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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 textwrap
from sympy.core.compatibility import (exec_, is_sequence, iterable,
NotIterable, string_types, range, builtins)
from sympy.utilities.decorator import doctest_depends_on
# These are the namespaces the lambda functions will use.
MATH = {}
NUMPY = {}
SYMPY = {}
# Default namespaces, letting us define translations that can't be defined
# by simple variable maps, like I => 1j
# These are separate from the names above because the above names are modified
# throughout this file, whereas these should remain unmodified.
# Mappings between sympy and other modules function names.
"ceiling": "ceil",
"E": "e",
"ln": "log",
"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",
"oo": "inf",
"LambertW": "lambertw",
"MutableDenseMatrix": "matrix",
"ImmutableMatrix": "matrix",
"conjugate": "conj",
"dirichlet_eta": "altzeta",
"Ei": "ei",
"Shi": "shi",
"Chi": "chi",
"Si": "si",
"Ci": "ci"
"acos": "arccos",
"acosh": "arccosh",
"arg": "angle",
"asin": "arcsin",
"asinh": "arcsinh",
"atan": "arctan",
"atan2": "arctan2",
"atanh": "arctanh",
"ceiling": "ceil",
"E": "e",
"im": "imag",
"ln": "log",
"Mod": "mod",
"oo": "inf",
"re": "real",
"SparseMatrix": "array",
"ImmutableSparseMatrix": "array",
"Matrix": "array",
"MutableDenseMatrix": "array",
"ImmutableMatrix": "array",
"ImmutableDenseMatrix": "array",
"Abs": "abs",
"ceiling": "ceil",
"im": "imag",
"ln": "log",
"Mod": "mod",
"conjugate": "conj",
"re": "real",
# Available 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_module('numpy')",)),
"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",)),
("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.
from sympy.external import import_module
namespace, namespace_default, translations, import_commands = MODULES[
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:
for import_command in import_commands:
if import_command.startswith('import_module'):
module = eval(import_command)
if module is not None:
exec_(import_command, {}, namespace)
except ImportError:
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
def lambdify(args, expr, modules=None, printer=None, use_imps=True,
Returns a lambda function for fast calculation of numerical values.
If not specified differently by the user, ``modules`` defaults to
``["numpy"]`` if NumPy is installed, and ``["math", "mpmath", "sympy"]``
if it isn't, that is, SymPy functions are replaced as far as possible by
either ``numpy`` functions if available, and Python's standard library
``math``, or ``mpmath`` functions otherwise. To change this behavior, the
"modules" argument can be used. It accepts:
- the strings "math", "mpmath", "numpy", "numexpr", "sympy", "tensorflow"
- any modules (e.g. math)
- dictionaries that map names of sympy functions to arbitrary functions
- lists that contain a mix of the arguments above, with higher priority
given to entries appearing first.
The default behavior is to substitute all arguments in the provided
expression with dummy symbols. This allows for applied functions (e.g.
f(t)) to be supplied as arguments. Call the function with dummify=False if
dummy substitution is unwanted (and `args` is not a string). If you want
to view the lambdified function or provide "sympy" as the module, you
should probably set dummify=False.
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:
In previous releases ``lambdify`` replaced ``Matrix`` with ``numpy.matrix``
by default. As of release 1.0 ``numpy.array`` is the default.
To get the old default behavior you must pass in ``[{'ImmutableMatrix':
numpy.matrix}, 'numpy']`` to the ``modules`` kwarg.
>>> from sympy import lambdify, Matrix
>>> from import x, y
>>> import numpy
>>> array2mat = [{'ImmutableMatrix': numpy.matrix}, 'numpy']
>>> f = lambdify((x, y), Matrix([x, y]), modules=array2mat)
>>> f(1, 2)
(1) Use one of the provided modules:
>>> from sympy import sin, tan, gamma
>>> from sympy.utilities.lambdify import lambdastr
>>> from import x, y
>>> f = lambdify(x, sin(x), "math")
Attention: Functions that are not in the math module will throw a name
error when the lambda function is evaluated! So this would
be better:
>>> f = lambdify(x, sin(x)*gamma(x), ("math", "mpmath", "sympy"))
(2) Use some other module:
>>> import numpy
>>> f = lambdify((x,y), tan(x*y), numpy)
Attention: There are naming differences between numpy and sympy. So if
you simply take the numpy module, e.g. sympy.atan will not be
translated to numpy.arctan. Use the modified module instead
by passing the string "numpy":
>>> f = lambdify((x,y), tan(x*y), "numpy")
>>> f(1, 2)
>>> from numpy import array
>>> f(array([1, 2, 3]), array([2, 3, 5]))
[-2.18503986 -0.29100619 -0.8559934 ]
(3) Use a dictionary defining custom functions:
>>> def my_cool_function(x): return 'sin(%s) is cool' % x
>>> myfuncs = {"sin" : my_cool_function}
>>> f = lambdify(x, sin(x), myfuncs); f(1)
'sin(1) is cool'
>>> from sympy.utilities.lambdify import implemented_function
>>> from sympy import sqrt, sin, Matrix
>>> from sympy import Function
>>> from import w, x, y, z
>>> f = lambdify(x, x**2)
>>> f(2)
>>> f = lambdify((x, y, z), [z, y, x])
>>> f(1,2,3)
[3, 2, 1]
>>> f = lambdify(x, sqrt(x))
>>> f(4)
>>> f = lambdify((x, y), sin(x*y)**2)
>>> f(0, 5)
>>> row = lambdify((x, y), Matrix((x, x + y)).T, modules='sympy')
>>> row(1, 2)
Matrix([[1, 3]])
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))
A more robust way of handling this is to always work with flattened
>>> 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))
Functions present in `expr` can also carry their own numerical
implementations, in a callable attached to the ``_imp_``
attribute. Usually you attach this using the
``implemented_function`` factory:
>>> f = implemented_function(Function('f'), lambda x: x+1)
>>> func = lambdify(x, f(x))
>>> func(4)
``lambdify`` always prefers ``_imp_`` implementations to implementations
in other namespaces, unless the ``use_imps`` input parameter is False.
