Source code for sympy.core.expr

from core import C
from basic import Basic, Atom
from singleton import S
from evalf import EvalfMixin
from decorators import _sympifyit, call_highest_priority
from cache import cacheit
from compatibility import reduce

[docs]class Expr(Basic, EvalfMixin): __slots__ = [] def sort_key(self, order=None): # XXX: The order argument does not actually work from sympy.core import S def key_inner(arg): if isinstance(arg, Basic): return arg.sort_key(order=order) elif hasattr(arg, '__iter__'): return tuple(key_inner(arg) for arg in args) else: return arg coeff, expr = self.as_coeff_Mul() if expr.is_Pow: expr, exp = expr.args else: expr, exp = expr, S.One if expr.is_Atom: if expr.is_Symbol: args = (str(expr),) else: args = (expr,) else: if expr.is_Add: args = expr.as_ordered_terms(order=order) else: args = expr.args args = tuple(key_inner(arg) for arg in args) if expr.is_Mul: args = sorted(args) args = (len(args), args) exp = exp.sort_key(order=order) return expr.class_key(), args, exp, coeff # *************** # * Arithmetics * # *************** # Expr and its sublcasses use _op_priority to determine which object # passed to a binary special method (__mul__, etc.) will handle the # operation. In general, the 'call_highest_priority' decorator will choose # the object with the highest _op_priority to handle the call. # Custom subclasses that want to define their own binary special methods # should set an _op_priority value that is higher than the default. # # **NOTE**: # This is a temporary fix, and will eventually be replaced with # something better and more powerful. See issue 2411. _op_priority = 10.0 def __pos__(self): return self def __neg__(self): return Mul(S.NegativeOne, self) def __abs__(self): return C.Abs(self) @_sympifyit('other', NotImplemented) @call_highest_priority('__radd__') def __add__(self, other): return Add(self, other) @_sympifyit('other', NotImplemented) @call_highest_priority('__add__') def __radd__(self, other): return Add(other, self) @_sympifyit('other', NotImplemented) @call_highest_priority('__rsub__') def __sub__(self, other): return Add(self, -other) @_sympifyit('other', NotImplemented) @call_highest_priority('__sub__') def __rsub__(self, other): return Add(other, -self) @_sympifyit('other', NotImplemented) @call_highest_priority('__rmul__') def __mul__(self, other): return Mul(self, other) @_sympifyit('other', NotImplemented) @call_highest_priority('__mul__') def __rmul__(self, other): return Mul(other, self) @_sympifyit('other', NotImplemented) @call_highest_priority('__rpow__') def __pow__(self, other): return Pow(self, other) @_sympifyit('other', NotImplemented) @call_highest_priority('__pow__') def __rpow__(self, other): return Pow(other, self) @_sympifyit('other', NotImplemented) @call_highest_priority('__rdiv__') def __div__(self, other): return Mul(self, Pow(other, S.NegativeOne)) @_sympifyit('other', NotImplemented) @call_highest_priority('__div__') def __rdiv__(self, other): return Mul(other, Pow(self, S.NegativeOne)) __truediv__ = __div__ __rtruediv__ = __rdiv__ def __float__(self): result = self.evalf() if result.is_Number: return float(result) else: raise ValueError("Symbolic value, can't compute") def __complex__(self): result = self.evalf() re, im = result.as_real_imag() return complex(float(re), float(im)) @_sympifyit('other', False) # sympy > other def __lt__(self, other): dif = self - other if dif.is_negative != dif.is_nonnegative: return dif.is_negative return C.StrictInequality(self, other) @_sympifyit('other', True) # sympy > other def __gt__(self, other): dif = self - other if dif.is_positive != dif.is_nonpositive: return dif.is_positive return C.StrictInequality(other, self) @_sympifyit('other', False) # sympy > other def __le__(self, other): dif = self - other if dif.is_nonpositive != dif.is_positive: return dif.is_nonpositive return C.Inequality(self, other) @_sympifyit('other', True) # sympy > other def __ge__(self, other): dif = self - other if dif.is_nonnegative != dif.is_negative: return dif.is_nonnegative return C.Inequality(other, self) @staticmethod def _from_mpmath(x, prec): if hasattr(x, "_mpf_"): return C.Float._new(x._mpf_, prec) elif hasattr(x, "_mpc_"): re, im = x._mpc_ re = C.Float._new(re, prec) im = C.Float._new(im, prec)*S.ImaginaryUnit return re+im else: raise TypeError("expected mpmath number (mpf or mpc)") @property
[docs] def is_number(self): """Returns True if 'self' is a number. >>> from sympy import log, Integral >>> from sympy.abc import x, y >>> x.is_number False >>> (2*x).is_number False >>> (2 + log(2)).is_number True >>> (2 + Integral(2, x)).is_number False >>> (2 + Integral(2, (x, 1, 2))).is_number True """ if not self.args: return False return all(obj.is_number for obj in self.iter_basic_args())
def _eval_interval(self, x, a, b): """ Returns evaluation over an interval. For most functions this is: self.subs(x, b) - self.subs(x, a), possibly using limit() if NaN is returned from subs. If b or a is None, it only evaluates -self.subs(x, a) or self.subs(b, x), respectively. """ from sympy.series import limit if (a is None and b is None): raise ValueError('Both interval ends cannot be None.') if a is None: A = 0 else: A = self.subs(x, a) if A is S.NaN: A = limit(self, x, a) if A is S.NaN: return A if b is None: B = 0 else: B = self.subs(x, b) if B is S.NaN: B = limit(self, x, b) return B - A def _eval_power(self, other): return None def _eval_conjugate(self): if self.is_real: return self def conjugate(self): from sympy.functions.elementary.complexes import conjugate as c return c(self) @classmethod def _parse_order(cls, order): """Parse and configure the ordering of terms. """ from sympy.polys.monomialtools import monomial_key try: reverse = order.startswith('rev-') except AttributeError: reverse = False else: if reverse: order = order[4:] monom_key = monomial_key(order) def neg(monom): result = [] for m in monom: if isinstance(m, tuple): result.append(neg(m)) else: result.append(-m) return tuple(result) def key(term): _, ((re, im), monom, ncpart) = term monom = neg(monom_key(monom)) ncpart = tuple([ e.sort_key(order=order) for e in ncpart ]) coeff = ((bool(im), im), (re, im)) return monom, ncpart, coeff return key, reverse
