# Source code for sympy.core.function

"""
There are three types of functions implemented in SymPy:

1) defined functions (in the sense that they can be evaluated) like
exp or sin; they have a name and a body:
f = exp
2) undefined function which have a name but no body. Undefined
functions can be defined using a Function class as follows:
f = Function('f')
(the result will be a Function instance)
3) anonymous function (or lambda function) which have a body (defined
with dummy variables) but have no name:
f = Lambda(x, exp(x)*x)
f = Lambda((x, y), exp(x)*y)
The fourth type of functions are composites, like (sin + cos)(x); these work in
SymPy core, but are not yet part of SymPy.

Examples
========

>>> import sympy
>>> f = sympy.Function("f")
>>> from sympy.abc import x
>>> f(x)
f(x)
>>> print(sympy.srepr(f(x).func))
Function('f')
>>> f(x).args
(x,)

"""
from __future__ import print_function, division

from .assumptions import ManagedProperties
from .basic import Basic
from .cache import cacheit
from .compatibility import iterable, is_sequence, as_int, ordered
from .core import BasicMeta
from .decorators import _sympifyit
from .expr import Expr, AtomicExpr
from .numbers import Rational, Float
from .operations import LatticeOp
from .rules import Transform
from .singleton import S
from .sympify import sympify

from sympy.core.containers import Tuple, Dict
from sympy.core.logic import fuzzy_and
from sympy.core.compatibility import string_types, with_metaclass, range
from sympy.utilities import default_sort_key
from sympy.utilities.misc import filldedent
from sympy.utilities.iterables import uniq
from sympy.core.evaluate import global_evaluate

import mpmath
import mpmath.libmp as mlib

import inspect

def _coeff_isneg(a):
"""Return True if the leading Number is negative.

Examples
========

>>> from sympy.core.function import _coeff_isneg
>>> from sympy import S, Symbol, oo, pi
>>> _coeff_isneg(-3*pi)
True
>>> _coeff_isneg(S(3))
False
>>> _coeff_isneg(-oo)
True
>>> _coeff_isneg(Symbol('n', negative=True)) # coeff is 1
False

"""

if a.is_Mul:
a = a.args[0]
return a.is_Number and a.is_negative

[docs]class PoleError(Exception):
pass

class ArgumentIndexError(ValueError):
def __str__(self):
return ("Invalid operation with argument number %s for Function %s" %
(self.args[1], self.args[0]))

[docs]class FunctionClass(ManagedProperties):
"""
Base class for function classes. FunctionClass is a subclass of type.

Use Function('<function name>' [ , signature ]) to create
undefined function classes.
"""
_new = type.__new__

def __init__(cls, *args, **kwargs):
if hasattr(cls, 'eval'):
evalargspec = inspect.getargspec(cls.eval)
if evalargspec.varargs:
evalargs = None
else:
evalargs = len(evalargspec.args) - 1  # subtract 1 for cls
if evalargspec.defaults:
# if there are default args then they are optional; the
# fewest args will occur when all defaults are used and
# the most when none are used (i.e. all args are given)
evalargs = tuple(range(
evalargs - len(evalargspec.defaults), evalargs + 1))
else:
evalargs = None
# honor kwarg value or class-defined value before using
# the number of arguments in the eval function (if present)
nargs = kwargs.pop('nargs', cls.__dict__.get('nargs', evalargs))
super(FunctionClass, cls).__init__(args, kwargs)

# Canonicalize nargs here; change to set in nargs.
if is_sequence(nargs):
if not nargs:
raise ValueError(filldedent('''
Incorrectly specified nargs as %s:
if there are no arguments, it should be
nargs = 0;
if there are any number of arguments,
it should be
nargs = None''' % str(nargs)))
nargs = tuple(ordered(set(nargs)))
elif nargs is not None:
nargs = (as_int(nargs),)
cls._nargs = nargs

@property
[docs]    def nargs(self):
"""Return a set of the allowed number of arguments for the function.

Examples
========

>>> from sympy.core.function import Function
>>> from sympy.abc import x, y
>>> f = Function('f')

If the function can take any number of arguments, the set of whole
numbers is returned:

>>> Function('f').nargs
Naturals0()

If the function was initialized to accept one or more arguments, a
corresponding set will be returned:

>>> Function('f', nargs=1).nargs
{1}
>>> Function('f', nargs=(2, 1)).nargs
{1, 2}

The undefined function, after application, also has the nargs
attribute; the actual number of arguments is always available by
checking the args attribute:

>>> f = Function('f')
>>> f(1).nargs
Naturals0()
>>> len(f(1).args)
1
"""
from sympy.sets.sets import FiniteSet
# XXX it would be nice to handle this in __init__ but there are import
# problems with trying to import FiniteSet there
return FiniteSet(*self._nargs) if self._nargs else S.Naturals0

def __repr__(cls):
return cls.__name__

class Application(with_metaclass(FunctionClass, Basic)):
"""
Base class for applied functions.

Instances of Application represent the result of applying an application of
any type to any object.
"""

is_Function = True

@cacheit
def __new__(cls, *args, **options):
from sympy.sets.fancysets import Naturals0
from sympy.sets.sets import FiniteSet

args = list(map(sympify, args))
evaluate = options.pop('evaluate', global_evaluate[0])
# WildFunction (and anything else like it) may have nargs defined
# and we throw that value away here
options.pop('nargs', None)

if options:
raise ValueError("Unknown options: %s" % options)

if evaluate:
evaluated = cls.eval(*args)
if evaluated is not None:
return evaluated

obj = super(Application, cls).__new__(cls, *args, **options)

# make nargs uniform here
try:
# things passing through here:
#  - functions subclassed from Function (e.g. myfunc(1).nargs)
#  - functions like cos(1).nargs
#  - AppliedUndef with given nargs like Function('f', nargs=1)(1).nargs
# Canonicalize nargs here
if is_sequence(obj.nargs):
nargs = tuple(ordered(set(obj.nargs)))
elif obj.nargs is not None:
nargs = (as_int(obj.nargs),)
else:
nargs = None
except AttributeError:
# things passing through here:
#  - WildFunction('f').nargs
#  - AppliedUndef with no nargs like Function('f')(1).nargs
nargs = obj._nargs  # note the underscore here
# convert to FiniteSet
obj.nargs = FiniteSet(*nargs) if nargs else Naturals0()
return obj

@classmethod
def eval(cls, *args):
"""
Returns a canonical form of cls applied to arguments args.

The eval() method is called when the class cls is about to be
instantiated and it should return either some simplified instance
(possible of some other class), or if the class cls should be
unmodified, return None.

Examples of eval() for the function "sign"
---------------------------------------------

@classmethod
def eval(cls, arg):
if arg is S.NaN:
return S.NaN
if arg is S.Zero: return S.Zero
if arg.is_positive: return S.One
if arg.is_negative: return S.NegativeOne
if isinstance(arg, Mul):
coeff, terms = arg.as_coeff_Mul(rational=True)
if coeff is not S.One:
return cls(coeff) * cls(terms)

"""
return

@property
def func(self):
return self.__class__

def _eval_subs(self, old, new):
if (old.is_Function and new.is_Function and old == self.func and
len(self.args) in new.nargs):
return new(*self.args)

[docs]class Function(Application, Expr):
"""Base class for applied mathematical functions.

It also serves as a constructor for undefined function classes.

Examples
========

First example shows how to use Function as a constructor for undefined
function classes:

>>> from sympy import Function, Symbol
>>> x = Symbol('x')
>>> f = Function('f')
>>> g = Function('g')(x)
>>> f
f
>>> f(x)
f(x)
>>> g
g(x)
>>> f(x).diff(x)
Derivative(f(x), x)
>>> g.diff(x)
Derivative(g(x), x)

In the following example Function is used as a base class for
my_func that represents a mathematical function *my_func*. Suppose
that it is well known, that *my_func(0)* is *1* and *my_func* at infinity
goes to *0*, so we want those two simplifications to occur automatically.
Suppose also that *my_func(x)* is real exactly when *x* is real. Here is
an implementation that honours those requirements:

>>> from sympy import Function, S, oo, I, sin
>>> class my_func(Function):
...
...     @classmethod
...     def eval(cls, x):
...         if x.is_Number:
...             if x is S.Zero:
...                 return S.One
...             elif x is S.Infinity:
...                 return S.Zero
...
...     def _eval_is_real(self):
...         return self.args[0].is_real
...
>>> x = S('x')
>>> my_func(0) + sin(0)
1
>>> my_func(oo)
0
>>> my_func(3.54).n() # Not yet implemented for my_func.
my_func(3.54)
>>> my_func(I).is_real
False

In order for my_func to become useful, several other methods would
need to be implemented. See source code of some of the already
implemented functions for more complete examples.

