"""
This module provides convenient functions to transform sympy expressions to
lambda functions which can be used to calculate numerical values very fast.
"""

from __future__ import print_function, division

import inspect
import keyword
import re
import textwrap
import linecache

from sympy.core.compatibility import (exec_, is_sequence, iterable,
    NotIterable, string_types, range, builtins, PY3)
from sympy.utilities.decorator import doctest_depends_on

__doctest_requires__ = {('lambdify',): ['numpy', 'tensorflow']}

# Default namespaces, letting us define translations that can't be defined
# by simple variable maps, like I => 1j
MATH_DEFAULT = {}
MPMATH_DEFAULT = {}
NUMPY_DEFAULT = {"I": 1j}
SCIPY_DEFAULT = {"I": 1j}
TENSORFLOW_DEFAULT = {}
SYMPY_DEFAULT = {}
NUMEXPR_DEFAULT = {}

# These are the namespaces the lambda functions will use.
# These are separate from the names above because they are modified
# throughout this file, whereas the defaults should remain unmodified.

MATH = MATH_DEFAULT.copy()
MPMATH = MPMATH_DEFAULT.copy()
NUMPY = NUMPY_DEFAULT.copy()
SCIPY = SCIPY_DEFAULT.copy()
TENSORFLOW = TENSORFLOW_DEFAULT.copy()
SYMPY = SYMPY_DEFAULT.copy()
NUMEXPR = NUMEXPR_DEFAULT.copy()


# Mappings between sympy and other modules function names.
MATH_TRANSLATIONS = {
    "ceiling": "ceil",
    "E": "e",
    "ln": "log",
}

# NOTE: This dictionary is reused in Function._eval_evalf to allow subclasses
# of Function to automatically evalf.
MPMATH_TRANSLATIONS = {
    "Abs": "fabs",
    "elliptic_k": "ellipk",
    "elliptic_f": "ellipf",
    "elliptic_e": "ellipe",
    "elliptic_pi": "ellippi",
    "ceiling": "ceil",
    "chebyshevt": "chebyt",
    "chebyshevu": "chebyu",
    "E": "e",
    "I": "j",
    "ln": "log",
    #"lowergamma":"lower_gamma",
    "oo": "inf",
    #"uppergamma":"upper_gamma",
    "LambertW": "lambertw",
    "MutableDenseMatrix": "matrix",
    "ImmutableDenseMatrix": "matrix",
    "conjugate": "conj",
    "dirichlet_eta": "altzeta",
    "Ei": "ei",
    "Shi": "shi",
    "Chi": "chi",
    "Si": "si",
    "Ci": "ci",
    "RisingFactorial": "rf",
    "FallingFactorial": "ff",
}

NUMPY_TRANSLATIONS = {}
SCIPY_TRANSLATIONS = {}

TENSORFLOW_TRANSLATIONS = {
    "Abs": "abs",
    "ceiling": "ceil",
    "im": "imag",
    "ln": "log",
    "Mod": "mod",
    "conjugate": "conj",
    "re": "real",
}

NUMEXPR_TRANSLATIONS = {}

# Available modules:
MODULES = {
    "math": (MATH, MATH_DEFAULT, MATH_TRANSLATIONS, ("from math import *",)),
    "mpmath": (MPMATH, MPMATH_DEFAULT, MPMATH_TRANSLATIONS, ("from mpmath import *",)),
    "numpy": (NUMPY, NUMPY_DEFAULT, NUMPY_TRANSLATIONS, ("import numpy; from numpy import *; from numpy.linalg import *",)),
    "scipy": (SCIPY, SCIPY_DEFAULT, SCIPY_TRANSLATIONS, ("import numpy; import scipy; from scipy import *; from scipy.special import *",)),
    "tensorflow": (TENSORFLOW, TENSORFLOW_DEFAULT, TENSORFLOW_TRANSLATIONS, ("import_module('tensorflow')",)),
    "sympy": (SYMPY, SYMPY_DEFAULT, {}, (
        "from sympy.functions import *",
        "from sympy.matrices import *",
        "from sympy import Integral, pi, oo, nan, zoo, E, I",)),
    "numexpr" : (NUMEXPR, NUMEXPR_DEFAULT, NUMEXPR_TRANSLATIONS,
                 ("import_module('numexpr')", )),
}


def _import(module, reload=False):
    """
    Creates a global translation dictionary for module.

    The argument module has to be one of the following strings: "math",
    "mpmath", "numpy", "sympy", "tensorflow".
    These dictionaries map names of python functions to their equivalent in
    other modules.
    """
    # Required despite static analysis claiming it is not used
    from sympy.external import import_module
    try:
        namespace, namespace_default, translations, import_commands = MODULES[
            module]
    except KeyError:
        raise NameError(
            "'%s' module can't be used for lambdification" % module)

    # Clear namespace or exit
    if namespace != namespace_default:
        # The namespace was already generated, don't do it again if not forced.
        if reload:
            namespace.clear()
            namespace.update(namespace_default)
        else:
            return

    for import_command in import_commands:
        if import_command.startswith('import_module'):
            module = eval(import_command)

            if module is not None:
                namespace.update(module.__dict__)
                continue
        else:
            try:
                exec_(import_command, {}, namespace)
                continue
            except ImportError:
                pass

        raise ImportError(
            "can't import '%s' with '%s' command" % (module, import_command))

    # Add translated names to namespace
    for sympyname, translation in translations.items():
        namespace[sympyname] = namespace[translation]

