def test_log1p():
# Eval
assert log1p(0) == 0
d = S(10)
assert expand_log(log1p(d**-1000) - log(d**1000 + 1) + log(d**1000)) == 0
x = Symbol('x', real=True, finite=True)
# Expand and rewrite
assert log1p(x).expand(func=True) - log(x + 1) == 0
assert log1p(x).rewrite('tractable') - log(x + 1) == 0
assert log1p(x).rewrite('log') - log(x + 1) == 0
# Precision
assert not abs(log(1e-99 + 1).evalf() - 1e-99) < 1e-100 # for comparison
assert abs(expand_log(log1p(1e-99)).evalf() - 1e-99) < 1e-100
# Properties
assert log1p(-2**(-S(1)/2)).is_real
assert not log1p(-1).is_finite
assert log1p(pi).is_finite
assert not log1p(x).is_positive
assert log1p(Symbol('y', positive=True)).is_positive
assert not log1p(x).is_zero
assert log1p(Symbol('z', zero=True)).is_zero
assert not log1p(x).is_nonnegative
assert log1p(Symbol('o', nonnegative=True)).is_nonnegative
# Diff
assert log1p(42*x).diff(x) - 42/(42*x + 1) == 0
assert log1p(42*x).diff(x) - log1p(42*x).expand(func=True).diff(x) == 0
def _lambert(eq, x):
"""
Given an expression assumed to be in the form
``F(X, a..f) = a*log(b*X + c) + d*X + f = 0``
where X = g(x) and x = g^-1(X), return the Lambert solution if possible:
``x = g^-1(-c/b + (a/d)*W(d/(a*b)*exp(c*d/a/b)*exp(-f/a)))``.
"""
eq = _mexpand(expand_log(eq))
mainlog = _mostfunc(eq, log, x)
if not mainlog:
return [] # violated assumptions
other = eq.subs(mainlog, 0)
if (-other).func is log:
eq = (eq - other).subs(mainlog, mainlog.args[0])
mainlog = mainlog.args[0]
if mainlog.func is not log:
return [] # violated assumptions
other = -(-other).args[0]
eq += other
if not x in other.free_symbols:
return [] # violated assumptions
d, f, X2 = _linab(other, x)
logterm = collect(eq - other, mainlog)
a = logterm.as_coefficient(mainlog)
if a is None or x in a.free_symbols:
return [] # violated assumptions
logarg = mainlog.args[0]
b, c, X1 = _linab(logarg, x)
if X1 != X2:
return [] # violated assumptions
u = Dummy('rhs')
sol = []
# check only real solutions:
for k in [-1, 0]:
l = LambertW(d/(a*b)*exp(c*d/a/b)*exp(-f/a), k)
# if W's arg is between -1/e and 0 there is
# a -1 branch real solution, too.
if k and not l.is_real:
continue
rhs = -c/b + (a/d)*l
solns = solve(X1 - u, x)
for i, tmp in enumerate(solns):
solns[i] = tmp.subs(u, rhs)
sol.append(solns[i])
return sol
def _solve_lambert(f, symbol, gens):
"""Return solution to ``f`` if it is a Lambert-type expression
else raise NotImplementedError.
The equality, ``f(x, a..f) = a*log(b*X + c) + d*X - f = 0`` has the
solution, `X = -c/b + (a/d)*W(d/(a*b)*exp(c*d/a/b)*exp(f/a))`. There
are a variety of forms for `f(X, a..f)` as enumerated below:
1a1)
if B**B = R for R not [0, 1] then
log(B) + log(log(B)) = log(log(R))
X = log(B), a = 1, b = 1, c = 0, d = 1, f = log(log(R))
1a2)
if B*(b*log(B) + c)**a = R then
log(B) + a*log(b*log(B) + c) = log(R)
X = log(B); d=1, f=log(R)
1b)
if a*log(b*B + c) + d*B = R then
X = B, f = R
2a)
if (b*B + c)*exp(d*B + g) = R then
log(b*B + c) + d*B + g = log(R)
a = 1, f = log(R) - g, X = B
2b)
if -b*B + g*exp(d*B + h) = c then
log(g) + d*B + h - log(b*B + c) = 0
a = -1, f = -h - log(g), X = B
3)
if d*p**(a*B + g) - b*B = c then
log(d) + (a*B + g)*log(p) - log(c + b*B) = 0
a = -1, d = a*log(p), f = -log(d) - g*log(p)
"""
nrhs, lhs = f.as_independent(symbol, as_Add=True)
rhs = -nrhs
lamcheck = [tmp for tmp in gens
if (tmp.func in [exp, log] or
(tmp.is_Pow and symbol in tmp.exp.free_symbols))]
if not lamcheck:
raise NotImplementedError()
if lhs.is_Mul:
lhs = expand_log(log(lhs))
rhs = log(rhs)
lhs = factor(lhs, deep=True)
# make sure we are inverted as completely as possible
r = Dummy()
i, lhs = _invert(lhs - r, symbol)
rhs = i.xreplace({r: rhs})
# For the first ones:
# 1a1) B**B = R != 0 (when 0, there is only a solution if the base is 0,
# but if it is, the exp is 0 and 0**0=1
# comes back as B*log(B) = log(R)
# 1a2) B*(a + b*log(B))**p = R or with monomial expanded or with whole
# thing expanded comes back unchanged
# log(B) + p*log(a + b*log(B)) = log(R)
# lhs is Mul:
# expand log of both sides to give:
# log(B) + log(log(B)) = log(log(R))
# 1b) d*log(a*B + b) + c*B = R
# lhs is Add:
# isolate c*B and expand log of both sides:
# log(c) + log(B) = log(R - d*log(a*B + b))
soln = []
if not soln:
mainlog = _mostfunc(lhs, log, symbol)
if mainlog:
if lhs.is_Mul and rhs != 0:
soln = _lambert(log(lhs) - log(rhs), symbol)
elif lhs.is_Add:
other = lhs.subs(mainlog, 0)
if other and not other.is_Add and [
tmp for tmp in other.atoms(Pow)
if symbol in tmp.free_symbols]:
if not rhs:
diff = log(other) - log(other - lhs)
else:
diff = log(lhs - other) - log(rhs - other)
