1360 lines
48 KiB
Python
1360 lines
48 KiB
Python
"""
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This module provides convenient functions to transform sympy expressions to
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lambda functions which can be used to calculate numerical values very fast.
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"""
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from typing import Any, Dict, Iterable
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import builtins
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import inspect
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import keyword
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import textwrap
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import linecache
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from sympy.utilities.exceptions import SymPyDeprecationWarning
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from sympy.core.compatibility import (is_sequence, iterable,
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NotIterable)
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from sympy.utilities.misc import filldedent
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from sympy.utilities.decorator import doctest_depends_on
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__doctest_requires__ = {('lambdify',): ['numpy', 'tensorflow']}
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# Default namespaces, letting us define translations that can't be defined
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# by simple variable maps, like I => 1j
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MATH_DEFAULT = {} # type: Dict[str, Any]
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MPMATH_DEFAULT = {} # type: Dict[str, Any]
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NUMPY_DEFAULT = {"I": 1j} # type: Dict[str, Any]
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SCIPY_DEFAULT = {"I": 1j} # type: Dict[str, Any]
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CUPY_DEFAULT = {"I": 1j} # type: Dict[str, Any]
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TENSORFLOW_DEFAULT = {} # type: Dict[str, Any]
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SYMPY_DEFAULT = {} # type: Dict[str, Any]
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NUMEXPR_DEFAULT = {} # type: Dict[str, Any]
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# These are the namespaces the lambda functions will use.
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# These are separate from the names above because they are modified
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# throughout this file, whereas the defaults should remain unmodified.
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MATH = MATH_DEFAULT.copy()
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MPMATH = MPMATH_DEFAULT.copy()
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NUMPY = NUMPY_DEFAULT.copy()
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SCIPY = SCIPY_DEFAULT.copy()
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CUPY = CUPY_DEFAULT.copy()
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TENSORFLOW = TENSORFLOW_DEFAULT.copy()
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SYMPY = SYMPY_DEFAULT.copy()
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NUMEXPR = NUMEXPR_DEFAULT.copy()
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# Mappings between sympy and other modules function names.
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MATH_TRANSLATIONS = {
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"ceiling": "ceil",
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"E": "e",
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"ln": "log",
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}
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# NOTE: This dictionary is reused in Function._eval_evalf to allow subclasses
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# of Function to automatically evalf.
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MPMATH_TRANSLATIONS = {
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"Abs": "fabs",
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"elliptic_k": "ellipk",
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"elliptic_f": "ellipf",
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"elliptic_e": "ellipe",
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"elliptic_pi": "ellippi",
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"ceiling": "ceil",
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"chebyshevt": "chebyt",
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"chebyshevu": "chebyu",
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"E": "e",
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"I": "j",
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"ln": "log",
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#"lowergamma":"lower_gamma",
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"oo": "inf",
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#"uppergamma":"upper_gamma",
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"LambertW": "lambertw",
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"MutableDenseMatrix": "matrix",
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"ImmutableDenseMatrix": "matrix",
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"conjugate": "conj",
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"dirichlet_eta": "altzeta",
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"Ei": "ei",
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"Shi": "shi",
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"Chi": "chi",
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"Si": "si",
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"Ci": "ci",
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"RisingFactorial": "rf",
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"FallingFactorial": "ff",
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"betainc_regularized": "betainc",
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}
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NUMPY_TRANSLATIONS = {
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"Heaviside": "heaviside",
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} # type: Dict[str, str]
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SCIPY_TRANSLATIONS = {} # type: Dict[str, str]
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CUPY_TRANSLATIONS = {} # type: Dict[str, str]
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TENSORFLOW_TRANSLATIONS = {} # type: Dict[str, str]
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NUMEXPR_TRANSLATIONS = {} # type: Dict[str, str]
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# Available modules:
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MODULES = {
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"math": (MATH, MATH_DEFAULT, MATH_TRANSLATIONS, ("from math import *",)),
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"mpmath": (MPMATH, MPMATH_DEFAULT, MPMATH_TRANSLATIONS, ("from mpmath import *",)),
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"numpy": (NUMPY, NUMPY_DEFAULT, NUMPY_TRANSLATIONS, ("import numpy; from numpy import *; from numpy.linalg import *",)),
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"scipy": (SCIPY, SCIPY_DEFAULT, SCIPY_TRANSLATIONS, ("import numpy; import scipy; from scipy import *; from scipy.special import *",)),
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"cupy": (CUPY, CUPY_DEFAULT, CUPY_TRANSLATIONS, ("import cupy",)),
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"tensorflow": (TENSORFLOW, TENSORFLOW_DEFAULT, TENSORFLOW_TRANSLATIONS, ("import tensorflow",)),
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"sympy": (SYMPY, SYMPY_DEFAULT, {}, (
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"from sympy.functions import *",
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"from sympy.matrices import *",
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"from sympy import Integral, pi, oo, nan, zoo, E, I",)),
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"numexpr" : (NUMEXPR, NUMEXPR_DEFAULT, NUMEXPR_TRANSLATIONS,
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("import_module('numexpr')", )),
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}
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def _import(module, reload=False):
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"""
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Creates a global translation dictionary for module.