Also, if the function can take more than one argument, then nargs
must be defined, e.g. if my_func can take one or two arguments
then,

>>> class my_func(Function):
...     nargs = (1, 2)
...
>>>
"""

@property
def _diff_wrt(self):
"""Allow derivatives wrt functions.

Examples
========

>>> from sympy import Function, Symbol
>>> f = Function('f')
>>> x = Symbol('x')
>>> f(x)._diff_wrt
True

"""
return True

@cacheit
def __new__(cls, *args, **options):
# Handle calls like Function('f')
if cls is Function:
return UndefinedFunction(*args, **options)

n = len(args)
if n not in cls.nargs:
# XXX: exception message must be in exactly this format to
# make it work with NumPy's functions like vectorize(). See,
# for example, https://github.com/numpy/numpy/issues/1697.
# The ideal solution would be just to attach metadata to
# the exception and change NumPy to take advantage of this.
temp = ('%(name)s takes %(qual)s %(args)s '
'argument%(plural)s (%(given)s given)')
raise TypeError(temp % {
'name': cls,
'qual': 'exactly' if len(cls.nargs) == 1 else 'at least',
'args': min(cls.nargs),
'plural': 's'*(min(cls.nargs) != 1),
'given': n})

evaluate = options.get('evaluate', global_evaluate[0])
result = super(Function, cls).__new__(cls, *args, **options)
if not evaluate or not isinstance(result, cls):
return result

pr = max(cls._should_evalf(a) for a in result.args)
pr2 = min(cls._should_evalf(a) for a in result.args)
if pr2 > 0:
return result.evalf(mlib.libmpf.prec_to_dps(pr))
return result

@classmethod
def _should_evalf(cls, arg):
"""
Decide if the function should automatically evalf().

By default (in this implementation), this happens if (and only if) the
ARG is a floating point number.
This function is used by __new__.

Returns the precision to evalf to, or -1 if it shouldn't evalf.
"""
from sympy.core.symbol import Wild
if arg.is_Float:
return arg._prec
return -1
# Don't use as_real_imag() here, that's too much work
a, b = Wild('a'), Wild('b')
m = arg.match(a + b*S.ImaginaryUnit)
if not m or not (m[a].is_Float or m[b].is_Float):
return -1
l = [m[i]._prec for i in m if m[i].is_Float]
l.append(-1)
return max(l)

@classmethod
def class_key(cls):
from sympy.sets.fancysets import Naturals0
funcs = {
'exp': 10,
'log': 11,
'sin': 20,
'cos': 21,
'tan': 22,
'cot': 23,
'sinh': 30,
'cosh': 31,
'tanh': 32,
'coth': 33,
'conjugate': 40,
're': 41,
'im': 42,
'arg': 43,
}
name = cls.__name__

try:
i = funcs[name]
except KeyError:
i = 0 if isinstance(cls.nargs, Naturals0) else 10000

return 4, i, name

@property
[docs]    def is_commutative(self):
"""
Returns whether the functon is commutative.
"""
if all(getattr(t, 'is_commutative') for t in self.args):
return True
else:
return False

def _eval_evalf(self, prec):
# Lookup mpmath function based on name
fname = self.func.__name__
try:
if not hasattr(mpmath, fname):
from sympy.utilities.lambdify import MPMATH_TRANSLATIONS
fname = MPMATH_TRANSLATIONS[fname]
func = getattr(mpmath, fname)
except (AttributeError, KeyError):
try:
return Float(self._imp_(*self.args), prec)
except (AttributeError, TypeError):
return

# Convert all args to mpf or mpc
# Convert the arguments to *higher* precision than requested for the
# final result.
# XXX + 5 is a guess, it is similar to what is used in evalf.py. Should
#     we be more intelligent about it?
try:
args = [arg._to_mpmath(prec + 5) for arg in self.args]
from mpmath import mpf, mpc
# the precision of an mpf value is the last element
# if that is 1 (and m[1] is not 1 which would indicate a
# power of 2), then the eval failed; so check that none of
# the arguments failed to compute to a finite precision.
# Note: An mpc value has two parts, the re and imag tuple;
# check each of those parts, too. Anything else is allowed to
# pass
if isinstance(m, mpf):
m = m._mpf_
return m[1] !=1 and m[-1] == 1
elif isinstance(m, mpc):
m, n = m._mpc_
return m[1] !=1 and m[-1] == 1 and \
n[1] !=1 and n[-1] == 1
else:
return False
if any(bad(a) for a in args):
raise ValueError  # one or more args failed to compute with significance
except ValueError:
return

with mpmath.workprec(prec):
v = func(*args)

return Expr._from_mpmath(v, prec)

def _eval_derivative(self, s):
# f(x).diff(s) -> x.diff(s) * f.fdiff(1)(s)
i = 0
l = []
for a in self.args:
i += 1
da = a.diff(s)
if da is S.Zero:
continue
try:
df = self.fdiff(i)
except ArgumentIndexError:
df = Function.fdiff(self, i)
l.append(df * da)

def _eval_is_commutative(self):
return fuzzy_and(a.is_commutative for a in self.args)

def _eval_is_complex(self):
return fuzzy_and(a.is_complex for a in self.args)

[docs]    def as_base_exp(self):
"""
Returns the method as the 2-tuple (base, exponent).
"""
return self, S.One

def _eval_aseries(self, n, args0, x, logx):
"""
Compute an asymptotic expansion around args0, in terms of self.args.
This function is only used internally by _eval_nseries and should not
be called directly; derived classes can overwrite this to implement
asymptotic expansions.
"""
from sympy.utilities.misc import filldedent
raise PoleError(filldedent('''
Asymptotic expansion of %s around %s is
not implemented.''' % (type(self), args0)))

def _eval_nseries(self, x, n, logx):
"""
This function does compute series for multivariate functions,
but the expansion is always in terms of *one* variable.
Examples
========

>>> from sympy import atan2
>>> from sympy.abc import x, y
>>> atan2(x, y).series(x, n=2)
atan2(0, y) + x/y + O(x**2)
>>> atan2(x, y).series(y, n=2)
-y/x + atan2(x, 0) + O(y**2)

This function also computes asymptotic expansions, if necessary
and possible:

>>> from sympy import loggamma
>>> loggamma(1/x)._eval_nseries(x,0,None)
-1/x - log(x)/x + log(x)/2 + O(1)

"""
from sympy import Order
from sympy.sets.sets import FiniteSet
args = self.args
args0 = [t.limit(x, 0) for t in args]
if any(t.is_finite is False for t in args0):
from sympy import oo, zoo, nan
# XXX could use t.as_leading_term(x) here but it's a little
# slower
a = [t.compute_leading_term(x, logx=logx) for t in args]
a0 = [t.limit(x, 0) for t in a]
if any([t.has(oo, -oo, zoo, nan) for t in a0]):
return self._eval_aseries(n, args0, x, logx)
# Careful: the argument goes to oo, but only logarithmically so. We
# are supposed to do a power series expansion "around the
# logarithmic term". e.g.
#      f(1+x+log(x))
#     -> f(1+logx) + x*f'(1+logx) + O(x**2)
# where 'logx' is given in the argument
a = [t._eval_nseries(x, n, logx) for t in args]
z = [r - r0 for (r, r0) in zip(a, a0)]
p = [Dummy() for t in z]
q = []
v = None
for ai, zi, pi in zip(a0, z, p):
if zi.has(x):
if v is not None:
raise NotImplementedError
q.append(ai + pi)
v = pi
else:
q.append(ai)
e1 = self.func(*q)
if v is None:
return e1
s = e1._eval_nseries(v, n, logx)
o = s.getO()
s = s.removeO()
s = s.subs(v, zi).expand() + Order(o.expr.subs(v, zi), x)
return s
if (self.func.nargs is S.Naturals0
or (self.func.nargs == FiniteSet(1) and args0[0])
or any(c > 1 for c in self.func.nargs)):
e = self
e1 = e.expand()
if e == e1:
#for example when e = sin(x+1) or e = sin(cos(x))
#let's try the general algorithm
term = e.subs(x, S.Zero)
if term.is_finite is False or term is S.NaN:
raise PoleError("Cannot expand %s around 0" % (self))
series = term
fact = S.One
_x = Dummy('x')
e = e.subs(x, _x)
for i in range(n - 1):
i += 1
fact *= Rational(i)
e = e.diff(_x)
subs = e.subs(_x, S.Zero)
if subs is S.NaN:
# try to evaluate a limit if we have to
subs = e.limit(_x, S.Zero)
if subs.is_finite is False:
raise PoleError("Cannot expand %s around 0" % (self))
term = subs*(x**i)/fact
term = term.expand()
series += term
return series + Order(x**n, x)
return e1.nseries(x, n=n, logx=logx)
arg = self.args[0]
l = []
g = None
# try to predict a number of terms needed
nterms = n + 2
if cf != 0:
nterms = int(nterms / cf)
for i in range(nterms):
g = self.taylor_term(i, arg, g)
g = g.nseries(x, n=n, logx=logx)
l.append(g)