soln = _lambert(expand_log(diff), symbol)
else:
#it's ready to go
soln = _lambert(lhs - rhs, symbol)
# For the next two,
# collect on main exp
# 2a) (b*B + c)*exp(d*B + g) = R
# lhs is mul:
# log to give
# log(b*B + c) + d*B = log(R) - g
# 2b) -b*B + g*exp(d*B + h) = R
# lhs is add:
# add b*B
# log and rearrange
# log(R + b*B) - d*B = log(g) + h
if not soln:
mainexp = _mostfunc(lhs, exp, symbol)
if mainexp:
lhs = collect(lhs, mainexp)
if lhs.is_Mul and rhs != 0:
soln = _lambert(expand_log(log(lhs) - log(rhs)), symbol)
elif lhs.is_Add:
# move all but mainexp-containing term to rhs
other = lhs.subs(mainexp, 0)
mainterm = lhs - other
rhs=rhs - other
if (mainterm.could_extract_minus_sign() and
rhs.could_extract_minus_sign()):
mainterm *= -1
rhs *= -1
diff = log(mainterm) - log(rhs)
soln = _lambert(expand_log(diff), symbol)
# 3) d*p**(a*B + b) + c*B = R
# collect on main pow
# log(R - c*B) - a*B*log(p) = log(d) + b*log(p)
if not soln:
mainpow = _mostfunc(lhs, Pow, symbol)
if mainpow and symbol in mainpow.exp.free_symbols:
lhs = collect(lhs, mainpow)
if lhs.is_Mul and rhs != 0:
soln = _lambert(expand_log(log(lhs) - log(rhs)), symbol)
elif lhs.is_Add:
# move all but mainpow-containing term to rhs
other = lhs.subs(mainpow, 0)
mainterm = lhs - other
rhs = rhs - other
diff = log(mainterm) - log(rhs)
soln = _lambert(expand_log(diff), symbol)
if not soln:
raise NotImplementedError('%s does not appear to have a solution in '
'terms of LambertW' % f)
return list(ordered(soln))
def simplify(expr, ratio=1.7, measure=count_ops, fu=False):
"""
Simplifies the given expression.
Simplification is not a well defined term and the exact strategies
this function tries can change in the future versions of SymPy. If
your algorithm relies on "simplification" (whatever it is), try to
determine what you need exactly - is it powsimp()?, radsimp()?,
together()?, logcombine()?, or something else? And use this particular
function directly, because those are well defined and thus your algorithm
will be robust.
Nonetheless, especially for interactive use, or when you don't know
anything about the structure of the expression, simplify() tries to apply
intelligent heuristics to make the input expression "simpler". For
example:
>>> from sympy import simplify, cos, sin
>>> from sympy.abc import x, y
>>> a = (x + x**2)/(x*sin(y)**2 + x*cos(y)**2)
>>> a
(x**2 + x)/(x*sin(y)**2 + x*cos(y)**2)
>>> simplify(a)
x + 1
Note that we could have obtained the same result by using specific
simplification functions:
>>> from sympy import trigsimp, cancel
>>> trigsimp(a)
(x**2 + x)/x
>>> cancel(_)
x + 1
In some cases, applying :func:`simplify` may actually result in some more
complicated expression. The default ``ratio=1.7`` prevents more extreme
cases: if (result length)/(input length) > ratio, then input is returned
unmodified. The ``measure`` parameter lets you specify the function used
to determine how complex an expression is. The function should take a
single argument as an expression and return a number such that if
expression ``a`` is more complex than expression ``b``, then
``measure(a) > measure(b)``. The default measure function is
:func:`count_ops`, which returns the total number of operations in the
expression.
For example, if ``ratio=1``, ``simplify`` output can't be longer
than input.
::
>>> from sympy import sqrt, simplify, count_ops, oo
>>> root = 1/(sqrt(2)+3)
Since ``simplify(root)`` would result in a slightly longer expression,
root is returned unchanged instead::
>>> simplify(root, ratio=1) == root
True
If ``ratio=oo``, simplify will be applied anyway::
>>> count_ops(simplify(root, ratio=oo)) > count_ops(root)
True
Note that the shortest expression is not necessary the simplest, so
setting ``ratio`` to 1 may not be a good idea.
Heuristically, the default value ``ratio=1.7`` seems like a reasonable
choice.
You can easily define your own measure function based on what you feel
should represent the "size" or "complexity" of the input expression. Note
that some choices, such as ``lambda expr: len(str(expr))`` may appear to be
good metrics, but have other problems (in this case, the measure function
may slow down simplify too much for very large expressions). If you don't
know what a good metric would be, the default, ``count_ops``, is a good
one.
For example:
>>> from sympy import symbols, log
>>> a, b = symbols('a b', positive=True)
>>> g = log(a) + log(b) + log(a)*log(1/b)
>>> h = simplify(g)
>>> h
log(a*b**(-log(a) + 1))
>>> count_ops(g)
8
>>> count_ops(h)
5
So you can see that ``h`` is simpler than ``g`` using the count_ops metric.