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The argument module has to be one of the following strings: "math",
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"mpmath", "numpy", "sympy", "tensorflow".
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These dictionaries map names of python functions to their equivalent in
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other modules.
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"""
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# Required despite static analysis claiming it is not used
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from sympy.external import import_module # noqa:F401
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try:
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namespace, namespace_default, translations, import_commands = MODULES[
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module]
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except KeyError:
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raise NameError(
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"'%s' module can't be used for lambdification" % module)
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# Clear namespace or exit
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if namespace != namespace_default:
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# The namespace was already generated, don't do it again if not forced.
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if reload:
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namespace.clear()
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namespace.update(namespace_default)
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else:
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return
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for import_command in import_commands:
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if import_command.startswith('import_module'):
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module = eval(import_command)
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if module is not None:
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namespace.update(module.__dict__)
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continue
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else:
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try:
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exec(import_command, {}, namespace)
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continue
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except ImportError:
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pass
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raise ImportError(
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"can't import '%s' with '%s' command" % (module, import_command))
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# Add translated names to namespace
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for sympyname, translation in translations.items():
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namespace[sympyname] = namespace[translation]
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# For computing the modulus of a sympy expression we use the builtin abs
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# function, instead of the previously used fabs function for all
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# translation modules. This is because the fabs function in the math
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# module does not accept complex valued arguments. (see issue 9474). The
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# only exception, where we don't use the builtin abs function is the
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# mpmath translation module, because mpmath.fabs returns mpf objects in
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# contrast to abs().
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if 'Abs' not in namespace:
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namespace['Abs'] = abs
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# Used for dynamically generated filenames that are inserted into the
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# linecache.
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_lambdify_generated_counter = 1
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@doctest_depends_on(modules=('numpy', 'scipy', 'tensorflow',), python_version=(3,))
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def lambdify(args: Iterable, expr, modules=None, printer=None, use_imps=True,
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dummify=False, cse=False):
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"""Convert a SymPy expression into a function that allows for fast
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numeric evaluation.
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.. warning::
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This function uses ``exec``, and thus shouldn't be used on
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unsanitized input.
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.. versionchanged:: 1.7.0
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Passing a set for the *args* parameter is deprecated as sets are
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unordered. Use an ordered iterable such as a list or tuple.
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Explanation
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===========
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For example, to convert the SymPy expression ``sin(x) + cos(x)`` to an
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equivalent NumPy function that numerically evaluates it:
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>>> from sympy import sin, cos, symbols, lambdify
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>>> import numpy as np
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>>> x = symbols('x')
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>>> expr = sin(x) + cos(x)
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>>> expr
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sin(x) + cos(x)
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>>> f = lambdify(x, expr, 'numpy')
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>>> a = np.array([1, 2])
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>>> f(a)
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[1.38177329 0.49315059]
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The primary purpose of this function is to provide a bridge from SymPy
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expressions to numerical libraries such as NumPy, SciPy, NumExpr, mpmath,
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and tensorflow. In general, SymPy functions do not work with objects from
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other libraries, such as NumPy arrays, and functions from numeric
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libraries like NumPy or mpmath do not work on SymPy expressions.