[docs] def as_ordered_factors(self, order=None): """ Transform an expression to an ordered list of factors. **Examples** >>> from sympy import sin, cos >>> from sympy.abc import x, y >>> (2*x*y*sin(x)*cos(x)).as_ordered_factors() [2, x, y, sin(x), cos(x)] """ if not self.is_Mul: return [self] cpart = [] ncpart = [] for arg in self.args: if arg.is_commutative: cpart.append(arg) else: ncpart.append(arg) return sorted(cpart, key=lambda expr: expr.sort_key(order=order)) + ncpart
[docs] def as_ordered_terms(self, order=None, data=False): """ Transform an expression to an ordered list of terms. **Examples** >>> from sympy import sin, cos >>> from sympy.abc import x, y >>> (sin(x)**2*cos(x) + sin(x)**2 + 1).as_ordered_terms() [sin(x)**2*cos(x), sin(x)**2, 1] """ key, reverse = self._parse_order(order) terms, gens = self.as_terms() if not any(term.is_Order for term, _ in terms): ordered = sorted(terms, key=key, reverse=reverse) else: _terms, _order = [], [] for term, repr in terms: if not term.is_Order: _terms.append((term, repr)) else: _order.append((term, repr)) ordered = sorted(_terms, key=key, reverse=True) \ + sorted(_order, key=key, reverse=True) if data: return ordered, gens else: return [ term for term, _ in ordered ]
[docs] def as_terms(self): """Transform an expression to a list of terms. """ from sympy.core import Add, Mul, S from sympy.core.exprtools import decompose_power from sympy.utilities import default_sort_key gens, terms = set([]), [] for term in Add.make_args(self): coeff, _term = term.as_coeff_Mul() coeff = complex(coeff) cpart, ncpart = {}, [] if _term is not S.One: for factor in Mul.make_args(_term): if factor.is_number: try: coeff *= complex(factor) except ValueError: pass else: continue if factor.is_commutative: base, exp = decompose_power(factor) cpart[base] = exp gens.add(base) else: ncpart.append(factor) coeff = coeff.real, coeff.imag ncpart = tuple(ncpart) terms.append((term, (coeff, cpart, ncpart))) gens = sorted(gens, key=default_sort_key) k, indices = len(gens), {} for i, g in enumerate(gens): indices[g] = i result = [] for term, (coeff, cpart, ncpart) in terms: monom = [0]*k for base, exp in cpart.iteritems(): monom[indices[base]] = exp result.append((term, (coeff, tuple(monom), ncpart))) return result, gens
[docs] def removeO(self): """Removes the additive O(..) symbol if there is one""" return self
[docs] def getO(self): """Returns the additive O(..) symbol if there is one, else None.""" return None
[docs] def getn(self): """ Returns the order of the expression. The order is determined either from the O(...) term. If there is no O(...) term, it returns None. Example: >>> from sympy import O >>> from sympy.abc import x >>> (1 + x + O(x**2)).getn() 2 >>> (1 + x).getn() """ o = self.getO() if o is None: return None elif o.is_Order: o = o.expr if o is S.One: return S.Zero if o.is_Symbol: return S.One if o.is_Pow: return o.args[1] if o.is_Mul: # x**n*log(x)**n or x**n/log(x)**n for oi in o.args: if oi.is_Symbol: return S.One if oi.is_Pow: syms = oi.atoms(C.Symbol) if len(syms) == 1: x = syms.pop() oi = oi.subs(x, C.Dummy('x', positive=True)) if oi.base.is_Symbol and oi.exp.is_Rational: return abs(oi.exp) raise NotImplementedError('not sure of order of %s' % o)
[docs] def count_ops(self, visual=None): """wrapper for count_ops that returns the operation count.""" from sympy import count_ops return count_ops(self, visual)
[docs] def args_cnc(self): """treat self as Mul and split it into tuple (set, list) where ``set`` contains the commutative parts and ``list`` contains the ordered non-commutative args. A special treatment is that -1 is separated from a Rational: >>> from sympy import symbols >>> A, B = symbols('A B', commutative=0) >>> x, y = symbols('x y') >>> (-2*x*y).args_cnc() [set([-1, 2, x, y]), []] >>> (-2*x*A*B*y).args_cnc() [set([-1, 2, x, y]), [A, B]] The arg is treated as a Mul: >>> (-2 + x + A).args_cnc() [set(), [x - 2 + A]] """ if self.is_Mul: args = list(self.args) else: args = [self] for i, mi in enumerate(args): if not mi.is_commutative: c = args[:i] nc = args[i:] break else: c = args nc = [] if c and c[0].is_Rational and c[0].is_negative and c[0] != S.NegativeOne: c[:1] = [S.NegativeOne, -c[0]] return [set(c), nc]
[docs] def coeff(self, x, right=False): """ Returns the coefficient of the exact term "x" or None if there is no "x". When x is noncommutative, the coeff to the left (default) or right of x can be returned. The keyword 'right' is ignored when x is commutative. Examples:: >>> from sympy import symbols >>> from sympy.abc import x, y, z You can select terms that have an explicit negative in front of them: >>> (-x+2*y).coeff(-1) x >>> (x-2*y).coeff(-1) 2*y You can select terms with no rational coefficient: >>> (x+2*y).coeff(1) x >>> (3+2*x+4*x**2).coeff(1) You can select terms that have a numerical term in front of them: >>> (-x-2*y).coeff(2) -y >>> from sympy import sqrt >>> (x+sqrt(2)*x).coeff(sqrt(2)) x The matching is exact: >>> (3+2*x+4*x**2).coeff(x) 2 >>> (3+2*x+4*x**2).coeff(x**2) 4 >>> (3+2*x+4*x**2).coeff(x**3) >>> (z*(x+y)**2).coeff((x+y)**2) z >>> (z*(x+y)**2).coeff(x+y) In addition, no factoring is done, so 2 + y is not obtained from the following: >>> (2*x+2+(x+1)*y).coeff(x+1) y >>> n, m, o = symbols('n m o', commutative=False) >>> n.coeff(n) 1 >>> (3*n).coeff(n) 3 >>> (n*m + m*n*m).coeff(n) # = (1 + m)*n*m 1 + m >>> (n*m + m*n*m).coeff(n, right=True) # = (1 + m)*n*m m If there is more than one possible coefficient None is returned: >>> (n*m + m*n).coeff(n) If there is only one possible coefficient, it is returned: >>> (n*m + o*m*n).coeff(m*n) o >>> (n*m + o*m*n).coeff(m*n, right=1) 1 """ x = sympify(x) if not x: # 0 or None return None if x == self: return S.One if x is S.One: try: assert Add.make_args(S.Zero) and Mul.make_args(S.One) # replace try/except with this co = [a for a in Add.make_args(self) if not any(ai.is_number for ai in Mul.make_args(a))] except AssertionError: co = [a for a in (Add.make_args(self) or [S.Zero]) if not any(ai.is_number for ai in (Mul.make_args(a) or [S.One]))] if not co: return None return Add(*co) def incommon(l1, l2): if not l1 or not l2: return [] n = min(len(l1), len(l2)) for i in xrange(n): if l1[i] != l2[i]: return l1[:i] return l1[:] def arglist(x): """ Return list of x's args when treated as a Mul after checking to see if a negative Rational is present (in which case it is made positive and a -1 is added to the list). """ margs = list(Mul.make_args(x)) try: assert Mul.make_args(S.One) # replace try/except with the following if margs[0].is_Rational and margs[0].is_negative and margs[0] != S.NegativeOne: margs.append(S.NegativeOne) margs[0] *= -1 except AssertionError: if margs and margs[0].is_Rational and margs[0].is_negative and margs[0] != S.NegativeOne: margs.append(S.NegativeOne) margs[0] *= -1 return margs def find(l, sub, first=True): """ Find where list sub appears in list l. When ``first`` is True the first occurance from the left is returned, else the last occurance is returned. Return None if sub is not in l. >> l = range(5)*2 >> find(l, [2, 3]) 2 >> find(l, [2, 3], first=0) 7 >> find(l, [2, 4]) None """ if not sub or not l or len(sub) > len(l): return None n = len(sub) if not first: l.reverse() sub.reverse() for i in xrange(0, len(l) - n + 1): if all(l[i + j] == sub[j] for j in range(n)): break else: i = None if not first: l.reverse() sub.reverse() if i is not None and not first: i = len(l) - (i + n) return i co = [] try: assert Add.make_args(S.Zero) # replace try/except with this args = Add.make_args(self) except AssertionError: args = Add.make_args(self) or [S.Zero] self_c = self.is_commutative x_c = x.is_commutative if self_c and not x_c: return None if self_c: xargs = set(arglist(x)) for a in Add.make_args(self): margs = set(arglist(a)) if len(xargs) > len(margs): continue resid = margs.difference(xargs) if len(resid) + len(xargs) == len(margs): co.append(Mul(*resid)) if co == []: return None elif co: return Add(*co) elif x_c: xargs = set(arglist(x)) for a in Add.make_args(self): margs, nc = a.args_cnc() if len(xargs) > len(margs): continue resid = margs.difference(xargs) if len(resid) + len(xargs) == len(margs): co.append(Mul(*(list(resid) + nc))) if co == []: return None elif co: return Add(*co) else: # both nc xargs, nx = x.args_cnc() # find the parts that pass the commutative terms for a in Add.make_args(self): margs, nc = a.args_cnc() if len(xargs) > len(margs): continue resid = margs.difference(xargs) if len(resid) + len(xargs) == len(margs): co.append((resid, nc)) # now check the non-comm parts if not co: return None if all(n == co[0][1] for r, n in co): ii = find(co[0][1], nx, right) if not ii is None: if not right: return Mul(Add(*[Mul(*r) for r, c in co]), Mul(*co[0][1][:ii])) else: return Mul(*co[0][1][ii+len(nx):]) beg = reduce(incommon, (n[1] for n in co)) if beg: ii = find(beg, nx, right) if not ii is None: if not right: gcdc = co[0][0] for i in xrange(1, len(co)): gcdc = gcdc.intersection(co[i][0]) if not gcdc: break return Mul(*(list(gcdc) + beg[:ii])) else: m = ii + len(nx) return Add(*[Mul(*(list(r) + n[m:])) for r, n in co]) end = list(reversed(reduce(incommon, (list(reversed(n[1])) for n in co)))) if end: ii = find(end, nx, right) if not ii is None: if not right: return Add(*[Mul(*(list(r) + n[:-len(end)+ii])) for r, n in co]) else: return Mul(*end[ii+len(nx):]) # look for single match hit = None for i, (r, n) in enumerate(co): ii = find(n, nx, right) if not ii is None: if not hit: hit = ii, r, n else: break else: if hit: ii, r, n = hit if not right: return Mul(*(list(r) + n[:ii])) else: return Mul(*n[ii+len(nx):]) return None
[docs] def as_expr(self, *gens): """ Convert a polynomial to a SymPy expression. **Examples** >>> from sympy import sin >>> from sympy.abc import x, y >>> f = (x**2 + x*y).as_poly(x, y) >>> f.as_expr() x**2 + x*y >>> sin(x).as_expr() sin(x) """ return self
[docs] def as_coefficient(self, expr): """Extracts symbolic coefficient at the given expression. In other words, this functions separates 'self' into product of 'expr' and 'expr'-free coefficient. If such separation is not possible it will return None. >>> from sympy import E, pi, sin, I, symbols >>> from sympy.abc import x, y >>> E.as_coefficient(E) 1 >>> (2*E).as_coefficient(E) 2 >>> (2*sin(E)*E).as_coefficient(E) >>> (2*E + x*E).as_coefficient(E) x + 2 >>> (2*E*x + x).as_coefficient(E) >>> (E*(x + 1) + x).as_coefficient(E) >>> (2*pi*I).as_coefficient(pi*I) 2 >>> (2*I).as_coefficient(pi*I) """ r = self.extract_multiplicatively(expr) if r and not r.has(expr): return r
[docs] def as_independent(self, *deps, **hint): """ A mostly naive separation of a Mul or Add into arguments that are not are dependent on deps. To obtain as complete a separation of variables as possible, use a separation method first, e.g.: * separatevars() to change Mul, Add and Pow (including exp) into Mul * .expand(mul=True) to change Add or Mul into Add * .expand(log=True) to change log expr into an Add The only non-naive thing that is done here is to respect noncommutative ordering of variables. The returned tuple (i, d) has the following interpretation: * i will has no variable that appears in deps * d will be 1 or else have terms that contain variables that are in deps * if self is an Add then self = i + d * if self is a Mul then self = i*d * if self is anything else, either tuple (self, S.One) or (S.One, self) is returned. To force the expression to be treated as an Add, use the hint as_Add=True Examples: -- self is an Add >>> from sympy import sin, cos, exp >>> from sympy.abc import x, y, z >>> (x + x*y).as_independent(x) (0, x*y + x) >>> (x + x*y).as_independent(y) (x, x*y) >>> (2*x*sin(x) + y + x + z).as_independent(x) (y + z, 2*x*sin(x) + x) >>> (2*x*sin(x) + y + x + z).as_independent(x, y) (z, 2*x*sin(x) + x + y) -- self is a Mul >>> (x*sin(x)*cos(y)).as_independent(x) (cos(y), x*sin(x)) non-commutative terms cannot always be separated out when self is a Mul >>> from sympy import symbols >>> n1, n2, n3 = symbols('n1 n2 n3', commutative=False) >>> (n1 + n1*n2).as_independent(n2) (n1, n1*n2) >>> (n2*n1 + n1*n2).as_independent(n2) (0, n1*n2 + n2*n1) >>> (n1*n2*n3).as_independent(n1) (1, n1*n2*n3) >>> (n1*n2*n3).as_independent(n2) (n1, n2*n3) >>> ((x-n1)*(x-y)).as_independent(x) (1, (x - y)*(x - n1)) -- self is anything else: >>> (sin(x)).as_independent(x) (1, sin(x)) >>> (sin(x)).as_independent(y) (sin(x), 1) >>> exp(x+y).as_independent(x) (1, exp(x + y)) -- force self to be treated as an Add: >>> (3*x).as_independent(x, as_Add=1) (0, 3*x) -- force self to be treated as a Mul: >>> (3+x).as_independent(x, as_Add=0) (1, x + 3) >>> (-3+x).as_independent(x, as_Add=0) (1, x - 3) Note how the below differs from the above in making the constant on the dep term positive. >>> (y*(-3+x)).as_independent(x) (y, x - 3) Note: when trying to get independent terms, a separation method might need to be used first. In this case, it is important to keep track of what you send to this routine so you know how to interpret the returned values >>> from sympy import separatevars, log >>> separatevars(exp(x+y)).as_independent(x) (exp(y), exp(x)) >>> (x + x*y).as_independent(y) (x, x*y) >>> separatevars(x + x*y).as_independent(y) (x, y + 1) >>> (x*(1 + y)).as_independent(y) (x, y + 1) >>> (x*(1 + y)).expand(mul=True).as_independent(y) (x, x*y) >>> a, b=symbols('a b',positive=True) >>> (log(a*b).expand(log=True)).as_independent(b) (log(a), log(b)) See also: .separatevars(), .expand(log=True), .as_two_terms(), .as_coeff_add(), .as_coeff_mul() """ from sympy.utilities.iterables import sift func = self.func if hint.get('as_Add', func is Add): want = Add else: want = Mul if (want is not func or func is not Add and func is not Mul): if self.has(*deps): return (want.identity, self) else: return (self, want.identity) else: if func is Add: args = list(self.args) else: args, nc = self.args_cnc() d = sift(args, lambda x: x.has(*deps)) depend = d.pop(True, []) indep = d.pop(False, []) if func is Add: # all terms were treated as commutative return (Add(*indep), Add(*depend)) else: # handle noncommutative by stopping at first dependent term for i, n in enumerate(nc): if n.has(*deps): depend.extend(nc[i:]) break indep.append(n) return Mul(*indep), Mul(*depend)
[docs] def as_real_imag(self, deep=True): """Performs complex expansion on 'self' and returns a tuple containing collected both real and imaginary parts. This method can't be confused with re() and im() functions, which does not perform complex expansion at evaluation. However it is possible to expand both re() and im() functions and get exactly the same results as with a single call to this function. >>> from sympy import symbols, I >>> x, y = symbols('x,y', real=True) >>> (x + y*I).as_real_imag() (x, y) >>> from sympy.abc import z, w >>> (z + w*I).as_real_imag() (-im(w) + re(z), im(z) + re(w)) """ return (C.re(self), C.im(self))
def as_powers_dict(self): return dict([self.as_base_exp()]) def as_base_exp(self): # a -> b ** e return self, S.One def as_coeff_terms(self, *deps): import warnings warnings.warn("\nuse as_coeff_mul() instead of as_coeff_terms().", DeprecationWarning) def as_coeff_factors(self, *deps): import warnings warnings.warn("\nuse as_coeff_add() instead of as_coeff_factors().", DeprecationWarning)
[docs] def as_coeff_mul(self, *deps): """Return the tuple (c, args) where self is written as a Mul, ``m``. c should be a Rational multiplied by any terms of the Mul that are independent of deps. args should be a tuple of all other terms of m; args is empty if self is a Number or if self is independent of deps (when given). This should be used when you don't know if self is a Mul or not but you want to treat self as a Mul or if you want to process the individual arguments of the tail of self as a Mul. - if you know self is a Mul and want only the head, use self.args[0]; - if you don't want to process the arguments of the tail but need the tail then use self.as_two_terms() which gives the head and tail; - if you want to split self into an independent and dependent parts use self.as_independent(\*deps) >>> from sympy import S >>> from sympy.abc import x, y >>> (S(3)).as_coeff_mul() (3, ()) >>> (3*x*y).as_coeff_mul() (3, (x, y)) >>> (3*x*y).as_coeff_mul(x) (3*y, (x,)) >>> (3*y).as_coeff_mul(x) (3*y, ()) """ if deps: if not self.has(*deps): return self, tuple() return S.One, (self,)
[docs] def as_coeff_add(self, *deps): """Return the tuple (c, args) where self is written as an Add, ``a``. c should be a Rational added to any terms of the Add that are independent of deps. args should be a tuple of all other terms of ``a``; args is empty if self is a Number or if self is independent of deps (when given). This should be used when you don't know if self is an Add or not but you want to treat self as an Add or if you want to process the individual arguments of the tail of self as an Add. - if you know self is an Add and want only the head, use self.args[0]; - if you don't want to process the arguments of the tail but need the tail then use self.as_two_terms() which gives the head and tail. - if you want to split self into an independent and dependent parts use self.as_independent(\*deps) >>> from sympy import S >>> from sympy.abc import x, y >>> (S(3)).as_coeff_add() (3, ()) >>> (3 + x + y).as_coeff_add() (3, (y, x)) >>> (3 + x +y).as_coeff_add(x) (y + 3, (x,)) >>> (3 + y).as_coeff_add(x) (y + 3, ()) """ if deps: if not self.has(*deps): return self, tuple() return S.Zero, (self,)
[docs] def as_numer_denom(self): """ a/b -> a,b This is just a stub that should be defined by an object's class methods to get anything else.""" return self, S.One
def normal(self): n, d = self.as_numer_denom() if d is S.One: return n return n/d