However, we may not like how ``simplify`` (in this case, using
``logcombine``) has created the ``b**(log(1/a) + 1)`` term. A simple way
to reduce this would be to give more weight to powers as operations in
``count_ops``. We can do this by using the ``visual=True`` option:
>>> print(count_ops(g, visual=True))
2*ADD + DIV + 4*LOG + MUL
>>> print(count_ops(h, visual=True))
2*LOG + MUL + POW + SUB
>>> from sympy import Symbol, S
>>> def my_measure(expr):
... POW = Symbol('POW')
... # Discourage powers by giving POW a weight of 10
... count = count_ops(expr, visual=True).subs(POW, 10)
... # Every other operation gets a weight of 1 (the default)
... count = count.replace(Symbol, type(S.One))
... return count
>>> my_measure(g)
8
>>> my_measure(h)
14
>>> 15./8 > 1.7 # 1.7 is the default ratio
True
>>> simplify(g, measure=my_measure)
-log(a)*log(b) + log(a) + log(b)
Note that because ``simplify()`` internally tries many different
simplification strategies and then compares them using the measure
function, we get a completely different result that is still different
from the input expression by doing this.
"""
expr = sympify(expr)
try:
return expr._eval_simplify(ratio=ratio, measure=measure)
except AttributeError:
pass
original_expr = expr = signsimp(expr)
from sympy.simplify.hyperexpand import hyperexpand
from sympy.functions.special.bessel import BesselBase
from sympy import Sum, Product
if not isinstance(expr, Basic) or not expr.args: # XXX: temporary hack
return expr
if not isinstance(expr, (Add, Mul, Pow, ExpBase)):
if isinstance(expr, Function) and hasattr(expr, "inverse"):
if len(expr.args) == 1 and len(expr.args[0].args) == 1 and \
isinstance(expr.args[0], expr.inverse(argindex=1)):
return simplify(expr.args[0].args[0], ratio=ratio,
measure=measure, fu=fu)
return expr.func(*[simplify(x, ratio=ratio, measure=measure, fu=fu)
for x in expr.args])
# TODO: Apply different strategies, considering expression pattern:
# is it a purely rational function? Is there any trigonometric function?...
# See also https://github.com/sympy/sympy/pull/185.
def shorter(*choices):
'''Return the choice that has the fewest ops. In case of a tie,
the expression listed first is selected.'''
if not has_variety(choices):
return choices[0]
return min(choices, key=measure)
expr = bottom_up(expr, lambda w: w.normal())
expr = Mul(*powsimp(expr).as_content_primitive())
_e = cancel(expr)
expr1 = shorter(_e, _mexpand(_e).cancel()) # issue 6829
expr2 = shorter(together(expr, deep=True), together(expr1, deep=True))
if ratio is S.Infinity:
expr = expr2
else:
expr = shorter(expr2, expr1, expr)
if not isinstance(expr, Basic): # XXX: temporary hack
return expr
expr = factor_terms(expr, sign=False)
# hyperexpand automatically only works on hypergeometric terms
expr = hyperexpand(expr)
expr = piecewise_fold(expr)
if expr.has(BesselBase):
expr = besselsimp(expr)
if expr.has(TrigonometricFunction) and not fu or expr.has(
HyperbolicFunction):
expr = trigsimp(expr, deep=True)
if expr.has(log):
expr = shorter(expand_log(expr, deep=True), logcombine(expr))
if expr.has(CombinatorialFunction, gamma):
expr = combsimp(expr)
if expr.has(Sum):
expr = sum_simplify(expr)
if expr.has(Product):
expr = product_simplify(expr)
short = shorter(powsimp(expr, combine='exp', deep=True), powsimp(expr), expr)
short = shorter(short, factor_terms(short), expand_power_exp(expand_mul(short)))
if short.has(TrigonometricFunction, HyperbolicFunction, ExpBase):
short = exptrigsimp(short, simplify=False)
# get rid of hollow 2-arg Mul factorization
hollow_mul = Transform(
lambda x: Mul(*x.args),
lambda x:
x.is_Mul and
len(x.args) == 2 and
x.args[0].is_Number and
x.args[1].is_Add and
x.is_commutative)
expr = short.xreplace(hollow_mul)
numer, denom = expr.as_numer_denom()
if denom.is_Add:
n, d = fraction(radsimp(1/denom, symbolic=False, max_terms=1))
if n is not S.One:
expr = (numer*n).expand()/d
if expr.could_extract_minus_sign():
n, d = fraction(expr)
if d != 0:
expr = signsimp(-n/(-d))
if measure(expr) > ratio*measure(original_expr):
expr = original_expr
return expr
def _denest_pow(eq):
"""
Denest powers.
This is a helper function for powdenest that performs the actual
transformation.
"""
from sympy.simplify.simplify import logcombine
b, e = eq.as_base_exp()
if b.is_Pow or isinstance(b.func, exp) and e != 1:
new = b._eval_power(e)
if new is not None:
eq = new
b, e = new.as_base_exp()
# denest exp with log terms in exponent
if b is S.Exp1 and e.is_Mul:
logs = []
other = []
for ei in e.args:
if any(isinstance(ai, log) for ai in Add.make_args(ei)):
logs.append(ei)
else:
other.append(ei)
logs = logcombine(Mul(*logs))
return Pow(exp(logs), Mul(*other))
_, be = b.as_base_exp()
if be is S.One and not (b.is_Mul or
b.is_Rational and b.q != 1 or
b.is_positive):
return eq
# denest eq which is either pos**e or Pow**e or Mul**e or
# Mul(b1**e1, b2**e2)
# handle polar numbers specially
polars, nonpolars = [], []
for bb in Mul.make_args(b):
if bb.is_polar:
polars.append(bb.as_base_exp())
else:
nonpolars.append(bb)
if len(polars) == 1 and not polars[0][0].is_Mul:
return Pow(polars[0][0], polars[0][1]*e)*powdenest(Mul(*nonpolars)**e)
elif polars:
return Mul(*[powdenest(bb**(ee*e)) for (bb, ee) in polars]) \
*powdenest(Mul(*nonpolars)**e)
if b.is_Integer:
# use log to see if there is a power here
logb = expand_log(log(b))
if logb.is_Mul:
c, logb = logb.args
e *= c
base = logb.args[0]
return Pow(base, e)
# if b is not a Mul or any factor is an atom then there is nothing to do
if not b.is_Mul or any(s.is_Atom for s in Mul.make_args(b)):