    # For computing the modulus of a sympy expression we use the builtin abs
    # function, instead of the previously used fabs function for all
    # translation modules. This is because the fabs function in the math
    # module does not accept complex valued arguments. (see issue 9474). The
    # only exception, where we don't use the builtin abs function is the
    # mpmath translation module, because mpmath.fabs returns mpf objects in
    # contrast to abs().
    if 'Abs' not in namespace:
        namespace['Abs'] = abs


# Used for dynamically generated filenames that are inserted into the
# linecache.
_lambdify_generated_counter = 1

@doctest_depends_on(modules=('numpy', 'tensorflow', ), python_version=(3,))
def lambdify(args, expr, modules=None, printer=None, use_imps=True,
             dummify=False):
    """
    Translates a SymPy expression into an equivalent numeric function

    For example, to convert the SymPy expression ``sin(x) + cos(x)`` to an
    equivalent NumPy function that numerically evaluates it:

    >>> from sympy import sin, cos, symbols, lambdify
    >>> import numpy as np
    >>> x = symbols('x')
    >>> expr = sin(x) + cos(x)
    >>> expr
    sin(x) + cos(x)
    >>> f = lambdify(x, expr, 'numpy')
    >>> a = np.array([1, 2])
    >>> f(a)
    [1.38177329 0.49315059]

    The primary purpose of this function is to provide a bridge from SymPy
    expressions to numerical libraries such as NumPy, SciPy, NumExpr, mpmath,
    and tensorflow. In general, SymPy functions do not work with objects from
    other libraries, such as NumPy arrays, and functions from numeric
    libraries like NumPy or mpmath do not work on SymPy expressions.
    ``lambdify`` bridges the two by converting a SymPy expression to an
    equivalent numeric function.

    The basic workflow with ``lambdify`` is to first create a SymPy expression
    representing whatever mathematical function you wish to evaluate. This
    should be done using only SymPy functions and expressions. Then, use
    ``lambdify`` to convert this to an equivalent function for numerical
    evaluation. For instance, above we created ``expr`` using the SymPy symbol
    ``x`` and SymPy functions ``sin`` and ``cos``, then converted it to an
    equivalent NumPy function ``f``, and called it on a NumPy array ``a``.

    .. warning::
       This function uses ``exec``, and thus shouldn't be used on unsanitized
       input.

    Arguments
    =========

    The first argument of ``lambdify`` is a variable or list of variables in
    the expression. Variable lists may be nested. Variables can be Symbols,
    undefined functions, or matrix symbols. The order and nesting of the
    variables corresponds to the order and nesting of the parameters passed to
    the lambdified function. For instance,

    >>> from sympy.abc import x, y, z
    >>> f = lambdify([x, (y, z)], x + y + z)
    >>> f(1, (2, 3))
    6

    The second argument of ``lambdify`` is the expression, list of
    expressions, or matrix to be evaluated. Lists may be nested. If the
    expression is a list, the output will also be a list.

    >>> f = lambdify(x, [x, [x + 1, x + 2]])
    >>> f(1)
    [1, [2, 3]]

    If it is a matrix, an array will be returned (for the NumPy module).

    >>> from sympy import Matrix
    >>> f = lambdify(x, Matrix([x, x + 1]))
    >>> f(1)
    [[1]
     [2]]

    Note that the argument order here, variables then expression, is used to
    emulate the Python ``lambda`` keyword. ``lambdify(x, expr)`` works
    (roughly) like ``lambda x: expr`` (see :ref:`lambdify-how-it-works` below).

    The third argument, ``modules`` is optional. If not specified, ``modules``
    defaults to ``["scipy", "numpy"]`` if SciPy is installed, ``["numpy"]`` if
    only NumPy is installed, and ``["math", "mpmath", "sympy"]`` if neither is
    installed. That is, SymPy functions are replaced as far as possible by
    either ``scipy`` or ``numpy`` functions if available, and Python's
    standard library ``math``, or ``mpmath`` functions otherwise.

    ``modules`` can be one of the following types

     - the strings ``"math"``, ``"mpmath"``, ``"numpy"``, ``"numexpr"``,
       ``"scipy"``, ``"sympy"``, or ``"tensorflow"``. This uses the
       corresponding printer and namespace mapping for that module.
     - a module (e.g., ``math``). This uses the global namespace of the
       module. If the module is one of the above known modules, it will also
       use the corresponding printer and namespace mapping (i.e.,
       ``modules=numpy`` is equivalent to ``modules="numpy"``).
     - a dictionary that maps names of SymPy functions to arbitrary functions
       (e.g., ``{'sin': custom_sin}``).
     - a list that contains a mix of the arguments above, with higher priority
       given to entries appearing first (e.g., to use the NumPy module but
       override the ``sin`` function with a custom version, you can use
       ``[{'sin': custom_sin}, 'numpy']``).

    The ``dummify`` keyword argument controls whether or not the variables in
    the provided expression that are not valid Python identifiers are
    substituted with dummy symbols. This allows for undefined functions like
    ``Function('f')(t)`` to be supplied as arguments. By default, the
    variables are only dummified if they are not valid Python identifiers. Set
    ``dummify=True`` to replace all arguments with dummy symbols (if ``args``
    is not a string) - for example, to ensure that the arguments do not
    redefine any built-in names.

    .. _lambdify-how-it-works:

    How it works
    ============

    When using this function, it helps a great deal to have an idea of what it
    is doing. At its core, lambdify is nothing more than a namespace
    translation, on top of a special printer that makes some corner cases work
    properly.

