def test_issue_5596(): assert str(S("Q & C", locals=_clash1)) == 'And(C, Q)' assert str(S('pi(x)', locals=_clash2)) == 'pi(x)' assert str(S('pi(C, Q)', locals=_clash)) == 'pi(C, Q)' locals = {} exec_("from sympy.abc import Q, C", locals) assert str(S('C&Q', locals)) == 'And(C, Q)'
def parse_expr(s, local_dict=None, transformations=standard_transformations, global_dict=None): """Converts the string ``s`` to a SymPy expression, in ``local_dict`` Parameters ========== s : str The string to parse. local_dict : dict, optional A dictionary of local variables to use when parsing. global_dict : dict, optional A dictionary of global variables. By default, this is initialized with ``from sympy import *``; provide this parameter to override this behavior (for instance, to parse ``"Q & S"``). transformations : tuple, optional A tuple of transformation functions used to modify the tokens of the parsed expression before evaluation. The default transformations convert numeric literals into their SymPy equivalents, convert undefined variables into SymPy symbols, and allow the use of standard mathematical factorial notation (e.g. ``x!``). Examples ======== >>> from sympy.parsing.sympy_parser import parse_expr >>> parse_expr("1/2") 1/2 >>> type(_) <class 'sympy.core.numbers.Half'> >>> from sympy.parsing.sympy_parser import standard_transformations,\\ ... implicit_multiplication_application >>> transformations = (standard_transformations + ... (implicit_multiplication_application,)) >>> parse_expr("2x", transformations=transformations) 2*x See Also ======== stringify_expr, eval_expr, standard_transformations, implicit_multiplication_application """ if local_dict is None: local_dict = {} if global_dict is None: global_dict = {} exec_('from sympy import *', global_dict) code = stringify_expr(s, local_dict, global_dict, transformations) return eval_expr(code, local_dict, global_dict)
def test_newtons_method_function__pycode(): x = sp.Symbol('x', real=True) expr = sp.cos(x) - x**3 func = newtons_method_function(expr, x) py_mod = py_module(func) namespace = {} exec_(py_mod, namespace, namespace) res = eval('newton(0.5)', namespace) assert abs(res - 0.865474033102) < 1e-12
def __init__(self, args, expr, print_lambda=False, use_evalf=False, float_wrap_evalf=False, complex_wrap_evalf=False, use_np=False, use_python_math=False, use_python_cmath=False, use_interval=False): self.print_lambda = print_lambda self.use_evalf = use_evalf self.float_wrap_evalf = float_wrap_evalf self.complex_wrap_evalf = complex_wrap_evalf self.use_np = use_np self.use_python_math = use_python_math self.use_python_cmath = use_python_cmath self.use_interval = use_interval # Constructing the argument string if not all([isinstance(a, Symbol) for a in args]): raise ValueError('The arguments must be Symbols.') else: argstr = ', '.join([str(a) for a in args]) # Constructing the translation dictionaries and making the translation self.dict_str = self.get_dict_str() self.dict_fun = self.get_dict_fun() exprstr = str(expr) newexpr = self.tree2str_translate(self.str2tree(exprstr)) # Constructing the namespaces namespace = {} namespace.update(self.sympy_atoms_namespace(expr)) namespace.update(self.sympy_expression_namespace(expr)) # XXX Workaround # Ugly workaround because Pow(a,Half) prints as sqrt(a) # and sympy_expression_namespace can not catch it. from sympy import sqrt namespace.update({'sqrt': sqrt}) # End workaround. if use_python_math: namespace.update({'math': __import__('math')}) if use_python_cmath: namespace.update({'cmath': __import__('cmath')}) if use_np: try: namespace.update({'np': __import__('numpy')}) except ImportError: raise ImportError( 'experimental_lambdify failed to import numpy.') if use_interval: namespace.update({'imath': __import__( 'sympy.plotting.intervalmath', fromlist=['intervalmath'])}) namespace.update({'math': __import__('math')}) # Construct the lambda if self.print_lambda: print(newexpr) eval_str = 'lambda %s : ( %s )' % (argstr, newexpr) exec_("from __future__ import division; MYNEWLAMBDA = %s" % eval_str, namespace) self.lambda_func = namespace['MYNEWLAMBDA']
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