return eq
# let log handle the case of the base of the argument being a Mul, e.g.
# sqrt(x**(2*i)*y**(6*i)) -> x**i*y**(3**i) if x and y are positive; we
# will take the log, expand it, and then factor out the common powers that
# now appear as coefficient. We do this manually since terms_gcd pulls out
# fractions, terms_gcd(x+x*y/2) -> x*(y + 2)/2 and we don't want the 1/2;
# gcd won't pull out numerators from a fraction: gcd(3*x, 9*x/2) -> x but
# we want 3*x. Neither work with noncommutatives.
def nc_gcd(aa, bb):
a, b = [i.as_coeff_Mul() for i in [aa, bb]]
c = gcd(a[0], b[0]).as_numer_denom()[0]
g = Mul(*(a[1].args_cnc(cset=True)[0] & b[1].args_cnc(cset=True)[0]))
return _keep_coeff(c, g)
glogb = expand_log(log(b))
if glogb.is_Add:
args = glogb.args
g = reduce(nc_gcd, args)
if g != 1:
cg, rg = g.as_coeff_Mul()
glogb = _keep_coeff(cg, rg*Add(*[a/g for a in args]))
# now put the log back together again
if isinstance(glogb, log) or not glogb.is_Mul:
if glogb.args[0].is_Pow or isinstance(glogb.args[0], exp):
glogb = _denest_pow(glogb.args[0])
if (abs(glogb.exp) < 1) == True:
return Pow(glogb.base, glogb.exp*e)
return eq
# the log(b) was a Mul so join any adds with logcombine
add = []
other = []
for a in glogb.args:
if a.is_Add:
add.append(a)
else:
other.append(a)
return Pow(exp(logcombine(Mul(*add))), e*Mul(*other))
def _eval_nseries(self, x, x0, n):
from sympy import powsimp
arg = self.args[0]
k, l = Wild("k"), Wild("l")
r = arg.match(k*x**l)
if r is not None:
k, l = r[k], r[l]
if l != 0 and not l.has(x) and not k.has(x):
r = log(k) + l*log(x)
return r
order = C.Order(x**n, x)
arg = self.args[0]
x = order.symbols[0]
ln = C.log
use_lt = not C.Order(1,x).contains(arg)
if not use_lt:
arg0 = arg.limit(x, 0)
use_lt = (arg0 is S.Zero)
if use_lt: # singularity, #example: self = log(sin(x))
# arg = (arg / lt) * lt
lt = arg.as_leading_term(x) # arg = sin(x); lt = x
a = powsimp((arg/lt).expand(), deep=True, combine='exp') # a = sin(x)/x
# the idea is to recursively call ln(a).series(), but one needs to
# make sure that ln(sin(x)/x) doesn't get "simplified" to
# -log(x)+ln(sin(x)) and an infinite recursion occurs, see also the
# issue 252.
obj = ln(lt) + ln(a)._eval_nseries(x, x0, n)
else:
# arg -> arg0 + (arg - arg0) -> arg0 * (1 + (arg/arg0 - 1))
z = (arg/arg0 - 1)
x = order.symbols[0]
ln = C.log
o = C.Order(z, x)
if o is S.Zero:
return ln(1+z)+ ln(arg0)
if o.expr.is_number:
e = ln(order.expr*x)/ln(x)
else:
e = ln(order.expr)/ln(o.expr)
n = e.limit(x,0) + 1
if n.is_unbounded:
# requested accuracy gives infinite series,
# order is probably nonpolynomial e.g. O(exp(-1/x), x).
return ln(1+z)+ ln(arg0)
try:
n = int(n)
except TypeError:
#well, the n is something more complicated (like 1+log(2))
n = int(n.evalf()) + 1
assert n>=0,`n`
l = []
g = None
for i in xrange(n+2):
g = ln.taylor_term(i, z, g)
g = g.nseries(x, x0, n)
l.append(g)
obj = C.Add(*l) + ln(arg0)
obj2 = expand_log(powsimp(obj, deep=True, combine='exp'))
if obj2 != obj:
r = obj2.nseries(x, x0, n)
else:
r = obj
if r == self:
return self
return r + order
def _solve_lambert(f, symbol, gens):
"""Return solution to ``f`` if it is a Lambert-type expression
else raise NotImplementedError.
