"""Base class for all the objects in SymPy""" from __future__ import annotations from collections import Counter from collections.abc import Mapping, Iterable from itertools import zip_longest from functools import cmp_to_key from typing import TYPE_CHECKING, overload from .assumptions import _prepare_class_assumptions from .cache import cacheit from .sympify import _sympify, sympify, SympifyError, _external_converter from .sorting import ordered from .kind import Kind, UndefinedKind from ._print_helpers import Printable from sympy.utilities.decorator import deprecated from sympy.utilities.exceptions import sympy_deprecation_warning from sympy.utilities.iterables import iterable, numbered_symbols from sympy.utilities.misc import filldedent, func_name if TYPE_CHECKING: from typing import ClassVar, TypeVar, Any from typing_extensions import Self from .assumptions import StdFactKB from .symbol import Symbol Tbasic = TypeVar("Tbasic", bound='Basic') def as_Basic(expr): """Return expr as a Basic instance using strict sympify or raise a TypeError; this is just a wrapper to _sympify, raising a TypeError instead of a SympifyError.""" try: return _sympify(expr) except SympifyError: raise TypeError( 'Argument must be a Basic object, not `%s`' % func_name( expr)) # Key for sorting commutative args in canonical order # by name. This is used for canonical ordering of the # args for Add and Mul *if* the names of both classes # being compared appear here. Some things in this list # are not spelled the same as their name so they do not, # in effect, appear here. See Basic.compare. ordering_of_classes = [ # singleton numbers 'Zero', 'One', 'Half', 'Infinity', 'NaN', 'NegativeOne', 'NegativeInfinity', # numbers 'Integer', 'Rational', 'Float', # singleton symbols 'Exp1', 'Pi', 'ImaginaryUnit', # symbols 'Symbol', 'Wild', # arithmetic operations 'Pow', 'Mul', 'Add', # function values 'Derivative', 'Integral', # defined singleton functions 'Abs', 'Sign', 'Sqrt', 'Floor', 'Ceiling', 'Re', 'Im', 'Arg', 'Conjugate', 'Exp', 'Log', 'Sin', 'Cos', 'Tan', 'Cot', 'ASin', 'ACos', 'ATan', 'ACot', 'Sinh', 'Cosh', 'Tanh', 'Coth', 'ASinh', 'ACosh', 'ATanh', 'ACoth', 'RisingFactorial', 'FallingFactorial', 'factorial', 'binomial', 'Gamma', 'LowerGamma', 'UpperGamma', 'PolyGamma', 'Erf', # special polynomials 'Chebyshev', 'Chebyshev2', # undefined functions 'Function', 'WildFunction', # anonymous functions 'Lambda', # Landau O symbol 'Order', # relational operations 'Equality', 'Unequality', 'StrictGreaterThan', 'StrictLessThan', 'GreaterThan', 'LessThan', ] def _cmp_name(x: type, y: type) -> int: """return -1, 0, 1 if the name of x is before that of y. A string comparison is done if either name does not appear in `ordering_of_classes`. This is the helper for ``Basic.compare`` Examples ======== >>> from sympy import cos, tan, sin >>> from sympy.core import basic >>> save = basic.ordering_of_classes >>> basic.ordering_of_classes = () >>> basic._cmp_name(cos, tan) -1 >>> basic.ordering_of_classes = ["tan", "sin", "cos"] >>> basic._cmp_name(cos, tan) 1 >>> basic._cmp_name(sin, cos) -1 >>> basic.ordering_of_classes = save """ n1 = x.__name__ n2 = y.__name__ if n1 == n2: return 0 # If the other object is not a Basic subclass, then we are not equal to it. if not issubclass(y, Basic): return -1 UNKNOWN = len(ordering_of_classes) + 1 try: i1 = ordering_of_classes.index(n1) except ValueError: i1 = UNKNOWN try: i2 = ordering_of_classes.index(n2) except ValueError: i2 = UNKNOWN if i1 == UNKNOWN and i2 == UNKNOWN: return (n1 > n2) - (n1 < n2) return (i1 > i2) - (i1 < i2) @cacheit def _get_postprocessors(clsname, arg_type): # Since only Add, Mul, Pow can be clsname, this cache # is not quadratic. postprocessors = set() mappings = _get_postprocessors_for_type(arg_type) for mapping in mappings: f = mapping.get(clsname, None) if f is not None: postprocessors.update(f) return postprocessors @cacheit def _get_postprocessors_for_type(arg_type): return tuple( Basic._constructor_postprocessor_mapping[cls] for cls in arg_type.mro() if cls in Basic._constructor_postprocessor_mapping ) class Basic(Printable): """ Base class for all SymPy objects. Notes and conventions ===================== 1) Always use ``.args``, when accessing parameters of some instance: >>> from sympy import cot >>> from sympy.abc import x, y >>> cot(x).args (x,) >>> cot(x).args[0] x >>> (x*y).args (x, y) >>> (x*y).args[1] y 2) Never use internal methods or variables (the ones prefixed with ``_``): >>> cot(x)._args # do not use this, use cot(x).args instead (x,) 3) By "SymPy object" we mean something that can be returned by ``sympify``. But not all objects one encounters using SymPy are subclasses of Basic. For example, mutable objects are not: >>> from sympy import Basic, Matrix, sympify >>> A = Matrix([[1, 2], [3, 4]]).as_mutable() >>> isinstance(A, Basic) False >>> B = sympify(A) >>> isinstance(B, Basic) True """ __slots__ = ('_mhash', # hash value '_args', # arguments '_assumptions' ) _args: tuple[Basic, ...] _mhash: int | None @property def __sympy__(self): return True def __init_subclass__(cls): # Initialize the default_assumptions FactKB and also any assumptions # property methods. This method will only be called for subclasses of # Basic but not for Basic itself so we call # _prepare_class_assumptions(Basic) below the class definition. super().__init_subclass__() _prepare_class_assumptions(cls) # To be overridden with True in the appropriate subclasses is_number = False is_Atom = False is_Symbol = False is_symbol = False is_Indexed = False is_Dummy = False is_Wild = False is_Function = False is_Add = False is_Mul = False is_Pow = False is_Number = False is_Float = False is_Rational = False is_Integer = False is_NumberSymbol = False is_Order = False is_Derivative = False is_Piecewise = False is_Poly = False is_AlgebraicNumber = False is_Relational = False is_Equality = False is_Boolean = False is_Not = False is_Matrix = False is_Vector = False is_Point = False is_MatAdd = False is_MatMul = False default_assumptions: ClassVar[StdFactKB] is_composite: bool | None is_noninteger: bool | None is_extended_positive: bool | None is_negative: bool | None is_complex: bool | None is_extended_nonpositive: bool | None is_integer: bool | None is_positive: bool | None is_rational: bool | None is_extended_nonnegative: bool | None is_infinite: bool | None is_antihermitian: bool | None is_extended_negative: bool | None is_extended_real: bool | None is_finite: bool | None is_polar: bool | None is_imaginary: bool | None is_transcendental: bool | None is_extended_nonzero: bool | None is_nonzero: bool | None is_odd: bool | None is_algebraic: bool | None is_prime: bool | None is_commutative: bool | None is_nonnegative: bool | None is_nonpositive: bool | None is_hermitian: bool | None is_irrational: bool | None is_real: bool | None is_zero: bool | None is_even: bool | None kind: Kind = UndefinedKind def __new__(cls, *args): obj = object.__new__(cls) obj._assumptions = cls.default_assumptions obj._mhash = None # will be set by __hash__ method. obj._args = args # all items in args must be Basic objects return obj def copy(self): return self.func(*self.args) def __getnewargs__(self): return self.args def __getstate__(self): return None def __setstate__(self, state): for name, value in state.items(): setattr(self, name, value) def __reduce_ex__(self, protocol): if protocol < 2: msg = "Only pickle protocol 2 or higher is supported by SymPy" raise NotImplementedError(msg) return super().