[docs]    def fdiff(self, argindex=1):
"""
Returns the first derivative of the function.
"""
if not (1 <= argindex <= len(self.args)):
raise ArgumentIndexError(self, argindex)

if self.args[argindex - 1].is_Symbol:
for i in range(len(self.args)):
if i == argindex - 1:
continue
# See issue 8510
if self.args[argindex - 1] in self.args[i].free_symbols:
break
else:
return Derivative(self, self.args[argindex - 1], evaluate=False)
# See issue 4624 and issue 4719 and issue 5600
arg_dummy = Dummy('xi_%i' % argindex)
arg_dummy.dummy_index = hash(self.args[argindex - 1])
new_args = [arg for arg in self.args]
new_args[argindex-1] = arg_dummy
return Subs(Derivative(self.func(*new_args), arg_dummy),
arg_dummy, self.args[argindex - 1])

"""Stub that should be overridden by new Functions to return
the first non-zero term in a series if ever an x-dependent
argument whose leading term vanishes as x -> 0 might be encountered.
"""
from sympy import Order
args = [a.as_leading_term(x) for a in self.args]
o = Order(1, x)
if any(x in a.free_symbols and o.contains(a) for a in args):
# Whereas x and any finite number are contained in O(1, x),
# expressions like 1/x are not. If any arg simplified to a
# vanishing expression as x -> 0 (like x or x**2, but not
# 3, 1/x, etc...) then the _eval_as_leading_term is needed
# to supply the first non-zero term of the series,
#
#      ----------    ------------
#      cos(1/x)      cos(1/x)
#      cos(cos(x))   cos(1)
#      cos(x)        1        <- _eval_as_leading_term needed
#      sin(x)        x        <- _eval_as_leading_term needed
#
raise NotImplementedError(
'%s has no _eval_as_leading_term routine' % self.func)
else:
return self.func(*args)

def _sage_(self):
import sage.all as sage
fname = self.func.__name__
func = getattr(sage, fname)
args = [arg._sage_() for arg in self.args]
return func(*args)

class AppliedUndef(Function):
"""
Base class for expressions resulting from the application of an undefined
function.
"""

def __new__(cls, *args, **options):
args = list(map(sympify, args))
obj = super(AppliedUndef, cls).__new__(cls, *args, **options)
return obj

return self

def _sage_(self):
import sage.all as sage
fname = str(self.func)
args = [arg._sage_() for arg in self.args]
func = sage.function(fname)(*args)
return func

class UndefinedFunction(FunctionClass):
"""
The (meta)class of undefined functions.
"""
def __new__(mcl, name, **kwargs):
ret = BasicMeta.__new__(mcl, name, (AppliedUndef,), kwargs)
ret.__module__ = None
return ret

def __instancecheck__(cls, instance):
return cls in type(instance).__mro__

UndefinedFunction.__eq__ = lambda s, o: (isinstance(o, s.__class__) and
(s.class_key() == o.class_key()))

[docs]class WildFunction(Function, AtomicExpr):
"""
A WildFunction function matches any function (with its arguments).

Examples
========

>>> from sympy import WildFunction, Function, cos
>>> from sympy.abc import x, y
>>> F = WildFunction('F')
>>> f = Function('f')
>>> F.nargs
Naturals0()
>>> x.match(F)
>>> F.match(F)
{F_: F_}
>>> f(x).match(F)
{F_: f(x)}
>>> cos(x).match(F)
{F_: cos(x)}
>>> f(x, y).match(F)
{F_: f(x, y)}

To match functions with a given number of arguments, set nargs to the
desired value at instantiation:

>>> F = WildFunction('F', nargs=2)
>>> F.nargs
{2}
>>> f(x).match(F)
>>> f(x, y).match(F)
{F_: f(x, y)}

To match functions with a range of arguments, set nargs to a tuple
containing the desired number of arguments, e.g. if nargs = (1, 2)
then functions with 1 or 2 arguments will be matched.

>>> F = WildFunction('F', nargs=(1, 2))
>>> F.nargs
{1, 2}
>>> f(x).match(F)
{F_: f(x)}
>>> f(x, y).match(F)
{F_: f(x, y)}
>>> f(x, y, 1).match(F)

"""

include = set()

def __init__(cls, name, **assumptions):
from sympy.sets.sets import Set, FiniteSet
cls.name = name
nargs = assumptions.pop('nargs', S.Naturals0)
if not isinstance(nargs, Set):
if is_sequence(nargs):
nargs = tuple(ordered(set(nargs)))
elif nargs is not None:
nargs = (as_int(nargs),)
nargs = FiniteSet(*nargs)
cls.nargs = nargs

def matches(self, expr, repl_dict={}, old=False):
if not isinstance(expr, (AppliedUndef, Function)):
return None
if len(expr.args) not in self.nargs:
return None

repl_dict = repl_dict.copy()
repl_dict[self] = expr
return repl_dict

[docs]class Derivative(Expr):
"""
Carries out differentiation of the given expression with respect to symbols.

expr must define ._eval_derivative(symbol) method that returns
the differentiation result. This function only needs to consider the
non-trivial case where expr contains symbol and it should call the diff()
method internally (not _eval_derivative); Derivative should be the only
one to call _eval_derivative.

Simplification of high-order derivatives:

Because there can be a significant amount of simplification that can be
done when multiple differentiations are performed, results will be
automatically simplified in a fairly conservative fashion unless the
keyword simplify is set to False.

>>> from sympy import sqrt, diff
>>> from sympy.abc import x
>>> e = sqrt((x + 1)**2 + x)
>>> diff(e, x, 5, simplify=False).count_ops()
136
>>> diff(e, x, 5).count_ops()
30

Ordering of variables:

If evaluate is set to True and the expression can not be evaluated, the
list of differentiation symbols will be sorted, that is, the expression is
assumed to have continuous derivatives up to the order asked. This sorting
assumes that derivatives wrt Symbols commute, derivatives wrt non-Symbols
commute, but Symbol and non-Symbol derivatives don't commute with each
other.

Derivative wrt non-Symbols:

This class also allows derivatives wrt non-Symbols that have _diff_wrt
set to True, such as Function and Derivative. When a derivative wrt a non-
Symbol is attempted, the non-Symbol is temporarily converted to a Symbol
while the differentiation is performed.

Note that this may seem strange, that Derivative allows things like
f(g(x)).diff(g(x)), or even f(cos(x)).diff(cos(x)).  The motivation for
allowing this syntax is to make it easier to work with variational calculus
(i.e., the Euler-Lagrange method).  The best way to understand this is that
the action of derivative with respect to a non-Symbol is defined by the
above description:  the object is substituted for a Symbol and the
derivative is taken with respect to that.  This action is only allowed for
objects for which this can be done unambiguously, for example Function and
Derivative objects.  Note that this leads to what may appear to be
mathematically inconsistent results.  For example::

>>> from sympy import cos, sin, sqrt
>>> from sympy.abc import x
>>> (2*cos(x)).diff(cos(x))
2
>>> (2*sqrt(1 - sin(x)**2)).diff(cos(x))
0

This appears wrong because in fact 2*cos(x) and 2*sqrt(1 - sin(x)**2) are
identically equal.  However this is the wrong way to think of this.  Think
of it instead as if we have something like this::

>>> from sympy.abc import c, s
>>> def F(u):
...     return 2*u
...
>>> def G(u):
...     return 2*sqrt(1 - u**2)
...
>>> F(cos(x))
2*cos(x)
>>> G(sin(x))
2*sqrt(-sin(x)**2 + 1)
>>> F(c).diff(c)
2
>>> F(c).diff(c)
2
>>> G(s).diff(c)
0
>>> G(sin(x)).diff(cos(x))
0

Here, the Symbols c and s act just like the functions cos(x) and sin(x),
respectively. Think of 2*cos(x) as f(c).subs(c, cos(x)) (or f(c) *at*
c = cos(x)) and 2*sqrt(1 - sin(x)**2) as g(s).subs(s, sin(x)) (or g(s) *at*
s = sin(x)), where f(u) == 2*u and g(u) == 2*sqrt(1 - u**2).  Here, we
define the function first and evaluate it at the function, but we can
actually unambiguously do this in reverse in SymPy, because
expr.subs(Function, Symbol) is well-defined:  just structurally replace the
function everywhere it appears in the expression.