The equality, ``f(x, a..f) = a*log(b*X + c) + d*X - f = 0`` has the
solution, `X = -c/b + (a/d)*W(d/(a*b)*exp(c*d/a/b)*exp(f/a))`. There
are a variety of forms for `f(X, a..f)` as enumerated below:
1a1)
if B**B = R for R not [0, 1] then
log(B) + log(log(B)) = log(log(R))
X = log(B), a = 1, b = 1, c = 0, d = 1, f = log(log(R))
1a2)
if B*(b*log(B) + c)**a = R then
log(B) + a*log(b*log(B) + c) = log(R)
X = log(B); d=1, f=log(R)
1b)
if a*log(b*B + c) + d*B = R then
X = B, f = R
2a)
if (b*B + c)*exp(d*B + g) = R then
log(b*B + c) + d*B + g = log(R)
a = 1, f = log(R) - g, X = B
2b)
if -b*B + g*exp(d*B + h) = c then
log(g) + d*B + h - log(b*B + c) = 0
a = -1, f = -h - log(g), X = B
3)
if d*p**(a*B + g) - b*B = c then
log(d) + (a*B + g)*log(p) - log(c + b*B) = 0
a = -1, d = a*log(p), f = -log(d) - g*log(p)
"""
nrhs, lhs = f.as_independent(symbol, as_Add=True)
rhs = -nrhs
lamcheck = [
tmp for tmp in gens if (tmp.func in [exp, log] or (
tmp.is_Pow and symbol in tmp.exp.free_symbols))
]
if not lamcheck:
raise NotImplementedError()
if lhs.is_Mul:
lhs = expand_log(log(lhs))
rhs = log(rhs)
lhs = factor(lhs, deep=True)
# make sure we are inverted as completely as possible
r = Dummy()
i, lhs = _invert(lhs - r, symbol)
rhs = i.xreplace({r: rhs})
# For the first ones:
# 1a1) B**B = R != 0 (when 0, there is only a solution if the base is 0,
# but if it is, the exp is 0 and 0**0=1
# comes back as B*log(B) = log(R)
# 1a2) B*(a + b*log(B))**p = R or with monomial expanded or with whole
# thing expanded comes back unchanged
# log(B) + p*log(a + b*log(B)) = log(R)
# lhs is Mul:
# expand log of both sides to give:
# log(B) + log(log(B)) = log(log(R))
# 1b) d*log(a*B + b) + c*B = R
# lhs is Add:
# isolate c*B and expand log of both sides:
# log(c) + log(B) = log(R - d*log(a*B + b))
soln = []
if not soln:
mainlog = _mostfunc(lhs, log, symbol)
if mainlog:
if lhs.is_Mul and rhs != 0:
soln = _lambert(log(lhs) - log(rhs), symbol)
elif lhs.is_Add:
other = lhs.subs(mainlog, 0)
if other and not other.is_Add and [
tmp for tmp in other.atoms(Pow)
if symbol in tmp.free_symbols
]:
if not rhs:
diff = log(other) - log(other - lhs)
else:
diff = log(lhs - other) - log(rhs - other)
soln = _lambert(expand_log(diff), symbol)
else:
#it's ready to go
soln = _lambert(lhs - rhs, symbol)
# For the next two,
# collect on main exp
# 2a) (b*B + c)*exp(d*B + g) = R
# lhs is mul:
# log to give
# log(b*B + c) + d*B = log(R) - g
# 2b) -b*B + g*exp(d*B + h) = R
# lhs is add:
# add b*B
# log and rearrange
# log(R + b*B) - d*B = log(g) + h
if not soln:
mainexp = _mostfunc(lhs, exp, symbol)
if mainexp:
lhs = collect(lhs, mainexp)
if lhs.is_Mul and rhs != 0:
soln = _lambert(expand_log(log(lhs) - log(rhs)), symbol)
elif lhs.is_Add:
# move all but mainexp-containing term to rhs
other = lhs.subs(mainexp, 0)
mainterm = lhs - other
rhs = rhs - other
if (mainterm.could_extract_minus_sign()
and rhs.could_extract_minus_sign()):
mainterm *= -1
rhs *= -1
diff = log(mainterm) - log(rhs)
soln = _lambert(expand_log(diff), symbol)
# 3) d*p**(a*B + b) + c*B = R
# collect on main pow
# log(R - c*B) - a*B*log(p) = log(d) + b*log(p)
if not soln:
mainpow = _mostfunc(lhs, Pow, symbol)
if mainpow and symbol in mainpow.exp.free_symbols:
lhs = collect(lhs, mainpow)
if lhs.is_Mul and rhs != 0:
soln = _lambert(expand_log(log(lhs) - log(rhs)), symbol)
elif lhs.is_Add:
# move all but mainpow-containing term to rhs
other = lhs.subs(mainpow, 0)
mainterm = lhs - other
rhs = rhs - other
diff = log(mainterm) - log(rhs)
soln = _lambert(expand_log(diff), symbol)
if not soln:
raise NotImplementedError('%s does not appear to have a solution in '
'terms of LambertW' % f)
return list(ordered(soln))
def _denest_pow(eq):
"""
Denest powers.
This is a helper function for powdenest that performs the actual
transformation.
"""
from sympy.simplify.simplify import logcombine
b, e = eq.as_base_exp()
if b.is_Pow or isinstance(b, exp) and e != 1:
new = b._eval_power(e)
if new is not None:
eq = new
b, e = new.as_base_exp()
# denest exp with log terms in exponent
if b is S.Exp1 and e.is_Mul:
logs = []
other = []
for ei in e.args:
if any(isinstance(ai, log) for ai in Add.make_args(ei)):
logs.append(ei)
else:
other.append(ei)
logs = logcombine(Mul(*logs))
return Pow(exp(logs), Mul(*other))
_, be = b.as_base_exp()
if be is S.One and not (b.is_Mul or b.is_Rational and b.q != 1
or b.is_positive):
return eq
# denest eq which is either pos**e or Pow**e or Mul**e or
# Mul(b1**e1, b2**e2)
# handle polar numbers specially
polars, nonpolars = [], []
for bb in Mul.make_args(b):
if bb.is_polar:
polars.append(bb.as_base_exp())
else:
nonpolars.append(bb)
if len(polars) == 1 and not polars[0][0].is_Mul:
return Pow(polars[0][0], polars[0][1] * e) * powdenest(
Mul(*nonpolars)**e)
elif polars:
return Mul(*[powdenest(bb**(ee*e)) for (bb, ee) in polars]) \
*powdenest(Mul(*nonpolars)**e)
if b.is_Integer:
# use log to see if there is a power here
logb = expand_log(log(b))
if logb.is_Mul:
c, logb = logb.args
e *= c
base = logb.args[0]
return Pow(base, e)
# if b is not a Mul or any factor is an atom then there is nothing to do
if not b.is_Mul or any(s.is_Atom for s in Mul.make_args(b)):
