PythonMPQ, check divide by zero and canonicalize signs"""
+        if not denominator:
+            raise ZeroDivisionError(f'Zero divisor {numerator}/{denominator}')
+        elif denominator < 0:
+            numerator = -numerator
+            denominator = -denominator
+        return cls._new(numerator, denominator)
+
+    @classmethod
+    def _new(cls, numerator, denominator):
+        """Construct PythonMPQ efficiently (no checks)"""
+        obj = super().__new__(cls)
+        obj.numerator = numerator
+        obj.denominator = denominator
+        return obj
+
+    def __int__(self):
+        """Convert to int (truncates towards zero)"""
+        p, q = self.numerator, self.denominator
+        if p < 0:
+            return -(-p//q)
+        return p//q
+
+    def __float__(self):
+        """Convert to float (approximately)"""
+        return self.numerator / self.denominator
+
+    def __bool__(self):
+        """True/False if nonzero/zero"""
+        return bool(self.numerator)
+
+    def __eq__(self, other):
+        """Compare equal with PythonMPQ, int, float, Decimal or Fraction"""
+        if isinstance(other, PythonMPQ):
+            return (self.numerator == other.numerator
+                and self.denominator == other.denominator)
+        elif isinstance(other, self._compatible_types):
+            return self.__eq__(PythonMPQ(other))
+        else:
+            return NotImplemented
+
+    def __hash__(self):
+        """hash - same as mpq/Fraction"""
+        try:
+            dinv = pow(self.denominator, -1, _PyHASH_MODULUS)
+        except ValueError:
+            hash_ = _PyHASH_INF
+        else:
+            hash_ = hash(hash(abs(self.numerator)) * dinv)
+        result = hash_ if self.numerator >= 0 else -hash_
+        return -2 if result == -1 else result
+
+    def __reduce__(self):
+        """Deconstruct for pickling"""
+        return type(self), (self.numerator, self.denominator)
+
+    def __str__(self):
+        """Convert to string"""
+        if self.denominator != 1:
+            return f"{self.numerator}/{self.denominator}"
+        else:
+            return f"{self.numerator}"
+
+    def __repr__(self):
+        """Convert to string"""
+        return f"MPQ({self.numerator},{self.denominator})"
+
+    def _cmp(self, other, op):
+        """Helper for lt/le/gt/ge"""
+        if not isinstance(other, self._compatible_types):
+            return NotImplemented
+        lhs = self.numerator * other.denominator
+        rhs = other.numerator * self.denominator
+        return op(lhs, rhs)
+
+    def __lt__(self, other):
+        """self < other"""
+        return self._cmp(other, operator.lt)
+
+    def __le__(self, other):
+        """self <= other"""
+        return self._cmp(other, operator.le)
+
+    def __gt__(self, other):
+        """self > other"""
+        return self._cmp(other, operator.gt)
+
+    def __ge__(self, other):
+        """self >= other"""
+        return self._cmp(other, operator.ge)
+
+    def __abs__(self):
+        """abs(q)"""
+        return self._new(abs(self.numerator), self.denominator)
+
+    def __pos__(self):
+        """+q"""
+        return self
+
+    def __neg__(self):
+        """-q"""
+        return self._new(-self.numerator, self.denominator)
+
+    def __add__(self, other):
+        """q1 + q2"""
+        if isinstance(other, PythonMPQ):
+            #
+            # This is much faster than the naive method used in the stdlib
+            # fractions module. Not sure where this method comes from
+            # though...
+            #
+            # Compare timings for something like:
+            #   nums = range(1000)
+            #   rats = [PythonMPQ(n, d) for n, d in zip(nums[:-5], nums[5:])]
+            #   sum(rats) # <-- time this
+            #
+            ap, aq = self.numerator, self.denominator
+            bp, bq = other.numerator, other.denominator
+            g = gcd(aq, bq)
+            if g == 1:
+                p = ap*bq + aq*bp
+                q = bq*aq
+            else:
+                q1, q2 = aq//g, bq//g
+                p, q = ap*q2 + bp*q1, q1*q2
+                g2 = gcd(p, g)
+                p, q = (p // g2), q * (g // g2)
+
+        elif isinstance(other, int):
+            p = self.numerator + self.denominator * other
+            q = self.denominator
+        else:
+            return NotImplemented
+
+        return self._new(p, q)
+
+    def __radd__(self, other):
+        """z1 + q2"""
+        if isinstance(other, int):
+            p = self.numerator + self.denominator * other
+            q = self.denominator
+            return self._new(p, q)
+        else:
+            return NotImplemented
+
+    def __sub__(self ,other):
+        """q1 - q2"""
+        if isinstance(other, PythonMPQ):
+            ap, aq = self.numerator, self.denominator
+            bp, bq = other.numerator, other.denominator
+            g = gcd(aq, bq)
+            if g == 1:
+                p = ap*bq - aq*bp
+                q = bq*aq
+            else:
+                q1, q2 = aq//g, bq//g
+                p, q = ap*q2 - bp*q1, q1*q2
+                g2 = gcd(p, g)
+                p, q = (p // g2), q * (g // g2)
+        elif isinstance(other, int):
+            p = self.numerator - self.denominator*other
+            q = self.denominator
+        else:
+            return NotImplemented
+
+        return self._new(p, q)
+
+    def __rsub__(self, other):
+        """z1 - q2"""
+        if isinstance(other, int):
+            p = self.denominator * other - self.numerator
+            q = self.denominator
+            return self._new(p, q)
+        else:
+            return NotImplemented
+
+    def __mul__(self, other):
+        """q1 * q2"""
+        if isinstance(other, PythonMPQ):
+            ap, aq = self.numerator, self.denominator
+            bp, bq = other.numerator, other.denominator
+            x1 = gcd(ap, bq)
+            x2 = gcd(bp, aq)
+            p, q = ((ap//x1)*(bp//x2), (aq//x2)*(bq//x1))
+        elif isinstance(other, int):
+            x = gcd(other, self.denominator)
+            p = self.numerator*(other//x)
+            q = self.denominator//x
+        else:
+            return NotImplemented
+
+        return self._new(p, q)
+
+    def __rmul__(self, other):
+        """z1 * q2"""
+        if isinstance(other, int):
+            x = gcd(self.denominator, other)
+            p = self.numerator*(other//x)
+            q = self.denominator//x
+            return self._new(p, q)
+        else:
+            return NotImplemented
+
+    def __pow__(self, exp):
+        """q ** z"""
+        p, q = self.numerator, self.denominator
+
+        if exp < 0:
+            p, q, exp = q, p, -exp
+
+        return self._new_check(p**exp, q**exp)
+
+    def __truediv__(self, other):
+        """q1 / q2"""
+        if isinstance(other, PythonMPQ):
+            ap, aq = self.numerator, self.denominator
+            bp, bq = other.numerator, other.denominator
+            x1 = gcd(ap, bp)
+            x2 = gcd(bq, aq)
+            p, q = ((ap//x1)*(bq//x2), (aq//x2)*(bp//x1))
+        elif isinstance(other, int):
+            x = gcd(other, self.numerator)
+            p = self.numerator//x
+            q = self.denominator*(other//x)
+        else:
+            return NotImplemented
+
+        return self._new_check(p, q)
+
+    def __rtruediv__(self, other):
+        """z / q"""
+        if isinstance(other, int):
+            x = gcd(self.numerator, other)
+            p = self.denominator*(other//x)
+            q = self.numerator//x
+            return self._new_check(p, q)
+        else:
+            return NotImplemented
+
+    _compatible_types: tTuple[Type, ...] = ()
+
+#
+# These are the types that PythonMPQ will interoperate with for operations
+# and comparisons such as ==, + etc. We define this down here so that we can
+# include PythonMPQ in the list as well.
+#
+PythonMPQ._compatible_types = (PythonMPQ, int, Decimal, Fraction)