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``lambdify`` bridges the two by converting a SymPy expression to an
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equivalent numeric function.
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The basic workflow with ``lambdify`` is to first create a SymPy expression
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representing whatever mathematical function you wish to evaluate. This
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should be done using only SymPy functions and expressions. Then, use
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``lambdify`` to convert this to an equivalent function for numerical
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evaluation. For instance, above we created ``expr`` using the SymPy symbol
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``x`` and SymPy functions ``sin`` and ``cos``, then converted it to an
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equivalent NumPy function ``f``, and called it on a NumPy array ``a``.
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Parameters
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==========
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args : List[Symbol]
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A variable or a list of variables whose nesting represents the
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nesting of the arguments that will be passed to the function.
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Variables can be symbols, undefined functions, or matrix symbols.
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>>> from sympy import Eq
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>>> from sympy.abc import x, y, z
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The list of variables should match the structure of how the
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arguments will be passed to the function. Simply enclose the
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parameters as they will be passed in a list.
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To call a function like ``f(x)`` then ``[x]``
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should be the first argument to ``lambdify``; for this
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case a single ``x`` can also be used:
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>>> f = lambdify(x, x + 1)
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>>> f(1)
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2
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>>> f = lambdify([x], x + 1)
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>>> f(1)
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2
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To call a function like ``f(x, y)`` then ``[x, y]`` will
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be the first argument of the ``lambdify``:
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>>> f = lambdify([x, y], x + y)
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>>> f(1, 1)
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2
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To call a function with a single 3-element tuple like
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``f((x, y, z))`` then ``[(x, y, z)]`` will be the first
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argument of the ``lambdify``:
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>>> f = lambdify([(x, y, z)], Eq(z**2, x**2 + y**2))
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>>> f((3, 4, 5))
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True
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If two args will be passed and the first is a scalar but
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the second is a tuple with two arguments then the items
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in the list should match that structure:
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>>> f = lambdify([x, (y, z)], x + y + z)
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>>> f(1, (2, 3))
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6
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expr : Expr
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An expression, list of expressions, or matrix to be evaluated.
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Lists may be nested.
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If the expression is a list, the output will also be a list.
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>>> f = lambdify(x, [x, [x + 1, x + 2]])
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>>> f(1)
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[1, [2, 3]]
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If it is a matrix, an array will be returned (for the NumPy module).
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>>> from sympy import Matrix
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>>> f = lambdify(x, Matrix([x, x + 1]))
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>>> f(1)
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[[1]
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[2]]
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Note that the argument order here (variables then expression) is used
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to emulate the Python ``lambda`` keyword. ``lambdify(x, expr)`` works
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(roughly) like ``lambda x: expr``
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(see :ref:`lambdify-how-it-works` below).
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modules : str, optional
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Specifies the numeric library to use.
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If not specified, *modules* defaults to:
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- ``["scipy", "numpy"]`` if SciPy is installed
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- ``["numpy"]`` if only NumPy is installed
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- ``["math", "mpmath", "sympy"]`` if neither is installed.
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That is, SymPy functions are replaced as far as possible by
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either ``scipy`` or ``numpy`` functions if available, and Python's
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standard library ``math``, or ``mpmath`` functions otherwise.
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*modules* can be one of the following types:
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- The strings ``"math"``, ``"mpmath"``, ``"numpy"``, ``"numexpr"``,
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``"scipy"``, ``"sympy"``, or ``"tensorflow"``. This uses the
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corresponding printer and namespace mapping for that module.
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- A module (e.g., ``math``). This uses the global namespace of the
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module. If the module is one of the above known modules, it will
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also use the corresponding printer and namespace mapping
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(i.e., ``modules=numpy`` is equivalent to ``modules="numpy"``).
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- A dictionary that maps names of SymPy functions to arbitrary
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functions
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(e.g., ``{'sin': custom_sin}``).
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- A list that contains a mix of the arguments above, with higher
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priority given to entries appearing first
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(e.g., to use the NumPy module but override the ``sin`` function
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with a custom version, you can use
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``[{'sin': custom_sin}, 'numpy']``).