return eq
# let log handle the case of the base of the argument being a Mul, e.g.
# sqrt(x**(2*i)*y**(6*i)) -> x**i*y**(3**i) if x and y are positive; we
# will take the log, expand it, and then factor out the common powers that
# now appear as coefficient. We do this manually since terms_gcd pulls out
# fractions, terms_gcd(x+x*y/2) -> x*(y + 2)/2 and we don't want the 1/2;
# gcd won't pull out numerators from a fraction: gcd(3*x, 9*x/2) -> x but
# we want 3*x. Neither work with noncommutatives.
def nc_gcd(aa, bb):
a, b = [i.as_coeff_Mul() for i in [aa, bb]]
c = gcd(a[0], b[0]).as_numer_denom()[0]
g = Mul(*(a[1].args_cnc(cset=True)[0] & b[1].args_cnc(cset=True)[0]))
return _keep_coeff(c, g)
glogb = expand_log(log(b))
if glogb.is_Add:
args = glogb.args
g = reduce(nc_gcd, args)
if g != 1:
cg, rg = g.as_coeff_Mul()
glogb = _keep_coeff(cg, rg * Add(*[a / g for a in args]))
# now put the log back together again
if isinstance(glogb, log) or not glogb.is_Mul:
if glogb.args[0].is_Pow or isinstance(glogb.args[0], exp):
glogb = _denest_pow(glogb.args[0])
if (abs(glogb.exp) < 1) == True:
return Pow(glogb.base, glogb.exp * e)
return eq
# the log(b) was a Mul so join any adds with logcombine
add = []
other = []
for a in glogb.args:
if a.is_Add:
add.append(a)
else:
other.append(a)
return Pow(exp(logcombine(Mul(*add))), e * Mul(*other))
def simplify(expr, ratio=1.7, measure=count_ops, fu=False):
"""
Simplifies the given expression.
Simplification is not a well defined term and the exact strategies
this function tries can change in the future versions of SymPy. If
your algorithm relies on "simplification" (whatever it is), try to
determine what you need exactly - is it powsimp()?, radsimp()?,
together()?, logcombine()?, or something else? And use this particular
function directly, because those are well defined and thus your algorithm
will be robust.
Nonetheless, especially for interactive use, or when you don't know
anything about the structure of the expression, simplify() tries to apply
intelligent heuristics to make the input expression "simpler". For
example:
>>> from sympy import simplify, cos, sin
>>> from sympy.abc import x, y
>>> a = (x + x**2)/(x*sin(y)**2 + x*cos(y)**2)
>>> a
(x**2 + x)/(x*sin(y)**2 + x*cos(y)**2)
>>> simplify(a)
x + 1
Note that we could have obtained the same result by using specific
simplification functions:
>>> from sympy import trigsimp, cancel
>>> trigsimp(a)
(x**2 + x)/x
>>> cancel(_)
x + 1
In some cases, applying :func:`simplify` may actually result in some more
complicated expression. The default ``ratio=1.7`` prevents more extreme
cases: if (result length)/(input length) > ratio, then input is returned
unmodified. The ``measure`` parameter lets you specify the function used
to determine how complex an expression is. The function should take a
single argument as an expression and return a number such that if
expression ``a`` is more complex than expression ``b``, then
``measure(a) > measure(b)``. The default measure function is
:func:`count_ops`, which returns the total number of operations in the
expression.
For example, if ``ratio=1``, ``simplify`` output can't be longer
than input.
::
>>> from sympy import sqrt, simplify, count_ops, oo
>>> root = 1/(sqrt(2)+3)
Since ``simplify(root)`` would result in a slightly longer expression,
root is returned unchanged instead::
>>> simplify(root, ratio=1) == root
True
If ``ratio=oo``, simplify will be applied anyway::
>>> count_ops(simplify(root, ratio=oo)) > count_ops(root)
True
Note that the shortest expression is not necessary the simplest, so
setting ``ratio`` to 1 may not be a good idea.
Heuristically, the default value ``ratio=1.7`` seems like a reasonable
choice.
You can easily define your own measure function based on what you feel
should represent the "size" or "complexity" of the input expression. Note
that some choices, such as ``lambda expr: len(str(expr))`` may appear to be
good metrics, but have other problems (in this case, the measure function
may slow down simplify too much for very large expressions). If you don't
know what a good metric would be, the default, ``count_ops``, is a good
one.
For example:
>>> from sympy import symbols, log
>>> a, b = symbols('a b', positive=True)
>>> g = log(a) + log(b) + log(a)*log(1/b)
>>> h = simplify(g)
>>> h
log(a*b**(-log(a) + 1))
>>> count_ops(g)
8
>>> count_ops(h)
5
So you can see that ``h`` is simpler than ``g`` using the count_ops metric.