rl/Lib/site-packages/sympy/external/tests/test_autowrap.py

@@ -0,0 +1,313 @@
+import sympy
+import tempfile
+import os
+from sympy.core.mod import Mod
+from sympy.core.relational import Eq
+from sympy.core.symbol import symbols
+from sympy.external import import_module
+from sympy.tensor import IndexedBase, Idx
+from sympy.utilities.autowrap import autowrap, ufuncify, CodeWrapError
+from sympy.testing.pytest import skip
+
+numpy = import_module('numpy', min_module_version='1.6.1')
+Cython = import_module('Cython', min_module_version='0.15.1')
+f2py = import_module('numpy.f2py', import_kwargs={'fromlist': ['f2py']})
+
+f2pyworks = False
+if f2py:
+    try:
+        autowrap(symbols('x'), 'f95', 'f2py')
+    except (CodeWrapError, ImportError, OSError):
+        f2pyworks = False
+    else:
+        f2pyworks = True
+
+a, b, c = symbols('a b c')
+n, m, d = symbols('n m d', integer=True)
+A, B, C = symbols('A B C', cls=IndexedBase)
+i = Idx('i', m)
+j = Idx('j', n)
+k = Idx('k', d)
+
+
+def has_module(module):
+    """
+    Return True if module exists, otherwise run skip().
+
+    module should be a string.
+    """
+    # To give a string of the module name to skip(), this function takes a
+    # string.  So we don't waste time running import_module() more than once,
+    # just map the three modules tested here in this dict.
+    modnames = {'numpy': numpy, 'Cython': Cython, 'f2py': f2py}
+
+    if modnames[module]:
+        if module == 'f2py' and not f2pyworks:
+            skip("Couldn't run f2py.")
+        return True
+    skip("Couldn't import %s." % module)
+
+#
+# test runners used by several language-backend combinations
+#
+
+def runtest_autowrap_twice(language, backend):
+    f = autowrap((((a + b)/c)**5).expand(), language, backend)
+    g = autowrap((((a + b)/c)**4).expand(), language, backend)
+
+    # check that autowrap updates the module name.  Else, g gives the same as f
+    assert f(1, -2, 1) == -1.0
+    assert g(1, -2, 1) == 1.0
+
+
+def runtest_autowrap_trace(language, backend):
+    has_module('numpy')
+    trace = autowrap(A[i, i], language, backend)
+    assert trace(numpy.eye(100)) == 100
+
+
+def runtest_autowrap_matrix_vector(language, backend):
+    has_module('numpy')
+    x, y = symbols('x y', cls=IndexedBase)
+    expr = Eq(y[i], A[i, j]*x[j])
+    mv = autowrap(expr, language, backend)
+
+    # compare with numpy's dot product
+    M = numpy.random.rand(10, 20)
+    x = numpy.random.rand(20)
+    y = numpy.dot(M, x)
+    assert numpy.sum(numpy.abs(y - mv(M, x))) < 1e-13
+
+
+def runtest_autowrap_matrix_matrix(language, backend):
+    has_module('numpy')
+    expr = Eq(C[i, j], A[i, k]*B[k, j])
+    matmat = autowrap(expr, language, backend)
+
+    # compare with numpy's dot product
+    M1 = numpy.random.rand(10, 20)
+    M2 = numpy.random.rand(20, 15)
+    M3 = numpy.dot(M1, M2)
+    assert numpy.sum(numpy.abs(M3 - matmat(M1, M2))) < 1e-13
+
+
+def runtest_ufuncify(language, backend):
+    has_module('numpy')
+    a, b, c = symbols('a b c')
+    fabc = ufuncify([a, b, c], a*b + c, backend=backend)
+    facb = ufuncify([a, c, b], a*b + c, backend=backend)
+    grid = numpy.linspace(-2, 2, 50)
+    b = numpy.linspace(-5, 4, 50)
+    c = numpy.linspace(-1, 1, 50)
+    expected = grid*b + c
+    numpy.testing.assert_allclose(fabc(grid, b, c), expected)
+    numpy.testing.assert_allclose(facb(grid, c, b), expected)
+
+
+def runtest_issue_10274(language, backend):
+    expr = (a - b + c)**(13)
+    tmp = tempfile.mkdtemp()
+    f = autowrap(expr, language, backend, tempdir=tmp,
+                 helpers=('helper', a - b + c, (a, b, c)))
+    assert f(1, 1, 1) == 1
+
+    for file in os.listdir(tmp):
+        if not (file.startswith("wrapped_code_") and file.endswith(".c")):
+            continue
+
+        with open(tmp + '/' + file) as fil:
+            lines = fil.readlines()
+            assert lines[0] == "/******************************************************************************\n"
+            assert "Code generated with SymPy " + sympy.__version__ in lines[1]
+            assert lines[2:] == [
+                " *                                                                            *\n",
+                " *              See http://www.sympy.org/ for more information.               *\n",
+                " *                                                                            *\n",
+                " *                      This file is part of 'autowrap'                       *\n",
+                " ******************************************************************************/\n",
+                "#include " + '"' + file[:-1]+ 'h"' + "\n",
+                "#include <math.h>\n",
+                "\n",
+                "double helper(double a, double b, double c) {\n",
+                "\n",
+                "   double helper_result;\n",
+                "   helper_result = a - b + c;\n",
+                "   return helper_result;\n",
+                "\n",
+                "}\n",
+                "\n",
+                "double autofunc(double a, double b, double c) {\n",
+                "\n",
+                "   double autofunc_result;\n",
+                "   autofunc_result = pow(helper(a, b, c), 13);\n",
+                "   return autofunc_result;\n",
+                "\n",
+                "}\n",
+                ]
+
+
+def runtest_issue_15337(language, backend):
+    has_module('numpy')
+    # NOTE : autowrap was originally designed to only accept an iterable for
+    # the kwarg "helpers", but in issue 10274 the user mistakenly thought that
+    # if there was only a single helper it did not need to be passed via an
+    # iterable that wrapped the helper tuple. There were no tests for this
+    # behavior so when the code was changed to accept a single tuple it broke
+    # the original behavior. These tests below ensure that both now work.
+    a, b, c, d, e = symbols('a, b, c, d, e')
+    expr = (a - b + c - d + e)**13
+    exp_res = (1. - 2. + 3. - 4. + 5.)**13
+
+    f = autowrap(expr, language, backend, args=(a, b, c, d, e),
+                 helpers=('f1', a - b + c, (a, b, c)))
+    numpy.testing.assert_allclose(f(1, 2, 3, 4, 5), exp_res)
+
+    f = autowrap(expr, language, backend, args=(a, b, c, d, e),
+                 helpers=(('f1', a - b, (a, b)), ('f2', c - d, (c, d))))
+    numpy.testing.assert_allclose(f(1, 2, 3, 4, 5), exp_res)
+
+
+def test_issue_15230():
+    has_module('f2py')
+
+    x, y = symbols('x, y')
+    expr = Mod(x, 3.0) - Mod(y, -2.0)
+    f = autowrap(expr, args=[x, y], language='F95')
+    exp_res = float(expr.xreplace({x: 3.5, y: 2.7}).evalf())
+    assert abs(f(3.5, 2.7) - exp_res) < 1e-14
+
+    x, y = symbols('x, y', integer=True)
+    expr = Mod(x, 3) - Mod(y, -2)
+    f = autowrap(expr, args=[x, y], language='F95')
+    assert f(3, 2) == expr.xreplace({x: 3, y: 2})
+
+#
+# tests of language-backend combinations
+#
+
+# f2py
+
+
+def test_wrap_twice_f95_f2py():
+    has_module('f2py')
+    runtest_autowrap_twice('f95', 'f2py')
+
+
+def test_autowrap_trace_f95_f2py():
+    has_module('f2py')
+    runtest_autowrap_trace('f95', 'f2py')
+
+
+def test_autowrap_matrix_vector_f95_f2py():
+    has_module('f2py')
+    runtest_autowrap_matrix_vector('f95', 'f2py')
+
+
+def test_autowrap_matrix_matrix_f95_f2py():
+    has_module('f2py')
+    runtest_autowrap_matrix_matrix('f95', 'f2py')
+
+
+def test_ufuncify_f95_f2py():
+    has_module('f2py')
+    runtest_ufuncify('f95', 'f2py')
+
+
+def test_issue_15337_f95_f2py():
+    has_module('f2py')
+    runtest_issue_15337('f95', 'f2py')
+
+# Cython
+
+
+def test_wrap_twice_c_cython():
+    has_module('Cython')
+    runtest_autowrap_twice('C', 'cython')
+
+
+def test_autowrap_trace_C_Cython():
+    has_module('Cython')
+    runtest_autowrap_trace('C99', 'cython')
+
+
+def test_autowrap_matrix_vector_C_cython():
+    has_module('Cython')
+    runtest_autowrap_matrix_vector('C99', 'cython')
+
+
+def test_autowrap_matrix_matrix_C_cython():
+    has_module('Cython')
+    runtest_autowrap_matrix_matrix('C99', 'cython')
+
+
+def test_ufuncify_C_Cython():
+    has_module('Cython')
+    runtest_ufuncify('C99', 'cython')
+
+
+def test_issue_10274_C_cython():
+    has_module('Cython')
+    runtest_issue_10274('C89', 'cython')
+
+
+def test_issue_15337_C_cython():
+    has_module('Cython')
+    runtest_issue_15337('C89', 'cython')
+
+
+def test_autowrap_custom_printer():
+    has_module('Cython')
+
+    from sympy.core.numbers import pi
+    from sympy.utilities.codegen import C99CodeGen
+    from sympy.printing.c import C99CodePrinter
+
+    class PiPrinter(C99CodePrinter):
+        def _print_Pi(self, expr):
+            return "S_PI"
+
+    printer = PiPrinter()
+    gen = C99CodeGen(printer=printer)
+    gen.preprocessor_statements.append('#include "shortpi.h"')
+
+    expr = pi * a
+
+    expected = (
+        '#include "%s"\n'
+        '#include <math.h>\n'
+        '#include "shortpi.h"\n'
+        '\n'
+        'double autofunc(double a) {\n'
+        '\n'
+        '   double autofunc_result;\n'
+        '   autofunc_result = S_PI*a;\n'
+        '   return autofunc_result;\n'
+        '\n'
+        '}\n'
+    )
+
+    tmpdir = tempfile.mkdtemp()
+    # write a trivial header file to use in the generated code
+    with open(os.path.join(tmpdir, 'shortpi.h'), 'w') as f:
+        f.write('#define S_PI 3.14')
+
+    func = autowrap(expr, backend='cython', tempdir=tmpdir, code_gen=gen)
+
+    assert func(4.2) == 3.14 * 4.2
+
+    # check that the generated code is correct
+    for filename in os.listdir(tmpdir):
+        if filename.startswith('wrapped_code') and filename.endswith('.c'):
+            with open(os.path.join(tmpdir, filename)) as f:
+                lines = f.readlines()
+                expected = expected % filename.replace('.c', '.h')
+                assert ''.join(lines[7:]) == expected
+
+
+# Numpy
+
+def test_ufuncify_numpy():
+    # This test doesn't use Cython, but if Cython works, then there is a valid
+    # C compiler, which is needed.
+    has_module('Cython')
+    runtest_ufuncify('C99', 'numpy')