    To understand lambdify, first we must properly understand how Python
    namespaces work. Say we had two files. One called ``sin_cos_sympy.py``,
    with

    .. code:: python

        # sin_cos_sympy.py

        from sympy import sin, cos

        def sin_cos(x):
            return sin(x) + cos(x)


    and one called ``sin_cos_numpy.py`` with

    .. code:: python

        # sin_cos_numpy.py

        from numpy import sin, cos

        def sin_cos(x):
            return sin(x) + cos(x)

    The two files define an identical function ``sin_cos``. However, in the
    first file, ``sin`` and ``cos`` are defined as the SymPy ``sin`` and
    ``cos``. In the second, they are defined as the NumPy versions.

    If we were to import the first file and use the ``sin_cos`` function, we
    would get something like

    >>> from sin_cos_sympy import sin_cos # doctest: +SKIP
    >>> sin_cos(1) # doctest: +SKIP
    cos(1) + sin(1)

    On the other hand, if we imported ``sin_cos`` from the second file, we
    would get

    >>> from sin_cos_numpy import sin_cos # doctest: +SKIP
    >>> sin_cos(1) # doctest: +SKIP
    1.38177329068

    In the first case we got a symbolic output, because it used the symbolic
    ``sin`` and ``cos`` functions from SymPy. In the second, we got a numeric
    result, because ``sin_cos`` used the numeric ``sin`` and ``cos`` functions
    from NumPy. But notice that the versions of ``sin`` and ``cos`` that were
    used was not inherent to the ``sin_cos`` function definition. Both
    ``sin_cos`` definitions are exactly the same. Rather, it was based on the
    names defined at the module where the ``sin_cos`` function was defined.

    The key point here is that when function in Python references a name that
    is not defined in the function, that name is looked up in the "global"
    namespace of the module where that function is defined.

    Now, in Python, we can emulate this behavior without actually writing a
    file to disk using the ``exec`` function. ``exec`` takes a string
    containing a block of Python code, and a dictionary that should contain
    the global variables of the module. It then executes the code "in" that
    dictionary, as if it were the module globals. The following is equivalent
    to the ``sin_cos`` defined in ``sin_cos_sympy.py``:

    >>> import sympy
    >>> module_dictionary = {'sin': sympy.sin, 'cos': sympy.cos}
    >>> exec('''
    ... def sin_cos(x):
    ...     return sin(x) + cos(x)
    ... ''', module_dictionary)
    >>> sin_cos = module_dictionary['sin_cos']
    >>> sin_cos(1)
    cos(1) + sin(1)

    and similarly with ``sin_cos_numpy``:

    >>> import numpy
    >>> module_dictionary = {'sin': numpy.sin, 'cos': numpy.cos}
    >>> exec('''
    ... def sin_cos(x):
    ...     return sin(x) + cos(x)
    ... ''', module_dictionary)
    >>> sin_cos = module_dictionary['sin_cos']
    >>> sin_cos(1)
    1.38177329068

    So now we can get an idea of how ``lambdify`` works. The name "lambdify"
    comes from the fact that we can think of something like ``lambdify(x,
    sin(x) + cos(x), 'numpy')`` as ``lambda x: sin(x) + cos(x)``, where
    ``sin`` and ``cos`` come from the ``numpy`` namespace. This is also why
    the symbols argument is first in ``lambdify``, as opposed to most SymPy
    functions where it comes after the expression: to better mimic the
    ``lambda`` keyword.

    ``lambdify`` takes the input expression (like ``sin(x) + cos(x)``) and

    1. Converts it to a string
    2. Creates a module globals dictionary based on the modules that are
       passed in (by default, it uses the NumPy module)
    3. Creates the string ``"def func({vars}): return {expr}"``, where ``{vars}`` is the
       list of variables separated by commas, and ``{expr}`` is the string
       created in step 1., then ``exec``s that string with the module globals
       namespace and returns ``func``.

    In fact, functions returned by ``lambdify`` support inspection. So you can
    see exactly how they are defined by using ``inspect.getsource``, or ``??`` if you
    are using IPython or the Jupyter notebook.

    >>> f = lambdify(x, sin(x) + cos(x))
    >>> import inspect
    >>> print(inspect.getsource(f))
    def _lambdifygenerated(x):
        return (sin(x) + cos(x))

    This shows us the source code of the function, but not the namespace it
    was defined in. We can inspect that by looking at the ``__globals__``
    attribute of ``f``:

    >>> f.__globals__['sin']
    <ufunc 'sin'>
    >>> f.__globals__['cos']
    <ufunc 'cos'>
    >>> f.__globals__['sin'] is numpy.sin
    True

    This shows us that ``sin`` and ``cos`` in the namespace of ``f`` will be
    ``numpy.sin`` and ``numpy.cos``.

    Note that there are some convenience layers in each of these steps, but at
    the core, this is how ``lambdify`` works. Step 1 is done using the
    ``LambdaPrinter`` printers defined in the printing module (see
    :mod:`sympy.printing.lambdarepr`). This allows different SymPy expressions
    to define how they should be converted to a string for different modules.
    You can change which printer ``lambdify`` uses by passing a custom printer
    in to the ``printer`` argument.

    Step 2 is augmented by certain translations. There are default
    translations for each module, but you can provide your own by passing a
    list to the ``modules`` argument. For instance,

    >>> def mysin(x):
    ...     print('taking the sin of', x)
    ...     return numpy.sin(x)
    ...
    >>> f = lambdify(x, sin(x), [{'sin': mysin}, 'numpy'])
    >>> f(1)
    taking the sin of 1
    0.8414709848078965

    The globals dictionary is generated from the list by merging the
    dictionary ``{'sin': mysin}`` and the module dictionary for NumPy. The
    merging is done so that earlier items take precedence, which is why
    ``mysin`` is used above instead of ``numpy.sin``.