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dummify : bool, optional
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Whether or not the variables in the provided expression that are not
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valid Python identifiers are substituted with dummy symbols.
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This allows for undefined functions like ``Function('f')(t)`` to be
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supplied as arguments. By default, the variables are only dummified
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if they are not valid Python identifiers.
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Set ``dummify=True`` to replace all arguments with dummy symbols
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(if ``args`` is not a string) - for example, to ensure that the
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arguments do not redefine any built-in names.
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cse : bool, or callable, optional
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Large expressions can be computed more efficiently when
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common subexpressions are identified and precomputed before
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being used multiple time. Finding the subexpressions will make
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creation of the 'lambdify' function slower, however.
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When ``True``, ``sympy.simplify.cse`` is used, otherwise (the default)
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the user may pass a function matching the ``cse`` signature.
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Examples
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========
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>>> from sympy.utilities.lambdify import implemented_function
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>>> from sympy import sqrt, sin, Matrix
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>>> from sympy import Function
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>>> from sympy.abc import w, x, y, z
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>>> f = lambdify(x, x**2)
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>>> f(2)
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4
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>>> f = lambdify((x, y, z), [z, y, x])
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>>> f(1,2,3)
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[3, 2, 1]
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>>> f = lambdify(x, sqrt(x))
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>>> f(4)
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2.0
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>>> f = lambdify((x, y), sin(x*y)**2)
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>>> f(0, 5)
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0.0
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>>> row = lambdify((x, y), Matrix((x, x + y)).T, modules='sympy')
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>>> row(1, 2)
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Matrix([[1, 3]])
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``lambdify`` can be used to translate SymPy expressions into mpmath
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functions. This may be preferable to using ``evalf`` (which uses mpmath on
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the backend) in some cases.
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>>> f = lambdify(x, sin(x), 'mpmath')
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>>> f(1)
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0.8414709848078965
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Tuple arguments are handled and the lambdified function should
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be called with the same type of arguments as were used to create
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the function:
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>>> f = lambdify((x, (y, z)), x + y)
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>>> f(1, (2, 4))
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3
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The ``flatten`` function can be used to always work with flattened
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arguments:
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>>> from sympy.utilities.iterables import flatten
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>>> args = w, (x, (y, z))
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>>> vals = 1, (2, (3, 4))
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>>> f = lambdify(flatten(args), w + x + y + z)
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>>> f(*flatten(vals))
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10
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Functions present in ``expr`` can also carry their own numerical
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implementations, in a callable attached to the ``_imp_`` attribute. This
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can be used with undefined functions using the ``implemented_function``
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factory:
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>>> f = implemented_function(Function('f'), lambda x: x+1)
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>>> func = lambdify(x, f(x))
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>>> func(4)
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5
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``lambdify`` always prefers ``_imp_`` implementations to implementations
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in other namespaces, unless the ``use_imps`` input parameter is False.
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Usage with Tensorflow:
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>>> import tensorflow as tf
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>>> from sympy import Max, sin, lambdify
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>>> from sympy.abc import x
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>>> f = Max(x, sin(x))
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>>> func = lambdify(x, f, 'tensorflow')
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After tensorflow v2, eager execution is enabled by default.
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If you want to get the compatible result across tensorflow v1 and v2
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as same as this tutorial, run this line.
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>>> tf.compat.v1.enable_eager_execution()
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If you have eager execution enabled, you can get the result out
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immediately as you can use numpy.
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If you pass tensorflow objects, you may get an ``EagerTensor``
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object instead of value.
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>>> result = func(tf.constant(1.0))
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>>> print(result)
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tf.Tensor(1.0, shape=(), dtype=float32)
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>>> print(result.__class__)
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<class 'tensorflow.python.framework.ops.EagerTensor'>
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You can use ``.numpy()`` to get the numpy value of the tensor.
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>>> result.numpy()
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1.0
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>>> var = tf.Variable(2.0)
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>>> result = func(var) # also works for tf.Variable and tf.Placeholder
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>>> result.numpy()
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2.0
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And it works with any shape array.