[docs] def extract_multiplicatively(self, c): """Return None if it's not possible to make self in the form c * something in a nice way, i.e. preserving the properties of arguments of self. >>> from sympy import symbols, Rational >>> x, y = symbols('x,y', real=True) >>> ((x*y)**3).extract_multiplicatively(x**2 * y) x*y**2 >>> ((x*y)**3).extract_multiplicatively(x**4 * y) >>> (2*x).extract_multiplicatively(2) x >>> (2*x).extract_multiplicatively(3) >>> (Rational(1,2)*x).extract_multiplicatively(3) x/6 """ c = sympify(c) if c is S.One: return self elif c == self: return S.One elif c.is_Mul: x = self.extract_multiplicatively(c.as_two_terms()[0]) if x != None: return x.extract_multiplicatively(c.as_two_terms()[1]) quotient = self / c if self.is_Number: if self is S.Infinity: if c.is_positive: return S.Infinity elif self is S.NegativeInfinity: if c.is_negative: return S.Infinity elif c.is_positive: return S.NegativeInfinity elif self is S.ComplexInfinity: if not c.is_zero: return S.ComplexInfinity elif self is S.NaN: return S.NaN elif self.is_Integer: if not quotient.is_Integer: return None elif self.is_positive and quotient.is_negative: return None else: return quotient elif self.is_Rational: if not quotient.is_Rational: return None elif self.is_positive and quotient.is_negative: return None else: return quotient elif self.is_Float: if not quotient.is_Float: return None elif self.is_positive and quotient.is_negative: return None else: return quotient elif self.is_NumberSymbol or self.is_Symbol or self is S.ImaginaryUnit: if quotient.is_Mul and len(quotient.args) == 2: if quotient.args[0].is_Integer and quotient.args[0].is_positive and quotient.args[1] == self: return quotient elif quotient.is_Integer: return quotient elif self.is_Add: newargs = [] for arg in self.args: newarg = arg.extract_multiplicatively(c) if newarg != None: newargs.append(newarg) else: return None return Add(*newargs) elif self.is_Mul: for i in xrange(len(self.args)): newargs = list(self.args) del(newargs[i]) tmp = self._new_rawargs(*newargs).extract_multiplicatively(c) if tmp != None: return tmp * self.args[i] elif self.is_Pow: if c.is_Pow and c.base == self.base: new_exp = self.exp.extract_additively(c.exp) if new_exp != None: return self.base ** (new_exp) elif c == self.base: new_exp = self.exp.extract_additively(1) if new_exp != None: return self.base ** (new_exp)
[docs] def extract_additively(self, c): """Return None if it's not possible to make self in the form something + c in a nice way, i.e. preserving the properties of arguments of self. >>> from sympy import symbols >>> x, y = symbols('x,y', real=True) >>> ((x*y)**3).extract_additively(1) >>> (x+1).extract_additively(x) 1 >>> (x+1).extract_additively(2*x) >>> (x+1).extract_additively(-x) 2*x + 1 >>> (-x+1).extract_additively(2*x) -3*x + 1 """ c = sympify(c) if c is S.Zero: return self elif c == self: return S.Zero elif self is S.Zero: return None elif c.is_Add: x = self.extract_additively(c.as_two_terms()[0]) if x != None: return x.extract_additively(c.as_two_terms()[1]) sub = self - c if self.is_Number: if self.is_Integer: if not sub.is_Integer: return None elif self.is_positive and sub.is_negative: return None else: return sub elif self.is_Rational: if not sub.is_Rational: return None elif self.is_positive and sub.is_negative: return None else: return sub elif self.is_Float: if not sub.is_Float: return None elif self.is_positive and sub.is_negative: return None else: return sub elif self.is_NumberSymbol or self.is_Symbol or self is S.ImaginaryUnit: if sub.is_Mul and len(sub.args) == 2: if sub.args[0].is_Integer and sub.args[0].is_positive and sub.args[1] == self: return sub elif sub.is_Integer: return sub elif self.is_Add: terms = self.as_two_terms() subs0 = terms[0].extract_additively(c) if subs0 != None: return subs0 + terms[1] else: subs1 = terms[1].extract_additively(c) if subs1 != None: return subs1 + terms[0] elif self.is_Mul: self_coeff, self_terms = self.as_coeff_mul() if c.is_Mul: c_coeff, c_terms = c.as_coeff_mul() if c_terms == self_terms: new_coeff = self_coeff.extract_additively(c_coeff) if new_coeff != None: return new_coeff * c._new_rawargs(*c_terms) elif c == self_terms: new_coeff = self_coeff.extract_additively(1) if new_coeff != None: return new_coeff * c
[docs] def could_extract_minus_sign(self): """Canonical way to choose an element in the set {e, -e} where e is any expression. If the canonical element is e, we have e.could_extract_minus_sign() == True, else e.could_extract_minus_sign() == False. For any expression, the set ``{e.could_extract_minus_sign(), (-e).could_extract_minus_sign()}`` must be ``{True, False}``. >>> from sympy.abc import x, y >>> (x-y).could_extract_minus_sign() != (y-x).could_extract_minus_sign() True """ negative_self = -self self_has_minus = (self.extract_multiplicatively(-1) != None) negative_self_has_minus = ((negative_self).extract_multiplicatively(-1) != None) if self_has_minus != negative_self_has_minus: return self_has_minus else: if self.is_Add: # We choose the one with less arguments with minus signs all_args = len(self.args) negative_args = len([False for arg in self.args if arg.could_extract_minus_sign()]) positive_args = all_args - negative_args if positive_args > negative_args: return False elif positive_args < negative_args: return True elif self.is_Mul: # We choose the one with an odd number of minus signs num, den = self.as_numer_denom() args = Mul.make_args(num) + Mul.make_args(den) arg_signs = [arg.could_extract_minus_sign() for arg in args] negative_args = filter(None, arg_signs) return len(negative_args) % 2 == 1 # As a last resort, we choose the one with greater hash return hash(self) < hash(negative_self)
def _eval_is_polynomial(self, syms): if self.free_symbols.intersection(syms) == set([]): return True return False
[docs] def is_polynomial(self, *syms): """ Return True if self is a polynomial in syms and False otherwise. This checks if self is an exact polynomial in syms. This function returns False for expressions that are "polynomials" with symbolic exponents. Thus, you should be able to apply polynomial algorithms to expressions for which this returns True, and Poly(expr, \*syms) should work only if and only if expr.is_polynomial(\*syms) returns True. The polynomial does not have to be in expanded form. If no symbols are given, all free symbols in the expression will be used. This is not part of the assumptions system. You cannot do Symbol('z', polynomial=True). **Examples** >>> from sympy import Symbol >>> x = Symbol('x') >>> ((x**2 + 1)**4).is_polynomial(x) True >>> ((x**2 + 1)**4).is_polynomial() True >>> (2**x + 1).is_polynomial(x) False >>> n = Symbol('n', nonnegative=True, integer=True) >>> (x**n + 1).is_polynomial(x) False This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be a polynomial to become one. >>> from sympy import sqrt, factor, cancel >>> y = Symbol('y', positive=True) >>> a = sqrt(y**2 + 2*y + 1) >>> a.is_polynomial(y) False >>> factor(a) y + 1 >>> factor(a).is_polynomial(y) True >>> b = (y**2 + 2*y + 1)/(y + 1) >>> b.is_polynomial(y) False >>> cancel(b) y + 1 >>> cancel(b).is_polynomial(y) True See also .is_rational_function() """ if syms: syms = set(map(sympify, syms)) else: syms = self.free_symbols if syms.intersection(self.free_symbols) == set([]): # constant polynomial return True else: return self._eval_is_polynomial(syms)
