def _eval_subs(self, old, new):
if self == old: return new
arg = self.args[0]
o = old
if old.is_Pow: # handle (exp(3*log(x))).subs(x**2, z) -> z**(3/2)
old = exp(old.exp * log(old.base))
if old.func is exp:
# exp(a*expr) .subs( exp(b*expr), y ) -> y ** (a/b)
a, expr_terms = self.args[0].as_coeff_terms()
b, expr_terms_ = old.args[0].as_coeff_terms()
if expr_terms == expr_terms_:
return new**(a / b)
if arg.is_Add: # exp(2*x+a).subs(exp(3*x),y) -> y**(2/3) * exp(a)
# exp(exp(x) + exp(x**2)).subs(exp(exp(x)), w) -> w * exp(exp(x**2))
oarg = old.args[0]
new_l = []
old_al = []
coeff2, terms2 = oarg.as_coeff_terms()
for a in arg.args:
a = a._eval_subs(old, new)
coeff1, terms1 = a.as_coeff_terms()
if terms1 == terms2:
new_l.append(new**(coeff1 / coeff2))
else:
old_al.append(a._eval_subs(old, new))
if new_l:
new_l.append(self.func(C.Add(*old_al)))
r = C.Mul(*new_l)
return r
old = o
return Function._eval_subs(self, old, new)
def _eval_as_leading_term(self, x):
arg = self.args[0]
if arg.is_Add:
return C.Mul(*[exp(f).as_leading_term(x) for f in arg.args])
arg = self.args[0].as_leading_term(x)
if C.Order(1, x).contains(arg):
return S.One
return exp(arg)
def _print_Mul(self, expr):
coeff, terms = expr.as_coeff_terms()
if not coeff.is_negative:
tex = ""
else:
coeff = -coeff
tex = "- "
numer, denom = fraction(C.Mul(*terms))
def convert(terms):
product = []
if not terms.is_Mul:
return str(self._print(terms))
else:
for term in terms.args:
pretty = self._print(term)
if term.is_Add:
product.append(r"\left(%s\right)" % pretty)
else:
product.append(str(pretty))
return r" ".join(product)
if denom is S.One:
if coeff is not S.One:
tex += str(self._print(coeff)) + " "
if numer.is_Add:
tex += r"\left(%s\right)" % convert(numer)
else:
tex += r"%s" % convert(numer)
else:
if numer is S.One:
if coeff.is_Integer:
numer *= coeff.p
elif coeff.is_Rational:
if coeff.p != 1:
numer *= coeff.p
denom *= coeff.q
elif coeff is not S.One:
tex += str(self._print(coeff)) + " "
else:
if coeff.is_Rational and coeff.p == 1:
denom *= coeff.q
elif coeff is not S.One:
tex += str(self._print(coeff)) + " "
tex += r"\frac{%s}{%s}" % \
(convert(numer), convert(denom))
return tex
def parse_term(expr):
"""Parses expression expr and outputs tuple (sexpr, rat_expo, sym_expo, deriv)
where:
- sexpr is the base expression
- rat_expo is the rational exponent that sexpr is raised to
- sym_expo is the symbolic exponent that sexpr is raised to
- deriv contains the derivatives the the expression
for example, the output of x would be (x, 1, None, None)
the output of 2**x would be (2, 1, x, None)
"""
rat_expo, sym_expo = S.One, None
sexpr, deriv = expr, None
if expr.is_Pow:
if isinstance(expr.base, Derivative):
sexpr, deriv = parse_derivative(expr.base)
else:
sexpr = expr.base
if expr.exp.is_Rational:
rat_expo = expr.exp
elif expr.exp.is_Mul:
coeff, tail = expr.exp.as_coeff_terms()
if coeff.is_Rational:
rat_expo, sym_expo = coeff, C.Mul(*tail)
else:
sym_expo = expr.exp
else:
sym_expo = expr.exp
elif expr.func is C.exp:
if expr.args[0].is_Rational:
sexpr, rat_expo = S.Exp1, expr.args[0]
elif expr.args[0].is_Mul:
coeff, tail = expr.args[0].as_coeff_terms()
if coeff.is_Rational:
sexpr, rat_expo = C.exp(C.Mul(*tail)), coeff
elif isinstance(expr, Derivative):
sexpr, deriv = parse_derivative(expr)
return sexpr, rat_expo, sym_expo, deriv
def separate(expr, deep=False):
"""Rewrite or separate a power of product to a product of powers
but without any expanding, ie. rewriting products to summations.