__reduce_ex__(protocol) def __hash__(self) -> int: # hash cannot be cached using cache_it because infinite recurrence # occurs as hash is needed for setting cache dictionary keys h = self._mhash if h is None: h = hash((type(self).__name__,) + self._hashable_content()) self._mhash = h return h def _hashable_content(self): """Return a tuple of information about self that can be used to compute the hash. If a class defines additional attributes, like ``name`` in Symbol, then this method should be updated accordingly to return such relevant attributes. Defining more than _hashable_content is necessary if __eq__ has been defined by a class. See note about this in Basic.__eq__.""" return self._args @property def assumptions0(self): """ Return object `type` assumptions. For example: Symbol('x', real=True) Symbol('x', integer=True) are different objects. In other words, besides Python type (Symbol in this case), the initial assumptions are also forming their typeinfo. Examples ======== >>> from sympy import Symbol >>> from sympy.abc import x >>> x.assumptions0 {'commutative': True} >>> x = Symbol("x", positive=True) >>> x.assumptions0 {'commutative': True, 'complex': True, 'extended_negative': False, 'extended_nonnegative': True, 'extended_nonpositive': False, 'extended_nonzero': True, 'extended_positive': True, 'extended_real': True, 'finite': True, 'hermitian': True, 'imaginary': False, 'infinite': False, 'negative': False, 'nonnegative': True, 'nonpositive': False, 'nonzero': True, 'positive': True, 'real': True, 'zero': False} """ return {} def compare(self, other): """ Return -1, 0, 1 if the object is less than, equal, or greater than other in a canonical sense. Non-Basic are always greater than Basic. If both names of the classes being compared appear in the `ordering_of_classes` then the ordering will depend on the appearance of the names there. If either does not appear in that list, then the comparison is based on the class name. If the names are the same then a comparison is made on the length of the hashable content. Items of the equal-lengthed contents are then successively compared using the same rules. If there is never a difference then 0 is returned. Examples ======== >>> from sympy.abc import x, y >>> x.compare(y) -1 >>> x.compare(x) 0 >>> y.compare(x) 1 """ # all redefinitions of __cmp__ method should start with the # following lines: if self is other: return 0 n1 = self.__class__ n2 = other.__class__ c = _cmp_name(n1, n2) if c: return c # st = self._hashable_content() ot = other._hashable_content() len_st = len(st) len_ot = len(ot) c = (len_st > len_ot) - (len_st < len_ot) if c: return c for l, r in zip(st, ot): if isinstance(l, Basic): c = l.compare(r) elif isinstance(l, frozenset): l = Basic(*l) if isinstance(l, frozenset) else l r = Basic(*r) if isinstance(r, frozenset) else r c = l.compare(r) else: c = (l > r) - (l < r) if c: return c return 0 @classmethod def fromiter(cls, args, **assumptions): """ Create a new object from an iterable. This is a convenience function that allows one to create objects from any iterable, without having to convert to a list or tuple first. Examples ======== >>> from sympy import Tuple >>> Tuple.fromiter(i for i in range(5)) (0, 1, 2, 3, 4) """ return cls(*tuple(args), **assumptions) @classmethod def class_key(cls) -> tuple[int, int, str]: """Nice order of classes.""" return 5, 0, cls.__name__ @cacheit def sort_key(self, order=None): """ Return a sort key. Examples ======== >>> from sympy import S, I >>> sorted([S(1)/2, I, -I], key=lambda x: x.sort_key()) [1/2, -I, I] >>> S("[x, 1/x, 1/x**2, x**2, x**(1/2), x**(1/4), x**(3/2)]") [x, 1/x, x**(-2), x**2, sqrt(x), x**(1/4), x**(3/2)] >>> sorted(_, key=lambda x: x.sort_key()) [x**(-2), 1/x, x**(1/4), sqrt(x), x, x**(3/2), x**2] """ # XXX: remove this when issue 5169 is fixed def inner_key(arg): if isinstance(arg, Basic): return arg.sort_key(order) else: return arg args = self._sorted_args args = len(args), tuple([inner_key(arg) for arg in args]) return self.class_key(), args, S.One.sort_key(), S.One def _do_eq_sympify(self, other): """Returns a boolean indicating whether a == b when either a or b is not a Basic. This is only done for types that were either added to `converter` by a 3rd party or when the object has `_sympy_` defined. This essentially reuses the code in `_sympify` that is specific for this use case. Non-user defined types that are meant to work with SymPy should be handled directly in the __eq__ methods of the `Basic` classes it could equate to and not be converted. Note that after conversion, `==` is used again since it is not necessarily clear whether `self` or `other`'s __eq__ method needs to be used.""" for superclass in type(other).__mro__: conv = _external_converter.get(superclass) if conv is not None: return self == conv(other) if hasattr(other, '_sympy_'): return self == other._sympy_() return NotImplemented def __eq__(self, other): """Return a boolean indicating whether a == b on the basis of their symbolic trees. This is the same as a.compare(b) == 0 but faster. Notes ===== If a class that overrides __eq__() needs to retain the implementation of __hash__() from a parent class, the interpreter must be told this explicitly by setting __hash__ : Callable[[object], int] = .__hash__. Otherwise the inheritance of __hash__() will be blocked, just as if __hash__ had been explicitly set to None. References ========== from https://docs.python.org/dev/reference/datamodel.html#object.__hash__ """ if self is other: return True if not isinstance(other, Basic): return self._do_eq_sympify(other) # check for pure number expr if not (self.is_Number and other.is_Number) and ( type(self) != type(other)): return False a, b = self._hashable_content(), other._hashable_content() if a != b: return False # check number *in* an expression for a, b in zip(a, b): if not isinstance(a, Basic): continue if a.is_Number and type(a) != type(b): return False return True def __ne__(self, other): """``a != b`` -> Compare two symbolic trees and see whether they are different this is the same as: ``a.compare(b) != 0`` but faster """ return not self == other def dummy_eq(self, other, symbol=None): """ Compare two expressions and handle dummy symbols. Examples ======== >>> from sympy import Dummy >>> from sympy.abc import x, y >>> u = Dummy('u') >>> (u**2 + 1).dummy_eq(x**2 + 1) True >>> (u**2 + 1) == (x**2 + 1) False >>> (u**2 + y).dummy_eq(x**2 + y, x) True >>> (u**2 + y).dummy_eq(x**2 + y, y) False """ s = self.as_dummy() o = _sympify(other) o = o.as_dummy() dummy_symbols = [i for i in s.free_symbols if i.is_Dummy] if len(dummy_symbols) == 1: dummy = dummy_symbols.pop() else: return s == o if symbol is None: symbols = o.free_symbols if len(symbols) == 1: symbol = symbols.pop() else: return s == o tmp = dummy.