This is the same notational convenience used in the Euler-Lagrange method
when one says F(t, f(t), f'(t)).diff(f(t)).  What is actually meant is
that the expression in question is represented by some F(t, u, v) at u =
f(t) and v = f'(t), and F(t, f(t), f'(t)).diff(f(t)) simply means F(t, u,
v).diff(u) at u = f(t).

We do not allow derivatives to be taken with respect to expressions where this
is not so well defined.  For example, we do not allow expr.diff(x*y)
because there are multiple ways of structurally defining where x*y appears
in an expression, some of which may surprise the reader (for example, a
very strict definition would have that (x*y*z).diff(x*y) == 0).

>>> from sympy.abc import x, y, z
>>> (x*y*z).diff(x*y)
Traceback (most recent call last):
...
ValueError: Can't differentiate wrt the variable: x*y, 1

Note that this definition also fits in nicely with the definition of the
chain rule.  Note how the chain rule in SymPy is defined using unevaluated
Subs objects::

>>> from sympy import symbols, Function
>>> f, g = symbols('f g', cls=Function)
>>> f(2*g(x)).diff(x)
2*Derivative(g(x), x)*Subs(Derivative(f(_xi_1), _xi_1),
(_xi_1,), (2*g(x),))
>>> f(g(x)).diff(x)
Derivative(g(x), x)*Subs(Derivative(f(_xi_1), _xi_1),
(_xi_1,), (g(x),))

Finally, note that, to be consistent with variational calculus, and to
ensure that the definition of substituting a Function for a Symbol in an
expression is well-defined, derivatives of functions are assumed to not be
related to the function.  In other words, we have::

>>> from sympy import diff
>>> diff(f(x), x).diff(f(x))
0

The same is true for derivatives of different orders::

>>> diff(f(x), x, 2).diff(diff(f(x), x, 1))
0
>>> diff(f(x), x, 1).diff(diff(f(x), x, 2))
0

Note, any class can allow derivatives to be taken with respect to itself.
See the docstring of Expr._diff_wrt.

Examples
========

Some basic examples:

>>> from sympy import Derivative, Symbol, Function
>>> f = Function('f')
>>> g = Function('g')
>>> x = Symbol('x')
>>> y = Symbol('y')

>>> Derivative(x**2, x, evaluate=True)
2*x
>>> Derivative(Derivative(f(x,y), x), y)
Derivative(f(x, y), x, y)
>>> Derivative(f(x), x, 3)
Derivative(f(x), x, x, x)
>>> Derivative(f(x, y), y, x, evaluate=True)
Derivative(f(x, y), x, y)

Now some derivatives wrt functions:

>>> Derivative(f(x)**2, f(x), evaluate=True)
2*f(x)
>>> Derivative(f(g(x)), x, evaluate=True)
Derivative(g(x), x)*Subs(Derivative(f(_xi_1), _xi_1),
(_xi_1,), (g(x),))

"""

is_Derivative = True

@property
def _diff_wrt(self):
"""Allow derivatives wrt Derivatives if it contains a function.

Examples
========

>>> from sympy import Function, Symbol, Derivative
>>> f = Function('f')
>>> x = Symbol('x')
>>> Derivative(f(x),x)._diff_wrt
True
>>> Derivative(x**2,x)._diff_wrt
False
"""
if self.expr.is_Function:
return True
else:
return False

def __new__(cls, expr, *variables, **assumptions):

expr = sympify(expr)

# There are no variables, we differentiate wrt all of the free symbols
# in expr.
if not variables:
variables = expr.free_symbols
if len(variables) != 1:
if expr.is_number:
return S.Zero
from sympy.utilities.misc import filldedent
if len(variables) == 0:
raise ValueError(filldedent('''
Since there are no variables in the expression,
the variable(s) of differentiation must be supplied
to differentiate %s''' % expr))
else:
raise ValueError(filldedent('''
Since there is more than one variable in the
expression, the variable(s) of differentiation
must be supplied to differentiate %s''' % expr))

# Standardize the variables by sympifying them and making appending a
# count of 1 if there is only one variable: diff(e,x)->diff(e,x,1).
variables = list(sympify(variables))
if not variables[-1].is_Integer or len(variables) == 1:
variables.append(S.One)

# Split the list of variables into a list of the variables we are diff
# wrt, where each element of the list has the form (s, count) where
# s is the entity to diff wrt and count is the order of the
# derivative.
variable_count = []
all_zero = True
i = 0
while i < len(variables) - 1:  # process up to final Integer
v, count = variables[i: i + 2]
iwas = i
if v._diff_wrt:
# We need to test the more specific case of count being an
# Integer first.
if count.is_Integer:
count = int(count)
i += 2
elif count._diff_wrt:
count = 1
i += 1

if i == iwas:  # didn't get an update because of bad input
from sympy.utilities.misc import filldedent
last_digit = int(str(count)[-1])
ordinal = 'st' if last_digit == 1 else 'nd' if last_digit == 2 else 'rd' if last_digit == 3 else 'th'
raise ValueError(filldedent('''
Can\'t calculate %s%s derivative wrt %s.''' % (count, ordinal, v)))

if all_zero and not count == 0:
all_zero = False

if count:
variable_count.append((v, count))

# We make a special case for 0th derivative, because there is no
# good way to unambiguously print this.
if all_zero:
return expr

# Pop evaluate because it is not really an assumption and we will need
# to track it carefully below.
evaluate = assumptions.pop('evaluate', False)

# Look for a quick exit if there are symbols that don't appear in
# expression at all. Note, this cannnot check non-symbols like
# functions and Derivatives as those can be created by intermediate
# derivatives.
if evaluate:
symbol_set = set(sc[0] for sc in variable_count if sc[0].is_Symbol)
if symbol_set.difference(expr.free_symbols):
return S.Zero

# We make a generator so as to only generate a variable when necessary.
# If a high order of derivative is requested and the expr becomes 0
# after a few differentiations, then we won't need the other variables.
variablegen = (v for v, count in variable_count for i in range(count))

# If we can't compute the derivative of expr (but we wanted to) and
# expr is itself not a Derivative, finish building an unevaluated
# derivative class by calling Expr.__new__.
if (not (hasattr(expr, '_eval_derivative') and evaluate) and
(not isinstance(expr, Derivative))):
variables = list(variablegen)
# If we wanted to evaluate, we sort the variables into standard
# order for later comparisons. This is too aggressive if evaluate
# is False, so we don't do it in that case.
if evaluate:
#TODO: check if assumption of discontinuous derivatives exist
variables = cls._sort_variables(variables)
# Here we *don't* need to reinject evaluate into assumptions
# because we are done with it and it is not an assumption that
obj = Expr.__new__(cls, expr, *variables, **assumptions)
return obj

# Compute the derivative now by repeatedly calling the
# _eval_derivative method of expr for each variable. When this method
# returns None, the derivative couldn't be computed wrt that variable
# and we save the variable for later.
unhandled_variables = []

# Once we encouter a non_symbol that is unhandled, we stop taking
# derivatives entirely. This is because derivatives wrt functions
# don't commute with derivatives wrt symbols and we can't safely
# continue.
unhandled_non_symbol = False
nderivs = 0  # how many derivatives were performed
for v in variablegen:
is_symbol = v.is_Symbol

if unhandled_non_symbol:
obj = None
else:
if not is_symbol:
new_v = Dummy('xi_%i' % i)
new_v.dummy_index = hash(v)
expr = expr.xreplace({v: new_v})
old_v = v
v = new_v
obj = expr._eval_derivative(v)
nderivs += 1
if not is_symbol:
if obj is not None:
if not old_v.is_Symbol and obj.is_Derivative:
# Derivative evaluated at a point that is not a
# symbol
obj = Subs(obj, v, old_v)
else:
obj = obj.xreplace({v: old_v})
v = old_v

if obj is None:
unhandled_variables.append(v)
if not is_symbol:
unhandled_non_symbol = True
elif obj is S.Zero:
return S.Zero
else:
expr = obj

if unhandled_variables:
unhandled_variables = cls._sort_variables(unhandled_variables)
expr = Expr.__new__(cls, expr, *unhandled_variables, **assumptions)
else:
# We got a Derivative at the end of it all, and we rebuild it by
# sorting its variables.
if isinstance(expr, Derivative):
expr = cls(
expr.args[0], *cls._sort_variables(expr.args[1:])
)

if nderivs > 1 and assumptions.get('simplify', True):
from sympy.core.exprtools import factor_terms
from sympy.simplify.simplify import signsimp
expr = factor_terms(signsimp(expr))
return expr

@classmethod
def _sort_variables(cls, vars):
"""Sort variables, but disallow sorting of non-symbols.

When taking derivatives, the following rules usually hold:

* Derivative wrt different symbols commute.
* Derivative wrt different non-symbols commute.
* Derivatives wrt symbols and non-symbols don't commute.