rl/Lib/site-packages/sympy/external/tests/test_codegen.py

@@ -0,0 +1,379 @@
+# This tests the compilation and execution of the source code generated with
+# utilities.codegen. The compilation takes place in a temporary directory that
+# is removed after the test. By default the test directory is always removed,
+# but this behavior can be changed by setting the environment variable
+# SYMPY_TEST_CLEAN_TEMP to:
+#   export SYMPY_TEST_CLEAN_TEMP=always   : the default behavior.
+#   export SYMPY_TEST_CLEAN_TEMP=success  : only remove the directories of working tests.
+#   export SYMPY_TEST_CLEAN_TEMP=never    : never remove the directories with the test code.
+# When a directory is not removed, the necessary information is printed on
+# screen to find the files that belong to the (failed) tests. If a test does
+# not fail, py.test captures all the output and you will not see the directories
+# corresponding to the successful tests. Use the --nocapture option to see all
+# the output.
+
+# All tests below have a counterpart in utilities/test/test_codegen.py. In the
+# latter file, the resulting code is compared with predefined strings, without
+# compilation or execution.
+
+# All the generated Fortran code should conform with the Fortran 95 standard,
+# and all the generated C code should be ANSI C, which facilitates the
+# incorporation in various projects. The tests below assume that the binary cc
+# is somewhere in the path and that it can compile ANSI C code.
+
+from sympy.abc import x, y, z
+from sympy.external import import_module
+from sympy.testing.pytest import skip
+from sympy.utilities.codegen import codegen, make_routine, get_code_generator
+import sys
+import os
+import tempfile
+import subprocess
+
+
+pyodide_js = import_module('pyodide_js')
+
+# templates for the main program that will test the generated code.
+
+main_template = {}
+main_template['F95'] = """
+program main
+  include "codegen.h"
+  integer :: result;
+  result = 0
+
+  %(statements)s
+
+  call exit(result)
+end program
+"""
+
+main_template['C89'] = """
+#include "codegen.h"
+#include <stdio.h>
+#include <math.h>
+
+int main() {
+  int result = 0;
+
+  %(statements)s
+
+  return result;
+}
+"""
+main_template['C99'] = main_template['C89']
+# templates for the numerical tests
+
+numerical_test_template = {}
+numerical_test_template['C89'] = """
+  if (fabs(%(call)s)>%(threshold)s) {
+    printf("Numerical validation failed: %(call)s=%%e threshold=%(threshold)s\\n", %(call)s);
+    result = -1;
+  }
+"""
+numerical_test_template['C99'] = numerical_test_template['C89']
+
+numerical_test_template['F95'] = """
+  if (abs(%(call)s)>%(threshold)s) then
+    write(6,"('Numerical validation failed:')")
+    write(6,"('%(call)s=',e15.5,'threshold=',e15.5)") %(call)s, %(threshold)s
+    result = -1;
+  end if
+"""
+# command sequences for supported compilers
+
+compile_commands = {}
+compile_commands['cc'] = [
+    "cc -c codegen.c -o codegen.o",
+    "cc -c main.c -o main.o",
+    "cc main.o codegen.o -lm -o test.exe"
+]
+
+compile_commands['gfortran'] = [
+    "gfortran -c codegen.f90 -o codegen.o",
+    "gfortran -ffree-line-length-none -c main.f90 -o main.o",
+    "gfortran main.o codegen.o -o test.exe"
+]
+
+compile_commands['g95'] = [
+    "g95 -c codegen.f90 -o codegen.o",
+    "g95 -ffree-line-length-huge -c main.f90 -o main.o",
+    "g95 main.o codegen.o -o test.exe"
+]
+
+compile_commands['ifort'] = [
+    "ifort -c codegen.f90 -o codegen.o",
+    "ifort -c main.f90 -o main.o",
+    "ifort main.o codegen.o -o test.exe"
+]
+
+combinations_lang_compiler = [
+    ('C89', 'cc'),
+    ('C99', 'cc'),
+    ('F95', 'ifort'),
+    ('F95', 'gfortran'),
+    ('F95', 'g95')
+]
+
+
+def try_run(commands):
+    """Run a series of commands and only return True if all ran fine."""
+    if pyodide_js:
+        return False
+    with open(os.devnull, 'w') as null:
+        for command in commands:
+            retcode = subprocess.call(command, stdout=null, shell=True,
+                    stderr=subprocess.STDOUT)
+            if retcode != 0:
+                return False
+    return True
+
+
+def run_test(label, routines, numerical_tests, language, commands, friendly=True):
+    """A driver for the codegen tests.
+
+       This driver assumes that a compiler ifort is present in the PATH and that
+       ifort is (at least) a Fortran 90 compiler. The generated code is written in
+       a temporary directory, together with a main program that validates the
+       generated code. The test passes when the compilation and the validation
+       run correctly.
+    """
+
+    # Check input arguments before touching the file system
+    language = language.upper()
+    assert language in main_template
+    assert language in numerical_test_template
+
+    # Check that environment variable makes sense
+    clean = os.getenv('SYMPY_TEST_CLEAN_TEMP', 'always').lower()
+    if clean not in ('always', 'success', 'never'):
+        raise ValueError("SYMPY_TEST_CLEAN_TEMP must be one of the following: 'always', 'success' or 'never'.")
+
+    # Do all the magic to compile, run and validate the test code
+    # 1) prepare the temporary working directory, switch to that dir
+    work = tempfile.mkdtemp("_sympy_%s_test" % language, "%s_" % label)
+    oldwork = os.getcwd()
+    os.chdir(work)
+
+    # 2) write the generated code
+    if friendly:
+        # interpret the routines as a name_expr list and call the friendly
+        # function codegen
+        codegen(routines, language, "codegen", to_files=True)
+    else:
+        code_gen = get_code_generator(language, "codegen")
+        code_gen.write(routines, "codegen", to_files=True)
+
+    # 3) write a simple main program that links to the generated code, and that
+    #    includes the numerical tests
+    test_strings = []
+    for fn_name, args, expected, threshold in numerical_tests:
+        call_string = "%s(%s)-(%s)" % (
+            fn_name, ",".join(str(arg) for arg in args), expected)
+        if language == "F95":
+            call_string = fortranize_double_constants(call_string)
+            threshold = fortranize_double_constants(str(threshold))
+        test_strings.append(numerical_test_template[language] % {
+            "call": call_string,
+            "threshold": threshold,
+        })
+
+    if language == "F95":
+        f_name = "main.f90"
+    elif language.startswith("C"):
+        f_name = "main.c"
+    else:
+        raise NotImplementedError(
+            "FIXME: filename extension unknown for language: %s" % language)