However, we may not like how ``simplify`` (in this case, using
``logcombine``) has created the ``b**(log(1/a) + 1)`` term. A simple way
to reduce this would be to give more weight to powers as operations in
``count_ops``. We can do this by using the ``visual=True`` option:
>>> print(count_ops(g, visual=True))
2*ADD + DIV + 4*LOG + MUL
>>> print(count_ops(h, visual=True))
2*LOG + MUL + POW + SUB
>>> from sympy import Symbol, S
>>> def my_measure(expr):
... POW = Symbol('POW')
... # Discourage powers by giving POW a weight of 10
... count = count_ops(expr, visual=True).subs(POW, 10)
... # Every other operation gets a weight of 1 (the default)
... count = count.replace(Symbol, type(S.One))
... return count
>>> my_measure(g)
8
>>> my_measure(h)
14
>>> 15./8 > 1.7 # 1.7 is the default ratio
True
>>> simplify(g, measure=my_measure)
-log(a)*log(b) + log(a) + log(b)
Note that because ``simplify()`` internally tries many different
simplification strategies and then compares them using the measure
function, we get a completely different result that is still different
from the input expression by doing this.
"""
expr = sympify(expr)
try:
return expr._eval_simplify(ratio=ratio, measure=measure)
except AttributeError:
pass
original_expr = expr = signsimp(expr)
from sympy.simplify.hyperexpand import hyperexpand
from sympy.functions.special.bessel import BesselBase
from sympy import Sum, Product
if not isinstance(expr, Basic) or not expr.args: # XXX: temporary hack
return expr
if not isinstance(expr, (Add, Mul, Pow, ExpBase)):
if isinstance(expr, Function) and hasattr(expr, "inverse"):
if len(expr.args) == 1 and len(expr.args[0].args) == 1 and \
isinstance(expr.args[0], expr.inverse(argindex=1)):
return simplify(expr.args[0].args[0],
ratio=ratio,
measure=measure,
fu=fu)
return expr.func(*[
simplify(x, ratio=ratio, measure=measure, fu=fu) for x in expr.args
])
# TODO: Apply different strategies, considering expression pattern:
# is it a purely rational function? Is there any trigonometric function?...
# See also https://github.com/sympy/sympy/pull/185.
def shorter(*choices):
'''Return the choice that has the fewest ops. In case of a tie,
the expression listed first is selected.'''
if not has_variety(choices):
return choices[0]
return min(choices, key=measure)
expr = bottom_up(expr, lambda w: w.normal())
expr = Mul(*powsimp(expr).as_content_primitive())
_e = cancel(expr)
expr1 = shorter(_e, _mexpand(_e).cancel()) # issue 6829
expr2 = shorter(together(expr, deep=True), together(expr1, deep=True))
if ratio is S.Infinity:
expr = expr2
else:
expr = shorter(expr2, expr1, expr)
if not isinstance(expr, Basic): # XXX: temporary hack
return expr
expr = factor_terms(expr, sign=False)
# hyperexpand automatically only works on hypergeometric terms
expr = hyperexpand(expr)
expr = piecewise_fold(expr)
if expr.has(BesselBase):
expr = besselsimp(expr)
if expr.has(TrigonometricFunction) and not fu or expr.has(
HyperbolicFunction):
expr = trigsimp(expr, deep=True)
if expr.has(log):
expr = shorter(expand_log(expr, deep=True), logcombine(expr))
if expr.has(CombinatorialFunction, gamma):
expr = combsimp(expr)
if expr.has(Sum):
expr = sum_simplify(expr)
if expr.has(Product):
expr = product_simplify(expr)
short = shorter(powsimp(expr, combine='exp', deep=True), powsimp(expr),
expr)
short = shorter(short, factor_terms(short),
expand_power_exp(expand_mul(short)))
if short.has(TrigonometricFunction, HyperbolicFunction, ExpBase):
short = exptrigsimp(short, simplify=False)
# get rid of hollow 2-arg Mul factorization
hollow_mul = Transform(
lambda x: Mul(*x.args), lambda x: x.is_Mul and len(x.args) == 2 and x.
args[0].is_Number and x.args[1].is_Add and x.is_commutative)
expr = short.xreplace(hollow_mul)
numer, denom = expr.as_numer_denom()
if denom.is_Add:
n, d = fraction(radsimp(1 / denom, symbolic=False, max_terms=1))
if n is not S.One:
expr = (numer * n).expand() / d
if expr.could_extract_minus_sign():
n, d = fraction(expr)
if d != 0:
expr = signsimp(-n / (-d))
if measure(expr) > ratio * measure(original_expr):
expr = original_expr
return expr
def _eval_expand_log(self, deep=True, **hints):
from sympy.functions.elementary.complexes import unpolarify
from sympy.ntheory.factor_ import factorint
from sympy.concrete import Sum, Product
force = hints.get('force', False)
factor = hints.get('factor', False)
if (len(self.args) == 2):
return expand_log(self.func(*self.args), deep=deep, force=force)
arg = self.args[0]
if arg.is_Integer:
# remove perfect powers
p = perfect_power(arg)
logarg = None
coeff = 1
if p is not False:
arg, coeff = p
logarg = self.func(arg)
# expand as product of its prime factors if factor=True
if factor:
p = factorint(arg)
if arg not in p.keys():
logarg = sum(n * log(val) for val, n in p.items())
if logarg is not None:
return coeff * logarg
elif arg.is_Rational:
return log(arg.p) - log(arg.q)
elif arg.is_Mul:
expr = []
nonpos = []
for x in arg.args:
if force or x.is_positive or x.is_polar:
a = self.func(x)
if isinstance(a, log):
expr.append(self.func(x)._eval_expand_log(**hints))
else:
expr.append(a)
elif x.is_negative:
a = self.func(-x)
expr.append(a)
nonpos.append(S.NegativeOne)
else:
nonpos.append(x)
return Add(*expr) + log(Mul(*nonpos))
elif arg.is_Pow or isinstance(arg, exp):
if force or (
arg.exp.is_extended_real and
(arg.base.is_positive or
((arg.exp + 1).is_positive and
(arg.exp - 1).is_nonpositive))) or arg.base.is_polar:
b = arg.base
e = arg.exp
a = self.func(b)
if isinstance(a, log):
return unpolarify(e) * a._eval_expand_log(**hints)
else:
return unpolarify(e) * a
elif isinstance(arg, Product):
if force or arg.function.is_positive:
return Sum(log(arg.function), *arg.limits)
return self.func(arg)
def __new__(cls, expr, *args, **kwargs):
expr = sympify(expr)
if not args:
if expr.is_Order:
variables = expr.variables
point = expr.point
else:
variables = list(expr.free_symbols)
point = [S.Zero]*len(variables)
else:
args = list(args if is_sequence(args) else [args])
variables, point = [], []
if is_sequence(args[0]):
for a in args:
v, p = list(map(sympify, a))
variables.append(v)
point.append(p)
else:
variables = list(map(sympify, args))
point = [S.Zero]*len(variables)
if not all(v.is_symbol for v in variables):
raise TypeError('Variables are not symbols, got %s' % variables)
if len(list(uniq(variables))) != len(variables):
raise ValueError('Variables are supposed to be unique symbols, got %s' % variables)
if expr.is_Order:
expr_vp = dict(expr.args[1:])
new_vp = dict(expr_vp)
vp = dict(zip(variables, point))
for v, p in vp.items():
if v in new_vp.keys():
if p != new_vp[v]:
raise NotImplementedError(
"Mixing Order at different points is not supported.")