Examples
========

>>> from sympy import Derivative, Function, symbols
>>> vsort = Derivative._sort_variables
>>> x, y, z = symbols('x y z')
>>> f, g, h = symbols('f g h', cls=Function)

>>> vsort((x,y,z))
[x, y, z]

>>> vsort((h(x),g(x),f(x)))
[f(x), g(x), h(x)]

>>> vsort((z,y,x,h(x),g(x),f(x)))
[x, y, z, f(x), g(x), h(x)]

>>> vsort((x,f(x),y,f(y)))
[x, f(x), y, f(y)]

>>> vsort((y,x,g(x),f(x),z,h(x),y,x))
[x, y, f(x), g(x), z, h(x), x, y]

>>> vsort((z,y,f(x),x,f(x),g(x)))
[y, z, f(x), x, f(x), g(x)]

>>> vsort((z,y,f(x),x,f(x),g(x),z,z,y,x))
[y, z, f(x), x, f(x), g(x), x, y, z, z]
"""

sorted_vars = []
symbol_part = []
non_symbol_part = []
for v in vars:
if not v.is_Symbol:
if len(symbol_part) > 0:
sorted_vars.extend(sorted(symbol_part,
key=default_sort_key))
symbol_part = []
non_symbol_part.append(v)
else:
if len(non_symbol_part) > 0:
sorted_vars.extend(sorted(non_symbol_part,
key=default_sort_key))
non_symbol_part = []
symbol_part.append(v)
if len(non_symbol_part) > 0:
sorted_vars.extend(sorted(non_symbol_part,
key=default_sort_key))
if len(symbol_part) > 0:
sorted_vars.extend(sorted(symbol_part,
key=default_sort_key))
return sorted_vars

def _eval_is_commutative(self):
return self.expr.is_commutative

def _eval_derivative(self, v):
# If the variable s we are diff wrt is not in self.variables, we
# assume that we might be able to take the derivative.
if v not in self.variables:
obj = self.expr.diff(v)
if obj is S.Zero:
return S.Zero
if isinstance(obj, Derivative):
return obj.func(obj.expr, *(self.variables + obj.variables))
# The derivative wrt s could have simplified things such that the
# derivative wrt things in self.variables can now be done. Thus,
# we set evaluate=True to see if there are any other derivatives
# that can be done. The most common case is when obj is a simple
# number so that the derivative wrt anything else will vanish.
return self.func(obj, *self.variables, evaluate=True)
# In this case s was in self.variables so the derivatve wrt s has
# already been attempted and was not computed, either because it
# couldn't be or evaluate=False originally.
return self.func(self.expr, *(self.variables + (v, )), evaluate=False)

def doit(self, **hints):
expr = self.expr
if hints.get('deep', True):
expr = expr.doit(**hints)
hints['evaluate'] = True
return self.func(expr, *self.variables, **hints)

@_sympifyit('z0', NotImplementedError)
[docs]    def doit_numerically(self, z0):
"""
Evaluate the derivative at z numerically.

When we can represent derivatives at a point, this should be folded
into the normal evalf. For now, we need a special method.
"""
import mpmath
from sympy.core.expr import Expr
if len(self.free_symbols) != 1 or len(self.variables) != 1:
raise NotImplementedError('partials and higher order derivatives')
z = list(self.free_symbols)[0]

def eval(x):
f0 = self.expr.subs(z, Expr._from_mpmath(x, prec=mpmath.mp.prec))
f0 = f0.evalf(mlib.libmpf.prec_to_dps(mpmath.mp.prec))
return f0._to_mpmath(mpmath.mp.prec)
return Expr._from_mpmath(mpmath.diff(eval,
z0._to_mpmath(mpmath.mp.prec)),
mpmath.mp.prec)

@property
def expr(self):
return self._args[0]

@property
def variables(self):
return self._args[1:]

@property
def free_symbols(self):
return self.expr.free_symbols

def _eval_subs(self, old, new):
if old in self.variables and not new._diff_wrt:
# issue 4719
return Subs(self, old, new)
# If both are Derivatives with the same expr, check if old is
# equivalent to self or if old is a subderivative of self.
if old.is_Derivative and old.expr == self.args[0]:
# Check if canonnical order of variables is equal.
old_vars = Derivative._sort_variables(old.variables)
self_vars = Derivative._sort_variables(self.args[1:])
if old_vars == self_vars:
return new

# Check if olf is a subderivative of self.
if len(old_vars) < len(self_vars):
self_vars_front = []
match = True
while old_vars and self_vars and match:
if old_vars[0] == self_vars[0]:
old_vars.pop(0)
self_vars.pop(0)
else:
# If self_v does not match old_v, we need to check if
# the types are the same (symbol vs non-symbol). If
# they are, we can continue checking self_vars for a
# match.
if old_vars[0].is_Symbol != self_vars[0].is_Symbol:
match = False
else:
self_vars_front.append(self_vars.pop(0))
if match:
variables = self_vars_front + self_vars
return Derivative(new, *variables)
return Derivative(*(x._subs(old, new) for x in self.args))

def _eval_lseries(self, x, logx):
dx = self.args[1:]
for term in self.args[0].lseries(x, logx=logx):
yield self.func(term, *dx)

def _eval_nseries(self, x, n, logx):
arg = self.args[0].nseries(x, n=n, logx=logx)
o = arg.getO()
dx = self.args[1:]
rv = [self.func(a, *dx) for a in Add.make_args(arg.removeO())]
if o:
rv.append(o/x)

def _sage_(self):
import sage.all as sage
args = [arg._sage_() for arg in self.args]
return sage.derivative(*args)

[docs]class Lambda(Expr):
"""
Lambda(x, expr) represents a lambda function similar to Python's
'lambda x: expr'. A function of several variables is written as
Lambda((x, y, ...), expr).

A simple example:

>>> from sympy import Lambda
>>> from sympy.abc import x
>>> f = Lambda(x, x**2)
>>> f(4)
16

For multivariate functions, use:

>>> from sympy.abc import y, z, t
>>> f2 = Lambda((x, y, z, t), x + y**z + t**z)
>>> f2(1, 2, 3, 4)
73

A handy shortcut for lots of arguments:

>>> p = x, y, z
>>> f = Lambda(p, x + y*z)
>>> f(*p)
x + y*z

"""
is_Function = True

def __new__(cls, variables, expr):
from sympy.sets.sets import FiniteSet
v = list(variables) if iterable(variables) else [variables]
for i in v:
if not getattr(i, 'is_Symbol', False):
raise TypeError('variable is not a symbol: %s' % i)
if len(v) == 1 and v[0] == expr:
return S.IdentityFunction

obj = Expr.__new__(cls, Tuple(*v), sympify(expr))
obj.nargs = FiniteSet(len(v))
return obj

@property
[docs]    def variables(self):
"""The variables used in the internal representation of the function"""
return self._args[0]

@property
[docs]    def expr(self):
"""The return value of the function"""
return self._args[1]

@property
def free_symbols(self):
return self.expr.free_symbols - set(self.variables)

def __call__(self, *args):
n = len(args)
if n not in self.nargs:  # Lambda only ever has 1 value in nargs
# XXX: exception message must be in exactly this format to
# make it work with NumPy's functions like vectorize(). See,
# for example, https://github.com/numpy/numpy/issues/1697.
# The ideal solution would be just to attach metadata to
# the exception and change NumPy to take advantage of this.
## XXX does this apply to Lambda? If not, remove this comment.
temp = ('%(name)s takes exactly %(args)s '
'argument%(plural)s (%(given)s given)')
raise TypeError(temp % {
'name': self,
'args': list(self.nargs)[0],
'plural': 's'*(list(self.nargs)[0] != 1),
'given': n})
return self.expr.xreplace(dict(list(zip(self.variables, args))))

def __eq__(self, other):
if not isinstance(other, Lambda):
return False
if self.nargs != other.nargs:
return False

selfexpr = self.args[1]
otherexpr = other.args[1]
otherexpr = otherexpr.xreplace(dict(list(zip(other.args[0], self.args[0]))))
return selfexpr == otherexpr

def __ne__(self, other):
return not(self == other)

def __hash__(self):
return super(Lambda, self).__hash__()

def _hashable_content(self):
return (self.expr.xreplace(self.canonical_variables),)

@property
[docs]    def is_identity(self):
"""Return True if this Lambda is an identity function. """
if len(self.args) == 2:
return self.args[0] == self.args[1]
else:
return None

[docs]class Subs(Expr):
"""
Represents unevaluated substitutions of an expression.

Subs(expr, x, x0) receives 3 arguments: an expression, a variable or
list of distinct variables and a point or list of evaluation points
corresponding to those variables.

Subs objects are generally useful to represent unevaluated derivatives
calculated at a point.