+
+    with open(f_name, "w") as f:
+        f.write(
+            main_template[language] % {'statements': "".join(test_strings)})
+
+    # 4) Compile and link
+    compiled = try_run(commands)
+
+    # 5) Run if compiled
+    if compiled:
+        executed = try_run(["./test.exe"])
+    else:
+        executed = False
+
+    # 6) Clean up stuff
+    if clean == 'always' or (clean == 'success' and compiled and executed):
+        def safe_remove(filename):
+            if os.path.isfile(filename):
+                os.remove(filename)
+        safe_remove("codegen.f90")
+        safe_remove("codegen.c")
+        safe_remove("codegen.h")
+        safe_remove("codegen.o")
+        safe_remove("main.f90")
+        safe_remove("main.c")
+        safe_remove("main.o")
+        safe_remove("test.exe")
+        os.chdir(oldwork)
+        os.rmdir(work)
+    else:
+        print("TEST NOT REMOVED: %s" % work, file=sys.stderr)
+        os.chdir(oldwork)
+
+    # 7) Do the assertions in the end
+    assert compiled, "failed to compile %s code with:\n%s" % (
+        language, "\n".join(commands))
+    assert executed, "failed to execute %s code from:\n%s" % (
+        language, "\n".join(commands))
+
+
+def fortranize_double_constants(code_string):
+    """
+    Replaces every literal float with literal doubles
+    """
+    import re
+    pattern_exp = re.compile(r'\d+(\.)?\d*[eE]-?\d+')
+    pattern_float = re.compile(r'\d+\.\d*(?!\d*d)')
+
+    def subs_exp(matchobj):
+        return re.sub('[eE]', 'd', matchobj.group(0))
+
+    def subs_float(matchobj):
+        return "%sd0" % matchobj.group(0)
+
+    code_string = pattern_exp.sub(subs_exp, code_string)
+    code_string = pattern_float.sub(subs_float, code_string)
+
+    return code_string
+
+
+def is_feasible(language, commands):
+    # This test should always work, otherwise the compiler is not present.
+    routine = make_routine("test", x)
+    numerical_tests = [
+        ("test", ( 1.0,), 1.0, 1e-15),
+        ("test", (-1.0,), -1.0, 1e-15),
+    ]
+    try:
+        run_test("is_feasible", [routine], numerical_tests, language, commands,
+                 friendly=False)
+        return True
+    except AssertionError:
+        return False
+
+valid_lang_commands = []
+invalid_lang_compilers = []
+for lang, compiler in combinations_lang_compiler:
+    commands = compile_commands[compiler]
+    if is_feasible(lang, commands):
+        valid_lang_commands.append((lang, commands))
+    else:
+        invalid_lang_compilers.append((lang, compiler))
+
+# We test all language-compiler combinations, just to report what is skipped
+
+def test_C89_cc():
+    if ("C89", 'cc') in invalid_lang_compilers:
+        skip("`cc' command didn't work as expected (C89)")
+
+
+def test_C99_cc():
+    if ("C99", 'cc') in invalid_lang_compilers:
+        skip("`cc' command didn't work as expected (C99)")
+
+
+def test_F95_ifort():
+    if ("F95", 'ifort') in invalid_lang_compilers:
+        skip("`ifort' command didn't work as expected")
+
+
+def test_F95_gfortran():
+    if ("F95", 'gfortran') in invalid_lang_compilers:
+        skip("`gfortran' command didn't work as expected")
+
+
+def test_F95_g95():
+    if ("F95", 'g95') in invalid_lang_compilers:
+        skip("`g95' command didn't work as expected")
+
+# Here comes the actual tests
+
+
+def test_basic_codegen():
+    numerical_tests = [
+        ("test", (1.0, 6.0, 3.0), 21.0, 1e-15),
+        ("test", (-1.0, 2.0, -2.5), -2.5, 1e-15),
+    ]
+    name_expr = [("test", (x + y)*z)]
+    for lang, commands in valid_lang_commands:
+        run_test("basic_codegen", name_expr, numerical_tests, lang, commands)
+
+
+def test_intrinsic_math1_codegen():
+    # not included: log10
+    from sympy.core.evalf import N
+    from sympy.functions import ln
+    from sympy.functions.elementary.exponential import log
+    from sympy.functions.elementary.hyperbolic import (cosh, sinh, tanh)
+    from sympy.functions.elementary.integers import (ceiling, floor)
+    from sympy.functions.elementary.miscellaneous import sqrt
+    from sympy.functions.elementary.trigonometric import (acos, asin, atan, cos, sin, tan)
+    name_expr = [
+        ("test_fabs", abs(x)),
+        ("test_acos", acos(x)),
+        ("test_asin", asin(x)),
+        ("test_atan", atan(x)),
+        ("test_cos", cos(x)),
+        ("test_cosh", cosh(x)),
+        ("test_log", log(x)),
+        ("test_ln", ln(x)),
+        ("test_sin", sin(x)),
+        ("test_sinh", sinh(x)),
+        ("test_sqrt", sqrt(x)),
+        ("test_tan", tan(x)),
+        ("test_tanh", tanh(x)),
+    ]
+    numerical_tests = []
+    for name, expr in name_expr:
+        for xval in 0.2, 0.5, 0.8:
+            expected = N(expr.subs(x, xval))
+            numerical_tests.append((name, (xval,), expected, 1e-14))
+    for lang, commands in valid_lang_commands:
+        if lang.startswith("C"):
+            name_expr_C = [("test_floor", floor(x)), ("test_ceil", ceiling(x))]
+        else:
+            name_expr_C = []
+        run_test("intrinsic_math1", name_expr + name_expr_C,
+                 numerical_tests, lang, commands)
+
+
+def test_instrinsic_math2_codegen():
+    # not included: frexp, ldexp, modf, fmod
+    from sympy.core.evalf import N
+    from sympy.functions.elementary.trigonometric import atan2
+    name_expr = [
+        ("test_atan2", atan2(x, y)),
+        ("test_pow", x**y),
+    ]
+    numerical_tests = []
+    for name, expr in name_expr:
+        for xval, yval in (0.2, 1.3), (0.5, -0.2), (0.8, 0.8):
+            expected = N(expr.subs(x, xval).subs(y, yval))
+            numerical_tests.append((name, (xval, yval), expected, 1e-14))
+    for lang, commands in valid_lang_commands:
+        run_test("intrinsic_math2", name_expr, numerical_tests, lang, commands)
+
+
+def test_complicated_codegen():
+    from sympy.core.evalf import N
+    from sympy.functions.elementary.trigonometric import (cos, sin, tan)
+    name_expr = [
+        ("test1", ((sin(x) + cos(y) + tan(z))**7).expand()),
+        ("test2", cos(cos(cos(cos(cos(cos(cos(cos(x + y + z))))))))),
+    ]
+    numerical_tests = []
+    for name, expr in name_expr:
+        for xval, yval, zval in (0.2, 1.3, -0.3), (0.5, -0.2, 0.0), (0.8, 2.1, 0.8):
+            expected = N(expr.subs(x, xval).subs(y, yval).subs(z, zval))
+            numerical_tests.append((name, (xval, yval, zval), expected, 1e-12))
+    for lang, commands in valid_lang_commands:
+        run_test(
+            "complicated_codegen", name_expr, numerical_tests, lang, commands)