__class__() return s.xreplace({dummy: tmp}) == o.xreplace({symbol: tmp}) @overload def atoms(self) -> set[Basic]: ... @overload def atoms(self, *types: Tbasic | type[Tbasic]) -> set[Tbasic]: ... def atoms(self, *types: Tbasic | type[Tbasic]) -> set[Basic] | set[Tbasic]: """Returns the atoms that form the current object. By default, only objects that are truly atomic and cannot be divided into smaller pieces are returned: symbols, numbers, and number symbols like I and pi. It is possible to request atoms of any type, however, as demonstrated below. Examples ======== >>> from sympy import I, pi, sin >>> from sympy.abc import x, y >>> (1 + x + 2*sin(y + I*pi)).atoms() {1, 2, I, pi, x, y} If one or more types are given, the results will contain only those types of atoms. >>> from sympy import Number, NumberSymbol, Symbol >>> (1 + x + 2*sin(y + I*pi)).atoms(Symbol) {x, y} >>> (1 + x + 2*sin(y + I*pi)).atoms(Number) {1, 2} >>> (1 + x + 2*sin(y + I*pi)).atoms(Number, NumberSymbol) {1, 2, pi} >>> (1 + x + 2*sin(y + I*pi)).atoms(Number, NumberSymbol, I) {1, 2, I, pi} Note that I (imaginary unit) and zoo (complex infinity) are special types of number symbols and are not part of the NumberSymbol class. The type can be given implicitly, too: >>> (1 + x + 2*sin(y + I*pi)).atoms(x) # x is a Symbol {x, y} Be careful to check your assumptions when using the implicit option since ``S(1).is_Integer = True`` but ``type(S(1))`` is ``One``, a special type of SymPy atom, while ``type(S(2))`` is type ``Integer`` and will find all integers in an expression: >>> from sympy import S >>> (1 + x + 2*sin(y + I*pi)).atoms(S(1)) {1} >>> (1 + x + 2*sin(y + I*pi)).atoms(S(2)) {1, 2} Finally, arguments to atoms() can select more than atomic atoms: any SymPy type (loaded in core/__init__.py) can be listed as an argument and those types of "atoms" as found in scanning the arguments of the expression recursively: >>> from sympy import Function, Mul >>> from sympy.core.function import AppliedUndef >>> f = Function('f') >>> (1 + f(x) + 2*sin(y + I*pi)).atoms(Function) {f(x), sin(y + I*pi)} >>> (1 + f(x) + 2*sin(y + I*pi)).atoms(AppliedUndef) {f(x)} >>> (1 + x + 2*sin(y + I*pi)).atoms(Mul) {I*pi, 2*sin(y + I*pi)} """ nodes = _preorder_traversal(self) if types: types2 = tuple([t if isinstance(t, type) else type(t) for t in types]) return {node for node in nodes if isinstance(node, types2)} else: return {node for node in nodes if not node.args} @property def free_symbols(self) -> set[Basic]: """Return from the atoms of self those which are free symbols. Not all free symbols are ``Symbol`` (see examples) For most expressions, all symbols are free symbols. For some classes this is not true. e.g. Integrals use Symbols for the dummy variables which are bound variables, so Integral has a method to return all symbols except those. Derivative keeps track of symbols with respect to which it will perform a derivative; those are bound variables, too, so it has its own free_symbols method. Any other method that uses bound variables should implement a free_symbols method. Examples ======== >>> from sympy import Derivative, Integral, IndexedBase >>> from sympy.abc import x, y, n >>> (x + 1).free_symbols {x} >>> Integral(x, y).free_symbols {x, y} Not all free symbols are actually symbols: >>> IndexedBase('F')[0].free_symbols {F, F[0]} The symbols of differentiation are not included unless they appear in the expression being differentiated. >>> Derivative(x + y, y).free_symbols {x, y} >>> Derivative(x, y).free_symbols {x} >>> Derivative(x, (y, n)).free_symbols {n, x} If you want to know if a symbol is in the variables of the Derivative you can do so as follows: >>> Derivative(x, y).has_free(y) True """ empty: set[Basic] = set() return empty.union(*(a.free_symbols for a in self.args)) @property def expr_free_symbols(self): sympy_deprecation_warning(""" The expr_free_symbols property is deprecated. Use free_symbols to get the free symbols of an expression. """, deprecated_since_version="1.9", active_deprecations_target="deprecated-expr-free-symbols") return set() def as_dummy(self) -> "Self": """Return the expression with any objects having structurally bound symbols replaced with unique, canonical symbols within the object in which they appear and having only the default assumption for commutativity being True. When applied to a symbol a new symbol having only the same commutativity will be returned. Examples ======== >>> from sympy import Integral, Symbol >>> from sympy.abc import x >>> r = Symbol('r', real=True) >>> Integral(r, (r, x)).as_dummy() Integral(_0, (_0, x)) >>> _.variables[0].is_real is None True >>> r.as_dummy() _r Notes ===== Any object that has structurally bound variables should have a property, ``bound_symbols`` that returns those symbols appearing in the object. """ from .symbol import Dummy, Symbol def can(x): # mask free that shadow bound free = x.free_symbols bound = set(x.bound_symbols) d = {i: Dummy() for i in bound & free} x = x.subs(d) # replace bound with canonical names x = x.xreplace(x.canonical_variables) # return after undoing masking return x.xreplace({v: k for k, v in d.items()}) if not self.has(Symbol): return self return self.replace( lambda x: hasattr(x, 'bound_symbols'), can, simultaneous=False) # type:ignore @property def canonical_variables(self) -> dict[Basic, Symbol]: """Return a dictionary mapping any variable defined in ``self.bound_symbols`` to Symbols that do not clash with any free symbols in the expression. Examples ======== >>> from sympy import Lambda >>> from sympy.abc import x >>> Lambda(x, 2*x).canonical_variables {x: _0} """ bound: list[Basic] | None = getattr(self, 'bound_symbols', None) if bound is None: return {} dums = numbered_symbols('_') reps = {} # watch out for free symbol that are not in bound symbols; # those that are in bound symbols are about to get changed # XXX: free_symbols only returns particular kinds of expressions that # generally have a .name attribute. There is not a proper class/type # that represents this. names = {i.name for i in self.free_symbols - set(bound)} # type: ignore for b in bound: d = next(dums) if b.is_Symbol: while d.name in names: d = next(dums) reps[b] = d return reps def rcall(self, *args): """Apply on the argument recursively through the expression tree. This method is used to simulate a common abuse of notation for operators. For instance, in SymPy the following will not work: ``(x+Lambda(y, 2*y))(z) == x+2*z``, however, you can use: >>> from sympy import Lambda >>> from sympy.abc import x, y, z >>> (x + Lambda(y, 2*y)).rcall(z) x + 2*z """ if callable(self): return self(*args) elif self.args: newargs = [sub.rcall(*args) for sub in self.args] return self.func(*newargs) else: return self def is_hypergeometric(self, k): from sympy.simplify.simplify import hypersimp from sympy.functions.elementary.piecewise import Piecewise if self.has(Piecewise): return None return hypersimp(self, k) is not None @property def is_comparable(self): """Return True if self can be computed to a real number (or already is a real number) with precision, else False. Examples ======== >>> from sympy import exp_polar, pi, I >>> (I*exp_polar(I*pi/2)).is_comparable True >>> (I*exp_polar(I*pi*2)).is_comparable False A False result does not mean that `self` cannot be rewritten into a form that would be comparable. For example, the difference computed below is zero but without simplification it does not evaluate to a zero with precision: >>> e = 2**pi*(1 + 2**pi) >>> dif = e - e.expand() >>> dif.is_comparable False >>> dif.n(2)._prec 1 """ return self._eval_is_comparable() def _eval_is_comparable(self) -> bool: # Expr.is_comparable overrides this return False @property def func(self): """ The top-level function in an expression. The following should hold for all objects:: >> x == x.func(*x.args) Examples ======== >>> from sympy.abc import x >>> a = 2*x >>> a.func >>> a.args (2, x) >>> a.func(*a.args) 2*x >>> a == a.func(*a.args) True """ return self.