def _eval_is_rational_function(self, syms): if self.free_symbols.intersection(syms) == set([]): return True return False
[docs] def is_rational_function(self, *syms): """ Test whether function is a ratio of two polynomials in the given symbols, syms. When syms is not given, all free symbols will be used. The rational function does not have to be in expanded or in any kind of canonical form. This function returns False for expressions that are "rational functions" with symbolic exponents. Thus, you should be able to call .as_numer_denom() and apply polynomial algorithms to the result for expressions for which this returns True. This is not part of the assumptions system. You cannot do Symbol('z', rational_function=True). Example: >>> from sympy import Symbol, sin >>> from sympy.abc import x, y >>> (x/y).is_rational_function() True >>> (x**2).is_rational_function() True >>> (x/sin(y)).is_rational_function(y) False >>> n = Symbol('n', integer=True) >>> (x**n + 1).is_rational_function(x) False This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be a rational function to become one. >>> from sympy import sqrt, factor, cancel >>> y = Symbol('y', positive=True) >>> a = sqrt(y**2 + 2*y + 1)/y >>> a.is_rational_function(y) False >>> factor(a) (y + 1)/y >>> factor(a).is_rational_function(y) True See also is_rational_function(). """ if syms: syms = set(map(sympify, syms)) else: syms = self.free_symbols if syms.intersection(self.free_symbols) == set([]): # constant rational function return True else: return self._eval_is_rational_function(syms) ################################################################################### ##################### SERIES, LEADING TERM, LIMIT, ORDER METHODS ################## ###################################################################################
[docs] def series(self, x=None, x0=0, n=6, dir="+"): """ Series expansion of "self" around ``x = x0`` yielding either terms of the series one by one (the lazy series given when n=None), else all the terms at once when n != None. Note: when n != None, if an O() term is returned then the x in the in it and the entire expression represents x - x0, the displacement from x0. (If there is no O() term then the series was exact and x has it's normal meaning.) This is currently necessary since sympy's O() can only represent terms at x0=0. So instead of:: cos(x).series(x0=1, n=2) --> (1 - x)*sin(1) + cos(1) + O((x - 1)**2) which graphically looks like this:: | .|. . . . | \ . . ---+---------------------- | . . . . | \ x=0 the following is returned instead:: -x*sin(1) + cos(1) + O(x**2) whose graph is this:: \ | . .| . . . \ . . -----+\------------------. | . . . . | \ x=0 which is identical to ``cos(x + 1).series(n=2)``. Usage: Returns the series expansion of "self" around the point ``x = x0`` with respect to ``x`` up to O(x**n) (default n is 6). If ``x=None`` and ``self`` is univariate, the univariate symbol will be supplied, otherwise an error will be raised. >>> from sympy import cos, exp >>> from sympy.abc import x, y >>> cos(x).series() 1 - x**2/2 + x**4/24 + O(x**6) >>> cos(x).series(n=4) 1 - x**2/2 + O(x**4) >>> e = cos(x + exp(y)) >>> e.series(y, n=2) cos(x + 1) - y*sin(x + 1) + O(y**2) >>> e.series(x, n=2) cos(exp(y)) - x*sin(exp(y)) + O(x**2) If ``n=None`` then an iterator of the series terms will be returned. >>> term=cos(x).series(n=None) >>> [term.next() for i in range(2)] [1, -x**2/2] For ``dir=+`` (default) the series is calculated from the right and for ``dir=-`` the series from the left. For smooth functions this flag will not alter the results. >>> abs(x).series(dir="+") x >>> abs(x).series(dir="-") -x """ if x is None: syms = self.atoms(C.Symbol) if len(syms) > 1: raise ValueError('x must be given for multivariate functions.') x = syms.pop() if not self.has(x): if n is None: return (s for s in [self]) else: return self ## it seems like the following should be doable, but several failures ## then occur. Is this related to issue 1747 et al See also XPOS below. #if x.is_positive is x.is_negative is None: # # replace x with an x that has a positive assumption # xpos = C.Dummy('x', positive=True) # rv = self.subs(x, xpos).series(xpos, x0, n, dir) # if n is None: # return (s.subs(xpos, x) for s in rv) # else: # return rv.subs(xpos, x) if len(dir) != 1 or dir not in '+-': raise ValueError("Dir must be '+' or '-'") if x0 in [S.Infinity, S.NegativeInfinity]: dir = {S.Infinity: '+', S.NegativeInfinity: '-'}[x0] s = self.subs(x, 1/x).series(x, n=n, dir=dir) if n is None: return (si.subs(x, 1/x) for si in s) # don't include the order term since it will eat the larger terms return s.removeO().subs(x, 1/x) # use rep to shift origin to x0 and change sign (if dir is negative) # and undo the process with rep2 if x0 or dir == '-': if dir == '-': rep = -x + x0 rep2 = -x rep2b = x0 else: rep = x + x0 rep2 = x rep2b = -x0 s = self.subs(x, rep).series(x, x0=0, n=n, dir='+') if n is None: # lseries... return (si.subs(x, rep2 + rep2b) for si in s) # nseries... o = s.getO() or S.Zero s = s.removeO() if o and x0: rep2b = 0 # when O() can handle x0 != 0 this can be removed return s.subs(x, rep2 + rep2b) + o # from here on it's x0=0 and dir='+' handling if n != None: # nseries handling s1 = self._eval_nseries(x, n=n, logx=None) o = s1.getO() or S.Zero if o: # make sure the requested order is returned ngot = o.getn() if ngot > n: # leave o in its current form (e.g. with x*log(x)) so # it eats terms