rl/Lib/site-packages/sympy/external/tests/test_gmpy.py

@@ -0,0 +1,12 @@
+from sympy.external.gmpy import LONG_MAX, iroot
+from sympy.testing.pytest import raises
+
+
+def test_iroot():
+    assert iroot(2, LONG_MAX) == (1, False)
+    assert iroot(2, LONG_MAX + 1) == (1, False)
+    for x in range(3):
+        assert iroot(x, 1) == (x, True)
+    raises(ValueError, lambda: iroot(-1, 1))
+    raises(ValueError, lambda: iroot(0, 0))
+    raises(ValueError, lambda: iroot(0, -1))

rl/Lib/site-packages/sympy/external/tests/test_importtools.py

@@ -0,0 +1,40 @@
+from sympy.external import import_module
+from sympy.testing.pytest import warns
+
+# fixes issue that arose in addressing issue 6533
+def test_no_stdlib_collections():
+    '''
+    make sure we get the right collections when it is not part of a
+    larger list
+    '''
+    import collections
+    matplotlib = import_module('matplotlib',
+        import_kwargs={'fromlist': ['cm', 'collections']},
+        min_module_version='1.1.0', catch=(RuntimeError,))
+    if matplotlib:
+        assert collections != matplotlib.collections
+
+def test_no_stdlib_collections2():
+    '''
+    make sure we get the right collections when it is not part of a
+    larger list
+    '''
+    import collections
+    matplotlib = import_module('matplotlib',
+        import_kwargs={'fromlist': ['collections']},
+        min_module_version='1.1.0', catch=(RuntimeError,))
+    if matplotlib:
+        assert collections != matplotlib.collections
+
+def test_no_stdlib_collections3():
+    '''make sure we get the right collections with no catch'''
+    import collections
+    matplotlib = import_module('matplotlib',
+        import_kwargs={'fromlist': ['cm', 'collections']},
+        min_module_version='1.1.0')
+    if matplotlib:
+        assert collections != matplotlib.collections
+
+def test_min_module_version_python3_basestring_error():
+    with warns(UserWarning):
+        import_module('mpmath', min_module_version='1000.0.1')