def __init__(self, stmt, setup='pass', timer=timeit.default_timer, globals=globals()): # copy of timeit.Timer.__init__ # similarity index 95% self.timer = timer stmt = timeit.reindent(stmt, 8) setup = timeit.reindent(setup, 4) src = timeit.template % {'stmt': stmt, 'setup': setup} self.src = src # Save for traceback display code = compile(src, timeit.dummy_src_name, "exec") ns = {} #exec code in globals(), ns -- original timeit code exec_(code, globals, ns) # -- we use caller-provided globals instead self.inner = ns["inner"]
def parse_expr(s, local_dict): """ Converts the string "s" to a SymPy expression, in local_dict. It converts all numbers to Integers before feeding it to Python and automatically creates Symbols. """ global_dict = {} exec_('from sympy import *', global_dict) try: a = parse(s.strip(), mode="eval") except SyntaxError: raise SympifyError("Cannot parse %s." % repr(s)) a = Transform(local_dict, global_dict).visit(a) e = compile(a, "<string>", "eval") return eval(e, global_dict, local_dict)
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". These dictionaries map names of python functions to their equivalent in other modules. """ 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]
def test_names_in_namespace(): # Every singleton name should be accessible from the 'from sympy import *' # namespace in addition to the S object. However, it does not need to be # by the same name (e.g., oo instead of S.Infinity). # As a general rule, things should only be added to the singleton registry # if they are used often enough that code can benefit either from the # performance benefit of being able to use 'is' (this only matters in very # tight loops), or from the memory savings of having exactly one instance # (this matters for the numbers singletons, but very little else). The # singleton registry is already a bit overpopulated, and things cannot be # removed from it without breaking backwards compatibility. So if you got # here by adding something new to the singletons, ask yourself if it # really needs to be singletonized. Note that SymPy classes compare to one # another just fine, so Class() == Class() will give True even if each # Class() returns a new instance. Having unique instances is only # necessary for the above noted performance gains. It should not be needed # for any behavioral purposes. # If you determine that something really should be a singleton, it must be # accessible to sympify() without using 'S' (hence this test). Also, its # str printer should print a form that does not use S. This is because # sympify() disables attribute lookups by default for safety purposes. d = {} exec_('from sympy import *', d) for name in dir(S) + list(S._classes_to_install): if name.startswith('_'): continue if name == 'register': continue if isinstance(getattr(S, name), Rational): continue if getattr(S, name).__module__.startswith('sympy.physics'): continue if name in ['MySingleton', 'MySingleton_sub']: # From the test above continue if name == 'NegativeInfinity': # Accessible by -oo continue # Use is here because of complications with == assert any(getattr(S, name) is i or type(getattr(S, name)) is i for i in d.values()), name
def parse_expr(s, local_dict=None, transformations=standard_transformations, global_dict=None, evaluate=True): """Converts the string ``s`` to a SymPy expression, in ``local_dict`` Parameters ========== s : str The string to parse. local_dict : dict, optional A dictionary of local variables to use when parsing. global_dict : dict, optional A dictionary of global variables. By default, this is initialized with ``from sympy import *``; provide this parameter to override this behavior (for instance, to parse ``"Q & S"``). transformations : tuple, optional A tuple of transformation functions used to modify the tokens of the parsed expression before evaluation. The default transformations convert numeric literals into their SymPy equivalents, convert undefined variables into SymPy symbols, and allow the use of standard mathematical factorial notation (e.g. ``x!