__class__ @property def args(self) -> tuple[Basic, ...]: """Returns a tuple of arguments of 'self'. Examples ======== >>> from sympy import cot >>> from sympy.abc import x, y >>> cot(x).args (x,) >>> cot(x).args[0] x >>> (x*y).args (x, y) >>> (x*y).args[1] y Notes ===== Never use self._args, always use self.args. Only use _args in __new__ when creating a new function. Do not override .args() from Basic (so that it is easy to change the interface in the future if needed). """ return self._args @property def _sorted_args(self): """ The same as ``args``. Derived classes which do not fix an order on their arguments should override this method to produce the sorted representation. """ return self.args def as_content_primitive(self, radical=False, clear=True): """A stub to allow Basic args (like Tuple) to be skipped when computing the content and primitive components of an expression. See Also ======== sympy.core.expr.Expr.as_content_primitive """ return S.One, self @overload def subs(self, arg1: Mapping[Basic | complex, Basic | complex], arg2: None=None, **kwargs: Any) -> Basic: ... @overload def subs(self, arg1: Iterable[tuple[Basic | complex, Basic | complex]], arg2: None=None, **kwargs: Any) -> Basic: ... @overload def subs(self, arg1: Basic | complex, arg2: Basic | complex, **kwargs: Any) -> Basic: ... def subs(self, arg1: Mapping[Basic | complex, Basic | complex] | Iterable[tuple[Basic | complex, Basic | complex]] | Basic | complex, arg2: Basic | complex | None = None, **kwargs: Any) -> Basic: """ Substitutes old for new in an expression after sympifying args. `args` is either: - two arguments, e.g. foo.subs(old, new) - one iterable argument, e.g. foo.subs(iterable). The iterable may be o an iterable container with (old, new) pairs. In this case the replacements are processed in the order given with successive patterns possibly affecting replacements already made. o a dict or set whose key/value items correspond to old/new pairs. In this case the old/new pairs will be sorted by op count and in case of a tie, by number of args and the default_sort_key. The resulting sorted list is then processed as an iterable container (see previous). If the keyword ``simultaneous`` is True, the subexpressions will not be evaluated until all the substitutions have been made. Examples ======== >>> from sympy import pi, exp, limit, oo >>> from sympy.abc import x, y >>> (1 + x*y).subs(x, pi) pi*y + 1 >>> (1 + x*y).subs({x:pi, y:2}) 1 + 2*pi >>> (1 + x*y).subs([(x, pi), (y, 2)]) 1 + 2*pi >>> reps = [(y, x**2), (x, 2)] >>> (x + y).subs(reps) 6 >>> (x + y).subs(reversed(reps)) x**2 + 2 >>> (x**2 + x**4).subs(x**2, y) y**2 + y To replace only the x**2 but not the x**4, use xreplace: >>> (x**2 + x**4).xreplace({x**2: y}) x**4 + y To delay evaluation until all substitutions have been made, set the keyword ``simultaneous`` to True: >>> (x/y).subs([(x, 0), (y, 0)]) 0 >>> (x/y).subs([(x, 0), (y, 0)], simultaneous=True) nan This has the added feature of not allowing subsequent substitutions to affect those already made: >>> ((x + y)/y).subs({x + y: y, y: x + y}) 1 >>> ((x + y)/y).subs({x + y: y, y: x + y}, simultaneous=True) y/(x + y) In order to obtain a canonical result, unordered iterables are sorted by count_op length, number of arguments and by the default_sort_key to break any ties. All other iterables are left unsorted. >>> from sympy import sqrt, sin, cos >>> from sympy.abc import a, b, c, d, e >>> A = (sqrt(sin(2*x)), a) >>> B = (sin(2*x), b) >>> C = (cos(2*x), c) >>> D = (x, d) >>> E = (exp(x), e) >>> expr = sqrt(sin(2*x))*sin(exp(x)*x)*cos(2*x) + sin(2*x) >>> expr.subs(dict([A, B, C, D, E])) a*c*sin(d*e) + b The resulting expression represents a literal replacement of the old arguments with the new arguments. This may not reflect the limiting behavior of the expression: >>> (x**3 - 3*x).subs({x: oo}) nan >>> limit(x**3 - 3*x, x, oo) oo If the substitution will be followed by numerical evaluation, it is better to pass the substitution to evalf as >>> (1/x).evalf(subs={x: 3.0}, n=21) 0.333333333333333333333 rather than >>> (1/x).subs({x: 3.0}).evalf(21) 0.333333333333333314830 as the former will ensure that the desired level of precision is obtained. See Also ======== replace: replacement capable of doing wildcard-like matching, parsing of match, and conditional replacements xreplace: exact node replacement in expr tree; also capable of using matching rules sympy.core.evalf.EvalfMixin.evalf: calculates the given formula to a desired level of precision """ from .containers import Dict from .symbol import Dummy, Symbol from .numbers import _illegal items: Iterable[tuple[Basic | complex, Basic | complex]] unordered = False if arg2 is None: if isinstance(arg1, set): items = arg1 unordered = True elif isinstance(arg1, (Dict, Mapping)): unordered = True items = arg1.items() # type: ignore elif not iterable(arg1): raise ValueError(filldedent(""" When a single argument is passed to subs it should be a dictionary of old: new pairs or an iterable of (old, new) tuples.""")) else: items = arg1 # type: ignore else: items = [(arg1, arg2)] # type: ignore def sympify_old(old) -> Basic: if isinstance(old, str): # Use Symbol rather than parse_expr for old return Symbol(old) elif isinstance(old, type): # Allow a type e.g. Function('f') or sin return sympify(old, strict=False) else: return sympify(old, strict=True) def sympify_new(new) -> Basic: if isinstance(new, (str, type)): # Allow a type or parse a string input return sympify(new, strict=False) else: return sympify(new, strict=True) sequence = [(sympify_old(s1), sympify_new(s2)) for s1, s2 in items] # skip if there is no change sequence = [(s1, s2) for s1, s2 in sequence if not _aresame(s1, s2)] simultaneous = kwargs.pop('simultaneous', False) if unordered: from .sorting import _nodes, default_sort_key sequence_dict = dict(sequence) # order so more complex items are first and items # of identical complexity are ordered so # f(x) < f(y) < x < y # \___ 2 __/ \_1_/ <- number of nodes # # For more complex ordering use an unordered sequence. k = list(ordered(sequence_dict, default=False, keys=( lambda x: -_nodes(x), default_sort_key, ))) sequence = [(k, sequence_dict[k]) for k in k] # do infinities first if not simultaneous: redo = [i for i, seq in enumerate(sequence) if seq[1] in _illegal] for i in reversed(redo): sequence.insert(0, sequence.pop(i)) if simultaneous: # XXX should this be the default for dict subs? reps = {} rv = self kwargs['hack2'] = True m = Dummy('subs_m') for old, new in sequence: com = new.is_commutative if com is None: com = True d = Dummy('subs_d', commutative=com) # using d*m so Subs will be used on dummy variables # in things like Derivative(f(x, y), x) in which x # is both free and bound rv = rv._subs(old, d*m, **kwargs) if not isinstance(rv, Basic): break reps[d] = new reps[m] = S.One # get rid of m return rv.xreplace(reps) else: rv = self for old, new in sequence: rv = rv._subs(old, new, **kwargs) if not isinstance(rv, Basic): break return rv @cacheit def _subs(self, old, new, **hints): """Substitutes an expression old -> new. If self is not equal to old then _eval_subs is called. If _eval_subs does not want to make any special replacement then a None is received which indicates that the fallback should be applied wherein a