rl/Lib/site-packages/sympy/external/tests/test_ntheory.py

@@ -0,0 +1,307 @@
+from itertools import permutations
+
+from sympy.external.ntheory import (bit_scan1, remove, bit_scan0, is_fermat_prp,
+                                    is_euler_prp, is_strong_prp, gcdext, _lucas_sequence,
+                                    is_fibonacci_prp, is_lucas_prp, is_selfridge_prp,
+                                    is_strong_lucas_prp, is_strong_selfridge_prp,
+                                    is_bpsw_prp, is_strong_bpsw_prp)
+from sympy.testing.pytest import raises
+
+
+def test_bit_scan1():
+    assert bit_scan1(0) is None
+    assert bit_scan1(1) == 0
+    assert bit_scan1(-1) == 0
+    assert bit_scan1(2) == 1
+    assert bit_scan1(7) == 0
+    assert bit_scan1(-7) == 0
+    for i in range(100):
+        assert bit_scan1(1 << i) == i
+        assert bit_scan1((1 << i) * 31337) == i
+    for i in range(500):
+        n = (1 << 500) + (1 << i)
+        assert bit_scan1(n) == i
+    assert bit_scan1(1 << 1000001) == 1000001
+    assert bit_scan1((1 << 273956)*7**37) == 273956
+    # issue 12709
+    for i in range(1, 10):
+        big = 1 << i
+        assert bit_scan1(-big) == bit_scan1(big)
+
+
+def test_bit_scan0():
+    assert bit_scan0(-1) is None
+    assert bit_scan0(0) == 0
+    assert bit_scan0(1) == 1
+    assert bit_scan0(-2) == 0
+
+
+def test_remove():
+    raises(ValueError, lambda: remove(1, 1))
+    assert remove(0, 3) == (0, 0)
+    for f in range(2, 10):
+        for y in range(2, 1000):
+            for z in [1, 17, 101, 1009]:
+                assert remove(z*f**y, f) == (z, y)
+
+
+def test_gcdext():
+    assert gcdext(0, 0) == (0, 0, 0)
+    assert gcdext(3, 0) == (3, 1, 0)
+    assert gcdext(0, 4) == (4, 0, 1)
+
+    for n in range(1, 10):
+        assert gcdext(n, 1) == gcdext(-n, 1) == (1, 0, 1)
+        assert gcdext(n, -1) == gcdext(-n, -1) == (1, 0, -1)
+        assert gcdext(n, n) == gcdext(-n, n) == (n, 0, 1)
+        assert gcdext(n, -n) == gcdext(-n, -n) == (n, 0, -1)
+
+    for n in range(2, 10):
+        assert gcdext(1, n) == gcdext(1, -n) == (1, 1, 0)
+        assert gcdext(-1, n) == gcdext(-1, -n) == (1, -1, 0)
+
+    for a, b in permutations([2**5, 3, 5, 7**2, 11], 2):
+        g, x, y = gcdext(a, b)
+        assert g == a*x + b*y == 1
+
+
+def test_is_fermat_prp():
+    # invalid input
+    raises(ValueError, lambda: is_fermat_prp(0, 10))
+    raises(ValueError, lambda: is_fermat_prp(5, 1))
+
+    # n = 1
+    assert not is_fermat_prp(1, 3)
+
+    # n is prime
+    assert is_fermat_prp(2, 4)
+    assert is_fermat_prp(3, 2)
+    assert is_fermat_prp(11, 3)
+    assert is_fermat_prp(2**31-1, 5)
+
+    # A001567
+    pseudorpime = [341, 561, 645, 1105, 1387, 1729, 1905, 2047,
+                   2465, 2701, 2821, 3277, 4033, 4369, 4371, 4681]
+    for n in pseudorpime:
+        assert is_fermat_prp(n, 2)
+
+    # A020136
+    pseudorpime = [15, 85, 91, 341, 435, 451, 561, 645, 703, 1105,
+                   1247, 1271, 1387, 1581, 1695, 1729, 1891, 1905]
+    for n in pseudorpime:
+        assert is_fermat_prp(n, 4)
+
+
+def test_is_euler_prp():
+    # invalid input
+    raises(ValueError, lambda: is_euler_prp(0, 10))
+    raises(ValueError, lambda: is_euler_prp(5, 1))
+
+    # n = 1
+    assert not is_euler_prp(1, 3)
+
+    # n is prime
+    assert is_euler_prp(2, 4)
+    assert is_euler_prp(3, 2)
+    assert is_euler_prp(11, 3)
+    assert is_euler_prp(2**31-1, 5)
+
+    # A047713
+    pseudorpime = [561, 1105, 1729, 1905, 2047, 2465, 3277, 4033,
+                   4681, 6601, 8321, 8481, 10585, 12801, 15841]
+    for n in pseudorpime:
+        assert is_euler_prp(n, 2)
+
+    # A048950
+    pseudorpime = [121, 703, 1729, 1891, 2821, 3281, 7381, 8401,
+                   8911, 10585, 12403, 15457, 15841, 16531, 18721]
+    for n in pseudorpime:
+        assert is_euler_prp(n, 3)
+
+
+def test_is_strong_prp():
+    # invalid input
+    raises(ValueError, lambda: is_strong_prp(0, 10))
+    raises(ValueError, lambda: is_strong_prp(5, 1))
+
+    # n = 1
+    assert not is_strong_prp(1, 3)
+
+    # n is prime
+    assert is_strong_prp(2, 4)
+    assert is_strong_prp(3, 2)
+    assert is_strong_prp(11, 3)
+    assert is_strong_prp(2**31-1, 5)
+
+    # A001262
+    pseudorpime = [2047, 3277, 4033, 4681, 8321, 15841, 29341,
+                   42799, 49141, 52633, 65281, 74665, 80581]
+    for n in pseudorpime:
+        assert is_strong_prp(n, 2)
+
+    # A020229
+    pseudorpime = [121, 703, 1891, 3281, 8401, 8911, 10585, 12403,
+                   16531, 18721, 19345, 23521, 31621, 44287, 47197]
+    for n in pseudorpime:
+        assert is_strong_prp(n, 3)
+
+
+def test_lucas_sequence():
+    def lucas_u(P, Q, length):
+        array = [0] * length
+        array[1] = 1
+        for k in range(2, length):
+            array[k] = P * array[k - 1] - Q * array[k - 2]
+        return array
+
+    def lucas_v(P, Q, length):
+        array = [0] * length
+        array[0] = 2
+        array[1] = P
+        for k in range(2, length):
+            array[k] = P * array[k - 1] - Q * array[k - 2]
+        return array
+
+    length = 20
+    for P in range(-10, 10):
+        for Q in range(-10, 10):
+            D = P**2 - 4*Q
+            if D == 0:
+                continue
+            us = lucas_u(P, Q, length)
+            vs = lucas_v(P, Q, length)
+            for n in range(3, 100, 2):
+                for k in range(length):
+                    U, V, Qk = _lucas_sequence(n, P, Q, k)
+                    assert U == us[k] % n
+                    assert V == vs[k] % n
+                    assert pow(Q, k, n) == Qk
+
+
+def test_is_fibonacci_prp():
+    # invalid input
+    raises(ValueError, lambda: is_fibonacci_prp(3, 2, 1))
+    raises(ValueError, lambda: is_fibonacci_prp(3, -5, 1))
+    raises(ValueError, lambda: is_fibonacci_prp(3, 5, 2))
+    raises(ValueError, lambda: is_fibonacci_prp(0, 5, -1))
+
+    # n = 1
+    assert not is_fibonacci_prp(1, 3, 1)
+
+    # n is prime
+    assert is_fibonacci_prp(2, 5, 1)
+    assert is_fibonacci_prp(3, 6, -1)
+    assert is_fibonacci_prp(11, 7, 1)
+    assert is_fibonacci_prp(2**31-1, 8, -1)
+
+    # A005845
+    pseudorpime = [705, 2465, 2737, 3745, 4181, 5777, 6721,
+                   10877, 13201, 15251, 24465, 29281, 34561]
+    for n in pseudorpime:
+        assert is_fibonacci_prp(n, 1, -1)
+
+
+def test_is_lucas_prp():
+    # invalid input
+    raises(ValueError, lambda: is_lucas_prp(3, 2, 1))
+    raises(ValueError, lambda: is_lucas_prp(0, 5, -1))
+    raises(ValueError, lambda: is_lucas_prp(15, 3, 1))
+
+    # n = 1
+    assert not is_lucas_prp(1, 3, 1)
+
+    # n is prime
+    assert is_lucas_prp(2, 5, 2)
+    assert is_lucas_prp(3, 6, -1)
+    assert is_lucas_prp(11, 7, 5)
+    assert is_lucas_prp(2**31-1, 8, -3)
+
+    # A081264
+    pseudorpime = [323, 377, 1891, 3827, 4181, 5777, 6601, 6721,
+                   8149, 10877, 11663, 13201, 13981, 15251, 17119]
+    for n in pseudorpime:
+        assert is_lucas_prp(n, 1, -1)
+
+
+def test_is_selfridge_prp():
+    # invalid input
+    raises(ValueError, lambda: is_selfridge_prp(0))
+
+    # n = 1
+    assert not is_selfridge_prp(1)
+
+    # n is prime
+    assert is_selfridge_prp(2)
+    assert is_selfridge_prp(3)
+    assert is_selfridge_prp(11)
+    assert is_selfridge_prp(2**31-1)
+
+    # A217120
+    pseudorpime = [323, 377, 1159, 1829, 3827, 5459, 5777, 9071,
+                   9179, 10877, 11419, 11663, 13919, 14839, 16109]
+    for n in pseudorpime:
+        assert is_selfridge_prp(n)
+
+
+def test_is_strong_lucas_prp():
+    # invalid input
+    raises(ValueError, lambda: is_strong_lucas_prp(3, 2, 1))
+    raises(ValueError, lambda: is_strong_lucas_prp(0, 5, -1))
+    raises(ValueError, lambda: is_strong_lucas_prp(15, 3, 1))
+
+    # n = 1
+    assert not is_strong_lucas_prp(1, 3, 1)
+
+    # n is prime
+    assert is_strong_lucas_prp(2, 5, 2)
+    assert is_strong_lucas_prp(3, 6, -1)
+    assert is_strong_lucas_prp(11, 7, 5)
+    assert is_strong_lucas_prp(2**31-1, 8, -3)
+
+
+def test_is_strong_selfridge_prp():
+    # invalid input
+    raises(ValueError, lambda: is_strong_selfridge_prp(0))
+
+    # n = 1
+    assert not is_strong_selfridge_prp(1)
+
+    # n is prime
+    assert is_strong_selfridge_prp(2)
+    assert is_strong_selfridge_prp(3)
+    assert is_strong_selfridge_prp(11)
+    assert is_strong_selfridge_prp(2**31-1)
+
+    # A217255
+    pseudorpime = [5459, 5777, 10877, 16109, 18971, 22499, 24569,
+                   25199, 40309, 58519, 75077, 97439, 100127, 113573]
+    for n in pseudorpime:
+        assert is_strong_selfridge_prp(n)
+
+
+def test_is_bpsw_prp():
+    # invalid input
+    raises(ValueError, lambda: is_bpsw_prp(0))
+
+    # n = 1
+    assert not is_bpsw_prp(1)
+
+    # n is prime
+    assert is_bpsw_prp(2)
+    assert is_bpsw_prp(3)
+    assert is_bpsw_prp(11)
+    assert is_bpsw_prp(2**31-1)
+
+
+def test_is_strong_bpsw_prp():
+    # invalid input
+    raises(ValueError, lambda: is_strong_bpsw_prp(0))
+
+    # n = 1
+    assert not is_strong_bpsw_prp(1)
+
+    # n is prime
+    assert is_strong_bpsw_prp(2)
+    assert is_strong_bpsw_prp(3)
+    assert is_strong_bpsw_prp(11)
+    assert is_strong_bpsw_prp(2**31-1)