The variables may be expressions, but they are subjected to the limitations
of subs(), so it is usually a good practice to use only symbols for
variables, since in that case there can be no ambiguity.

There's no automatic expansion - use the method .doit() to effect all
possible substitutions of the object and also of objects inside the
expression.

When evaluating derivatives at a point that is not a symbol, a Subs object
is returned. One is also able to calculate derivatives of Subs objects - in
this case the expression is always expanded (for the unevaluated form, use
Derivative()).

A simple example:

>>> from sympy import Subs, Function, sin
>>> from sympy.abc import x, y, z
>>> f = Function('f')
>>> e = Subs(f(x).diff(x), x, y)
>>> e.subs(y, 0)
Subs(Derivative(f(x), x), (x,), (0,))
>>> e.subs(f, sin).doit()
cos(y)

An example with several variables:

>>> Subs(f(x)*sin(y) + z, (x, y), (0, 1))
Subs(z + f(x)*sin(y), (x, y), (0, 1))
>>> _.doit()
z + f(0)*sin(1)

"""
def __new__(cls, expr, variables, point, **assumptions):
from sympy import Symbol
if not is_sequence(variables, Tuple):
variables = [variables]
variables = list(sympify(variables))

if list(uniq(variables)) != variables:
repeated = [ v for v in set(variables) if variables.count(v) > 1 ]
raise ValueError('cannot substitute expressions %s more than '
'once.' % repeated)

point = Tuple(*(point if is_sequence(point, Tuple) else [point]))

if len(point) != len(variables):
raise ValueError('Number of point values must be the same as '
'the number of variables.')

expr = sympify(expr)

# use symbols with names equal to the point value (with preppended _)
# to give a variable-independent expression
pre = "_"
pts = sorted(set(point), key=default_sort_key)
from sympy.printing import StrPrinter
class CustomStrPrinter(StrPrinter):
def _print_Dummy(self, expr):
return str(expr) + str(expr.dummy_index)
def mystr(expr, **settings):
p = CustomStrPrinter(settings)
return p.doprint(expr)
while 1:
s_pts = dict([(p, Symbol(pre + mystr(p))) for p in pts])
reps = [(v, s_pts[p])
for v, p in zip(variables, point)]
# if any underscore-preppended symbol is already a free symbol
# and is a variable with a different point value, then there
# is a clash, e.g. _0 clashes in Subs(_0 + _1, (_0, _1), (1, 0))
# because the new symbol that would be created is _1 but _1
# is already mapped to 0 so __0 and __1 are used for the new
# symbols
if any(r in expr.free_symbols and
r in variables and
Symbol(pre + mystr(point[variables.index(r)])) != r
for _, r in reps):
pre += "_"
continue
break

obj = Expr.__new__(cls, expr, Tuple(*variables), point)
obj._expr = expr.subs(reps)
return obj

def _eval_is_commutative(self):
return self.expr.is_commutative

def doit(self):
return self.expr.doit().subs(list(zip(self.variables, self.point)))

def evalf(self, prec=None, **options):
return self.doit().evalf(prec, **options)

n = evalf

@property
[docs]    def variables(self):
"""The variables to be evaluated"""
return self._args[1]

@property
[docs]    def expr(self):
"""The expression on which the substitution operates"""
return self._args[0]

@property
[docs]    def point(self):
"""The values for which the variables are to be substituted"""
return self._args[2]

@property
def free_symbols(self):
return (self.expr.free_symbols - set(self.variables) |
set(self.point.free_symbols))

def __eq__(self, other):
if not isinstance(other, Subs):
return False
return self._expr == other._expr

def __ne__(self, other):
return not(self == other)

def __hash__(self):
return super(Subs, self).__hash__()

def _hashable_content(self):
return (self._expr.xreplace(self.canonical_variables),)

def _eval_subs(self, old, new):
if old in self.variables:
return self

def _eval_derivative(self, s):
if s not in self.free_symbols:
return S.Zero
return self.func(self.expr.diff(s), self.variables, self.point).doit() \
self.variables, self.point).doit() for arg,
point in zip(self.variables, self.point) ])

[docs]def diff(f, *symbols, **kwargs):
"""
Differentiate f with respect to symbols.

This is just a wrapper to unify .diff() and the Derivative class; its
interface is similar to that of integrate().  You can use the same
shortcuts for multiple variables as with Derivative.  For example,
diff(f(x), x, x, x) and diff(f(x), x, 3) both return the third derivative
of f(x).

You can pass evaluate=False to get an unevaluated Derivative class.  Note
that if there are 0 symbols (such as diff(f(x), x, 0), then the result will
be the function (the zeroth derivative), even if evaluate=False.

Examples
========

>>> from sympy import sin, cos, Function, diff
>>> from sympy.abc import x, y
>>> f = Function('f')

>>> diff(sin(x), x)
cos(x)
>>> diff(f(x), x, x, x)
Derivative(f(x), x, x, x)
>>> diff(f(x), x, 3)
Derivative(f(x), x, x, x)
>>> diff(sin(x)*cos(y), x, 2, y, 2)
sin(x)*cos(y)

>>> type(diff(sin(x), x))
cos
>>> type(diff(sin(x), x, evaluate=False))
<class 'sympy.core.function.Derivative'>
>>> type(diff(sin(x), x, 0))
sin
>>> type(diff(sin(x), x, 0, evaluate=False))
sin

>>> diff(sin(x))
cos(x)
>>> diff(sin(x*y))
Traceback (most recent call last):
...
ValueError: specify differentiation variables to differentiate sin(x*y)

Note that diff(sin(x)) syntax is meant only for convenience
in interactive sessions and should be avoided in library code.

References
==========

http://reference.wolfram.com/legacy/v5_2/Built-inFunctions/AlgebraicComputation/Calculus/D.html

========

Derivative
sympy.geometry.util.idiff: computes the derivative implicitly

"""
kwargs.setdefault('evaluate', True)
try:
return f._eval_diff(*symbols, **kwargs)
except AttributeError:
pass
return Derivative(f, *symbols, **kwargs)

[docs]def expand(e, deep=True, modulus=None, power_base=True, power_exp=True,
mul=True, log=True, multinomial=True, basic=True, **hints):
"""
Expand an expression using methods given as hints.

Hints evaluated unless explicitly set to False are:  basic, log,
multinomial, mul, power_base, and power_exp The following
hints are supported but not applied unless set to True:  complex,
func, and trig.  In addition, the following meta-hints are
supported by some or all of the other hints:  frac, numer,
denom, modulus, and force.  deep is supported by all
hints.  Additionally, subclasses of Expr may define their own hints or
meta-hints.

The basic hint is used for any special rewriting of an object that
should be done automatically (along with the other hints like mul)
when expand is called. This is a catch-all hint to handle any sort of
expansion that may not be described by the existing hint names. To use
this hint an object should override the _eval_expand_basic method.
Objects may also define their own expand methods, which are not run by
default.  See the API section below.

If deep is set to True (the default), things like arguments of
functions are recursively expanded.  Use deep=False to only expand on
the top level.

If the force hint is used, assumptions about variables will be ignored
in making the expansion.

Hints
=====

These hints are run by default

mul
---

>>> from sympy import cos, exp, sin
>>> from sympy.abc import x, y, z
>>> (y*(x + z)).expand(mul=True)
x*y + y*z

multinomial
-----------

Expand (x + y + ...)**n where n is a positive integer.

>>> ((x + y + z)**2).expand(multinomial=True)
x**2 + 2*x*y + 2*x*z + y**2 + 2*y*z + z**2

power_exp
---------

Expand addition in exponents into multiplied bases.

>>> exp(x + y).expand(power_exp=True)
exp(x)*exp(y)
>>> (2**(x + y)).expand(power_exp=True)
2**x*2**y

power_base
----------

Split powers of multiplied bases.

This only happens by default if assumptions allow, or if the
force meta-hint is used:

>>> ((x*y)**z).expand(power_base=True)
(x*y)**z
>>> ((x*y)**z).expand(power_base=True, force=True)
x**z*y**z
>>> ((2*y)**z).expand(power_base=True)
2**z*y**z

Note that in some cases where this expansion always holds, SymPy performs
it automatically:

>>> (x*y)**2
x**2*y**2

log
---

Pull out power of an argument as a coefficient and split logs products
into sums of logs.

Note that these only work if the arguments of the log function have the
proper assumptions--the arguments must be positive and the exponents must
be real--or else the force hint must be True:

>>> from sympy import log, symbols
>>> log(x**2*y).expand(log=True)
log(x**2*y)
>>> log(x**2*y).expand(log=True, force=True)
2*log(x) + log(y)
>>> x, y = symbols('x,y', positive=True)
>>> log(x**2*y).expand(log=True)
2*log(x) + log(y)

basic
-----

This hint is intended primarily as a way for custom subclasses to enable
expansion by default.