search for replacements is made amongst the arguments of self. >>> from sympy import Add >>> from sympy.abc import x, y, z Examples ======== Add's _eval_subs knows how to target x + y in the following so it makes the change: >>> (x + y + z).subs(x + y, 1) z + 1 Add's _eval_subs does not need to know how to find x + y in the following: >>> Add._eval_subs(z*(x + y) + 3, x + y, 1) is None True The returned None will cause the fallback routine to traverse the args and pass the z*(x + y) arg to Mul where the change will take place and the substitution will succeed: >>> (z*(x + y) + 3).subs(x + y, 1) z + 3 ** Developers Notes ** An _eval_subs routine for a class should be written if: 1) any arguments are not instances of Basic (e.g. bool, tuple); 2) some arguments should not be targeted (as in integration variables); 3) if there is something other than a literal replacement that should be attempted (as in Piecewise where the condition may be updated without doing a replacement). If it is overridden, here are some special cases that might arise: 1) If it turns out that no special change was made and all the original sub-arguments should be checked for replacements then None should be returned. 2) If it is necessary to do substitutions on a portion of the expression then _subs should be called. _subs will handle the case of any sub-expression being equal to old (which usually would not be the case) while its fallback will handle the recursion into the sub-arguments. For example, after Add's _eval_subs removes some matching terms it must process the remaining terms so it calls _subs on each of the un-matched terms and then adds them onto the terms previously obtained. 3) If the initial expression should remain unchanged then the original expression should be returned. (Whenever an expression is returned, modified or not, no further substitution of old -> new is attempted.) Sum's _eval_subs routine uses this strategy when a substitution is attempted on any of its summation variables. """ def fallback(self, old, new): """ Try to replace old with new in any of self's arguments. """ hit = False args = list(self.args) for i, arg in enumerate(args): if not hasattr(arg, '_eval_subs'): continue arg = arg._subs(old, new, **hints) if not _aresame(arg, args[i]): hit = True args[i] = arg if hit: rv = self.func(*args) hack2 = hints.get('hack2', False) if hack2 and self.is_Mul and not rv.is_Mul: # 2-arg hack coeff = S.One nonnumber = [] for i in args: if i.is_Number: coeff *= i else: nonnumber.append(i) nonnumber = self.func(*nonnumber) if coeff is S.One: return nonnumber else: return self.func(coeff, nonnumber, evaluate=False) return rv return self if _aresame(self, old): return new rv = self._eval_subs(old, new) if rv is None: rv = fallback(self, old, new) return rv def _eval_subs(self, old, new) -> Basic | None: """Override this stub if you want to do anything more than attempt a replacement of old with new in the arguments of self. See also ======== _subs """ return None def xreplace(self, rule): """ Replace occurrences of objects within the expression. Parameters ========== rule : dict-like Expresses a replacement rule Returns ======= xreplace : the result of the replacement Examples ======== >>> from sympy import symbols, pi, exp >>> x, y, z = symbols('x y z') >>> (1 + x*y).xreplace({x: pi}) pi*y + 1 >>> (1 + x*y).xreplace({x: pi, y: 2}) 1 + 2*pi Replacements occur only if an entire node in the expression tree is matched: >>> (x*y + z).xreplace({x*y: pi}) z + pi >>> (x*y*z).xreplace({x*y: pi}) x*y*z >>> (2*x).xreplace({2*x: y, x: z}) y >>> (2*2*x).xreplace({2*x: y, x: z}) 4*z >>> (x + y + 2).xreplace({x + y: 2}) x + y + 2 >>> (x + 2 + exp(x + 2)).xreplace({x + 2: y}) x + exp(y) + 2 xreplace does not differentiate between free and bound symbols. In the following, subs(x, y) would not change x since it is a bound symbol, but xreplace does: >>> from sympy import Integral >>> Integral(x, (x, 1, 2*x)).xreplace({x: y}) Integral(y, (y, 1, 2*y)) Trying to replace x with an expression raises an error: >>> Integral(x, (x, 1, 2*x)).xreplace({x: 2*y}) # doctest: +SKIP ValueError: Invalid limits given: ((2*y, 1, 4*y),) See Also ======== replace: replacement capable of doing wildcard-like matching, parsing of match, and conditional replacements subs: substitution of subexpressions as defined by the objects themselves. """ value, _ = self._xreplace(rule) return value def _xreplace(self, rule): """ Helper for xreplace. Tracks whether a replacement actually occurred. """ if self in rule: return rule[self], True elif rule: args = [] changed = False for a in self.args: _xreplace = getattr(a, '_xreplace', None) if _xreplace is not None: a_xr = _xreplace(rule) args.append(a_xr[0]) changed |= a_xr[1] else: args.append(a) args = tuple(args) if changed: return self.func(*args), True return self, False @cacheit def has(self, *patterns): """ Test whether any subexpression matches any of the patterns. Examples ======== >>> from sympy import sin >>> from sympy.abc import x, y, z >>> (x**2 + sin(x*y)).has(z) False >>> (x**2 + sin(x*y)).has(x, y, z) True >>> x.has(x) True Note ``has`` is a structural algorithm with no knowledge of mathematics. Consider the following half-open interval: >>> from sympy import Interval >>> i = Interval.Lopen(0, 5); i Interval.Lopen(0, 5) >>> i.args (0, 5, True, False) >>> i.has(4) # there is no "4" in the arguments False >>> i.has(0) # there *is* a "0" in the arguments True Instead, use ``contains`` to determine whether a number is in the interval or not: >>> i.contains(4) True >>> i.contains(0) False Note that ``expr.has(*patterns)`` is exactly equivalent to ``any(expr.has(p) for p in patterns)``. In particular, ``False`` is returned when the list of patterns is empty. >>> x.has() False """ return self._has(iterargs, *patterns) def has_xfree(self, s: set[Basic]): """Return True if self has any of the patterns in s as a free argument, else False. This is like `Basic.has_free` but this will only report exact argument matches. Examples ======== >>> from sympy import Function >>> from sympy.abc import x, y >>> f = Function('f') >>> f(x).has_xfree({f}) False >>> f(x).has_xfree({f(x)}) True >>> f(x + 1).has_xfree({x}) True >>> f(x + 1).has_xfree({x + 1}) True >>> f(x + y + 1).has_xfree({x + 1}) False """ # protect O(1) containment check by requiring: if type(s) is not set: raise TypeError('expecting set argument') return any(a in s for a in iterfreeargs(self)) @cacheit def has_free(self, *patterns): """Return True if self has object(s) ``x`` as a free expression else False. Examples ======== >>> from sympy import Integral, Function >>> from sympy.abc import x, y >>> f = Function('f') >>> g = Function('g') >>> expr = Integral(f(x), (f(x), 1, g(y))) >>> expr.free_symbols {y} >>> expr.has_free(g(y)) True >>> expr.has_free(*(x, f(x))) False This works for subexpressions and types, too: >>> expr.has_free(g) True >>> (x + y + 1).has_free(y + 1) True """ if not patterns: return False p0 = patterns[0] if len(patterns) == 1 and iterable(p0) and not isinstance(p0, Basic): # Basic can contain iterables (though not non-Basic, ideally) # but don't encourage mixed passing patterns raise TypeError(filldedent(''' Expecting 1 or more Basic args, not a single non-Basic iterable. Don't forget to unpack iterables: `eq.has_free(*patterns)`''')) # try quick test first s = set(patterns) rv = self.has_xfree(s) if