rl/Lib/site-packages/sympy/external/tests/test_numpy.py

@@ -0,0 +1,335 @@
+# This testfile tests SymPy <-> NumPy compatibility
+
+# Don't test any SymPy features here. Just pure interaction with NumPy.
+# Always write regular SymPy tests for anything, that can be tested in pure
+# Python (without numpy). Here we test everything, that a user may need when
+# using SymPy with NumPy
+from sympy.external.importtools import version_tuple
+from sympy.external import import_module
+
+numpy = import_module('numpy')
+if numpy:
+    array, matrix, ndarray = numpy.array, numpy.matrix, numpy.ndarray
+else:
+    #bin/test will not execute any tests now
+    disabled = True
+
+
+from sympy.core.numbers import (Float, Integer, Rational)
+from sympy.core.symbol import (Symbol, symbols)
+from sympy.functions.elementary.trigonometric import sin
+from sympy.matrices.dense import (Matrix, list2numpy, matrix2numpy, symarray)
+from sympy.utilities.lambdify import lambdify
+import sympy
+
+import mpmath
+from sympy.abc import x, y, z
+from sympy.utilities.decorator import conserve_mpmath_dps
+from sympy.utilities.exceptions import ignore_warnings
+from sympy.testing.pytest import raises
+
+
+# first, systematically check, that all operations are implemented and don't
+# raise an exception
+
+
+def test_systematic_basic():
+    def s(sympy_object, numpy_array):
+        _ = [sympy_object + numpy_array,
+        numpy_array + sympy_object,
+        sympy_object - numpy_array,
+        numpy_array - sympy_object,
+        sympy_object * numpy_array,
+        numpy_array * sympy_object,
+        sympy_object / numpy_array,
+        numpy_array / sympy_object,
+        sympy_object ** numpy_array,
+        numpy_array ** sympy_object]
+    x = Symbol("x")
+    y = Symbol("y")
+    sympy_objs = [
+        Rational(2, 3),
+        Float("1.3"),
+        x,
+        y,
+        pow(x, y)*y,
+        Integer(5),
+        Float(5.5),
+    ]
+    numpy_objs = [
+        array([1]),
+        array([3, 8, -1]),
+        array([x, x**2, Rational(5)]),
+        array([x/y*sin(y), 5, Rational(5)]),
+    ]
+    for x in sympy_objs:
+        for y in numpy_objs:
+            s(x, y)
+
+
+# now some random tests, that test particular problems and that also
+# check that the results of the operations are correct
+
+def test_basics():
+    one = Rational(1)
+    zero = Rational(0)
+    assert array(1) == array(one)
+    assert array([one]) == array([one])
+    assert array([x]) == array([x])
+    assert array(x) == array(Symbol("x"))
+    assert array(one + x) == array(1 + x)
+
+    X = array([one, zero, zero])
+    assert (X == array([one, zero, zero])).all()
+    assert (X == array([one, 0, 0])).all()
+
+
+def test_arrays():
+    one = Rational(1)
+    zero = Rational(0)
+    X = array([one, zero, zero])
+    Y = one*X
+    X = array([Symbol("a") + Rational(1, 2)])
+    Y = X + X
+    assert Y == array([1 + 2*Symbol("a")])
+    Y = Y + 1
+    assert Y == array([2 + 2*Symbol("a")])
+    Y = X - X
+    assert Y == array([0])
+
+
+def test_conversion1():
+    a = list2numpy([x**2, x])
+    #looks like an array?
+    assert isinstance(a, ndarray)
+    assert a[0] == x**2
+    assert a[1] == x
+    assert len(a) == 2
+    #yes, it's the array
+
+
+def test_conversion2():
+    a = 2*list2numpy([x**2, x])
+    b = list2numpy([2*x**2, 2*x])
+    assert (a == b).all()
+
+    one = Rational(1)
+    zero = Rational(0)
+    X = list2numpy([one, zero, zero])
+    Y = one*X
+    X = list2numpy([Symbol("a") + Rational(1, 2)])
+    Y = X + X
+    assert Y == array([1 + 2*Symbol("a")])
+    Y = Y + 1
+    assert Y == array([2 + 2*Symbol("a")])
+    Y = X - X
+    assert Y == array([0])
+
+
+def test_list2numpy():
+    assert (array([x**2, x]) == list2numpy([x**2, x])).all()
+
+
+def test_Matrix1():
+    m = Matrix([[x, x**2], [5, 2/x]])
+    assert (array(m.subs(x, 2)) == array([[2, 4], [5, 1]])).all()
+    m = Matrix([[sin(x), x**2], [5, 2/x]])
+    assert (array(m.subs(x, 2)) == array([[sin(2), 4], [5, 1]])).all()
+
+
+def test_Matrix2():
+    m = Matrix([[x, x**2], [5, 2/x]])
+    with ignore_warnings(PendingDeprecationWarning):
+        assert (matrix(m.subs(x, 2)) == matrix([[2, 4], [5, 1]])).all()
+    m = Matrix([[sin(x), x**2], [5, 2/x]])
+    with ignore_warnings(PendingDeprecationWarning):
+        assert (matrix(m.subs(x, 2)) == matrix([[sin(2), 4], [5, 1]])).all()
+
+
+def test_Matrix3():
+    a = array([[2, 4], [5, 1]])
+    assert Matrix(a) == Matrix([[2, 4], [5, 1]])
+    assert Matrix(a) != Matrix([[2, 4], [5, 2]])
+    a = array([[sin(2), 4], [5, 1]])
+    assert Matrix(a) == Matrix([[sin(2), 4], [5, 1]])
+    assert Matrix(a) != Matrix([[sin(0), 4], [5, 1]])
+
+
+def test_Matrix4():
+    with ignore_warnings(PendingDeprecationWarning):
+        a = matrix([[2, 4], [5, 1]])
+    assert Matrix(a) == Matrix([[2, 4], [5, 1]])
+    assert Matrix(a) != Matrix([[2, 4], [5, 2]])
+    with ignore_warnings(PendingDeprecationWarning):
+        a = matrix([[sin(2), 4], [5, 1]])
+    assert Matrix(a) == Matrix([[sin(2), 4], [5, 1]])
+    assert Matrix(a) != Matrix([[sin(0), 4], [5, 1]])
+
+
+def test_Matrix_sum():
+    M = Matrix([[1, 2, 3], [x, y, x], [2*y, -50, z*x]])
+    with ignore_warnings(PendingDeprecationWarning):
+        m = matrix([[2, 3, 4], [x, 5, 6], [x, y, z**2]])
+    assert M + m == Matrix([[3, 5, 7], [2*x, y + 5, x + 6], [2*y + x, y - 50, z*x + z**2]])
+    assert m + M == Matrix([[3, 5, 7], [2*x, y + 5, x + 6], [2*y + x, y - 50, z*x + z**2]])
+    assert M + m == M.add(m)
+
+
+def test_Matrix_mul():
+    M = Matrix([[1, 2, 3], [x, y, x]])
+    with ignore_warnings(PendingDeprecationWarning):
+        m = matrix([[2, 4], [x, 6], [x, z**2]])
+    assert M*m == Matrix([
+        [         2 + 5*x,        16 + 3*z**2],
+        [2*x + x*y + x**2, 4*x + 6*y + x*z**2],
+    ])
+
+    assert m*M == Matrix([
+        [   2 + 4*x,      4 + 4*y,      6 + 4*x],
+        [       7*x,    2*x + 6*y,          9*x],
+        [x + x*z**2, 2*x + y*z**2, 3*x + x*z**2],
+    ])
+    a = array([2])
+    assert a[0] * M == 2 * M
+    assert M * a[0] == 2 * M
+
+
+def test_Matrix_array():
+    class matarray:
+        def __array__(self, dtype=object, copy=None):
+            if copy is not None and not copy:
+                raise TypeError("Cannot implement copy=False when converting Matrix to ndarray")
+            from numpy import array
+            return array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
+    matarr = matarray()
+    assert Matrix(matarr) == Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
+
+
+def test_matrix2numpy():
+    a = matrix2numpy(Matrix([[1, x**2], [3*sin(x), 0]]))
+    assert isinstance(a, ndarray)
+    assert a.shape == (2, 2)
+    assert a[0, 0] == 1
+    assert a[0, 1] == x**2
+    assert a[1, 0] == 3*sin(x)
+    assert a[1, 1] == 0
+
+
+def test_matrix2numpy_conversion():
+    a = Matrix([[1, 2, sin(x)], [x**2, x, Rational(1, 2)]])
+    b = array([[1, 2, sin(x)], [x**2, x, Rational(1, 2)]])
+    assert (matrix2numpy(a) == b).all()
+    assert matrix2numpy(a).dtype == numpy.dtype('object')
+
+    c = matrix2numpy(Matrix([[1, 2], [10, 20]]), dtype='int8')
+    d = matrix2numpy(Matrix([[1, 2], [10, 20]]), dtype='float64')
+    assert c.dtype == numpy.dtype('int8')
+    assert d.dtype == numpy.dtype('float64')
+
+
+def test_issue_3728():
+    assert (Rational(1, 2)*array([2*x, 0]) == array([x, 0])).all()
+    assert (Rational(1, 2) + array(
+        [2*x, 0]) == array([2*x + Rational(1, 2), Rational(1, 2)])).all()
+    assert (Float("0.5")*array([2*x, 0]) == array([Float("1.0")*x, 0])).all()
+    assert (Float("0.5") + array(
+        [2*x, 0]) == array([2*x + Float("0.5"), Float("0.5")])).all()
+
+
+@conserve_mpmath_dps
+def test_lambdify():
+    mpmath.mp.dps = 16
+    sin02 = mpmath.mpf("0.198669330795061215459412627")
+    f = lambdify(x, sin(x), "numpy")
+    prec = 1e-15
+    assert -prec < f(0.2) - sin02 < prec
+
+    # if this succeeds, it can't be a numpy function
+
+    if version_tuple(numpy.__version__) >= version_tuple('1.17'):
+        with raises(TypeError):
+            f(x)
+    else:
+        with raises(AttributeError):
+            f(x)
+
+
+def test_lambdify_matrix():
+    f = lambdify(x, Matrix([[x, 2*x], [1, 2]]), [{'ImmutableMatrix': numpy.array}, "numpy"])
+    assert (f(1) == array([[1, 2], [1, 2]])).all()
+
+
+def test_lambdify_matrix_multi_input():
+    M = sympy.Matrix([[x**2, x*y, x*z],
+                      [y*x, y**2, y*z],
+                      [z*x, z*y, z**2]])
+    f = lambdify((x, y, z), M, [{'ImmutableMatrix': numpy.array}, "numpy"])
+
+    xh, yh, zh = 1.0, 2.0, 3.0
+    expected = array([[xh**2, xh*yh, xh*zh],
+                      [yh*xh, yh**2, yh*zh],
+                      [zh*xh, zh*yh, zh**2]])
+    actual = f(xh, yh, zh)
+    assert numpy.allclose(actual, expected)
+
+
+def test_lambdify_matrix_vec_input():
+    X = sympy.DeferredVector('X')
+    M = Matrix([
+        [X[0]**2, X[0]*X[1], X[0]*X[2]],
+        [X[1]*X[0], X[1]**2, X[1]*X[2]],
+        [X[2]*X[0], X[2]*X[1], X[2]**2]])
+    f = lambdify(X, M, [{'ImmutableMatrix': numpy.array}, "numpy"])
+
+    Xh = array([1.0, 2.0, 3.0])
+    expected = array([[Xh[0]**2, Xh[0]*Xh[1], Xh[0]*Xh[2]],
+                      [Xh[1]*Xh[0], Xh[1]**2, Xh[1]*Xh[2]],
+                      [Xh[2]*Xh[0], Xh[2]*Xh[1], Xh[2]**2]])
+    actual = f(Xh)
+    assert numpy.allclose(actual, expected)
+
+
+def test_lambdify_transl():
+    from sympy.utilities.lambdify import NUMPY_TRANSLATIONS
+    for sym, mat in NUMPY_TRANSLATIONS.items():
+        assert sym in sympy.__dict__
+        assert mat in numpy.__dict__
+
+
+def test_symarray():
+    """Test creation of numpy arrays of SymPy symbols."""
+
+    import numpy as np
+    import numpy.testing as npt
+
+    syms = symbols('_0,_1,_2')
+    s1 = symarray("", 3)
+    s2 = symarray("", 3)
+    npt.assert_array_equal(s1, np.array(syms, dtype=object))
+    assert s1[0] == s2[0]
+
+    a = symarray('a', 3)
+    b = symarray('b', 3)
+    assert not(a[0] == b[0])
+
+    asyms = symbols('a_0,a_1,a_2')
+    npt.assert_array_equal(a, np.array(asyms, dtype=object))
+
+    # Multidimensional checks
+    a2d = symarray('a', (2, 3))
+    assert a2d.shape == (2, 3)
+    a00, a12 = symbols('a_0_0,a_1_2')
+    assert a2d[0, 0] == a00
+    assert a2d[1, 2] == a12
+
+    a3d = symarray('a', (2, 3, 2))
+    assert a3d.shape == (2, 3, 2)
+    a000, a120, a121 = symbols('a_0_0_0,a_1_2_0,a_1_2_1')
+    assert a3d[0, 0, 0] == a000
+    assert a3d[1, 2, 0] == a120
+    assert a3d[1, 2, 1] == a121
+
+
+def test_vectorize():
+    assert (numpy.vectorize(
+        sin)([1, 2, 3]) == numpy.array([sin(1), sin(2), sin(3)])).all()

