def _solve_as_rational(f, symbol, domain):
""" solve rational functions"""
f = together(f, deep=True)
g, h = fraction(f)
if not h.has(symbol):
return _solve_as_poly(g, symbol, domain)
else:
valid_solns = _solveset(g, symbol, domain)
invalid_solns = _solveset(h, symbol, domain)
return valid_solns - invalid_solns
def _solve_as_rational(f, symbol, solveset_solver, as_poly_solver):
""" solve rational functions"""
f = together(f, deep=True)
g, h = fraction(f)
if not h.has(symbol):
return as_poly_solver(g, symbol)
else:
valid_solns = solveset_solver(g, symbol)
invalid_solns = solveset_solver(h, symbol)
return valid_solns - invalid_solns
def _solve_as_rational(f, symbol, solveset_solver, as_poly_solver):
""" solve rational functions"""
f = together(f, deep=True)
g, h = fraction(f)
if not h.has(symbol):
return as_poly_solver(g, symbol)
else:
valid_solns = solveset_solver(g, symbol)
invalid_solns = solveset_solver(h, symbol)
return valid_solns - invalid_solns
def _solve_real_trig(f, symbol):
""" Helper to solve trigonometric equations """
f = trigsimp(f)
f = f.rewrite(exp)
f = together(f)
g, h = fraction(f)
y = Dummy('y')
g, h = g.expand(), h.expand()
g, h = g.subs(exp(I * symbol), y), h.subs(exp(I * symbol), y)
if g.has(symbol) or h.has(symbol):
return ConditionSet(symbol, Eq(f, 0), S.Reals)
solns = solveset_complex(g, y) - solveset_complex(h, y)
if isinstance(solns, FiniteSet):
return Union(
*[invert_complex(exp(I * symbol), s, symbol)[1] for s in solns])
elif solns is S.EmptySet:
return S.EmptySet
else:
return ConditionSet(symbol, Eq(f, 0), S.Reals)
def _solve_real_trig(f, symbol):
""" Helper to solve trigonometric equations """
f = trigsimp(f)
f = f.rewrite(exp)
f = together(f)
g, h = fraction(f)
y = Dummy('y')
g, h = g.expand(), h.expand()
g, h = g.subs(exp(I*symbol), y), h.subs(exp(I*symbol), y)
if g.has(symbol) or h.has(symbol):
raise NotImplementedError
solns = solveset_complex(g, y) - solveset_complex(h, y)
if isinstance(solns, FiniteSet):
return Union(*[invert_complex(exp(I*symbol), s, symbol)[1]
for s in solns])
elif solns is S.EmptySet:
return S.EmptySet
else:
raise NotImplementedError
def _solve_trig(f, symbol, domain):
""" Helper to solve trigonometric equations """
f = trigsimp(f)
f_original = f
f = f.rewrite(exp)
f = together(f)
g, h = fraction(f)
y = Dummy("y")
g, h = g.expand(), h.expand()
g, h = g.subs(exp(I * symbol), y), h.subs(exp(I * symbol), y)
if g.has(symbol) or h.has(symbol):
return ConditionSet(symbol, Eq(f, 0), S.Reals)
solns = solveset_complex(g, y) - solveset_complex(h, y)
if isinstance(solns, FiniteSet):
result = Union(*[invert_complex(exp(I * symbol), s, symbol)[1] for s in solns])
return Intersection(result, domain)
elif solns is S.EmptySet:
return S.EmptySet
else:
return ConditionSet(symbol, Eq(f_original, 0), S.Reals)
def solveset_complex(f, symbol):
""" Solve a complex valued equation.
Parameters
==========
f : Expr
The target equation
symbol : Symbol
The variable for which the equation is solved
Returns
=======
Set
A set of values for `symbol` for which `f` equal to
zero. An `EmptySet` is returned if no solution is found.
A `ConditionSet` is returned as an unsolved object if algorithms
to evaluate complete solutions are not yet implemented.