rv: return rv # now try matching through slower _has return self._has(iterfreeargs, *patterns) def _has(self, iterargs, *patterns): # separate out types and unhashable objects type_set = set() # only types p_set = set() # hashable non-types for p in patterns: if isinstance(p, type) and issubclass(p, Basic): type_set.add(p) continue if not isinstance(p, Basic): try: p = _sympify(p) except SympifyError: continue # Basic won't have this in it p_set.add(p) # fails if object defines __eq__ but # doesn't define __hash__ types = tuple(type_set) # for i in iterargs(self): # if i in p_set: # <--- here, too return True if isinstance(i, types): return True # use matcher if defined, e.g. operations defines # matcher that checks for exact subset containment, # (x + y + 1).has(x + 1) -> True for i in p_set - type_set: # types don't have matchers if not hasattr(i, '_has_matcher'): continue match = i._has_matcher() if any(match(arg) for arg in iterargs(self)): return True # no success return False def replace(self, query, value, map=False, simultaneous=True, exact=None) -> Basic: """ Replace matching subexpressions of ``self`` with ``value``. If ``map = True`` then also return the mapping {old: new} where ``old`` was a sub-expression found with query and ``new`` is the replacement value for it. If the expression itself does not match the query, then the returned value will be ``self.xreplace(map)`` otherwise it should be ``self.subs(ordered(map.items()))``. Traverses an expression tree and performs replacement of matching subexpressions from the bottom to the top of the tree. The default approach is to do the replacement in a simultaneous fashion so changes made are targeted only once. If this is not desired or causes problems, ``simultaneous`` can be set to False. In addition, if an expression containing more than one Wild symbol is being used to match subexpressions and the ``exact`` flag is None it will be set to True so the match will only succeed if all non-zero values are received for each Wild that appears in the match pattern. Setting this to False accepts a match of 0; while setting it True accepts all matches that have a 0 in them. See example below for cautions. The list of possible combinations of queries and replacement values is listed below: Examples ======== Initial setup >>> from sympy import log, sin, cos, tan, Wild, Mul, Add >>> from sympy.abc import x, y >>> f = log(sin(x)) + tan(sin(x**2)) 1.1. type -> type obj.replace(type, newtype) When object of type ``type`` is found, replace it with the result of passing its argument(s) to ``newtype``. >>> f.replace(sin, cos) log(cos(x)) + tan(cos(x**2)) >>> sin(x).replace(sin, cos, map=True) (cos(x), {sin(x): cos(x)}) >>> (x*y).replace(Mul, Add) x + y 1.2. type -> func obj.replace(type, func) When object of type ``type`` is found, apply ``func`` to its argument(s). ``func`` must be written to handle the number of arguments of ``type``. >>> f.replace(sin, lambda arg: sin(2*arg)) log(sin(2*x)) + tan(sin(2*x**2)) >>> (x*y).replace(Mul, lambda *args: sin(2*Mul(*args))) sin(2*x*y) 2.1. pattern -> expr obj.replace(pattern(wild), expr(wild)) Replace subexpressions matching ``pattern`` with the expression written in terms of the Wild symbols in ``pattern``. >>> a, b = map(Wild, 'ab') >>> f.replace(sin(a), tan(a)) log(tan(x)) + tan(tan(x**2)) >>> f.replace(sin(a), tan(a/2)) log(tan(x/2)) + tan(tan(x**2/2)) >>> f.replace(sin(a), a) log(x) + tan(x**2) >>> (x*y).replace(a*x, a) y Matching is exact by default when more than one Wild symbol is used: matching fails unless the match gives non-zero values for all Wild symbols: >>> (2*x + y).replace(a*x + b, b - a) y - 2 >>> (2*x).replace(a*x + b, b - a) 2*x When set to False, the results may be non-intuitive: >>> (2*x).replace(a*x + b, b - a, exact=False) 2/x 2.2. pattern -> func obj.replace(pattern(wild), lambda wild: expr(wild)) All behavior is the same as in 2.1 but now a function in terms of pattern variables is used rather than an expression: >>> f.replace(sin(a), lambda a: sin(2*a)) log(sin(2*x)) + tan(sin(2*x**2)) 3.1. func -> func obj.replace(filter, func) Replace subexpression ``e`` with ``func(e)`` if ``filter(e)`` is True. >>> g = 2*sin(x**3) >>> g.replace(lambda expr: expr.is_Number, lambda expr: expr**2) 4*sin(x**9) The expression itself is also targeted by the query but is done in such a fashion that changes are not made twice. >>> e = x*(x*y + 1) >>> e.replace(lambda x: x.is_Mul, lambda x: 2*x) 2*x*(2*x*y + 1) When matching a single symbol, `exact` will default to True, but this may or may not be the behavior that is desired: Here, we want `exact=False`: >>> from sympy import Function >>> f = Function('f') >>> e = f(1) + f(0) >>> q = f(a), lambda a: f(a + 1) >>> e.replace(*q, exact=False) f(1) + f(2) >>> e.replace(*q, exact=True) f(0) + f(2) But here, the nature of matching makes selecting the right setting tricky: >>> e = x**(1 + y) >>> (x**(1 + y)).replace(x**(1 + a), lambda a: x**-a, exact=False) x >>> (x**(1 + y)).replace(x**(1 + a), lambda a: x**-a, exact=True) x**(-x - y + 1) >>> (x**y).replace(x**(1 + a), lambda a: x**-a, exact=False) x >>> (x**y).replace(x**(1 + a), lambda a: x**-a, exact=True) x**(1 - y) It is probably better to use a different form of the query that describes the target expression more precisely: >>> (1 + x**(1 + y)).replace( ... lambda x: x.is_Pow and x.exp.is_Add and x.exp.args[0] == 1, ... lambda x: x.base**(1 - (x.exp - 1))) ... x**(1 - y) + 1 See Also ======== subs: substitution of subexpressions as defined by the objects themselves. xreplace: exact node replacement in expr tree; also capable of using matching rules """ try: query = _sympify(query) except SympifyError: pass try: value = _sympify(value) except SympifyError: pass if isinstance(query, type): _query = lambda expr: isinstance(expr, query) if isinstance(value, type): _value = lambda expr, result: value(*expr.args) elif callable(value): _value = lambda expr, result: value(*expr.args) else: raise TypeError( "given a type, replace() expects another " "type or a callable") elif isinstance(query, Basic): _query = lambda expr: expr.match(query) if exact is None: from .symbol import Wild exact = (len(query.atoms(Wild)) > 1) if isinstance(value, Basic): if exact: _value = lambda expr, result: (value.subs(result) if all(result.values()) else expr) else: _value = lambda expr, result: value.subs(result) elif callable(value): # match dictionary keys get the trailing underscore stripped # from them and are then passed as keywords to the callable; # if ``exact`` is True, only accept match if there are no null # values amongst those matched. if exact: _value = lambda expr, result: (value(** {str(k)[:-1]: v for k, v in result.items()}) if all(val for val in result.values()) else expr) else: _value = lambda expr, result: value(** {str(k)[:-1]: v for k, v in result.items()}) else: raise TypeError( "given an expression, replace() expects " "another expression or a callable") elif callable(query): _query = query if callable(value): _value = lambda expr, result: value(expr) else: raise TypeError( "given a callable, replace() expects " "another callable") else: raise TypeError( "first argument to replace() must be a " "type, an expression or a callable") def walk(rv, F): """Apply ``F`` to args and then to result. """ args = getattr(rv, 'args', None) if args is not None: if args: newargs = tuple([walk(a, F) for