``). evaluate : bool, optional When False, the order of the arguments will remain as they were in the string and automatic simplification that would normally occur is suppressed. (see examples) Examples ======== >>> from sympy.parsing.sympy_parser import parse_expr >>> parse_expr("1/2") 1/2 >>> type(_) <class 'sympy.core.numbers.Half'> >>> from sympy.parsing.sympy_parser import standard_transformations,\\ ... implicit_multiplication_application >>> transformations = (standard_transformations + ... (implicit_multiplication_application,)) >>> parse_expr("2x", transformations=transformations) 2*x When evaluate=False, some automatic simplifications will not occur: >>> parse_expr("2**3"), parse_expr("2**3", evaluate=False) (8, 2**3) In addition the order of the arguments will not be made canonical. This feature allows one to tell exactly how the expression was entered: >>> a = parse_expr('1 + x', evaluate=False) >>> b = parse_expr('x + 1', evaluate=0) >>> a == b False >>> a.args (1, x) >>> b.args (x, 1) See Also ======== stringify_expr, eval_expr, standard_transformations, implicit_multiplication_application """ if local_dict is None: local_dict = {} if global_dict is None: global_dict = {} exec_('from sympy import *', global_dict) code = stringify_expr(s, local_dict, global_dict, transformations) if not evaluate: code = compile(evaluateFalse(code), '<string>', 'eval') return eval_expr(code, local_dict, global_dict)
def _build(self, code, name): ns = {} exec_(code, ns) return ns[name]
from sympy import (symbols, Function, Integer, Matrix, Abs, Rational, Float, S, WildFunction, ImmutableDenseMatrix, sin, true, false, ones, sqrt, root, AlgebraicNumber, Symbol, Dummy, Wild) from sympy.core.compatibility import exec_ from sympy.geometry import Point, Ellipse from sympy.printing import srepr from sympy.polys import ring, field, ZZ, QQ, lex, grlex, Poly from sympy.polys.polyclasses import DMP from sympy.polys.agca.extensions import FiniteExtension x, y = symbols('x,y') # eval(srepr(expr)) == expr has to succeed in the right environment. The right # environment is the scope of "from sympy import *" for most cases. ENV = {} exec_("from sympy import *", ENV) def sT(expr, string): """ sT := sreprTest Tests that srepr delivers the expected string and that the condition eval(srepr(expr))==expr holds. """ assert srepr(expr) == string assert eval(string, ENV) == expr def test_printmethod(): class R(Abs):
def lambdify(args, expr, modules=None, printer=None, use_imps=True, dummify=False): """ Returns an anonymous 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. .. warning:: Note that this function uses ``eval``, and thus shouldn't be used on unsanitized input. Arguments in the provided expression that are not valid Python identifiers are substitued with dummy symbols. This allows for applied functions (e.g. f(t)) to be supplied as arguments. Call the function with 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. 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://github.com/pydata/numexpr#supported-functions 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 ``[{'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) matrix([[1], [2]]) Usage ===== (1) Use one of the provided modules: >>> from sympy import sin, tan, gamma >>> from sympy.abc 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 function definition 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) -2.18503986326 >>> from numpy import array >>> f(array([1, 2, 3]), array([2, 3, 5])) [-2.18503986 -0.29100619 -0.8559934 ] 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 >>> f = lambdify(x, Piecewise((x, x <= 1), (1/x, x > 1)), "numpy") >>> f(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: >>> float(f(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 (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' 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]]) 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 A more robust way of handling this is 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. 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) 5 ``lambdify`` always prefers ``_imp_`` implementations to implementations in other namespaces, unless the ``use_imps`` input parameter is False. Usage with Tensorflow module: >>> import tensorflow as tf >>> f = Max(x, sin(x)) >>> func = lambdify(x, f, 'tensorflow') >>> result = func(tf.constant(1.0)) >>> result # a tf.Tensor representing the result of the calculation <tf.