from sympy.core.symbol import Symbol
from sympy.utilities.iterables import flatten
# If the user hasn't specified any modules, use what is available.
module_provided = True
if modules is None:
module_provided = False
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"]
modules = ["numpy"]
# Get the needed namespaces.
namespaces = []
# First find any function implementations
if use_imps:
# Check for dict before iterating
if isinstance(modules, (dict, str)) or not hasattr(modules, '__iter__'):
# 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)
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 _module_present('numpy',namespaces) and printer is None:
#XXX: This has to be done here because of circular imports
from sympy.printing.lambdarepr import NumPyPrinter as printer
if _module_present('numexpr',namespaces) and printer is None:
#XXX: This has to be done here because of circular imports
from sympy.printing.lambdarepr import NumExprPrinter as printer
if _module_present('tensorflow',namespaces) and printer is None:
#XXX: This has to be done here because of circular imports
from sympy.printing.lambdarepr import TensorflowPrinter as printer
# 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'):
# 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:
# Cannot infer name with certainty. arg_# will have to do.
names.append('arg_' + str(n))
# Create lambda function.
lstr = lambdastr(args, expr, printer=printer, dummify=dummify)
flat = '__flatten_args__'
if flat in lstr:
namespace.update({flat: flatten})
# Provide lambda expression with builtins, and compatible implementation of range
namespace.update({'builtins':builtins, 'range':range})
func = eval(lstr, namespace)
# For numpy lambdify, wrap all input arguments in arrays.
# This is a fix for gh-11306.
if module_provided and _module_present('numpy',namespaces):
def array_wrap(funcarg):
def wrapper(*argsx, **kwargsx):
return funcarg(*[namespace['asarray'](i) for i in argsx], **kwargsx)
return wrapper
func = array_wrap(func)
# 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}").format(sig=sig, expr=expr_str)
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, str):
return MODULES[m][0]
elif isinstance(m, dict):
return m
elif hasattr(m, "__dict__"):
return m.__dict__
raise TypeError("Argument must be either a string, dict or module but it is: %s" % m)
def lambdastr(args, expr, printer=None, dummify=False):
Returns a string that can be evaluated to a lambda function.
>>> from 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))(*list(__flatten_args__([_0,_1])))'
# Transforming everything to strings.
from sympy.matrices import DeferredVector
from sympy import Dummy, sympify, Symbol, Function, flatten
if printer is not None:
if inspect.isfunction(printer):
lambdarepr = printer
if inspect.isclass(printer):
lambdarepr = lambda expr: printer().doprint(expr)
lambdarepr = lambda expr: printer.doprint(expr)
#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, str):
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)
#Sub in dummy variables for functions or symbols
if isinstance(args, (Function, Symbol)):
dummies = Dummy()
dummies_dict.update({args : dummies})
return str(dummies)
return str(args)
def sub_expr(expr, dummies_dict):
expr = sympify(expr).xreplace(dummies_dict)
except Exception:
if isinstance(expr, DeferredVector):
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))
if isiter(args) and any(isiter(i) for i in args):
from sympy.utilities.iterables import flatten
import re
dum_args = [str(Dummy(str(i))) for i in range(len(args))]
iter_args = ','.join([i if isiter(a) else i
for i, a in zip(dum_args, args)])
lstr = lambdastr(flatten(args), expr, printer=printer, dummify=dummify)
flat = '__flatten_args__'
rv = 'lambda %s: (%s)(*list(%s([%s])))' % (
','.join(dum_args), lstr, flat, iter_args)
if len(re.findall(r'\b%s\b' % flat, rv)) > 1:
raise ValueError('the name %s is reserved by lambdastr' % flat)
return rv
dummies_dict = {}
if dummify:
args = sub_args(args, dummies_dict)
if isinstance(args, str):
elif iterable(args, exclude=DeferredVector):
args = ",".join(str(a) for a in args)
# Transform expr
if dummify:
if isinstance(expr, str):
expr = sub_expr(expr, dummies_dict)
expr = lambdarepr(expr)
return "lambda %s: (%s)" % (args, expr)
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.
expr : object
Something passed to lambdify, that will generate valid code from
namespace : None or mapping
Namespace to fill. None results in new empty dict
namespace : dict
dict with keys of implemented function names within `expr` and
corresponding values being the numerical implementation of
>>> from 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
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``
symfunc : ``str`` or ``UndefinedFunction`` instance
If ``str``, then create new ``UndefinedFunction`` with this as
name. If `symfunc` is a sympy function, attach implementation to it.
implementation : callable
numerical implementation to be called by ``evalf()`` or ``lambdify``
afunc : sympy.FunctionClass instance
function with attached implementation
>>> from import x
>>> from sympy.utilities.lambdify import lambdify, implemented_function
>>> from sympy import Function
>>> f = implemented_function(Function('f'), lambda x: x+1)
>>> lam_f = lambdify(x, f(x))
>>> lam_f(4)
# Delayed import to avoid circular imports
from sympy.core.function import UndefinedFunction
# if name, create function to hold implementation
if isinstance(symfunc, string_types):
symfunc = UndefinedFunction(symfunc)
elif not isinstance(symfunc, UndefinedFunction):
raise ValueError('symfunc should be either a string or'
' an UndefinedFunction instance.')
# We need to attach as a method because symfunc will be a class
symfunc._imp_ = staticmethod(implementation)
return symfunc