rl/Lib/site-packages/sympy/external/tests/test_pythonmpq.py

@@ -0,0 +1,176 @@
+"""
+test_pythonmpq.py
+
+Test the PythonMPQ class for consistency with gmpy2's mpq type. If gmpy2 is
+installed run the same tests for both.
+"""
+from fractions import Fraction
+from decimal import Decimal
+import pickle
+from typing import Callable, List, Tuple, Type
+
+from sympy.testing.pytest import raises
+
+from sympy.external.pythonmpq import PythonMPQ
+
+#
+# If gmpy2 is installed then run the tests for both mpq and PythonMPQ.
+# That should ensure consistency between the implementation here and mpq.
+#
+rational_types: List[Tuple[Callable, Type, Callable, Type]]
+rational_types = [(PythonMPQ, PythonMPQ, int, int)]
+try:
+    from gmpy2 import mpq, mpz
+    rational_types.append((mpq, type(mpq(1)), mpz, type(mpz(1))))
+except ImportError:
+    pass
+
+
+def test_PythonMPQ():
+    #
+    # Test PythonMPQ and also mpq if gmpy/gmpy2 is installed.
+    #
+    for Q, TQ, Z, TZ in rational_types:
+
+        def check_Q(q):
+            assert isinstance(q, TQ)
+            assert isinstance(q.numerator, TZ)
+            assert isinstance(q.denominator, TZ)
+            return q.numerator, q.denominator
+
+        # Check construction from different types
+        assert check_Q(Q(3)) == (3, 1)
+        assert check_Q(Q(3, 5)) == (3, 5)
+        assert check_Q(Q(Q(3, 5))) == (3, 5)
+        assert check_Q(Q(0.5)) == (1, 2)
+        assert check_Q(Q('0.5')) == (1, 2)
+        assert check_Q(Q(Fraction(3, 5))) == (3, 5)
+
+        # https://github.com/aleaxit/gmpy/issues/327
+        if Q is PythonMPQ:
+            assert check_Q(Q(Decimal('0.6'))) == (3, 5)
+
+        # Invalid types
+        raises(TypeError, lambda: Q([]))
+        raises(TypeError, lambda: Q([], []))
+
+        # Check normalisation of signs
+        assert check_Q(Q(2, 3)) == (2, 3)
+        assert check_Q(Q(-2, 3)) == (-2, 3)
+        assert check_Q(Q(2, -3)) == (-2, 3)
+        assert check_Q(Q(-2, -3)) == (2, 3)
+
+        # Check gcd calculation
+        assert check_Q(Q(12, 8)) == (3, 2)
+
+        # __int__/__float__
+        assert int(Q(5, 3)) == 1
+        assert int(Q(-5, 3)) == -1
+        assert float(Q(5, 2)) == 2.5
+        assert float(Q(-5, 2)) == -2.5
+
+        # __str__/__repr__
+        assert str(Q(2, 1)) == "2"
+        assert str(Q(1, 2)) == "1/2"
+        if Q is PythonMPQ:
+            assert repr(Q(2, 1)) == "MPQ(2,1)"
+            assert repr(Q(1, 2)) == "MPQ(1,2)"
+        else:
+            assert repr(Q(2, 1)) == "mpq(2,1)"
+            assert repr(Q(1, 2)) == "mpq(1,2)"
+
+        # __bool__
+        assert bool(Q(1, 2)) is True
+        assert bool(Q(0)) is False
+
+        # __eq__/__ne__
+        assert (Q(2, 3) == Q(2, 3)) is True
+        assert (Q(2, 3) == Q(2, 5)) is False
+        assert (Q(2, 3) != Q(2, 3)) is False
+        assert (Q(2, 3) != Q(2, 5)) is True
+
+        # __hash__
+        assert hash(Q(3, 5)) == hash(Fraction(3, 5))
+
+        # __reduce__
+        q = Q(2, 3)
+        assert pickle.loads(pickle.dumps(q)) == q
+
+        # __ge__/__gt__/__le__/__lt__
+        assert (Q(1, 3) < Q(2, 3)) is True
+        assert (Q(2, 3) < Q(2, 3)) is False
+        assert (Q(2, 3) < Q(1, 3)) is False
+        assert (Q(-2, 3) < Q(1, 3)) is True
+        assert (Q(1, 3) < Q(-2, 3)) is False
+
+        assert (Q(1, 3) <= Q(2, 3)) is True
+        assert (Q(2, 3) <= Q(2, 3)) is True
+        assert (Q(2, 3) <= Q(1, 3)) is False
+        assert (Q(-2, 3) <= Q(1, 3)) is True
+        assert (Q(1, 3) <= Q(-2, 3)) is False
+
+        assert (Q(1, 3) > Q(2, 3)) is False
+        assert (Q(2, 3) > Q(2, 3)) is False
+        assert (Q(2, 3) > Q(1, 3)) is True
+        assert (Q(-2, 3) > Q(1, 3)) is False
+        assert (Q(1, 3) > Q(-2, 3)) is True
+
+        assert (Q(1, 3) >= Q(2, 3)) is False
+        assert (Q(2, 3) >= Q(2, 3)) is True
+        assert (Q(2, 3) >= Q(1, 3)) is True
+        assert (Q(-2, 3) >= Q(1, 3)) is False
+        assert (Q(1, 3) >= Q(-2, 3)) is True
+
+        # __abs__/__pos__/__neg__
+        assert abs(Q(2, 3)) == abs(Q(-2, 3)) == Q(2, 3)
+        assert +Q(2, 3) == Q(2, 3)
+        assert -Q(2, 3) == Q(-2, 3)
+
+        # __add__/__radd__
+        assert Q(2, 3) + Q(5, 7) == Q(29, 21)
+        assert Q(2, 3) + 1 == Q(5, 3)
+        assert 1 + Q(2, 3) == Q(5, 3)
+        raises(TypeError, lambda: [] + Q(1))
+        raises(TypeError, lambda: Q(1) + [])
+
+        # __sub__/__rsub__
+        assert Q(2, 3) - Q(5, 7) == Q(-1, 21)
+        assert Q(2, 3) - 1 == Q(-1, 3)
+        assert 1 - Q(2, 3) == Q(1, 3)
+        raises(TypeError, lambda: [] - Q(1))
+        raises(TypeError, lambda: Q(1) - [])
+
+        # __mul__/__rmul__
+        assert Q(2, 3) * Q(5, 7) == Q(10, 21)
+        assert Q(2, 3) * 1 == Q(2, 3)
+        assert 1 * Q(2, 3) == Q(2, 3)
+        raises(TypeError, lambda: [] * Q(1))
+        raises(TypeError, lambda: Q(1) * [])
+
+        # __pow__/__rpow__
+        assert Q(2, 3) ** 2 == Q(4, 9)
+        assert Q(2, 3) ** 1 == Q(2, 3)
+        assert Q(-2, 3) ** 2 == Q(4, 9)
+        assert Q(-2, 3) ** -1 == Q(-3, 2)
+        if Q is PythonMPQ:
+            raises(TypeError, lambda: 1 ** Q(2, 3))
+            raises(TypeError, lambda: Q(1, 4) ** Q(1, 2))
+        raises(TypeError, lambda: [] ** Q(1))
+        raises(TypeError, lambda: Q(1) ** [])
+
+        # __div__/__rdiv__
+        assert Q(2, 3) / Q(5, 7) == Q(14, 15)
+        assert Q(2, 3) / 1 == Q(2, 3)
+        assert 1 / Q(2, 3) == Q(3, 2)
+        raises(TypeError, lambda: [] / Q(1))
+        raises(TypeError, lambda: Q(1) / [])
+        raises(ZeroDivisionError, lambda: Q(1, 2) / Q(0))
+
+        # __divmod__
+        if Q is PythonMPQ:
+            raises(TypeError, lambda: Q(2, 3) // Q(1, 3))
+            raises(TypeError, lambda: Q(2, 3) % Q(1, 3))
+            raises(TypeError, lambda: 1 // Q(1, 3))
+            raises(TypeError, lambda: 1 % Q(1, 3))
+            raises(TypeError, lambda: Q(2, 3) // 1)
+            raises(TypeError, lambda: Q(2, 3) % 1)

rl/Lib/site-packages/sympy/external/tests/test_scipy.py

@@ -0,0 +1,35 @@
+# This testfile tests SymPy <-> SciPy compatibility
+
+# Don't test any SymPy features here. Just pure interaction with SciPy.
+# Always write regular SymPy tests for anything, that can be tested in pure
+# Python (without scipy). Here we test everything, that a user may need when
+# using SymPy with SciPy
+
+from sympy.external import import_module
+
+scipy = import_module('scipy')
+if not scipy:
+    #bin/test will not execute any tests now
+    disabled = True
+
+from sympy.functions.special.bessel import jn_zeros
+
+
+def eq(a, b, tol=1e-6):
+    for x, y in zip(a, b):
+        if not (abs(x - y) < tol):
+            return False
+    return True
+
+
+def test_jn_zeros():
+    assert eq(jn_zeros(0, 4, method="scipy"),
+            [3.141592, 6.283185, 9.424777, 12.566370])
+    assert eq(jn_zeros(1, 4, method="scipy"),
+            [4.493409, 7.725251, 10.904121, 14.066193])
+    assert eq(jn_zeros(2, 4, method="scipy"),
+            [5.763459, 9.095011, 12.322940, 15.514603])
+    assert eq(jn_zeros(3, 4, method="scipy"),
+            [6.987932, 10.417118, 13.698023, 16.923621])
+    assert eq(jn_zeros(4, 4, method="scipy"),
+            [8.182561, 11.704907, 15.039664, 18.301255])