`solveset_complex` claims to be complete in the solution set that
it returns.
Raises
======
NotImplementedError
The algorithms to solve inequalities in complex domain are
not yet implemented.
ValueError
The input is not valid.
RuntimeError
It is a bug, please report to the github issue tracker.
See Also
========
solveset_real: solver for real domain
Examples
========
>>> from sympy import Symbol, exp
>>> from sympy.solvers.solveset import solveset_complex
>>> from sympy.abc import x, a, b, c
>>> solveset_complex(a*x**2 + b*x +c, x)
{-b/(2*a) - sqrt(-4*a*c + b**2)/(2*a), -b/(2*a) + sqrt(-4*a*c + b**2)/(2*a)}
* Due to the fact that complex extension of my real valued functions are
multivariate even some simple equations can have infinitely many
solution.
>>> solveset_complex(exp(x) - 1, x)
ImageSet(Lambda(_n, 2*_n*I*pi), Integers())
"""
if not symbol.is_Symbol:
raise ValueError(" %s is not a symbol" % (symbol))
f = sympify(f)
original_eq = f
if not isinstance(f, (Expr, Number)):
raise ValueError(" %s is not a valid sympy expression" % (f))
f = together(f)
# Without this equations like a + 4*x**2 - E keep oscillating
# into form a/4 + x**2 - E/4 and (a + 4*x**2 - E)/4
if not fraction(f)[1].has(symbol):
f = expand(f)
if f.is_zero:
return S.Complexes
elif not f.has(symbol):
result = EmptySet()
elif f.is_Mul and all([_is_finite_with_finite_vars(m) for m in f.args]):
result = Union(*[solveset_complex(m, symbol) for m in f.args])
else:
lhs, rhs_s = invert_complex(f, 0, symbol)
if lhs == symbol:
result = rhs_s
elif isinstance(rhs_s, FiniteSet):
equations = [lhs - rhs for rhs in rhs_s]
result = EmptySet()
for equation in equations:
if equation == f:
if any(
_has_rational_power(g, symbol)[0]
for g in equation.args):
result += _solve_radical(equation, symbol,
solveset_complex)
else:
result += _solve_as_rational(
equation,
symbol,
solveset_solver=solveset_complex,
as_poly_solver=_solve_as_poly_complex)
else:
result += solveset_complex(equation, symbol)
else:
result = ConditionSet(symbol, Eq(f, 0), S.Complexes)
if isinstance(result, FiniteSet):
result = [
s for s in result
if isinstance(s, RootOf) or domain_check(original_eq, symbol, s)
]
return FiniteSet(*result)
else:
return result
def solveset_real(f, symbol):
""" Solves a real valued equation.
Parameters
==========
f : Expr
The target equation
symbol : Symbol
The variable for which the equation is solved
Returns
=======
Set
A set of values for `symbol` for which `f` is equal to
zero. An `EmptySet` is returned if no solution is found.
A `ConditionSet` is returned as unsolved object if algorithms
to evaluate complete solutions are not yet implemented.
`solveset_real` claims to be complete in the set of the solution it
returns.
Raises
======
NotImplementedError
Algorithms to solve inequalities in complex domain are
not yet implemented.
ValueError
The input is not valid.
RuntimeError
It is a bug, please report to the github issue tracker.
See Also
=======
solveset_complex : solver for complex domain
Examples
========
>>> from sympy import Symbol, exp, sin, sqrt, I
>>> from sympy.solvers.solveset import solveset_real
>>> x = Symbol('x', real=True)
>>> a = Symbol('a', real=True, finite=True, positive=True)
>>> solveset_real(x**2 - 1, x)
{-1, 1}
>>> solveset_real(sqrt(5*x + 6) - 2 - x, x)
{-1, 2}
>>> solveset_real(x - I, x)
EmptySet()
>>> solveset_real(x - a, x)
{a}
>>> solveset_real(exp(x) - a, x)
{log(a)}
* In case the equation has infinitely many solutions an infinitely indexed
`ImageSet` is returned.