>>> from sympy import *
>>> x, y, z = symbols('x', 'y', 'z')
>>> separate((x*y)**2)
x**2*y**2
>>> separate((x*(y*z)**3)**2)
x**2*y**6*z**6
>>> separate((x*sin(x))**y + (x*cos(x))**y)
x**y*cos(x)**y + x**y*sin(x)**y
>>> separate((exp(x)*exp(y))**x)
exp(x**2)*exp(x*y)
>>> separate((sin(x)*cos(x))**y)
cos(x)**y*sin(x)**y
Notice that summations are left un touched. If this is not the
requested behaviour, apply 'expand' to input expression before:
>>> separate(((x+y)*z)**2)
z**2*(x + y)**2
>>> separate((x*y)**(1+z))
x**(1 + z)*y**(1 + z)
"""
expr = sympify(expr)
if expr.is_Pow:
terms, expo = [], separate(expr.exp, deep)
if expr.base.is_Mul:
t = [separate(C.Pow(t, expo), deep) for t in expr.base.args]
return C.Mul(*t)
elif expr.base.func is C.exp:
if deep == True:
return C.exp(separate(expr.base[0], deep) * expo)
else:
return C.exp(expr.base[0] * expo)
else:
return C.Pow(separate(expr.base, deep), expo)
elif expr.is_Add or expr.is_Mul:
return type(expr)(*[separate(t, deep) for t in expr.args])
elif expr.is_Function and deep:
return expr.func(*[separate(t) for t in expr.args])
else:
return expr
def eval(cls, arg):
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Zero:
return S.One
elif arg is S.One:
return S.Exp1
elif arg is S.Infinity:
return S.Infinity
elif arg is S.NegativeInfinity:
return S.Zero
elif arg.func is log:
return arg.args[0]
elif arg.is_Mul:
coeff = arg.as_coefficient(S.Pi * S.ImaginaryUnit)
if coeff is not None:
if (2 * coeff).is_integer:
if coeff.is_even:
return S.One
elif coeff.is_odd:
return S.NegativeOne
elif (coeff + S.Half).is_even:
return -S.ImaginaryUnit
elif (coeff + S.Half).is_odd:
return S.ImaginaryUnit
I = S.ImaginaryUnit
oo = S.Infinity
a = Wild("a", exclude=[I, oo])
r = arg.match(I * a * oo)
if r and r[a] != 0:
return S.NaN
if arg.is_Add:
args = arg.args
else:
args = [arg]
included, excluded = [], []
for arg in args:
coeff, terms = arg.as_coeff_terms()
if coeff is S.Infinity:
excluded.append(coeff**C.Mul(*terms))
else:
coeffs, log_term = [coeff], None
for term in terms:
if term.func is log:
if log_term is None:
log_term = term.args[0]
else:
log_term = None
break
elif term.is_comparable:
coeffs.append(term)
else:
log_term = None
break
if log_term is not None:
excluded.append(log_term**C.Mul(*coeffs))
else:
included.append(arg)
if excluded:
return C.Mul(*(excluded + [cls(C.Add(*included))]))
def as_base_exp(self):
return S.Exp1, C.Mul(*self.args)
def powsimp(expr, deep=False, combine='all'):
"""
== Usage ==
Reduces expression by combining powers with similar bases and exponents.
== Notes ==
If deep is True then powsimp() will also simplify arguments of
functions. By default deep is set to False.
You can make powsimp() only combine bases or only combine exponents by
changing combine='base' or combine='exp'. By default, combine='all',
which does both. combine='base' will only combine::
a a a 2x x
x * y => (x*y) as well as things like 2 => 4
and combine='exp' will only combine
::
a b (a + b)
x * x => x
combine='exp' will strictly only combine exponents in the way that used
to be automatic. Also use deep=True if you need the old behavior.