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)) array([[ 1., 2.], [ 3., 4.]], dtype=float32) """ 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 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, 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 = {} 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.lambdarepr 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 parse_expr(s, local_dict=None, transformations=standard_transformations, global_dict=None, evaluate=True): """Converts the string ``s`` to a SymPy expression, in ``local_dict`` Parameters ========== s : str The string to parse. local_dict : dict, optional A dictionary of local variables to use when parsing. global_dict : dict, optional A dictionary of global variables. By default, this is initialized with ``from sympy import *``; provide this parameter to override this behavior (for instance, to parse ``"Q & S"``). transformations : tuple, optional A tuple of transformation functions used to modify the tokens of the parsed expression before evaluation. The default transformations convert numeric literals into their SymPy equivalents, convert undefined variables into SymPy symbols, and allow the use of standard mathematical factorial notation (e.g. ``x!``). Examples ======== >>> from sympy.parsing.sympy_parser import parse_expr >>> parse_expr("1/2") 1/2 >>> type(_) <class 'sympy.core.numbers.Half'> >>> from sympy.parsing.sympy_parser import standard_transformations,\\ ... implicit_multiplication_application >>> transformations = (standard_transformations + ... (implicit_multiplication_application,)) >>> parse_expr("2x", transformations=transformations) 2*x See Also ======== stringify_expr, eval_expr, standard_transformations, implicit_multiplication_application """ if local_dict is None: local_dict = {} if global_dict is None: global_dict = {} exec_('from sympy import *', global_dict) code = stringify_expr(s, local_dict, global_dict, transformations) if evaluate is False: code = compile(evaluateFalse(code), '<string>', 'eval') return eval_expr(code, local_dict, global_dict)
# Imports used in srepr strings from sympy.physics.quantum.constants import HBar from sympy.physics.quantum.hilbert import DirectSumHilbertSpace, TensorProductHilbertSpace, TensorPowerHilbertSpace from sympy.physics.quantum.spin import JzOp, J2Op from sympy import Add, Integer, Mul, Rational, Tuple from sympy.printing import srepr from sympy.printing.pretty import pretty as xpretty from sympy.printing.latex import latex from sympy.core.compatibility import u_decode as u MutableDenseMatrix = Matrix ENV = {} exec_("from sympy import *", ENV) def sT(expr, string): """ sT := sreprTest from sympy/printing/tests/test_repr.py """ assert srepr(expr) == string assert eval(string) == expr def pretty(expr): """ASCII pretty-printing""" return xpretty(expr, use_unicode=False, wrap_line=False)
def parse_expr(s, local_dict=None, transformations=standard_transformations, global_dict=None, evaluate=True): """Converts the string ``s`` to a SymPy expression, in ``local_dict`` Parameters ========== s : str The string to parse. local_dict : dict, optional A dictionary of local variables to use when parsing. global_dict : dict, optional A dictionary of global variables. By default, this is initialized with ``from sympy import *``; provide this parameter to override this behavior (for instance, to parse ``"Q & S"``). transformations : tuple, optional A tuple of transformation functions used to modify the tokens of the parsed expression before evaluation. The default transformations convert numeric literals into their SymPy equivalents, convert undefined variables into SymPy symbols, and allow the use of standard mathematical factorial notation (e.g. ``x!``). evaluate : bool, optional When False, the order of the arguments will remain as they were in the string and automatic simplification that would normally occur is suppressed. (see examples) Examples ======== >>> from sympy.parsing.sympy_parser import parse_expr >>> parse_expr("1/2") 1/2 >>> type(_) <class 'sympy.core.numbers.Half'> >>> from sympy.parsing.sympy_parser import standard_transformations,\\ ... implicit_multiplication_application >>> transformations = (standard_transformations + ... (implicit_multiplication_application,)) >>> parse_expr("2x", transformations=transformations) 2*x When evaluate=False, some automatic simplifications will not occur: >>> parse_expr("2**3"), parse_expr("2**3", evaluate=False) (8, 2**3) In addition the order of the arguments will not be made canonical. This feature allows one to tell exactly how the expression was entered: >>> a = parse_expr('1 + x', evaluate=False) >>> b = parse_expr('x + 1', evaluate=0) >>> a == b False >>> a.args (1, x) >>> b.args (x, 1) See Also ======== stringify_expr, eval_expr, standard_transformations, implicit_multiplication_application """ if local_dict is None: local_dict = {} elif not isinstance(local_dict, dict): raise TypeError('expecting local_dict to be a dict') if global_dict is None: global_dict = {} exec_('from sympy import *', global_dict) elif not isinstance(global_dict, dict): raise TypeError('expecting global_dict to be a dict') transformations = transformations or () if transformations: if not iterable(transformations): raise TypeError('`transformations` should be a list of functions.') for _ in transformations: if not callable(_): raise TypeError( filldedent(''' expected a function in `transformations`, not %s''' % func_name(_))) if arity(_) != 3: raise TypeError( filldedent(''' a transformation should be function that takes 3 arguments''')) code = stringify_expr(s, local_dict, global_dict, transformations) if not evaluate: code = compile(evaluateFalse(code), '<string>', 'eval') return eval_expr(code, local_dict, global_dict)
def __init__(self, args, expr, print_lambda=False, use_evalf=False, float_wrap_evalf=False, complex_wrap_evalf=False, use_np=False, use_python_math=False, use_python_cmath=False, use_interval=False): self.print_lambda = print_lambda self.use_evalf = use_evalf self.float_wrap_evalf = float_wrap_evalf self.complex_wrap_evalf = complex_wrap_evalf self.use_np = use_np self.use_python_math = use_python_math self.use_python_cmath = use_python_cmath self.use_interval = use_interval # Constructing the argument string # - check if not all([isinstance(a, Symbol) for a in args]): raise ValueError('The arguments must be Symbols.') # - use numbered symbols syms = numbered_symbols(exclude=expr.free_symbols) newargs = [next(syms) for i in args] expr = expr.xreplace(dict(zip(args, newargs))) argstr = ', '.join([str(a) for a in newargs]) del syms, newargs, args # Constructing the translation dictionaries and making the translation self.dict_str = self.get_dict_str() self.dict_fun = self.get_dict_fun() exprstr = str(expr) # the & and | operators don't work on tuples, see discussion #12108 exprstr = exprstr.replace(" & "," and ").replace(" | "," or ") newexpr = self.tree2str_translate(self.str2tree(exprstr)) # Constructing the namespaces namespace = {} namespace.update(self.sympy_atoms_namespace(expr)) namespace.update(self.sympy_expression_namespace(expr)) # XXX Workaround # Ugly workaround because Pow(a,Half) prints as sqrt(a) # and sympy_expression_namespace can not catch it. from sympy import sqrt namespace.update({'sqrt': sqrt}) namespace.update({'Eq': lambda x, y: x == y}) # End workaround. if use_python_math: namespace.update({'math': __import__('math')}) if use_python_cmath: namespace.update({'cmath': __import__('cmath')}) if use_np: try: namespace.update({'np': __import__('numpy')}) except ImportError: raise ImportError( 'experimental_lambdify failed to import numpy.') if use_interval: namespace.update({'imath': __import__( 'sympy.plotting.intervalmath', fromlist=['intervalmath'])}) namespace.update({'math': __import__('math')}) # Construct the lambda if self.print_lambda: print(newexpr) eval_str = 'lambda %s : ( %s )' % (argstr, newexpr) self.eval_str = eval_str exec_("from __future__ import division; MYNEWLAMBDA = %s" % eval_str, namespace) self.lambda_func = namespace['MYNEWLAMBDA']
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 spstr2spexpr(spstr): from sympy.parsing.sympy_parser import parse_expr from sympy.core.compatibility import exec_ global_dict = {} exec_("from symplus import *", global_dict) return parse_expr(spstr, global_dict=global_dict, evaluate=False)
def lambdify(args, expr, modules=None, printer=None, use_imps=True, dummify=False): """ Returns an anonymous 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. .. warning:: Note that this function uses ``eval``, and thus shouldn't be used on unsanitized input. Arguments in the provided expression that are not valid Python identifiers are substitued with dummy symbols. This allows for applied functions (e.g. f(t)) to be supplied as arguments. Call the function with 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. 