>>> solveset_real(sin(x) - 1, x)
ImageSet(Lambda(_n, 2*_n*pi + pi/2), Integers())
* If the equation is true for any arbitrary value of the symbol a `S.Reals`
set is returned.
>>> solveset_real(x - x, x)
(-oo, oo)
"""
if not symbol.is_Symbol:
raise ValueError(" %s is not a symbol" % (symbol))
f = sympify(f)
if not isinstance(f, (Expr, Number)):
raise ValueError(" %s is not a valid sympy expression" % (f))
original_eq = f
f = together(f)
# In this, unlike in solveset_complex, expression should only
# be expanded when fraction(f)[1] does not contain the symbol
# for which we are solving
if not symbol in fraction(f)[1].free_symbols and f.is_rational_function():
f = expand(f)
if f.has(Piecewise):
f = piecewise_fold(f)
result = EmptySet()
if f.expand().is_zero:
return S.Reals
elif not f.has(symbol):
return EmptySet()
elif f.is_Mul and all([_is_finite_with_finite_vars(m) for m in f.args]):
# if f(x) and g(x) are both finite we can say that the solution of
# f(x)*g(x) == 0 is same as Union(f(x) == 0, g(x) == 0) is not true in
# general. g(x) can grow to infinitely large for the values where
# f(x) == 0. To be sure that we are not silently allowing any
# wrong solutions we are using this technique only if both f and g are
# finite for a finite input.
result = Union(*[solveset_real(m, symbol) for m in f.args])
elif _is_function_class_equation(TrigonometricFunction, f, symbol) or \
_is_function_class_equation(HyperbolicFunction, f, symbol):
result = _solve_real_trig(f, symbol)
elif f.is_Piecewise:
result = EmptySet()
expr_set_pairs = f.as_expr_set_pairs()
for (expr, in_set) in expr_set_pairs:
solns = solveset_real(expr, symbol).intersect(in_set)
result = result + solns
else:
lhs, rhs_s = invert_real(f, 0, symbol)
if lhs == symbol:
result = rhs_s
elif isinstance(rhs_s, FiniteSet):
equations = [lhs - rhs for rhs in rhs_s]
for equation in equations:
if equation == f:
if any(
_has_rational_power(g, symbol)[0]
for g in equation.args):
result += _solve_radical(equation, symbol,
solveset_real)
elif equation.has(Abs):
result += _solve_abs(f, symbol)
else:
result += _solve_as_rational(
equation,
symbol,
solveset_solver=solveset_real,
as_poly_solver=_solve_as_poly_real)
else:
result += solveset_real(equation, symbol)
else:
result = ConditionSet(symbol, Eq(f, 0), S.Reals)
if isinstance(result, FiniteSet):
result = [
s for s in result
if isinstance(s, RootOf) or domain_check(original_eq, symbol, s)
]
return FiniteSet(*result).intersect(S.Reals)
else:
return result.intersect(S.Reals)
def solveset_complex(f, symbol):
""" Solve a complex valued equation.
Parameters
==========
f : Expr
The target equation
symbol : Symbol
The variable for which the equation is solved
Returns
=======
Set
A set of values for `symbol` for which `f` equal to
zero. An `EmptySet` is returned if no solution is found.
`solveset_complex` claims to be complete in the solution set that
it returns.
Raises
======
NotImplementedError
The algorithms for to find the solution of the given equation are
not yet implemented.
ValueError
The input is not valid.
RuntimeError
It is a bug, please report to the github issue tracker.
See Also
========
solveset_real: solver for real domain
Examples
========
>>> from sympy import Symbol, exp
>>> from sympy.solvers.solveset import solveset_complex
>>> from sympy.abc import x, a, b, c
>>> solveset_complex(a*x**2 + b*x +c, x)
{-b/(2*a) - sqrt(-4*a*c + b**2)/(2*a), -b/(2*a) + sqrt(-4*a*c + b**2)/(2*a)}
Due to the fact that complex extension of my real valued functions are
multivariate even some simple equations can have infinitely many solution.