    If you want to modify the way ``lambdify`` works for a given function, it
    is usually easiest to do so by modifying the globals dictionary as such.
    In more complicated cases, it may be necessary to create and pass in a
    custom printer.

    Finally, step 3 is augmented with certain convenience operations, such as
    the addition of a docstring.

    Understanding how ``lambdify`` works can make it easier to avoid certain
    gotchas when using it. For instance, a common mistake is to create a
    lambdified function for one module (say, NumPy), and pass it objects from
    another (say, a SymPy expression).

    For instance, say we create

    >>> from sympy.abc import x
    >>> f = lambdify(x, x + 1, 'numpy')

    Now if we pass in a NumPy array, we get that array plus 1

    >>> import numpy
    >>> a = numpy.array([1, 2])
    >>> f(a)
    [2 3]

    But what happens if you make the mistake of passing in a SymPy expression
    instead of a NumPy array:

    >>> f(x + 1)
    x + 2

    This worked, but it was only by accident. Now take a different lambdified
    function:

    >>> from sympy import sin
    >>> g = lambdify(x, x + sin(x), 'numpy')

    This works as expected on NumPy arrays:

    >>> g(a)
    [1.84147098 2.90929743]

    But if we try to pass in a SymPy expression, it fails

    >>> g(x + 1)
    Traceback (most recent call last):
    ...
    AttributeError: 'Add' object has no attribute 'sin'

    Now, let's look at what happened. The reason this fails is that ``g``
    calls ``numpy.sin`` on the input expression, and ``numpy.sin`` does not
    know how to operate on a SymPy object. **As a general rule, NumPy
    functions do not know how to operate on SymPy expressions, and SymPy
    functions do not know how to operate on NumPy arrays. This is why lambdify
    exists: to provide a bridge between SymPy and NumPy.**

    However, why is it that ``f`` did work? That's because ``f`` doesn't call
    any functions, it only adds 1. So the resulting function that is created,
    ``def _lambdifygenerated(x): return x + 1`` does not depend on the globals
    namespace it is defined in. Thus it works, but only by accident. A future
    version of ``lambdify`` may remove this behavior.

    Be aware that certain implementation details described here may change in
    future versions of SymPy. The API of passing in custom modules and
    printers will not change, but the details of how a lambda function is
    created may change. However, the basic idea will remain the same, and
    understanding it will be helpful to understanding the behavior of
    lambdify.

    **In general: you should create lambdified functions for one module (say,
    NumPy), and only pass it input types that are compatible with that module
    (say, NumPy arrays).** Remember that by default, if the ``module``
    argument is not provided, ``lambdify`` creates functions using the NumPy
    and SciPy namespaces.

    Examples
    ========

    >>> from sympy.utilities.lambdify import implemented_function
    >>> from sympy import sqrt, sin, Matrix
    >>> from sympy import Function
    >>> from sympy.abc import w, x, y, z

    >>> f = lambdify(x, x**2)
    >>> f(2)
    4
    >>> f = lambdify((x, y, z), [z, y, x])
    >>> f(1,2,3)
    [3, 2, 1]
    >>> f = lambdify(x, sqrt(x))
    >>> f(4)
    2.0
    >>> f = lambdify((x, y), sin(x*y)**2)
    >>> f(0, 5)
    0.0
    >>> row = lambdify((x, y), Matrix((x, x + y)).T, modules='sympy')
    >>> row(1, 2)
    Matrix([[1, 3]])

    ``lambdify`` can be used to translate SymPy expressions into mpmath
    functions. This may be preferable to using ``evalf`` (which uses mpmath on
    the backend) in some cases.

    >>> import mpmath
    >>> f = lambdify(x, sin(x), 'mpmath')
    >>> f(1)
    0.8414709848078965

    Tuple arguments are handled and the lambdified function should
    be called with the same type of arguments as were used to create
    the function:

    >>> f = lambdify((x, (y, z)), x + y)
    >>> f(1, (2, 4))
    3

    The ``flatten`` function can be used to always work with flattened
    arguments:

    >>> from sympy.utilities.iterables import flatten
    >>> args = w, (x, (y, z))
    >>> vals = 1, (2, (3, 4))
    >>> f = lambdify(flatten(args), w + x + y + z)
    >>> f(*flatten(vals))
    10

    Functions present in ``expr`` can also carry their own numerical
    implementations, in a callable attached to the ``_imp_`` attribute. This
    can be used with undefined functions using the ``implemented_function``
    factory:

    >>> f = implemented_function(Function('f'), lambda x: x+1)
    >>> func = lambdify(x, f(x))
    >>> func(4)
    5

    ``lambdify`` always prefers ``_imp_`` implementations to implementations
    in other namespaces, unless the ``use_imps`` input parameter is False.

    Usage with Tensorflow:

    >>> import tensorflow as tf
    >>> from sympy import Max, sin
    >>> f = Max(x, sin(x))
    >>> func = lambdify(x, f, 'tensorflow')
    >>> result = func(tf.constant(1.0))
    >>> print(result) # a tf.Tensor representing the result of the calculation
    Tensor("Maximum:0", shape=(), dtype=float32)
    >>> sess = tf.Session()
    >>> sess.run(result) # compute result
    1.0
    >>> var = tf.Variable(1.0)
    >>> sess.run(tf.global_variables_initializer())
    >>> sess.run(func(var)) # also works for tf.Variable and tf.Placeholder
    1.0
    >>> tensor = tf.constant([[1.0, 2.0], [3.0, 4.0]]) # works with any shape tensor
    >>> sess.run(func(tensor))
    [[1. 2.]
     [3. 4.]]