properly, then replace it below s1 += o.subs(x, x**C.Rational(n, ngot)) elif ngot < n: # increase the requested number of terms to get the desired # number keep increasing (up to 9) until the received order # is different than the original order and then predict how # many additional terms are needed for more in range(1, 9): s1 = self._eval_nseries(x, n=n + more, logx=None) newn = s1.getn() if newn != ngot: ndo = n + (n - ngot)*more/(newn - ngot) s1 = self._eval_nseries(x, n=ndo, logx=None) # if this assertion fails then our ndo calculation # needs modification assert s1.getn() == n break else: raise ValueError('Could not calculate %s terms for %s' % (str(n), self)) o = s1.getO() s1 = s1.removeO() else: o = C.Order(x**n) if (s1 + o).removeO() == s1: o = S.Zero return s1 + o else: # lseries handling def yield_lseries(s): """Return terms of lseries one at a time.""" for si in s: if not si.is_Add: yield si continue # yield terms 1 at a time if possible # by increasing order until all the # terms have been returned yielded = 0 o = C.Order(si)*x ndid = 0 ndo = len(si.args) while 1: do = (si - yielded + o).removeO() o *= x if not do or do.is_Order: continue if do.is_Add: ndid += len(do.args) else: ndid += 1 yield do if ndid == ndo: raise StopIteration yielded += do return yield_lseries(self.removeO()._eval_lseries(x))
[docs] def lseries(self, x=None, x0=0, dir='+'): """ Wrapper for series yielding an iterator of the terms of the series. Note: an infinite series will yield an infinite iterator. The following, for exaxmple, will never terminate. It will just keep printing terms of the sin(x) series:: for term in sin(x).lseries(x): print term The advantage of lseries() over nseries() is that many times you are just interested in the next term in the series (i.e. the first term for example), but you don't know how many you should ask for in nseries() using the "n" parameter. See also nseries(). """ return self.series(x, x0, n=None, dir=dir)
def _eval_lseries(self, x): # default implementation of lseries is using nseries(), and adaptively # increasing the "n". As you can see, it is not very efficient, because # we are calculating the series over and over again. Subclasses should # override this method and implement much more efficient yielding of # terms. n = 0 series = self._eval_nseries(x, n=n, logx=None) if not series.is_Order: if series.is_Add: yield series.removeO() else: yield series raise StopIteration while series.is_Order: n += 1 series = self._eval_nseries(x, n=n, logx=None) e = series.removeO() yield e while 1: while 1: n += 1 series = self._eval_nseries(x, n=n, logx=None).removeO() if e != series: break yield series - e e = series
[docs] def nseries(self, x=None, x0=0, n=6, dir='+',logx=None): """ Wrapper to _eval_nseries if assumptions allow, else to series. If x is given, x0 is 0, dir='+', and self has x, then _eval_nseries is called. This calculates "n" terms in the innermost expressions and then builds up the final series just by "cross-multiplying" everything out. Advantage -- it's fast, because we don't have to determine how many terms we need to calculate in advance. Disadvantage -- you may end up with less terms than you may have expected, but the O(x**n) term appended will always be correct and so the result, though perhaps shorter, will also be correct. If any of those assumptions is not met, this is treated like a wrapper to series which will try harder to return the correct number of terms. See also lseries(). """ if x and not self.has(x): return self if x is None or x0 or dir != '+':#{see XPOS above} or (x.is_positive == x.is_negative == None): assert logx == None return self.series(x, x0, n, dir) else: return self._eval_nseries(x, n=n, logx=logx)
def _eval_nseries(self, x, n, logx): """ Return terms of series for self up to O(x**n) at x=0 from the positive direction. This is a method that should be overridden in subclasses. Users should never call this method directly (use .nseries() instead), so you don't have to write docstrings for _eval_nseries(). """ raise NotImplementedError("(%s).nseries(%s, %s, %s)" % (self, x, x0, n))
[docs] def limit(self, x, xlim, dir='+'): """ Compute limit x->xlim. """ from sympy.series.limits import limit return limit(self, x, xlim, dir)
[docs] def compute_leading_term(self, x, skip_abs=False, logx=None): """ as_leading_term is only allowed for results of .series() This is a wrapper to compute a series first. If skip_abs is true, the absolute term is assumed to be zero. (This is necessary because sometimes it cannot be simplified to zero without a lot of work, but is still known to be zero. See log._eval_nseries for an example.) If skip_log is true, log(x) is treated as an independent symbol. (This is needed for the gruntz algorithm.) """ from sympy.series.gruntz import calculate_series from sympy import cancel, expand_mul if self.removeO() == 0: return self if logx is None: d = C.Dummy('logx') s = calculate_series(self, x, skip_abs, d).subs(d, C.log(x)) else: s = calculate_series(self, x, skip_abs, logx) s = cancel(s) if skip_abs: s = expand_mul(s).as_independent(x)[1] return s.as_leading_term(x)
@cacheit
[docs] def as_leading_term(self, *symbols): """ Returns the leading term. Example: >>> from sympy.abc import x >>> (1+x+x**2).as_leading_term(x) 1 >>> (1/x**2+x+x**2).as_leading_term(x) x**(-2) Note: self is assumed to be the result returned by Basic.series(). """ from sympy import powsimp if len(symbols)>1: c = self for x in symbols: c = c.as_leading_term(x) return c elif not symbols: return self x = sympify(symbols[0]) assert x.is_Symbol, repr(x) if not self.has(x): return self obj = self._eval_as_leading_term(x) if obj is not None: return powsimp(obj, deep=True, combine='exp') raise NotImplementedError('as_leading_term(%s, %s)' % (self, x))
def _eval_as_leading_term(self, x): return self
[docs] def as_coeff_exponent(self, x): """ c*x**e -> c,e where x can be any symbolic expression. """ x = sympify(x) wc = Wild('wc') we = Wild('we') p = wc*x**we from sympy import collect s = collect(self, x) d = s.match(p) if d is not None and we in d: return d[wc], d[we] return s, S.Zero