>>> solveset_complex(exp(x) - 1, x)
ImageSet(Lambda(_n, 2*_n*I*pi), Integers())
"""
if not symbol.is_Symbol:
raise ValueError(" %s is not a symbol" % (symbol))
f = sympify(f)
original_eq = f
if not isinstance(f, (Expr, Number)):
raise ValueError(" %s is not a valid sympy expression" % (f))
f = together(f)
# Without this equations like a + 4*x**2 - E keep oscillating
# into form a/4 + x**2 - E/4 and (a + 4*x**2 - E)/4
if not fraction(f)[1].has(symbol):
f = expand(f)
if f.is_zero:
raise NotImplementedError("S.Complex set is not yet implemented")
elif not f.has(symbol):
result = EmptySet()
elif f.is_Mul and all([_is_finite_with_finite_vars(m) for m in f.args]):
result = Union(*[solveset_complex(m, symbol) for m in f.args])
else:
lhs, rhs_s = invert_complex(f, 0, symbol)
if lhs == symbol:
result = rhs_s
elif isinstance(rhs_s, FiniteSet):
equations = [lhs - rhs for rhs in rhs_s]
result = EmptySet()
for equation in equations:
if equation == f:
result += _solve_as_rational(equation, symbol,
solveset_solver=solveset_complex,
as_poly_solver=_solve_as_poly_complex)
else:
result += solveset_complex(equation, symbol)
else:
raise NotImplementedError
if isinstance(result, FiniteSet):
result = [s for s in result
if isinstance(s, RootOf)
or domain_check(original_eq, symbol, s)]
return FiniteSet(*result)
else:
return result
def count_ops(expr, visual=False):
"""
Return a representation (integer or expression) of the operations in expr.
If `visual` is False (default) then the sum of the coefficients of the
visual expression will be returned.
If `visual` is True then the number of each type of operation is shown
with the core class types (or their virtual equivalent) multiplied by the
number of times they occur.
If expr is an iterable, the sum of the op counts of the
items will be returned.
Examples:
>>> from sympy.abc import a, b, x, y
>>> from sympy import sin, count_ops
Although there isn't a SUB object, minus signs are interpreted as
either negations or subtractions:
>>> (x - y).count_ops(visual=True)
SUB
>>> (-x).count_ops(visual=True)
NEG
Here, there are two Adds and a Pow:
>>> (1 + a + b**2).count_ops(visual=True)
POW + 2*ADD
In the following, an Add, Mul, Pow and two functions:
>>> (sin(x)*x + sin(x)**2).count_ops(visual=True)
ADD + MUL + POW + 2*SIN
for a total of 5:
>>> (sin(x)*x + sin(x)**2).count_ops(visual=False)
5
Note that "what you type" is not always what you get. The expression
1/x/y is translated by sympy into 1/(x*y) so it gives a DIV and MUL rather
than two DIVs:
>>> (1/x/y).count_ops(visual=True)
DIV + MUL
The visual option can be used to demonstrate the difference in
operations for expressions in different forms. Here, the Horner
representation is compared with the expanded form of a polynomial:
>>> eq=x*(1 + x*(2 + x*(3 + x)))
>>> count_ops(eq.expand(), visual=True) - count_ops(eq, visual=True)
-MUL + 3*POW
The count_ops function also handles iterables:
>>> count_ops([x, sin(x), None, True, x + 2], visual=False)
2
>>> count_ops([x, sin(x), None, True, x + 2], visual=True)
ADD + SIN
>>> count_ops({x: sin(x), x + 2: y + 1}, visual=True)
SIN + 2*ADD
"""
from sympy.simplify.simplify import fraction
expr = sympify(expr)
if isinstance(expr, Expr):
ops = []
args = [expr]
NEG = C.Symbol('NEG')