    Notes
    =====

    - For functions involving large array calculations, numexpr can provide a
      significant speedup over numpy. Please note that the available functions
      for numexpr are more limited than numpy but can be expanded with
      ``implemented_function`` and user defined subclasses of Function. If
      specified, numexpr may be the only option in modules. The official list
      of numexpr functions can be found at:
      https://numexpr.readthedocs.io/en/latest/user_guide.html#supported-functions

    - In previous versions of SymPy, ``lambdify`` replaced ``Matrix`` with
      ``numpy.matrix`` by default. As of SymPy 1.0 ``numpy.array`` is the
      default. To get the old default behavior you must pass in
      ``[{'ImmutableDenseMatrix':  numpy.matrix}, 'numpy']`` to the
      ``modules`` kwarg.

      >>> from sympy import lambdify, Matrix
      >>> from sympy.abc import x, y
      >>> import numpy
      >>> array2mat = [{'ImmutableDenseMatrix': numpy.matrix}, 'numpy']
      >>> f = lambdify((x, y), Matrix([x, y]), modules=array2mat)
      >>> f(1, 2)
      [[1]
       [2]]

    - In the above examples, the generated functions can accept scalar
      values or numpy arrays as arguments.  However, in some cases
      the generated function relies on the input being a numpy array:

      >>> from sympy import Piecewise
      >>> from sympy.utilities.pytest import ignore_warnings
      >>> f = lambdify(x, Piecewise((x, x <= 1), (1/x, x > 1)), "numpy")

      >>> with ignore_warnings(RuntimeWarning):
      ...     f(numpy.array([-1, 0, 1, 2]))
      [-1.   0.   1.   0.5]

      >>> f(0)
      Traceback (most recent call last):
          ...
      ZeroDivisionError: division by zero

      In such cases, the input should be wrapped in a numpy array:

      >>> with ignore_warnings(RuntimeWarning):
      ...     float(f(numpy.array([0])))
      0.0

      Or if numpy functionality is not required another module can be used:

      >>> f = lambdify(x, Piecewise((x, x <= 1), (1/x, x > 1)), "math")
      >>> f(0)
      0

    """
    from sympy.core.symbol import Symbol

    # If the user hasn't specified any modules, use what is available.
    if modules is None:
        try:
            _import("scipy")
        except ImportError:
            try:
                _import("numpy")
            except ImportError:
                # Use either numpy (if available) or python.math where possible.
                # XXX: This leads to different behaviour on different systems and
                #      might be the reason for irreproducible errors.
                modules = ["math", "mpmath", "sympy"]
            else:
                modules = ["numpy"]
        else:
            modules = ["scipy", "numpy"]

    # Get the needed namespaces.
    namespaces = []
    # First find any function implementations
    if use_imps:
        namespaces.append(_imp_namespace(expr))
    # Check for dict before iterating
    if isinstance(modules, (dict, string_types)) or not hasattr(modules, '__iter__'):
        namespaces.append(modules)
    else:
        # consistency check
        if _module_present('numexpr', modules) and len(modules) > 1:
            raise TypeError("numexpr must be the only item in 'modules'")
        namespaces += list(modules)
    # fill namespace with first having highest priority
    namespace = {}
    for m in namespaces[::-1]:
        buf = _get_namespace(m)
        namespace.update(buf)

    if hasattr(expr, "atoms"):
        #Try if you can extract symbols from the expression.
        #Move on if expr.atoms in not implemented.
        syms = expr.atoms(Symbol)
        for term in syms:
            namespace.update({str(term): term})

    if printer is None:
        if _module_present('mpmath', namespaces):
            from sympy.printing.pycode import MpmathPrinter as Printer
        elif _module_present('scipy', namespaces):
            from sympy.printing.pycode import SciPyPrinter as Printer
        elif _module_present('numpy', namespaces):
            from sympy.printing.pycode import NumPyPrinter as Printer
        elif _module_present('numexpr', namespaces):
            from sympy.printing.lambdarepr import NumExprPrinter as Printer
        elif _module_present('tensorflow', namespaces):
            from sympy.printing.tensorflow import TensorflowPrinter as Printer
        elif _module_present('sympy', namespaces):
            from sympy.printing.pycode import SymPyPrinter as Printer
        else:
            from sympy.printing.pycode import PythonCodePrinter as Printer
        user_functions = {}
        for m in namespaces[::-1]:
            if isinstance(m, dict):
                for k in m:
                    user_functions[k] = k
        printer = Printer({'fully_qualified_modules': False, 'inline': True,
                           'allow_unknown_functions': True,
                           'user_functions': user_functions})

    # Get the names of the args, for creating a docstring
    if not iterable(args):
        args = (args,)
    names = []
    # Grab the callers frame, for getting the names by inspection (if needed)
    callers_local_vars = inspect.currentframe().f_back.f_locals.items()
    for n, var in enumerate(args):
        if hasattr(var, 'name'):
            names.append(var.name)
        else:
            # It's an iterable. Try to get name by inspection of calling frame.
            name_list = [var_name for var_name, var_val in callers_local_vars
                    if var_val is var]
            if len(name_list) == 1:
                names.append(name_list[0])
            else:
                # Cannot infer name with certainty. arg_# will have to do.
                names.append('arg_' + str(n))

    imp_mod_lines = []
    for mod, keys in (getattr(printer, 'module_imports', None) or {}).items():
        for k in keys:
            if k not in namespace:
                imp_mod_lines.append("from %s import %s" % (mod, k))
    for ln in imp_mod_lines:
        exec_(ln, {}, namespace)