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://github.com/pydata/numexpr#supported-functions 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 ``[{'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) matrix([[1], [2]]) Usage ===== (1) Use one of the provided modules: >>> from sympy import sin, tan, gamma >>> from sympy.abc 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 function definition 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) -2.18503986326 >>> from numpy import array >>> f(array([1, 2, 3]), array([2, 3, 5])) [-2.18503986 -0.29100619 -0.8559934 ] 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 >>> f = lambdify(x, Piecewise((x, x <= 1), (1/x, x > 1)), "numpy") >>> f(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: >>> float(f(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 (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' 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]]) 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 A more robust way of handling this is 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. 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) 5 ``lambdify`` always prefers ``_imp_`` implementations to implementations in other namespaces, unless the ``use_imps`` input parameter is False. Usage with Tensorflow module: >>> import tensorflow as tf >>> f = Max(x, sin(x)) >>> func = lambdify(x, f, 'tensorflow') >>> result = func(tf.constant(1.0)) >>> result # a tf.Tensor representing the result of the calculation <tf.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)) array([[ 1., 2.], [ 3., 4.]], dtype=float32) """ 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 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, 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 = {} 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.lambdarepr 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 lambdify(args: iterable, expr, modules=None, printer=None, use_imps=True, dummify=False): """Convert a SymPy expression into a function that allows for fast numeric evaluation. .. warning:: This function uses ``exec``, and thus shouldn't be used on unsanitized input. .. versionchanged:: 1.7.0 Passing a set for the *args* parameter is deprecated as sets are unordered. Use an ordered iterable such as a list or tuple. Explanation =========== 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``. Parameters ========== args : List[Symbol] A variable or a list of variables whose nesting represents the nesting of the arguments that will be passed to the function. Variables can be symbols, undefined functions, or matrix symbols. >>> from sympy import Eq >>> from sympy.abc import x, y, z The list of variables should match the structure of how the arguments will be passed to the function. Simply enclose the parameters as they will be passed in a list. To call a function like ``f(x)`` then ``[x]`` should be the first argument to ``lambdify``; for this case a single ``x`` can also be used: >>> f = lambdify(x, x + 1) >>> f(1) 2 >>> f = lambdify([x], x + 1) >>> f(1) 2 To call a function like ``f(x, y)`` then ``[x, y]`` will be the first argument of the ``lambdify``: >>> f = lambdify([x, y], x + y) >>> f(1, 1) 2 To call a function with a single 3-element tuple like ``f((x, y, z))`` then ``[(x, y, z)]`` will be the first argument of the ``lambdify``: >>> f = lambdify([(x, y, z)], Eq(z**2, x**2 + y**2)) >>> f((3, 4, 5)) True If two args will be passed and the first is a scalar but the second is a tuple with two arguments then the items in the list should match that structure: >>> f = lambdify([x, (y, z)], x + y + z) >>> f(1, (2, 3)) 6 expr : Expr An 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). modules : str, optional Specifies the numeric library to use. If not specified, *modules* defaults to: - ``["scipy", "numpy"]`` if SciPy is installed - ``["numpy"]`` if only NumPy is installed - ``["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']``). dummify : bool, optional 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. 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. >>> 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, lambdify >>> from sympy.abc import x >>> f = Max(x, sin(x)) >>> func = lambdify(x, f, 'tensorflow') After tensorflow v2, eager execution is enabled by default. If you want to get the compatible result across tensorflow v1 and v2 as same as this tutorial, run this line. >>> tf.compat.v1.enable_eager_execution() If you have eager execution enabled, you can get the result out immediately as you can use numpy. If you pass tensorflow objects, you may get an ``EagerTensor`` object instead of value. >>> result = func(tf.constant(1.0)) >>> print(result) tf.Tensor(1.0, shape=(), dtype=float32) >>> print(result.