DIV = C.Symbol('DIV')
SUB = C.Symbol('SUB')
ADD = C.Symbol('ADD')
def isneg(a):
c = a.as_coeff_mul()[0]
return c.is_Number and c.is_negative
while args:
a = args.pop()
if a.is_Rational:
#-1/3 = NEG + DIV
if a is not S.One:
if a.p < 0:
ops.append(NEG)
if a.q != 1:
ops.append(DIV)
continue
elif a.is_Mul:
if isneg(a):
ops.append(NEG)
if a.args[0] is S.NegativeOne:
a = a.as_two_terms()[1]
else:
a = -a
n, d = fraction(a)
if n.is_Integer:
ops.append(DIV)
if n < 0:
ops.append(NEG)
args.append(d)
continue # won't be -Mul but could be Add
elif d is not S.One:
if not d.is_Integer:
args.append(d)
ops.append(DIV)
args.append(n)
continue # could be -Mul
elif a.is_Add:
aargs = list(a.args)
negs = 0
for i, ai in enumerate(aargs):
if isneg(ai):
negs += 1
args.append(-ai)
if i > 0:
ops.append(SUB)
else:
args.append(ai)
if i > 0:
ops.append(ADD)
if negs == len(aargs): # -x - y = NEG + SUB
ops.append(NEG)
elif isneg(aargs[0]): # -x + y = SUB, but we already recorded an ADD
ops.append(SUB - ADD)
continue
if a.is_Pow and a.exp is S.NegativeOne:
ops.append(DIV)
args.append(a.base) # won't be -Mul but could be Add
continue
if (a.is_Mul or
a.is_Pow or
a.is_Function or
isinstance(a, Derivative) or
isinstance(a, C.Integral)):
o = C.Symbol(a.func.__name__.upper())
# count the args
if (a.is_Mul or
isinstance(a, C.LatticeOp)):
ops.append(o*(len(a.args) - 1))
else:
ops.append(o)
args.extend(a.args)
elif type(expr) is dict:
ops = [count_ops(k, visual=visual) +
count_ops(v, visual=visual) for k, v in expr.iteritems()]
elif hasattr(expr, '__iter__'):
ops = [count_ops(i, visual=visual) for i in expr]
elif not isinstance(expr, Basic):
ops = []
else: # it's Basic not isinstance(expr, Expr):
assert isinstance(expr, Basic)
ops = [count_ops(a, visual=visual) for a in expr.args]
if not ops:
if visual:
return S.Zero
return 0
ops = Add(*ops)
if visual:
return ops
if ops.is_Number:
return int(ops)
return sum(int((a.args or [1])[0]) for a in Add.make_args(ops))
def solveset_real(f, symbol):
""" Solves a real valued equation.
Parameters
==========
f : Expr
The target equation
symbol : Symbol
The variable for which the equation is solved
Returns
=======
Set
A set of values for `symbol` for which `f` is equal to
zero. An `EmptySet` is returned if no solution is found.
A `ConditionSet` is returned as unsolved object if algorithms
to evaluate complete solutions are not yet implemented.
`solveset_real` claims to be complete in the set of the solution it
returns.
Raises
======
NotImplementedError
Algorithms to solve inequalities in complex domain are
not yet implemented.
ValueError
The input is not valid.
RuntimeError
It is a bug, please report to the github issue tracker.
See Also
=======
solveset_complex : solver for complex domain
Examples
========
>>> from sympy import Symbol, exp, sin, sqrt, I
>>> from sympy.solvers.solveset import solveset_real
>>> x = Symbol('x', real=True)
>>> a = Symbol('a', real=True, finite=True, positive=True)
>>> solveset_real(x**2 - 1, x)
{-1, 1}
>>> solveset_real(sqrt(5*x + 6) - 2 - x, x)
{-1, 2}
>>> solveset_real(x - I, x)
EmptySet()
>>> solveset_real(x - a, x)
{a}
>>> solveset_real(exp(x) - a, x)
{log(a)}
* In case the equation has infinitely many solutions an infinitely indexed
`ImageSet` is returned.