    # Provide lambda expression with builtins, and compatible implementation of range
    namespace.update({'builtins':builtins, 'range':range})

    # Create the function definition code and execute it

    funcname = '_lambdifygenerated'

    if _module_present('tensorflow', namespaces):
        funcprinter = _TensorflowEvaluatorPrinter(printer, dummify)
    else:
        funcprinter = _EvaluatorPrinter(printer, dummify)

    funcstr = funcprinter.doprint(funcname, args, expr)

    funclocals = {}
    global _lambdify_generated_counter
    filename = '<lambdifygenerated-%s>' % _lambdify_generated_counter
    _lambdify_generated_counter += 1
    c = compile(funcstr, filename, 'exec')
    exec_(c, namespace, funclocals)
    # mtime has to be None or else linecache.checkcache will remove it
    linecache.cache[filename] = (len(funcstr), None, funcstr.splitlines(True), filename)

    func = funclocals[funcname]

    # Apply the docstring
    sig = "func({0})".format(", ".join(str(i) for i in names))
    sig = textwrap.fill(sig, subsequent_indent=' '*8)
    expr_str = str(expr)
    if len(expr_str) > 78:
        expr_str = textwrap.wrap(expr_str, 75)[0] + '...'
    func.__doc__ = (
        "Created with lambdify. Signature:\n\n"
        "{sig}\n\n"
        "Expression:\n\n"
        "{expr}\n\n"
        "Source code:\n\n"
        "{src}\n\n"
        "Imported modules:\n\n"
        "{imp_mods}"
        ).format(sig=sig, expr=expr_str, src=funcstr, imp_mods='\n'.join(imp_mod_lines))
    return func

def _module_present(modname, modlist):
    if modname in modlist:
        return True
    for m in modlist:
        if hasattr(m, '__name__') and m.__name__ == modname:
            return True
    return False


def _get_namespace(m):
    """
    This is used by _lambdify to parse its arguments.
    """
    if isinstance(m, string_types):
        _import(m)
        return MODULES[m][0]
    elif isinstance(m, dict):
        return m
    elif hasattr(m, "__dict__"):
        return m.__dict__
    else:
        raise TypeError("Argument must be either a string, dict or module but it is: %s" % m)

def lambdastr(args, expr, printer=None, dummify=None):
    """
    Returns a string that can be evaluated to a lambda function.

    Examples
    ========

    >>> from sympy.abc import x, y, z
    >>> from sympy.utilities.lambdify import lambdastr
    >>> lambdastr(x, x**2)
    'lambda x: (x**2)'
    >>> lambdastr((x,y,z), [z,y,x])
    'lambda x,y,z: ([z, y, x])'

    Although tuples may not appear as arguments to lambda in Python 3,
    lambdastr will create a lambda function that will unpack the original
    arguments so that nested arguments can be handled:

    >>> lambdastr((x, (y, z)), x + y)
    'lambda _0,_1: (lambda x,y,z: (x + y))(_0,_1[0],_1[1])'
    """
    # Transforming everything to strings.
    from sympy.matrices import DeferredVector
    from sympy import Dummy, sympify, Symbol, Function, flatten, Derivative, Basic

    if printer is not None:
        if inspect.isfunction(printer):
            lambdarepr = printer
        else:
            if inspect.isclass(printer):
                lambdarepr = lambda expr: printer().doprint(expr)
            else:
                lambdarepr = lambda expr: printer.doprint(expr)
    else:
        #XXX: This has to be done here because of circular imports
        from sympy.printing.lambdarepr import lambdarepr

    def sub_args(args, dummies_dict):
        if isinstance(args, string_types):
            return args
        elif isinstance(args, DeferredVector):
            return str(args)
        elif iterable(args):
            dummies = flatten([sub_args(a, dummies_dict) for a in args])
            return ",".join(str(a) for a in dummies)
        else:
            # replace these with Dummy symbols
            if isinstance(args, (Function, Symbol, Derivative)):
                dummies = Dummy()
                dummies_dict.update({args : dummies})
                return str(dummies)
            else:
                return str(args)

    def sub_expr(expr, dummies_dict):
        try:
            expr = sympify(expr).xreplace(dummies_dict)
        except Exception:
            if isinstance(expr, DeferredVector):
                pass
            elif isinstance(expr, dict):
                k = [sub_expr(sympify(a), dummies_dict) for a in expr.keys()]
                v = [sub_expr(sympify(a), dummies_dict) for a in expr.values()]
                expr = dict(zip(k, v))
            elif isinstance(expr, tuple):
                expr = tuple(sub_expr(sympify(a), dummies_dict) for a in expr)
            elif isinstance(expr, list):
                expr = [sub_expr(sympify(a), dummies_dict) for a in expr]
        return expr