[docs] def leadterm(self, x): """ Returns the leading term a*x**b as a tuple (a, b). Example: >>> from sympy.abc import x >>> (1+x+x**2).leadterm(x) (1, 0) >>> (1/x**2+x+x**2).leadterm(x) (1, -2) Note: self is assumed to be the result returned by Basic.series(). """ from sympy import powsimp x = sympify(x) c, e = self.as_leading_term(x).as_coeff_exponent(x) c = powsimp(c, deep=True, combine='exp') if not c.has(x): return c, e raise ValueError("cannot compute leadterm(%s, %s), got c=%s" % (self, x, c))
[docs] def as_coeff_Mul(self): """Efficiently extract the coefficient of a product. """ return S.One, self ################################################################################### ##################### DERIVATIVE, INTEGRAL, FUNCTIONAL METHODS #################### ###################################################################################
def diff(self, *symbols, **assumptions): new_symbols = map(sympify, symbols) # e.g. x, 2, y, z assumptions.setdefault("evaluate", True) return Derivative(self, *new_symbols, **assumptions) ########################################################################### ###################### EXPRESSION EXPANSION METHODS ####################### ########################################################################### # These should be overridden in subclasses def _eval_expand_basic(self, deep=True, **hints): return self def _eval_expand_power_exp(self, deep=True, **hints): return self def _eval_expand_power_base(self, deep=True, **hints): return self def _eval_expand_mul(self, deep=True, **hints): return self def _eval_expand_multinomial(self, deep=True, **hints): return self def _eval_expand_log(self, deep=True, **hints): return self def _eval_expand_complex(self, deep=True, **hints): return self def _eval_expand_trig(self, deep=True, **hints): return self def _eval_expand_func(self, deep=True, **hints): return self
[docs] def expand(self, deep=True, modulus=None, power_base=True, power_exp=True, \ mul=True, log=True, multinomial=True, basic=True, **hints): """ Expand an expression using hints. See the docstring in function.expand for more information. """ hints.update(power_base=power_base, power_exp=power_exp, mul=mul, \ log=log, multinomial=multinomial, basic=basic) expr = self for hint, use_hint in hints.iteritems(): if use_hint: func = getattr(expr, '_eval_expand_'+hint, None) if func is not None: expr = func(deep=deep, **hints) if modulus is not None: modulus = sympify(modulus) if not modulus.is_Integer or modulus <= 0: raise ValueError("modulus must be a positive integer, got %s" % modulus) terms = [] for term in Add.make_args(expr): coeff, tail = term.as_coeff_mul() coeff %= modulus if coeff: terms.append(Mul(*((coeff,) + tail))) expr = Add(*terms) return expr ########################################################################### ################### GLOBAL ACTION VERB WRAPPER METHODS #################### ###########################################################################
[docs] def integrate(self, *args, **kwargs): """See the integrate function in sympy.integrals""" from sympy.integrals import integrate return integrate(self, *args, **kwargs)
[docs] def simplify(self): """See the simplify function in sympy.simplify""" from sympy.simplify import simplify return simplify(self)
[docs] def nsimplify(self, constants=[], tolerance=None, full=False): """See the nsimplify function in sympy.simplify""" from sympy.simplify import nsimplify return nsimplify(self, constants, tolerance, full)
[docs] def separate(self, deep=False, force=False): """See the separate function in sympy.simplify""" from sympy.simplify import separate return separate(self, deep)
[docs] def collect(self, syms, evaluate=True, exact=False): """See the collect function in sympy.simplify""" from sympy.simplify import collect return collect(self, syms, evaluate, exact)
[docs] def together(self, *args, **kwargs): """See the together function in sympy.polys""" from sympy.polys import together return together(self, *args, **kwargs)
[docs] def apart(self, x=None, **args): """See the apart function in sympy.polys""" from sympy.polys import apart return apart(self, x, **args)
[docs] def ratsimp(self): """See the ratsimp function in sympy.simplify""" from sympy.simplify import ratsimp return ratsimp(self)
[docs] def trigsimp(self, deep=False, recursive=False): """See the trigsimp function in sympy.simplify""" from sympy.simplify import trigsimp return trigsimp(self, deep, recursive)
[docs] def radsimp(self): """See the radsimp function in sympy.simplify""" from sympy.simplify import radsimp return radsimp(self)
[docs] def powsimp(self, deep=False, combine='all'): """See the powsimp function in sympy.simplify""" from sympy.simplify import powsimp return powsimp(self, deep, combine)
[docs] def combsimp(self): """See the combsimp function in sympy.simplify""" from sympy.simplify import combsimp return combsimp(self)
[docs] def factor(self, *gens, **args): """See the factor() function in sympy.polys.polytools""" from sympy.polys import factor return factor(self, *gens, **args)
[docs] def refine(self, assumption=True): """See the refine function in sympy.assumptions""" from sympy.assumptions import refine return refine(self, assumption)
[docs] def cancel(self, *gens, **args): """See the cancel function in sympy.polys""" from sympy.polys import cancel return cancel(self, *gens, **args)
[docs] def invert(self, g): """See the invert function in sympy.polys""" from sympy.polys import invert return invert(self, g)
[docs]class AtomicExpr(Atom, Expr): """ A parent class for object which are both atoms and Exprs. Examples: Symbol, Number, Rational, Integer, ... But not: Add, Mul, Pow, ... """ is_Atom = True __slots__ = [] def _eval_derivative(self, s): if self == s: return S.One return S.Zero def as_numer_denom(self): return self, S.One def _eval_is_polynomial(self, syms): return True def _eval_is_rational_function(self, syms): return True def _eval_nseries(self, x, n, logx): return self
from mul import Mul from add import Add from power import Pow from function import Derivative from sympify import sympify from symbol import Wild