>>> solveset_real(sin(x) - 1, x)
ImageSet(Lambda(_n, 2*_n*pi + pi/2), Integers())
* If the equation is true for any arbitrary value of the symbol a `S.Reals`
set is returned.
>>> solveset_real(x - x, x)
(-oo, oo)
"""
if not symbol.is_Symbol:
raise ValueError(" %s is not a symbol" % (symbol))
f = sympify(f)
if not isinstance(f, (Expr, Number)):
raise ValueError(" %s is not a valid sympy expression" % (f))
original_eq = f
f = together(f)
# In this, unlike in solveset_complex, expression should only
# be expanded when fraction(f)[1] does not contain the symbol
# for which we are solving
if not symbol in fraction(f)[1].free_symbols and f.is_rational_function():
f = expand(f)
if f.has(Piecewise):
f = piecewise_fold(f)
result = EmptySet()
if f.expand().is_zero:
return S.Reals
elif not f.has(symbol):
return EmptySet()
elif f.is_Mul and all([_is_finite_with_finite_vars(m) for m in f.args]):
# if f(x) and g(x) are both finite we can say that the solution of
# f(x)*g(x) == 0 is same as Union(f(x) == 0, g(x) == 0) is not true in
# general. g(x) can grow to infinitely large for the values where
# f(x) == 0. To be sure that we are not silently allowing any
# wrong solutions we are using this technique only if both f and g are
# finite for a finite input.
result = Union(*[solveset_real(m, symbol) for m in f.args])
elif _is_function_class_equation(TrigonometricFunction, f, symbol) or \
_is_function_class_equation(HyperbolicFunction, f, symbol):
result = _solve_real_trig(f, symbol)
elif f.is_Piecewise:
result = EmptySet()
expr_set_pairs = f.as_expr_set_pairs()
for (expr, in_set) in expr_set_pairs:
solns = solveset_real(expr, symbol).intersect(in_set)
result = result + solns
else:
lhs, rhs_s = invert_real(f, 0, symbol)
if lhs == symbol:
result = rhs_s
elif isinstance(rhs_s, FiniteSet):
equations = [lhs - rhs for rhs in rhs_s]
for equation in equations:
if equation == f:
if any(_has_rational_power(g, symbol)[0]
for g in equation.args):
result += _solve_radical(equation,
symbol,
solveset_real)
elif equation.has(Abs):
result += _solve_abs(f, symbol)
else:
result += _solve_as_rational(equation, symbol,
solveset_solver=solveset_real,
as_poly_solver=_solve_as_poly_real)
else:
result += solveset_real(equation, symbol)
else:
result = ConditionSet(symbol, Eq(f, 0), S.Reals)
if isinstance(result, FiniteSet):
result = [s for s in result
if isinstance(s, RootOf)
or domain_check(original_eq, symbol, s)]
return FiniteSet(*result).intersect(S.Reals)
else:
return result.intersect(S.Reals)
def count_ops(expr, visual=False):
"""
Return a representation (integer or expression) of the operations in expr.
If `visual` is False (default) then the sum of the coefficients of the
visual expression will be returned.
If `visual` is True then the number of each type of operation is shown
with the core class types (or their virtual equivalent) multiplied by the
number of times they occur.
If expr is an iterable, the sum of the op counts of the
items will be returned.