== Examples ==
>>> from sympy import *
>>> x,n = map(Symbol, 'xn')
>>> e = x**n * (x*n)**(-n) * n
>>> powsimp(e)
n**(1 - n)
>>> powsimp(log(e))
log(n*x**n*(n*x)**(-n))
>>> powsimp(log(e), deep=True)
log(n**(1 - n))
>>> powsimp(e, combine='exp')
n*x**n*(n*x)**(-n)
>>> powsimp(e, combine='base')
n*(1/n)**n
>>> y, z = symbols('yz')
>>> a = x**y*x**z*n**z*n**y
>>> powsimp(a, combine='exp')
n**(y + z)*x**(y + z)
>>> powsimp(a, combine='base')
(n*x)**y*(n*x)**z
>>> powsimp(a, combine='all')
(n*x)**(y + z)
"""
if combine not in ['all', 'exp', 'base']:
raise ValueError, "combine must be one of ('all', 'exp', 'base')."
if combine in ('all', 'base'):
expr = separate(expr, deep)
y = Symbol('y', dummy=True)
if expr.is_Pow:
if deep:
return powsimp(y*powsimp(expr.base, deep, combine)**powsimp(\
expr.exp, deep, combine), deep, combine)/y
else:
return powsimp(y * expr, deep,
combine) / y # Trick it into being a Mul
elif expr.is_Function:
if expr.func == exp and deep:
# Exp should really be like Pow
return powsimp(y * exp(powsimp(expr.args[0], deep, combine)), deep,
combine) / y
elif expr.func == exp and not deep:
return powsimp(y * expr, deep, combine) / y
elif deep:
return expr.func(*[powsimp(t, deep, combine) for t in expr.args])
else:
return expr
elif expr.is_Add:
return C.Add(*[powsimp(t, deep, combine) for t in expr.args])
elif expr.is_Mul:
if combine in ('exp', 'all'):
# Collect base/exp data, while maintaining order in the
# non-commutative parts of the product
if combine is 'all' and deep and any(
(t.is_Add for t in expr.args)):
# Once we get to 'base', there is no more 'exp', so we need to
# distribute here.
return powsimp(expand_mul(expr, deep=False), deep, combine)
c_powers = {}
nc_part = []
newexpr = sympify(1)
for term in expr.args:
if term.is_Add and deep:
newexpr *= powsimp(term, deep, combine)
else:
if term.is_commutative:
b, e = term.as_base_exp()
c_powers[b] = c_powers.get(b, 0) + e
else:
nc_part.append(term)
newexpr = Mul(newexpr,
Mul(*(Pow(b, e) for b, e in c_powers.items())))
if combine is 'exp':
return Mul(newexpr, Mul(*nc_part))
else:
# combine is 'all', get stuff ready for 'base'
if deep:
newexpr = expand_mul(newexpr, deep=False)
if newexpr.is_Add:
return powsimp(Mul(*nc_part), deep, combine='base') * Add(
*(powsimp(i, deep, combine='base')
for i in newexpr.args))
else:
return powsimp(Mul(*nc_part), deep, combine='base')*\
powsimp(newexpr, deep, combine='base')
else:
# combine is 'base'
if deep:
expr = expand_mul(expr, deep=False)
if expr.is_Add:
return Add(*(powsimp(i, deep, combine) for i in expr.args))
else:
# Build c_powers and nc_part. These must both be lists not
# dicts because exp's are not combined.
c_powers = []
nc_part = []
for term in expr.args:
if term.is_commutative:
c_powers.append(list(term.as_base_exp()))
else:
nc_part.append(term)
# Pull out numerical coefficients from exponent
# e.g., 2**(2*x) => 4**x
for i in xrange(len(c_powers)):
b, e = c_powers[i]
exp_c, exp_t = e.as_coeff_terms()
if not (exp_c is S.One) and exp_t:
c_powers[i] = [C.Pow(b, exp_c), C.Mul(*exp_t)]
# Combine bases whenever they have the same exponent which is
# not numeric
c_exp = {}
for b, e in c_powers:
if e in c_exp:
c_exp[e].append(b)
else:
c_exp[e] = [b]
# Merge back in the results of the above to form a new product
for e in c_exp:
bases = c_exp[e]
if deep:
simpe = powsimp(e, deep, combine)
c_exp = {}
for b, ex in c_powers:
if ex in c_exp:
c_exp[ex].append(b)
else:
c_exp[ex] = [b]
del c_exp[e]
c_exp[simpe] = bases
else:
simpe = e
if len(bases) > 1:
for b in bases:
c_powers.remove([b, e])
new_base = Mul(*bases)
in_c_powers = False
for i in xrange(len(c_powers)):
if c_powers[i][0] == new_base:
if combine == 'all':
c_powers[i][1] += simpe
else:
c_powers.append([new_base, simpe])
in_c_powers = True
if not in_c_powers:
c_powers.append([new_base, simpe])
c_part = [C.Pow(b, e) for b, e in c_powers]
return C.Mul(*(c_part + nc_part))
else:
return expr
def _together(expr):
from sympy.core.function import Function
if expr.is_Add:
items, coeffs, basis = [], [], {}
for elem in expr.args:
numer, q = fraction(_together(elem))
denom = {}
for term in make_list(q.expand(), Mul):
expo = S.One
coeff = S.One
if term.is_Pow:
if term.exp.is_Rational:
term, expo = term.base, term.exp
elif term.exp.is_Mul:
coeff, tail = term.exp.as_coeff_terms()
if coeff.is_Rational:
tail = C.Mul(*tail)
term, expo = Pow(term.base, tail), coeff
coeff = S.One
elif term.func is C.exp:
if term.args[0].is_Rational:
term, expo = S.Exp1, term.args[0]
elif term.args[0].is_Mul:
coeff, tail = term.args[0].as_coeff_terms()
if coeff.is_Rational:
tail = C.Mul(*tail)
term, expo = C.exp(tail), coeff
coeff = S.One
elif term.is_Rational:
coeff = Integer(term.q)
term = Integer(term.p)
if term in denom:
denom[term] += expo
else:
denom[term] = expo
if term in basis:
total, maxi = basis[term]
n_total = total + expo
n_maxi = max(maxi, expo)
basis[term] = (n_total, n_maxi)
else:
basis[term] = (expo, expo)
coeffs.append(coeff)
items.append((numer, denom))
numerator, denominator = [], []
for (term, (total, maxi)) in basis.iteritems():
basis[term] = (total, total - maxi)
if term.func is C.exp:
denominator.append(C.exp(maxi * term.args[0]))
else:
if maxi is S.One:
denominator.append(term)
else:
denominator.append(Pow(term, maxi))
if all([c.is_integer for c in coeffs]):
gcds = lambda x, y: igcd(int(x), int(y))
common = Rational(reduce(gcds, coeffs))
else:
common = S.One
product = Mul(*coeffs) / common
for ((numer, denom), coeff) in zip(items, coeffs):
expr, coeff = [], product / (coeff * common)
for term in basis.iterkeys():
total, sub = basis[term]
if term in denom:
expo = total - denom[term] - sub
else:
expo = total - sub
if term.func is C.exp:
expr.append(C.exp(expo * term.args[0]))
else:
if expo is S.One:
expr.append(term)
else:
expr.append(Pow(term, expo))
numerator.append(coeff * Mul(*([numer] + expr)))
return Add(*numerator) / (product * Mul(*denominator))
elif expr.is_Mul or expr.is_Pow:
return type(expr)(*[_together(t) for t in expr.args])
elif expr.is_Function and deep:
return expr.func(*[_together(t) for t in expr.args])
else:
return expr
def _eval_rewrite_as_Heaviside(self, *args):
return C.Add(*[j*C.Mul(*[C.Heaviside(i-j) for i in args if i!=j]) \
for j in args])
def _print_Mul(self, expr):
coeff, terms = expr.as_coeff_terms()
if not coeff.is_negative:
tex = ""
else:
coeff = -coeff
tex = "- "
numer, denom = fraction(C.Mul(*terms))