a in args]) if args != newargs: rv = rv.func(*newargs) if simultaneous: # if rv is something that was already # matched (that was changed) then skip # applying F again for i, e in enumerate(args): if rv == e and e != newargs[i]: return rv rv = F(rv) return rv mapping = {} # changes that took place def rec_replace(expr): result = _query(expr) if result or result == {}: v = _value(expr, result) if v is not None and v != expr: if map: mapping[expr] = v expr = v return expr rv = walk(self, rec_replace) return (rv, mapping) if map else rv # type: ignore def find(self, query, group=False): """Find all subexpressions matching a query.""" query = _make_find_query(query) results = list(filter(query, _preorder_traversal(self))) if not group: return set(results) return dict(Counter(results)) def count(self, query): """Count the number of matching subexpressions.""" query = _make_find_query(query) return sum(bool(query(sub)) for sub in _preorder_traversal(self)) def matches(self, expr, repl_dict=None, old=False): """ Helper method for match() that looks for a match between Wild symbols in self and expressions in expr. Examples ======== >>> from sympy import symbols, Wild, Basic >>> a, b, c = symbols('a b c') >>> x = Wild('x') >>> Basic(a + x, x).matches(Basic(a + b, c)) is None True >>> Basic(a + x, x).matches(Basic(a + b + c, b + c)) {x_: b + c} """ expr = sympify(expr) if not isinstance(expr, self.__class__): return None if repl_dict is None: repl_dict = {} else: repl_dict = repl_dict.copy() if self == expr: return repl_dict if len(self.args) != len(expr.args): return None d = repl_dict # already a copy for arg, other_arg in zip(self.args, expr.args): if arg == other_arg: continue if arg.is_Relational: try: d = arg.xreplace(d).matches(other_arg, d, old=old) except TypeError: # Should be InvalidComparisonError when introduced d = None else: d = arg.xreplace(d).matches(other_arg, d, old=old) if d is None: return None return d def match(self, pattern, old=False): """ Pattern matching. Wild symbols match all. Return ``None`` when expression (self) does not match with pattern. Otherwise return a dictionary such that:: pattern.xreplace(self.match(pattern)) == self Examples ======== >>> from sympy import Wild, Sum >>> from sympy.abc import x, y >>> p = Wild("p") >>> q = Wild("q") >>> r = Wild("r") >>> e = (x+y)**(x+y) >>> e.match(p**p) {p_: x + y} >>> e.match(p**q) {p_: x + y, q_: x + y} >>> e = (2*x)**2 >>> e.match(p*q**r) {p_: 4, q_: x, r_: 2} >>> (p*q**r).xreplace(e.match(p*q**r)) 4*x**2 Since match is purely structural expressions that are equivalent up to bound symbols will not match: >>> print(Sum(x, (x, 1, 2)).match(Sum(y, (y, 1, p)))) None An expression with bound symbols can be matched if the pattern uses a distinct ``Wild`` for each bound symbol: >>> Sum(x, (x, 1, 2)).match(Sum(q, (q, 1, p))) {p_: 2, q_: x} The ``old`` flag will give the old-style pattern matching where expressions and patterns are essentially solved to give the match. Both of the following give None unless ``old=True``: >>> (x - 2).match(p - x, old=True) {p_: 2*x - 2} >>> (2/x).match(p*x, old=True) {p_: 2/x**2} See Also ======== matches: pattern.matches(expr) is the same as expr.match(pattern) xreplace: exact structural replacement replace: structural replacement with pattern matching Wild: symbolic placeholders for expressions in pattern matching """ pattern = sympify(pattern) return pattern.matches(self, old=old) def count_ops(self, visual=False): """Wrapper for count_ops that returns the operation count.""" from .function import count_ops return count_ops(self, visual) def doit(self, **hints): """Evaluate objects that are not evaluated by default like limits, integrals, sums and products. All objects of this kind will be evaluated recursively, unless some species were excluded via 'hints' or unless the 'deep' hint was set to 'False'. >>> from sympy import Integral >>> from sympy.abc import x >>> 2*Integral(x, x) 2*Integral(x, x) >>> (2*Integral(x, x)).doit() x**2 >>> (2*Integral(x, x)).doit(deep=False) 2*Integral(x, x) """ if hints.get('deep', True): terms = [term.doit(**hints) if isinstance(term, Basic) else term for term in self.args] return self.func(*terms) else: return self def simplify(self, **kwargs) -> Basic: """See the simplify function in sympy.simplify""" from sympy.simplify.simplify import simplify return simplify(self, **kwargs) def refine(self, assumption=True): """See the refine function in sympy.assumptions""" from sympy.assumptions.refine import refine return refine(self, assumption) def _eval_derivative_n_times(self, s, n): # This is the default evaluator for derivatives (as called by `diff` # and `Derivative`), it will attempt a loop to derive the expression # `n` times by calling the corresponding `_eval_derivative` method, # while leaving the derivative unevaluated if `n` is symbolic. This # method should be overridden if the object has a closed form for its # symbolic n-th derivative. from .numbers import Integer if isinstance(n, (int, Integer)): obj = self for i in range(n): prev = obj obj = obj._eval_derivative(s) if obj is None: return None elif obj == prev: break return obj else: return None def rewrite(self, *args, deep=True, **hints): """ Rewrite *self* using a defined rule. Rewriting transforms an expression to another, which is mathematically equivalent but structurally different. For example you can rewrite trigonometric functions as complex exponentials or combinatorial functions as gamma function. This method takes a *pattern* and a *rule* as positional arguments. *pattern* is optional parameter which defines the types of expressions that will be transformed. If it is not passed, all possible expressions will be rewritten. *rule* defines how the expression will be rewritten. Parameters ========== args : Expr A *rule*, or *pattern* and *rule*. - *pattern* is a type or an iterable of types. - *rule* can be any object. deep : bool, optional If ``True``, subexpressions are recursively transformed. Default is ``True``. Examples ======== If *pattern* is unspecified, all possible expressions are transformed. >>> from sympy import cos, sin, exp, I >>> from sympy.abc import x >>> expr = cos(x) + I*sin(x) >>> expr.rewrite(exp) exp(I*x) Pattern can be a type or an iterable of types. >>> expr.rewrite(sin, exp) exp(I*x)/2 + cos(x) - exp(-I*x)/2 >>> expr.rewrite([cos,], exp) exp(I*x)/2 + I*sin(x) + exp(-I*x)/2 >>> expr.rewrite([cos, sin], exp) exp(I*x) Rewriting behavior can be implemented by defining ``_eval_rewrite()`` method. >>> from sympy import Expr, sqrt, pi >>> class MySin(Expr): ... def _eval_rewrite(self, rule, args, **hints): ... x, = args ... if rule == cos: ... return cos(pi/2 - x, evaluate=False) ... if rule == sqrt: ... return sqrt(1 - cos(x)**2) >>> MySin(MySin(x)).rewrite(cos) cos(-cos(-x + pi/2) + pi/2) >>> MySin(x).rewrite(sqrt) sqrt(1 - cos(x)**2) Defining ``_eval_rewrite_as_[...]()`` method is supported for backwards compatibility reason. This may be removed in the future and using it is discouraged. >>> class MySin(Expr): ... def _eval_rewrite_as_cos(self, *args, **hints): ... x, = args ... return cos(pi/2 - x, evaluate=False) >>> MySin(x).rewrite(cos) cos(-x + pi/2) """ if not args: return self hints.update(deep=deep) pattern = args[:-1] rule = args[-1] # Special case: map `abs` to `Abs` if rule is abs: from sympy.functions.elementary.complexes import Abs rule = Abs # support old design by _eval_rewrite_as_[...] method if isinstance(rule, str): method = "_eval_rewrite_as_%s" % rule elif hasattr(rule, "__name__"): # rule is class or function clsname = rule.