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>>> tensor = tf.constant([[1.0, 2.0], [3.0, 4.0]])
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>>> result = func(tensor)
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>>> result.numpy()
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[[1. 2.]
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[3. 4.]]
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Notes
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=====
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- For functions involving large array calculations, numexpr can provide a
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significant speedup over numpy. Please note that the available functions
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for numexpr are more limited than numpy but can be expanded with
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``implemented_function`` and user defined subclasses of Function. If
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specified, numexpr may be the only option in modules. The official list
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of numexpr functions can be found at:
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https://numexpr.readthedocs.io/en/latest/user_guide.html#supported-functions
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|
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- In previous versions of SymPy, ``lambdify`` replaced ``Matrix`` with
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``numpy.matrix`` by default. As of SymPy 1.0 ``numpy.array`` is the
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default. To get the old default behavior you must pass in
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``[{'ImmutableDenseMatrix': numpy.matrix}, 'numpy']`` to the
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``modules`` kwarg.
|
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|
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>>> from sympy import lambdify, Matrix
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>>> from sympy.abc import x, y
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>>> import numpy
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>>> array2mat = [{'ImmutableDenseMatrix': numpy.matrix}, 'numpy']
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>>> f = lambdify((x, y), Matrix([x, y]), modules=array2mat)
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>>> f(1, 2)
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[[1]
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[2]]
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- In the above examples, the generated functions can accept scalar
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values or numpy arrays as arguments. However, in some cases
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|
the generated function relies on the input being a numpy array:
|
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|
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>>> from sympy import Piecewise
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|
>>> from sympy.testing.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
|
|
|
|
.. _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
|
|
|
|
>>> try:
|
|
... g(x + 1)
|
|
... # NumPy release after 1.17 raises TypeError instead of
|
|
... # AttributeError
|
|
... except (AttributeError, TypeError):
|
|
... raise AttributeError() # doctest: +IGNORE_EXCEPTION_DETAIL
|
|
Traceback (most recent call last):
|
|
...
|
|
AttributeError:
|
|
|
|
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.
|
|
"""
|
|
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 = ["numpy", "scipy"]
|
|
|
|
# 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, str)) 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 = {} # type: Dict[str, Any]
|
|
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 # type: ignore
|
|
elif _module_present('scipy', namespaces):
|
|
from sympy.printing.numpy import SciPyPrinter as Printer # type: ignore
|
|
elif _module_present('numpy', namespaces):
|
|
from sympy.printing.numpy import NumPyPrinter as Printer # type: ignore
|
|
elif _module_present('cupy', namespaces):
|
|
from sympy.printing.numpy import CuPyPrinter as Printer # type: ignore
|
|
elif _module_present('numexpr', namespaces):
|
|
from sympy.printing.lambdarepr import NumExprPrinter as Printer # type: ignore
|
|
elif _module_present('tensorflow', namespaces):
|
|
from sympy.printing.tensorflow import TensorflowPrinter as Printer # type: ignore
|
|
elif _module_present('sympy', namespaces):
|
|
from sympy.printing.pycode import SymPyPrinter as Printer # type: ignore
|
|
else:
|
|
from sympy.printing.pycode import PythonCodePrinter as Printer # type: ignore
|
|
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})
|
|
|
|
if isinstance(args, set):
|
|
SymPyDeprecationWarning(
|
|
feature="The list of arguments is a `set`. This leads to unpredictable results",
|
|
useinstead=": Convert set into list or tuple",
|
|
issue=20013,
|
|
deprecated_since_version="1.6.3"
|
|
).warn()
|
|
|
|
# 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() # type: ignore
|
|
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))
|
|
|
|
# Create the function definition code and execute it
|
|
funcname = '_lambdifygenerated'
|
|
if _module_present('tensorflow', namespaces):
|
|
funcprinter = _TensorflowEvaluatorPrinter(printer, dummify) # type: _EvaluatorPrinter
|
|
else:
|
|
funcprinter = _EvaluatorPrinter(printer, dummify)
|
|
|
|
if cse == True:
|
|
from sympy.simplify.cse_main import cse
|
|
cses, _expr = cse(expr, list=False)
|
|
elif callable(cse):
|
|
cses, _expr = cse(expr)
|
|
else:
|
|
cses, _expr = (), expr
|
|
funcstr = funcprinter.doprint(funcname, args, _expr, cses=cses)