__class__) <class 'tensorflow.python.framework.ops.EagerTensor'> You can use ``.numpy()`` to get the numpy value of the tensor. >>> result.numpy() 1.0 >>> var = tf.Variable(2.0) >>> result = func(var) # also works for tf.Variable and tf.Placeholder >>> result.numpy() 2.0 And it works with any shape array. >>> tensor = tf.constant([[1.0, 2.0], [3.0, 4.0]]) >>> result = func(tensor) >>> result.numpy() [[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.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 = {} 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 }) 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() 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) else: funcprinter = _EvaluatorPrinter(printer, dummify) funcstr = funcprinter.doprint(funcname, args, expr) # 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 = {} 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({})".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
from sympy.core.compatibility import exec_ from sympy.testing.pytest import XFAIL # Imports used in srepr strings from sympy.physics.quantum.spin import JzOp from sympy.printing import srepr from sympy.printing.pretty import pretty as xpretty from sympy.printing.latex import latex from sympy.core.compatibility import u_decode as u MutableDenseMatrix = Matrix ENV = {} # type: Dict[str, Any] exec_("from sympy import *", ENV) exec_("from sympy.physics.quantum import *", ENV) exec_("from sympy.physics.quantum.cg import *", ENV) exec_("from sympy.physics.quantum.spin import *", ENV) exec_("from sympy.physics.quantum.hilbert import *", ENV) exec_("from sympy.physics.quantum.qubit import *", ENV) exec_("from sympy.physics.quantum.qexpr import *", ENV) exec_("from sympy.physics.quantum.gate import *", ENV) exec_("from sympy.physics.quantum.constants import *", ENV) def sT(expr, string): """ sT := sreprTest from sympy/printing/tests/test_repr.py """
def __init__(self, args, expr, print_lambda=False, use_evalf=False, float_wrap_evalf=False, complex_wrap_evalf=False, use_np=False, use_python_math=False, use_python_cmath=False, use_interval=False): self.print_lambda = print_lambda self.use_evalf = use_evalf self.float_wrap_evalf = float_wrap_evalf self.complex_wrap_evalf = complex_wrap_evalf self.use_np = use_np self.use_python_math = use_python_math self.use_python_cmath = use_python_cmath self.use_interval = use_interval # Constructing the argument string # - check if not all([isinstance(a, Symbol) for a in args]): raise ValueError('The arguments must be Symbols.') # - use numbered symbols syms = numbered_symbols(exclude=expr.free_symbols) newargs = [next(syms) for i in args] expr = expr.xreplace(dict(zip(args, newargs))) argstr = ', '.join([str(a) for a in newargs]) del syms, newargs, args # Constructing the translation dictionaries and making the translation self.dict_str = self.get_dict_str() self.dict_fun = self.get_dict_fun() exprstr = str(expr) newexpr = self.tree2str_translate(self.str2tree(exprstr)) # Constructing the namespaces namespace = {} namespace.update(self.sympy_atoms_namespace(expr)) namespace.update(self.sympy_expression_namespace(expr)) # XXX Workaround # Ugly workaround because Pow(a,Half) prints as sqrt(a) # and sympy_expression_namespace can not catch it. from sympy import sqrt namespace.update({'sqrt': sqrt}) namespace.update({'Eq': lambda x, y: x == y}) # End workaround. if use_python_math: namespace.update({'math': __import__('math')}) if use_python_cmath: namespace.update({'cmath': __import__('cmath')}) if use_np: try: namespace.update({'np': __import__('numpy')}) except ImportError: raise ImportError( 'experimental_lambdify failed to import numpy.') if use_interval: namespace.update({'imath': __import__( 'sympy.plotting.intervalmath', fromlist=['intervalmath'])}) namespace.update({'math': __import__('math')}) # Construct the lambda if self.print_lambda: print(newexpr) eval_str = 'lambda %s : ( %s )' % (argstr, newexpr) self.eval_str = eval_str exec_("from __future__ import division; MYNEWLAMBDA = %s" % eval_str, namespace) self.lambda_func = namespace['MYNEWLAMBDA']