    # Transform args
    def isiter(l):
        return iterable(l, exclude=(str, DeferredVector, NotIterable))

    def flat_indexes(iterable):
        n = 0

        for el in iterable:
            if isiter(el):
                for ndeep in flat_indexes(el):
                    yield (n,) + ndeep
            else:
                yield (n,)

            n += 1

    if dummify is None:
        dummify = any(isinstance(a, Basic) and
            a.atoms(Function, Derivative) for a in (
            args if isiter(args) else [args]))

    if isiter(args) and any(isiter(i) for i in args):
        dum_args = [str(Dummy(str(i))) for i in range(len(args))]

        indexed_args = ','.join([
            dum_args[ind[0]] + ''.join(["[%s]" % k for k in ind[1:]])
                    for ind in flat_indexes(args)])

        lstr = lambdastr(flatten(args), expr, printer=printer, dummify=dummify)

        return 'lambda %s: (%s)(%s)' % (','.join(dum_args), lstr, indexed_args)

    dummies_dict = {}
    if dummify:
        args = sub_args(args, dummies_dict)
    else:
        if isinstance(args, string_types):
            pass
        elif iterable(args, exclude=DeferredVector):
            args = ",".join(str(a) for a in args)

    # Transform expr
    if dummify:
        if isinstance(expr, string_types):
            pass
        else:
            expr = sub_expr(expr, dummies_dict)
    expr = lambdarepr(expr)
    return "lambda %s: (%s)" % (args, expr)

class _EvaluatorPrinter(object):
    def __init__(self, printer=None, dummify=False):
        self._dummify = dummify

        #XXX: This has to be done here because of circular imports
        from sympy.printing.lambdarepr import LambdaPrinter

        if printer is None:
            printer = LambdaPrinter()

        if inspect.isfunction(printer):
            self._exprrepr = printer
        else:
            if inspect.isclass(printer):
                printer = printer()

            self._exprrepr = printer.doprint

            if hasattr(printer, '_print_Symbol'):
                symbolrepr = printer._print_Symbol

            if hasattr(printer, '_print_Dummy'):
                dummyrepr = printer._print_Dummy

        # Used to print the generated function arguments in a standard way
        self._argrepr = LambdaPrinter().doprint

    def doprint(self, funcname, args, expr):
        """Returns the function definition code as a string."""
        from sympy import Dummy

        funcbody = []

        if not iterable(args):
            args = [args]

        argstrs, expr = self._preprocess(args, expr)

        # Generate argument unpacking and final argument list
        funcargs = []
        unpackings = []

        for argstr in argstrs:
            if iterable(argstr):
                funcargs.append(self._argrepr(Dummy()))
                unpackings.extend(self._print_unpacking(argstr, funcargs[-1]))
            else:
                funcargs.append(argstr)

        funcsig = 'def {}({}):'.format(funcname, ', '.join(funcargs))

        # Wrap input arguments before unpacking
        funcbody.extend(self._print_funcargwrapping(funcargs))

        funcbody.extend(unpackings)

        funcbody.append('return ({})'.format(self._exprrepr(expr)))

        funclines = [funcsig]
        funclines.extend('    ' + line for line in funcbody)

        return '\n'.join(funclines) + '\n'

    if PY3:
        @classmethod
        def _is_safe_ident(cls, ident):
            return isinstance(ident, string_types) and ident.isidentifier() \
                    and not keyword.iskeyword(ident)
    else:
        _safe_ident_re = re.compile('^[a-zA-Z_][a-zA-Z0-9_]*$')

        @classmethod
        def _is_safe_ident(cls, ident):
            return isinstance(ident, string_types) and cls._safe_ident_re.match(ident) \
                and not (keyword.iskeyword(ident) or ident == 'None')

    def _preprocess(self, args, expr):
        """Preprocess args, expr to replace arguments that do not map
        to valid Python identifiers.

        Returns string form of args, and updated expr.
        """
        from sympy import Dummy, Function, flatten, Derivative, ordered, Basic
        from sympy.matrices import DeferredVector

        # Args of type Dummy can cause name collisions with args
        # of type Symbol.  Force dummify of everything in this
        # situation.
        dummify = self._dummify or any(
            isinstance(arg, Dummy) for arg in flatten(args))

        argstrs = [None]*len(args)
        for arg, i in reversed(list(ordered(zip(args, range(len(args)))))):
            if iterable(arg):
                s, expr = self._preprocess(arg, expr)
            elif isinstance(arg, DeferredVector):
                s = str(arg)
            elif isinstance(arg, Basic) and arg.is_symbol:
                s = self._argrepr(arg)
                if dummify or not self._is_safe_ident(s):
                    dummy = Dummy()
                    s = self._argrepr(dummy)
                    expr = self._subexpr(expr, {arg: dummy})
            elif dummify or isinstance(arg, (Function, Derivative)):
                dummy = Dummy()
                s = self._argrepr(dummy)
                expr = self._subexpr(expr, {arg: dummy})
            else:
                s = str(arg)
            argstrs[i] = s
        return argstrs, expr

    def _subexpr(self, expr, dummies_dict):
        from sympy.matrices import DeferredVector
        from sympy import sympify

        expr = sympify(expr)
        xreplace = getattr(expr, 'xreplace', None)
        if xreplace is not None:
            expr = xreplace(dummies_dict)
        else:
            if isinstance(expr, DeferredVector):
                pass
            elif isinstance(expr, dict):
                k = [self._subexpr(sympify(a), dummies_dict) for a in expr.keys()]
                v = [self._subexpr(sympify(a), dummies_dict) for a in expr.values()]
                expr = dict(zip(k, v))
            elif isinstance(expr, tuple):
                expr = tuple(self._subexpr(sympify(a), dummies_dict) for a in expr)
            elif isinstance(expr, list):
                expr = [self._subexpr(sympify(a), dummies_dict) for a in expr]
        return expr

    def _print_funcargwrapping(self, args):
        """Generate argument wrapping code.

        args is the argument list of the generated function (strings).