|
|
|
|
# Collect the module imports from the code printers.
|
|
imp_mod_lines = []
|
|
for mod, keys in (getattr(printer, 'module_imports', None) or {}).items():
|
|
for k in keys:
|
|
if k not in namespace:
|
|
ln = "from %s import %s" % (mod, k)
|
|
try:
|
|
exec(ln, {}, namespace)
|
|
except ImportError:
|
|
# Tensorflow 2.0 has issues with importing a specific
|
|
# function from its submodule.
|
|
# https://github.com/tensorflow/tensorflow/issues/33022
|
|
ln = "%s = %s.%s" % (k, mod, k)
|
|
exec(ln, {}, namespace)
|
|
imp_mod_lines.append(ln)
|
|
|
|
# Provide lambda expression with builtins, and compatible implementation of range
|
|
namespace.update({'builtins':builtins, 'range':range})
|
|
|
|
funclocals = {} # type: Dict[str, Any]
|
|
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) # type: ignore
|
|
|
|
func = funclocals[funcname]
|
|
|
|
# Apply the docstring
|
|
sig = "func({})".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, str):
|
|
_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, 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)
|
|
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):
|
|
expr = sympify(expr)
|
|
# dict/tuple are sympified to Basic
|
|
if isinstance(expr, Basic):
|
|
expr = expr.xreplace(dummies_dict)
|
|
# list is not sympified to Basic
|
|
elif isinstance(expr, list):
|
|
expr = [sub_expr(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, str):
|
|
pass
|
|
elif iterable(args, exclude=DeferredVector):
|
|
args = ",".join(str(a) for a in args)
|
|
|
|
# Transform expr
|
|
if dummify:
|
|
if isinstance(expr, str):
|
|
pass
|
|
else:
|
|
expr = sub_expr(expr, dummies_dict)
|
|
expr = lambdarepr(expr)
|
|
return "lambda %s: (%s)" % (args, expr)
|
|
|
|
class _EvaluatorPrinter:
|
|
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, *, cses=()):
|
|
"""
|
|
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.extend(['{} = {}'.format(s, self._exprrepr(e)) for s, e in cses])
|
|
|
|
str_expr = self._exprrepr(expr)
|
|
if '\n' in str_expr:
|
|
str_expr = '({})'.format(str_expr)
|
|
funcbody.append('return {}'.format(str_expr))
|
|
|
|
funclines = [funcsig]
|
|
funclines.extend(' ' + line for line in funcbody)
|
|
|
|
return '\n'.join(funclines) + '\n'
|
|
|
|
@classmethod
|
|
def _is_safe_ident(cls, ident):
|
|
return isinstance(ident, str) and ident.isidentifier() \
|
|
and not keyword.iskeyword(ident)
|
|
|
|
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
|
|
from sympy.core.symbol import uniquely_named_symbol
|
|
from sympy.core.expr import Expr
|
|
|
|
# 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()
|
|
if isinstance(expr, Expr):
|
|
dummy = uniquely_named_symbol(
|
|
dummy.name, expr, modify=lambda s: '_' + s)
|
|
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
|
|
>>> 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
|
|
kwargs = {}
|
|
if isinstance(symfunc, UndefinedFunction):
|
|
kwargs = symfunc._kwargs
|
|
symfunc = symfunc.__name__
|
|
if isinstance(symfunc, str):
|
|
# Keyword arguments to UndefinedFunction are added as attributes to
|
|
# the created class.
|
|
symfunc = UndefinedFunction(
|
|
symfunc, _imp_=staticmethod(implementation), **kwargs)
|
|
elif not isinstance(symfunc, UndefinedFunction):
|
|
raise ValueError(filldedent('''
|
|
symfunc should be either a string or
|
|
an UndefinedFunction instance.'''))
|
|
return symfunc
|