These hints are not run by default:

complex
-------

Split an expression into real and imaginary parts.

>>> x, y = symbols('x,y')
>>> (x + y).expand(complex=True)
re(x) + re(y) + I*im(x) + I*im(y)
>>> cos(x).expand(complex=True)
-I*sin(re(x))*sinh(im(x)) + cos(re(x))*cosh(im(x))

Note that this is just a wrapper around as_real_imag().  Most objects
that wish to redefine _eval_expand_complex() should consider
redefining as_real_imag() instead.

func
----

Expand other functions.

>>> from sympy import gamma
>>> gamma(x + 1).expand(func=True)
x*gamma(x)

trig
----

Do trigonometric expansions.

>>> cos(x + y).expand(trig=True)
-sin(x)*sin(y) + cos(x)*cos(y)
>>> sin(2*x).expand(trig=True)
2*sin(x)*cos(x)

Note that the forms of sin(n*x) and cos(n*x) in terms of sin(x)
and cos(x) are not unique, due to the identity \sin^2(x) + \cos^2(x)
= 1.  The current implementation uses the form obtained from Chebyshev
polynomials, but this may change.  See this MathWorld article
<http://mathworld.wolfram.com/Multiple-AngleFormulas.html>_ for more
information.

Notes
=====

- You can shut off unwanted methods::

>>> (exp(x + y)*(x + y)).expand()
x*exp(x)*exp(y) + y*exp(x)*exp(y)
>>> (exp(x + y)*(x + y)).expand(power_exp=False)
x*exp(x + y) + y*exp(x + y)
>>> (exp(x + y)*(x + y)).expand(mul=False)
(x + y)*exp(x)*exp(y)

- Use deep=False to only expand on the top level::

>>> exp(x + exp(x + y)).expand()
exp(x)*exp(exp(x)*exp(y))
>>> exp(x + exp(x + y)).expand(deep=False)
exp(x)*exp(exp(x + y))

- Hints are applied in an arbitrary, but consistent order (in the current
implementation, they are applied in alphabetical order, except
multinomial comes before mul, but this may change).  Because of this,
some hints may prevent expansion by other hints if they are applied
first. For example, mul may distribute multiplications and prevent
log and power_base from expanding them. Also, if mul is
applied before multinomial, the expression might not be fully
distributed. The solution is to use the various expand_hint helper
functions or to use hint=False to this function to finely control
which hints are applied. Here are some examples::

>>> from sympy import expand, expand_mul, expand_power_base
>>> x, y, z = symbols('x,y,z', positive=True)

>>> expand(log(x*(y + z)))
log(x) + log(y + z)

Here, we see that log was applied before mul.  To get the mul
expanded form, either of the following will work::

>>> expand_mul(log(x*(y + z)))
log(x*y + x*z)
>>> expand(log(x*(y + z)), log=False)
log(x*y + x*z)

A similar thing can happen with the power_base hint::

>>> expand((x*(y + z))**x)
(x*y + x*z)**x

To get the power_base expanded form, either of the following will
work::

>>> expand((x*(y + z))**x, mul=False)
x**x*(y + z)**x
>>> expand_power_base((x*(y + z))**x)
x**x*(y + z)**x

>>> expand((x + y)*y/x)
y + y**2/x

The parts of a rational expression can be targeted::

>>> expand((x + y)*y/x/(x + 1), frac=True)
(x*y + y**2)/(x**2 + x)
>>> expand((x + y)*y/x/(x + 1), numer=True)
(x*y + y**2)/(x*(x + 1))
>>> expand((x + y)*y/x/(x + 1), denom=True)
y*(x + y)/(x**2 + x)

- The modulus meta-hint can be used to reduce the coefficients of an
expression post-expansion::

>>> expand((3*x + 1)**2)
9*x**2 + 6*x + 1
>>> expand((3*x + 1)**2, modulus=5)
4*x**2 + x + 1

- Either expand() the function or .expand() the method can be
used.  Both are equivalent::

>>> expand((x + 1)**2)
x**2 + 2*x + 1
>>> ((x + 1)**2).expand()
x**2 + 2*x + 1

API
===

Objects can define their own expand hints by defining
_eval_expand_hint().  The function should take the form::

def _eval_expand_hint(self, **hints):
# Only apply the method to the top-level expression
...

See also the example below.  Objects should define _eval_expand_hint()
methods only if hint applies to that specific object.  The generic
_eval_expand_hint() method defined in Expr will handle the no-op case.

Each hint should be responsible for expanding that hint only.
Furthermore, the expansion should be applied to the top-level expression
only.  expand() takes care of the recursion that happens when
deep=True.

You should only call _eval_expand_hint() methods directly if you are
100% sure that the object has the method, as otherwise you are liable to
get unexpected AttributeErrors.  Note, again, that you do not need to
recursively apply the hint to args of your object: this is handled
automatically by expand().  _eval_expand_hint() should
generally not be used at all outside of an _eval_expand_hint() method.
If you want to apply a specific expansion from within another method, use
the public expand() function, method, or expand_hint() functions.

In order for expand to work, objects must be rebuildable by their args,
i.e., obj.func(*obj.args) == obj must hold.

Expand methods are passed **hints so that expand hints may use
'metahints'--hints that control how different expand methods are applied.
For example, the force=True hint described above that causes
expand(log=True) to ignore assumptions is such a metahint.  The
deep meta-hint is handled exclusively by expand() and is not
passed to _eval_expand_hint() methods.

Note that expansion hints should generally be methods that perform some
kind of 'expansion'.  For hints that simply rewrite an expression, use the
.rewrite() API.

Examples
========

>>> from sympy import Expr, sympify
>>> class MyClass(Expr):
...     def __new__(cls, *args):
...         args = sympify(args)
...         return Expr.__new__(cls, *args)
...
...     def _eval_expand_double(self, **hints):
...         '''
...         Doubles the args of MyClass.
...
...         If there more than four args, doubling is not performed,
...         unless force=True is also used (False by default).
...         '''
...         force = hints.pop('force', False)
...         if not force and len(self.args) > 4:
...             return self
...         return self.func(*(self.args + self.args))
...
>>> a = MyClass(1, 2, MyClass(3, 4))
>>> a
MyClass(1, 2, MyClass(3, 4))
>>> a.expand(double=True)
MyClass(1, 2, MyClass(3, 4, 3, 4), 1, 2, MyClass(3, 4, 3, 4))
>>> a.expand(double=True, deep=False)
MyClass(1, 2, MyClass(3, 4), 1, 2, MyClass(3, 4))

>>> b = MyClass(1, 2, 3, 4, 5)
>>> b.expand(double=True)
MyClass(1, 2, 3, 4, 5)
>>> b.expand(double=True, force=True)
MyClass(1, 2, 3, 4, 5, 1, 2, 3, 4, 5)

========

expand_log, expand_mul, expand_multinomial, expand_complex, expand_trig,
expand_power_base, expand_power_exp, expand_func, hyperexpand

"""
# don't modify this; modify the Expr.expand method
hints['power_base'] = power_base
hints['power_exp'] = power_exp
hints['mul'] = mul
hints['log'] = log
hints['multinomial'] = multinomial
hints['basic'] = basic
return sympify(e).expand(deep=deep, modulus=modulus, **hints)

# This is a special application of two hints

def _mexpand(expr, recursive=False):
# expand multinomials and then expand products; this may not always
# be sufficient to give a fully expanded expression (see
# test_issue_8247_8354 in test_arit)
if expr is None:
return
was = None
while was != expr:
was, expr = expr, expand_mul(expand_multinomial(expr))
if not recursive:
break
return expr

# These are simple wrappers around single hints.

[docs]def expand_mul(expr, deep=True):
"""
Wrapper around expand that only uses the mul hint.  See the expand

Examples
========

>>> from sympy import symbols, expand_mul, exp, log
>>> x, y = symbols('x,y', positive=True)
>>> expand_mul(exp(x+y)*(x+y)*log(x*y**2))
x*exp(x + y)*log(x*y**2) + y*exp(x + y)*log(x*y**2)

"""
return sympify(expr).expand(deep=deep, mul=True, power_exp=False,
power_base=False, basic=False, multinomial=False, log=False)

[docs]def expand_multinomial(expr, deep=True):
"""
Wrapper around expand that only uses the multinomial hint.  See the expand

Examples
========

>>> from sympy import symbols, expand_multinomial, exp
>>> x, y = symbols('x y', positive=True)
>>> expand_multinomial((x + exp(x + 1))**2)
x**2 + 2*x*exp(x + 1) + exp(2*x + 2)

"""
return sympify(expr).expand(deep=deep, mul=False, power_exp=False,
power_base=False, basic=False, multinomial=True, log=False)