        Return value is a list of lines of code that will be inserted  at
        the beginning of the function definition.
        """
        return []

    def _print_unpacking(self, unpackto, arg):
        """Generate argument unpacking code.

        arg is the function argument to be unpacked (a string), and
        unpackto is a list or nested lists of the variable names (strings) to
        unpack to.
        """
        def unpack_lhs(lvalues):
            return '[{}]'.format(', '.join(
                unpack_lhs(val) if iterable(val) else val for val in lvalues))

        return ['{} = {}'.format(unpack_lhs(unpackto), arg)]

class _TensorflowEvaluatorPrinter(_EvaluatorPrinter):
    def _print_unpacking(self, lvalues, rvalue):
        """Generate argument unpacking code.

        This method is used when the input value is not interable,
        but can be indexed (see issue #14655).
        """
        from sympy import flatten

        def flat_indexes(elems):
            n = 0

            for el in elems:
                if iterable(el):
                    for ndeep in flat_indexes(el):
                        yield (n,) + ndeep
                else:
                    yield (n,)

                n += 1

        indexed = ', '.join('{}[{}]'.format(rvalue, ']['.join(map(str, ind)))
                                for ind in flat_indexes(lvalues))

        return ['[{}] = [{}]'.format(', '.join(flatten(lvalues)), indexed)]

def _imp_namespace(expr, namespace=None):
    """ Return namespace dict with function implementations

    We need to search for functions in anything that can be thrown at
    us - that is - anything that could be passed as ``expr``.  Examples
    include sympy expressions, as well as tuples, lists and dicts that may
    contain sympy expressions.

    Parameters
    ----------
    expr : object
       Something passed to lambdify, that will generate valid code from
       ``str(expr)``.
    namespace : None or mapping
       Namespace to fill.  None results in new empty dict

    Returns
    -------
    namespace : dict
       dict with keys of implemented function names within ``expr`` and
       corresponding values being the numerical implementation of
       function

    Examples
    ========

    >>> from sympy.abc import x
    >>> from sympy.utilities.lambdify import implemented_function, _imp_namespace
    >>> from sympy import Function
    >>> f = implemented_function(Function('f'), lambda x: x+1)
    >>> g = implemented_function(Function('g'), lambda x: x*10)
    >>> namespace = _imp_namespace(f(g(x)))
    >>> sorted(namespace.keys())
    ['f', 'g']
    """
    # Delayed import to avoid circular imports
    from sympy.core.function import FunctionClass
    if namespace is None:
        namespace = {}
    # tuples, lists, dicts are valid expressions
    if is_sequence(expr):
        for arg in expr:
            _imp_namespace(arg, namespace)
        return namespace
    elif isinstance(expr, dict):
        for key, val in expr.items():
            # functions can be in dictionary keys
            _imp_namespace(key, namespace)
            _imp_namespace(val, namespace)
        return namespace
    # sympy expressions may be Functions themselves
    func = getattr(expr, 'func', None)
    if isinstance(func, FunctionClass):
        imp = getattr(func, '_imp_', None)
        if imp is not None:
            name = expr.func.__name__
            if name in namespace and namespace[name] != imp:
                raise ValueError('We found more than one '
                                 'implementation with name '
                                 '"%s"' % name)
            namespace[name] = imp
    # and / or they may take Functions as arguments
    if hasattr(expr, 'args'):
        for arg in expr.args:
            _imp_namespace(arg, namespace)
    return namespace


def implemented_function(symfunc, implementation):
    """ Add numerical ``implementation`` to function ``symfunc``.

    ``symfunc`` can be an ``UndefinedFunction`` instance, or a name string.
    In the latter case we create an ``UndefinedFunction`` instance with that
    name.

    Be aware that this is a quick workaround, not a general method to create
    special symbolic functions. If you want to create a symbolic function to be
    used by all the machinery of SymPy you should subclass the ``Function``
    class.

    Parameters
    ----------
    symfunc : ``str`` or ``UndefinedFunction`` instance
       If ``str``, then create new ``UndefinedFunction`` with this as
       name.  If ``symfunc`` is an Undefined function, create a new function
       with the same name and the implemented function attached.
    implementation : callable
       numerical implementation to be called by ``evalf()`` or ``lambdify``

    Returns
    -------
    afunc : sympy.FunctionClass instance
       function with attached implementation

    Examples
    ========

    >>> from sympy.abc import x
    >>> from sympy.utilities.lambdify import lambdify, implemented_function
    >>> from sympy import Function
    >>> f = implemented_function('f', lambda x: x+1)
    >>> lam_f = lambdify(x, f(x))
    >>> lam_f(4)
    5
    """
    # Delayed import to avoid circular imports
    from sympy.core.function import UndefinedFunction
    # if name, create function to hold implementation
    _extra_kwargs = {}
    if isinstance(symfunc, UndefinedFunction):
        _extra_kwargs = symfunc._extra_kwargs
        symfunc = symfunc.__name__
    if isinstance(symfunc, string_types):
        # Keyword arguments to UndefinedFunction are added as attributes to
        # the created class.
        symfunc = UndefinedFunction(symfunc, _imp_=staticmethod(implementation), **_extra_kwargs)
    elif not isinstance(symfunc, UndefinedFunction):
        raise ValueError('symfunc should be either a string or'
                         ' an UndefinedFunction instance.')
    return symfunc