Examples:
>>> from sympy.abc import a, b, x, y
>>> from sympy import sin, count_ops
Although there isn't a SUB object, minus signs are interpreted as
either negations or subtractions:
>>> (x - y).count_ops(visual=True)
SUB
>>> (-x).count_ops(visual=True)
NEG
Here, there are two Adds and a Pow:
>>> (1 + a + b**2).count_ops(visual=True)
POW + 2*ADD
In the following, an Add, Mul, Pow and two functions:
>>> (sin(x)*x + sin(x)**2).count_ops(visual=True)
ADD + MUL + POW + 2*SIN
for a total of 5:
>>> (sin(x)*x + sin(x)**2).count_ops(visual=False)
5
Note that "what you type" is not always what you get. The expression
1/x/y is translated by sympy into 1/(x*y) so it gives a DIV and MUL rather
than two DIVs:
>>> (1/x/y).count_ops(visual=True)
DIV + MUL
The visual option can be used to demonstrate the difference in
operations for expressions in different forms. Here, the Horner
representation is compared with the expanded form of a polynomial:
>>> eq=x*(1 + x*(2 + x*(3 + x)))
>>> count_ops(eq.expand(), visual=True) - count_ops(eq, visual=True)
-MUL + 3*POW
The count_ops function also handles iterables:
>>> count_ops([x, sin(x), None, True, x + 2], visual=False)
2
>>> count_ops([x, sin(x), None, True, x + 2], visual=True)
ADD + SIN
>>> count_ops({x: sin(x), x + 2: y + 1}, visual=True)
SIN + 2*ADD
"""
from sympy.simplify.simplify import fraction
expr = sympify(expr)
if isinstance(expr, Expr):
ops = []
args = [expr]
NEG = C.Symbol('NEG')
DIV = C.Symbol('DIV')
SUB = C.Symbol('SUB')
ADD = C.Symbol('ADD')
def isneg(a):
c = a.as_coeff_mul()[0]
return c.is_Number and c.is_negative
while args:
a = args.pop()
if a.is_Rational:
#-1/3 = NEG + DIV
if a is not S.One:
if a.p < 0:
ops.append(NEG)
if a.q != 1:
ops.append(DIV)
continue
elif a.is_Mul:
if isneg(a):
ops.append(NEG)
if a.args[0] is S.NegativeOne:
a = a.as_two_terms()[1]
else:
a = -a
n, d = fraction(a)
if n.is_Integer:
ops.append(DIV)
if n < 0:
ops.append(NEG)
args.append(d)
continue # won't be -Mul but could be Add
elif d is not S.One:
if not d.is_Integer:
args.append(d)
ops.append(DIV)
args.append(n)
continue # could be -Mul
elif a.is_Add:
aargs = list(a.args)
negs = 0
for i, ai in enumerate(aargs):
if isneg(ai):
negs += 1
args.append(-ai)
if i > 0:
ops.append(SUB)
else:
args.append(ai)
if i > 0:
ops.append(ADD)
if negs == len(aargs): # -x - y = NEG + SUB
ops.append(NEG)
elif isneg(aargs[0]
): # -x + y = SUB, but we already recorded an ADD
ops.append(SUB - ADD)
continue
if a.is_Pow and a.exp is S.NegativeOne:
ops.append(DIV)
args.append(a.base) # won't be -Mul but could be Add
continue
if (a.is_Mul or a.is_Pow or a.is_Function
or isinstance(a, Derivative) or isinstance(a, C.Integral)):
o = C.Symbol(a.func.__name__.upper())
# count the args
if (a.is_Mul or isinstance(a, C.LatticeOp)):
ops.append(o * (len(a.args) - 1))
else:
ops.append(o)
args.extend(a.args)
elif type(expr) is dict:
ops = [
count_ops(k, visual=visual) + count_ops(v, visual=visual)
for k, v in expr.iteritems()
]
elif hasattr(expr, '__iter__'):
ops = [count_ops(i, visual=visual) for i in expr]
elif not isinstance(expr, Basic):
ops = []
else: # it's Basic not isinstance(expr, Expr):
assert isinstance(expr, Basic)
ops = [count_ops(a, visual=visual) for a in expr.args]
if not ops:
if visual:
return S.Zero
return 0
ops = Add(*ops)
if visual:
return ops
if ops.is_Number:
return int(ops)
return sum(int((a.args or [1])[0]) for a in Add.make_args(ops))