[docs]def expand_log(expr, deep=True, force=False):
"""
Wrapper around expand that only uses the log hint.  See the expand

Examples
========

>>> from sympy import symbols, expand_log, exp, log
>>> x, y = symbols('x,y', positive=True)
>>> expand_log(exp(x+y)*(x+y)*log(x*y**2))
(x + y)*(log(x) + 2*log(y))*exp(x + y)

"""
return sympify(expr).expand(deep=deep, log=True, mul=False,
power_exp=False, power_base=False, multinomial=False,
basic=False, force=force)

[docs]def expand_func(expr, deep=True):
"""
Wrapper around expand that only uses the func hint.  See the expand

Examples
========

>>> from sympy import expand_func, gamma
>>> from sympy.abc import x
>>> expand_func(gamma(x + 2))
x*(x + 1)*gamma(x)

"""
return sympify(expr).expand(deep=deep, func=True, basic=False,
log=False, mul=False, power_exp=False, power_base=False, multinomial=False)

[docs]def expand_trig(expr, deep=True):
"""
Wrapper around expand that only uses the trig hint.  See the expand

Examples
========

>>> from sympy import expand_trig, sin
>>> from sympy.abc import x, y
>>> expand_trig(sin(x+y)*(x+y))
(x + y)*(sin(x)*cos(y) + sin(y)*cos(x))

"""
return sympify(expr).expand(deep=deep, trig=True, basic=False,
log=False, mul=False, power_exp=False, power_base=False, multinomial=False)

[docs]def expand_complex(expr, deep=True):
"""
Wrapper around expand that only uses the complex hint.  See the expand

Examples
========

>>> from sympy import expand_complex, exp, sqrt, I
>>> from sympy.abc import z
>>> expand_complex(exp(z))
I*exp(re(z))*sin(im(z)) + exp(re(z))*cos(im(z))
>>> expand_complex(sqrt(I))
sqrt(2)/2 + sqrt(2)*I/2

========
Expr.as_real_imag
"""
return sympify(expr).expand(deep=deep, complex=True, basic=False,
log=False, mul=False, power_exp=False, power_base=False, multinomial=False)

[docs]def expand_power_base(expr, deep=True, force=False):
"""
Wrapper around expand that only uses the power_base hint.

A wrapper to expand(power_base=True) which separates a power with a base
that is a Mul into a product of powers, without performing any other
expansions, provided that assumptions about the power's base and exponent
allow.

deep=False (default is True) will only apply to the top-level expression.

force=True (default is False) will cause the expansion to ignore
assumptions about the base and exponent. When False, the expansion will
only happen if the base is non-negative or the exponent is an integer.

>>> from sympy.abc import x, y, z
>>> from sympy import expand_power_base, sin, cos, exp

>>> (x*y)**2
x**2*y**2

>>> (2*x)**y
(2*x)**y
>>> expand_power_base(_)
2**y*x**y

>>> expand_power_base((x*y)**z)
(x*y)**z
>>> expand_power_base((x*y)**z, force=True)
x**z*y**z
>>> expand_power_base(sin((x*y)**z), deep=False)
sin((x*y)**z)
>>> expand_power_base(sin((x*y)**z), force=True)
sin(x**z*y**z)

>>> expand_power_base((2*sin(x))**y + (2*cos(x))**y)
2**y*sin(x)**y + 2**y*cos(x)**y

>>> expand_power_base((2*exp(y))**x)
2**x*exp(y)**x

>>> expand_power_base((2*cos(x))**y)
2**y*cos(x)**y

Notice that sums are left untouched. If this is not the desired behavior,
apply full expand() to the expression:

>>> expand_power_base(((x+y)*z)**2)
z**2*(x + y)**2
>>> (((x+y)*z)**2).expand()
x**2*z**2 + 2*x*y*z**2 + y**2*z**2

>>> expand_power_base((2*y)**(1+z))
2**(z + 1)*y**(z + 1)
>>> ((2*y)**(1+z)).expand()
2*2**z*y*y**z

"""
return sympify(expr).expand(deep=deep, log=False, mul=False,
power_exp=False, power_base=True, multinomial=False,
basic=False, force=force)

[docs]def expand_power_exp(expr, deep=True):
"""
Wrapper around expand that only uses the power_exp hint.

Examples
========

>>> from sympy import expand_power_exp
>>> from sympy.abc import x, y
>>> expand_power_exp(x**(y + 2))
x**2*x**y
"""
return sympify(expr).expand(deep=deep, complex=False, basic=False,
log=False, mul=False, power_exp=True, power_base=False, multinomial=False)

[docs]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)

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)
>>> count_ops({x: sin(x), x + 2: y + 1}, visual=True)

"""
from sympy import Integral, Symbol
from sympy.logic.boolalg import BooleanFunction

expr = sympify(expr)
if isinstance(expr, Expr):

ops = []
args = [expr]
NEG = Symbol('NEG')
DIV = Symbol('DIV')
SUB = Symbol('SUB')
while args:
a = args.pop()

# XXX: This is a hack to support non-Basic args
if isinstance(a, string_types):
continue

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 _coeff_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
aargs = list(a.args)
negs = 0
for i, ai in enumerate(aargs):
if _coeff_isneg(ai):
negs += 1
args.append(-ai)
if i > 0:
ops.append(SUB)
else:
args.append(ai)
if i > 0:
if negs == len(aargs):  # -x - y = NEG + SUB
ops.append(NEG)
elif _coeff_isneg(aargs[0]):  # -x + y = SUB, but already recorded 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, Integral)):

o = Symbol(a.func.__name__.upper())
# count the args
if (a.is_Mul or isinstance(a, LatticeOp)):
ops.append(o*(len(a.args) - 1))
else:
ops.append(o)
if not a.is_Symbol:
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.items()]
elif iterable(expr):
ops = [count_ops(i, visual=visual) for i in expr]
elif isinstance(expr, BooleanFunction):
ops = []
for arg in expr.args:
ops.append(count_ops(arg, visual=True))
o = Symbol(expr.func.__name__.upper())
ops.append(o)
elif not isinstance(expr, Basic):
ops = []
else:  # it's Basic not isinstance(expr, Expr):
if not isinstance(expr, Basic):
raise TypeError("Invalid type of expr")
else:
ops = []
args = [expr]
while args:
a = args.pop()

# XXX: This is a hack to support non-Basic args
if isinstance(a, string_types):
continue

if a.args:
o = Symbol(a.func.__name__.upper())
if a.is_Boolean:
ops.append(o*(len(a.args)-1))
else:
ops.append(o)
args.extend(a.args)

if not ops:
if visual:
return S.Zero
return 0

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))

[docs]def nfloat(expr, n=15, exponent=False):
"""Make all Rationals in expr Floats except those in exponents
(unless the exponents flag is set to True).

Examples
========

>>> from sympy.core.function import nfloat
>>> from sympy.abc import x, y
>>> from sympy import cos, pi, sqrt
>>> nfloat(x**4 + x/2 + cos(pi/3) + 1 + sqrt(y))
x**4 + 0.5*x + sqrt(y) + 1.5
>>> nfloat(x**4 + sqrt(y), exponent=True)
x**4.0 + y**0.5

"""
from sympy.core.power import Pow
from sympy.polys.rootoftools import RootOf

if iterable(expr, exclude=string_types):
if isinstance(expr, (dict, Dict)):
return type(expr)([(k, nfloat(v, n, exponent)) for k, v in
list(expr.items())])
return type(expr)([nfloat(a, n, exponent) for a in expr])
rv = sympify(expr)

if rv.is_Number:
return Float(rv, n)
elif rv.is_number:
# evalf doesn't always set the precision
rv = rv.n(n)
if rv.is_Number:
rv = Float(rv.n(n), n)
else:
pass  # pure_complex(rv) is likely True
return rv

# watch out for RootOf instances that don't like to have
# their exponents replaced with Dummies and also sometimes have
# problems with evaluating at low precision (issue 6393)
rv = rv.xreplace(dict([(ro, ro.n(n)) for ro in rv.atoms(RootOf)]))

if not exponent:
reps = [(p, Pow(p.base, Dummy())) for p in rv.atoms(Pow)]
rv = rv.xreplace(dict(reps))
rv = rv.n(n)
if not exponent:
rv = rv.xreplace(dict([(d.exp, p.exp) for p, d in reps]))
else:
# Pow._eval_evalf special cases Integer exponents so if
# exponent is suppose to be handled we have to do so here
rv = rv.xreplace(Transform(
lambda x: Pow(x.base, Float(x.exp, n)),
lambda x: x.is_Pow and x.exp.is_Integer))

return rv.xreplace(Transform(
lambda x: x.func(*nfloat(x.args, n, exponent)),
lambda x: isinstance(x, Function)))

from sympy.core.symbol import Dummy
`