__name__ method = "_eval_rewrite_as_%s" % clsname else: # rule is instance clsname = rule.__class__.__name__ method = "_eval_rewrite_as_%s" % clsname if pattern: if iterable(pattern[0]): pattern = pattern[0] pattern = tuple(p for p in pattern if self.has(p)) if not pattern: return self # hereafter, empty pattern is interpreted as all pattern. return self._rewrite(pattern, rule, method, **hints) def _rewrite(self, pattern, rule, method, **hints): deep = hints.pop('deep', True) if deep: args = [a._rewrite(pattern, rule, method, **hints) for a in self.args] else: args = self.args if not pattern or any(isinstance(self, p) for p in pattern): meth = getattr(self, method, None) if meth is not None: rewritten = meth(*args, **hints) else: rewritten = self._eval_rewrite(rule, args, **hints) if rewritten is not None: return rewritten if not args: return self return self.func(*args) def _eval_rewrite(self, rule, args, **hints): return None _constructor_postprocessor_mapping = {} # type: ignore @classmethod def _exec_constructor_postprocessors(cls, obj): # WARNING: This API is experimental. # This is an experimental API that introduces constructor # postprosessors for SymPy Core elements. If an argument of a SymPy # expression has a `_constructor_postprocessor_mapping` attribute, it will # be interpreted as a dictionary containing lists of postprocessing # functions for matching expression node names. clsname = obj.__class__.__name__ postprocessors = {f for i in obj.args for f in _get_postprocessors(clsname, type(i))} for f in postprocessors: obj = f(obj) return obj def _sage_(self): """ Convert *self* to a symbolic expression of SageMath. This version of the method is merely a placeholder. """ old_method = self._sage_ from sage.interfaces.sympy import sympy_init # type: ignore sympy_init() # may monkey-patch _sage_ method into self's class or superclasses if old_method == self._sage_: raise NotImplementedError('conversion to SageMath is not implemented') else: # call the freshly monkey-patched method return self._sage_() def could_extract_minus_sign(self) -> bool: return False # see Expr.could_extract_minus_sign def is_same(a, b, approx=None): """Return True if a and b are structurally the same, else False. If `approx` is supplied, it will be used to test whether two numbers are the same or not. By default, only numbers of the same type will compare equal, so S.Half != Float(0.5). Examples ======== In SymPy (unlike Python) two numbers do not compare the same if they are not of the same type: >>> from sympy import S >>> 2.0 == S(2) False >>> 0.5 == S.Half False By supplying a function with which to compare two numbers, such differences can be ignored. e.g. `equal_valued` will return True for decimal numbers having a denominator that is a power of 2, regardless of precision. >>> from sympy import Float >>> from sympy.core.numbers import equal_valued >>> (S.Half/4).is_same(Float(0.125, 1), equal_valued) True >>> Float(1, 2).is_same(Float(1, 10), equal_valued) True But decimals without a power of 2 denominator will compare as not being the same. >>> Float(0.1, 9).is_same(Float(0.1, 10), equal_valued) False But arbitrary differences can be ignored by supplying a function to test the equivalence of two numbers: >>> import math >>> Float(0.1, 9).is_same(Float(0.1, 10), math.isclose) True Other objects might compare the same even though types are not the same. This routine will only return True if two expressions are identical in terms of class types. >>> from sympy import eye, Basic >>> eye(1) == S(eye(1)) # mutable vs immutable True >>> Basic.is_same(eye(1), S(eye(1))) False """ from .numbers import Number from .traversal import postorder_traversal as pot for t in zip_longest(pot(a), pot(b)): if None in t: return False a, b = t if isinstance(a, Number): if not isinstance(b, Number): return False if approx: return approx(a, b) if not (a == b and a.__class__ == b.__class__): return False return True _aresame = Basic.is_same # for sake of others importing this # key used by Mul and Add to make canonical args _args_sortkey = cmp_to_key(Basic.compare) # For all Basic subclasses _prepare_class_assumptions is called by # Basic.__init_subclass__ but that method is not called for Basic itself so we # call the function here instead. _prepare_class_assumptions(Basic) class Atom(Basic): """ A parent class for atomic things. An atom is an expression with no subexpressions. Examples ======== Symbol, Number, Rational, Integer, ... But not: Add, Mul, Pow, ... """ is_Atom = True __slots__ = () def matches(self, expr, repl_dict=None, old=False): if self == expr: if repl_dict is None: return {} return repl_dict.copy() def xreplace(self, rule, hack2=False): return rule.get(self, self) def doit(self, **hints): return self @classmethod def class_key(cls): return 2, 0, cls.__name__ @cacheit def sort_key(self, order=None): return self.class_key(), (1, (str(self),)), S.One.sort_key(), S.One def _eval_simplify(self, **kwargs): return self @property def _sorted_args(self): # this is here as a safeguard against accidentally using _sorted_args # on Atoms -- they cannot be rebuilt as atom.func(*atom._sorted_args) # since there are no args. So the calling routine should be checking # to see that this property is not called for Atoms. raise AttributeError('Atoms have no args. It might be necessary' ' to make a check for Atoms in the calling code.') def _atomic(e, recursive=False): """Return atom-like quantities as far as substitution is concerned: Derivatives, Functions and Symbols. Do not return any 'atoms' that are inside such quantities unless they also appear outside, too, unless `recursive` is True. Examples ======== >>> from sympy import Derivative, Function, cos >>> from sympy.abc import x, y >>> from sympy.core.basic import _atomic >>> f = Function('f') >>> _atomic(x + y) {x, y} >>> _atomic(x + f(y)) {x, f(y)} >>> _atomic(Derivative(f(x), x) + cos(x) + y) {y, cos(x), Derivative(f(x), x)} """ pot = _preorder_traversal(e) seen = set() if isinstance(e, Basic): free = getattr(e, "free_symbols", None) if free is None: return {e} else: return set() from .symbol import Symbol from .function import Derivative, Function atoms = set() for p in pot: if p in seen: pot.skip() continue seen.add(p) if isinstance(p, Symbol) and p in free: atoms.add(p) elif isinstance(p, (Derivative, Function)): if not recursive: pot.skip() atoms.add(p) return atoms def _make_find_query(query): """Convert the argument of Basic.find() into a callable""" try: query = _sympify(query) except SympifyError: pass if isinstance(query, type): return lambda expr: isinstance(expr, query) elif isinstance(query, Basic): return lambda expr: expr.match(query) is not None return query # Delayed to avoid cyclic import from .singleton import S from .traversal import (preorder_traversal as _preorder_traversal, iterargs, iterfreeargs) preorder_traversal = deprecated( """ Using preorder_traversal from the sympy.core.basic submodule is deprecated. Instead, use preorder_traversal from the top-level sympy namespace, like sympy.preorder_traversal """, deprecated_since_version="1.10", active_deprecations_target="deprecated-traversal-functions-moved", )(_preorder_traversal)