seperator = self._settings['mul_symbol_latex']
def convert(terms):
if not terms.is_Mul:
return str(self._print(terms))
else:
_tex = last_term_tex = ""
for term in terms.args:
pretty = self._print(term)
if term.is_Add:
term_tex = (r"\left(%s\right)" % pretty)
else:
term_tex = str(pretty)
# between two digits, \times must always be used,
# to avoid confusion
if seperator == " " and \
re.search("[0-9][} ]*$", last_term_tex) and \
re.match("[{ ]*[-+0-9]", term_tex):
_tex += r" \times "
elif _tex:
_tex += seperator
_tex += term_tex
last_term_tex = term_tex
return _tex
if denom is S.One:
if numer.is_Add:
_tex = r"\left(%s\right)" % convert(numer)
else:
_tex = r"%s" % convert(numer)
if coeff is not S.One:
tex += str(self._print(coeff))
# between two digits, \times must always be used, to avoid
# confusion
if seperator == " " and re.search("[0-9][} ]*$", tex) and \
re.match("[{ ]*[-+0-9]", _tex):
tex += r" \times " + _tex
else:
tex += seperator + _tex
else:
tex += _tex
else:
if numer is S.One:
if coeff.is_Integer:
numer *= coeff.p
elif coeff.is_Rational:
if coeff.p != 1:
numer *= coeff.p
denom *= coeff.q
elif coeff is not S.One:
tex += str(self._print(coeff)) + " "
else:
if coeff.is_Rational and coeff.p == 1:
denom *= coeff.q
elif coeff is not S.One:
tex += str(self._print(coeff)) + " "
tex += r"\frac{%s}{%s}" % \
(convert(numer), convert(denom))
return tex
def __new__(cls, expr, *symbols, **assumptions):
expr = sympify(expr).expand()
if expr is S.NaN:
return S.NaN
if symbols:
symbols = map(sympify, symbols)
else:
symbols = list(expr.atoms(C.Symbol))
symbols.sort(Basic.compare)
if expr.is_Order:
new_symbols = list(expr.symbols)
for s in symbols:
if s not in new_symbols:
new_symbols.append(s)
if len(new_symbols)==len(expr.symbols):
return expr
symbols = new_symbols
elif symbols:
symbol_map = {}
new_symbols = []
for s in symbols:
if isinstance(s, C.Symbol):
new_symbols.append(s)
continue
z = C.Symbol('z',dummy=True)
x1,s1 = solve4linearsymbol(s, z)
expr = expr.subs(x1,s1)
symbol_map[z] = s
new_symbols.append(z)
if symbol_map:
r = Order(expr, *new_symbols, **assumptions)
expr = r.expr.subs(symbol_map)
symbols = []
for s in r.symbols:
if symbol_map.has_key(s):
symbols.append(symbol_map[s])
else:
symbols.append(s)
else:
if expr.is_Add:
lst = expr.extract_leading_order(*symbols)
expr = C.Add(*[f.expr for (e,f) in lst])
else:
expr = expr.as_leading_term(*symbols)
coeff, terms = expr.as_coeff_terms()
if coeff is S.Zero:
return coeff
expr = C.Mul(*[t for t in terms if t.has(*symbols)])
elif expr is not S.Zero:
expr = S.One
if expr is S.Zero:
return expr
# create Order instance:
obj = Expr.__new__(cls, expr, *symbols, **assumptions)
return obj
def _print_Mul(self, expr):
coeff, terms = expr.as_coeff_terms()
if not coeff.is_negative:
tex = ""
else:
coeff = -coeff
tex = "- "
numer, denom = fraction(C.Mul(*terms))
mul_symbol_table = {
None: r" ",
"ldot": r" \,.\, ",
"dot": r" \cdot ",
"times": r" \times "
}
seperator = mul_symbol_table[self._settings['mul_symbol']]
def convert(terms):
product = []
if not terms.is_Mul:
return str(self._print(terms))
else:
for term in terms.args:
pretty = self._print(term)
if term.is_Add:
product.append(r"\left(%s\right)" % pretty)
else:
product.append(str(pretty))
return seperator.join(product)
if denom is S.One:
if coeff is not S.One:
tex += str(self._print(coeff)) + seperator
if numer.is_Add:
tex += r"\left(%s\right)" % convert(numer)
else:
tex += r"%s" % convert(numer)
else:
if numer is S.One:
if coeff.is_Integer:
numer *= coeff.p
elif coeff.is_Rational:
if coeff.p != 1:
numer *= coeff.p
denom *= coeff.q
elif coeff is not S.One:
tex += str(self._print(coeff)) + " "
else:
if coeff.is_Rational and coeff.p == 1:
denom *= coeff.q
elif coeff is not S.One:
tex += str(self._print(coeff)) + " "
tex += r"\frac{%s}{%s}" % \
(convert(numer), convert(denom))
return tex