else:
new_vp[v] = p
if set(expr_vp.keys()) == set(new_vp.keys()):
return expr
else:
variables = list(new_vp.keys())
point = [new_vp[v] for v in variables]
if expr is S.NaN:
return S.NaN
if any(x in p.free_symbols for x in variables for p in point):
raise ValueError('Got %s as a point.' % point)
if variables:
if any(p != point[0] for p in point):
raise NotImplementedError(
"Multivariable orders at different points are not supported.")
if point[0] is S.Infinity:
s = {k: 1/Dummy() for k in variables}
rs = {1/v: 1/k for k, v in s.items()}
ps = [S.Zero for p in point]
elif point[0] is S.NegativeInfinity:
s = {k: -1/Dummy() for k in variables}
rs = {-1/v: -1/k for k, v in s.items()}
ps = [S.Zero for p in point]
elif point[0] is not S.Zero:
s = {k: Dummy() + point[0] for k in variables}
rs = {(v - point[0]).together(): k - point[0] for k, v in s.items()}
ps = [S.Zero for p in point]
else:
s = ()
rs = ()
ps = list(point)
expr = expr.subs(s)
if expr.is_Add:
expr = expr.factor()
if s:
args = tuple([r[0] for r in rs.items()])
else:
args = tuple(variables)
if len(variables) > 1:
# XXX: better way? We need this expand() to
# workaround e.g: expr = x*(x + y).
# (x*(x + y)).as_leading_term(x, y) currently returns
# x*y (wrong order term!). That's why we want to deal with
# expand()'ed expr (handled in "if expr.is_Add" branch below).
expr = expr.expand()
old_expr = None
while old_expr != expr:
old_expr = expr
if expr.is_Add:
lst = expr.extract_leading_order(args)
expr = Add(*[f.expr for (e, f) in lst])
elif expr:
try:
expr = expr.as_leading_term(*args)
except PoleError:
if isinstance(expr, Function) or\
all(isinstance(arg, Function) for arg in expr.args):
# It is not possible to simplify an expression
# containing only functions (which raise error on
# call to leading term) further
pass
else:
orders = []
pts = tuple(zip(args, ps))
for arg in expr.args:
try:
lt = arg.as_leading_term(*args)
except PoleError:
lt = arg
if lt not in args:
order = Order(lt)
else:
order = Order(lt, *pts)
orders.append(order)
if expr.is_Add:
new_expr = Order(Add(*orders), *pts)
if new_expr.is_Add:
new_expr = Order(Add(*[a.expr for a in new_expr.args]), *pts)
expr = new_expr.expr
elif expr.is_Mul:
expr = Mul(*[a.expr for a in orders])
elif expr.is_Pow:
e = expr.exp
b = expr.base
expr = exp(e * log(b))
# It would probably be better to handle this somewhere
# else. This is needed for a testcase in which there is a
# symbol with the assumptions zero=True.
if expr.is_zero:
expr = S.Zero
else:
expr = expr.as_independent(*args, as_Add=False)[1]
expr = expand_power_base(expr)
expr = expand_log(expr)
if len(args) == 1:
# The definition of O(f(x)) symbol explicitly stated that
# the argument of f(x) is irrelevant. That's why we can
# combine some power exponents (only "on top" of the
# expression tree for f(x)), e.g.:
# x**p * (-x)**q -> x**(p+q) for real p, q.
x = args[0]
margs = list(Mul.make_args(
expr.as_independent(x, as_Add=False)[1]))
for i, t in enumerate(margs):
if t.is_Pow:
b, q = t.args
if b in (x, -x) and q.is_real and not q.has(x):
margs[i] = x**q
elif b.is_Pow and not b.exp.has(x):
b, r = b.args
if b in (x, -x) and r.is_real:
margs[i] = x**(r*q)
elif b.is_Mul and b.args[0] is S.NegativeOne:
b = -b
if b.is_Pow and not b.exp.has(x):
b, r = b.args
if b in (x, -x) and r.is_real:
margs[i] = x**(r*q)
expr = Mul(*margs)
expr = expr.subs(rs)
if expr.is_Order:
expr = expr.expr
if not expr.has(*variables) and not expr.is_zero:
expr = S.One
# create Order instance:
vp = dict(zip(variables, point))
variables.sort(key=default_sort_key)
point = [vp[v] for v in variables]
args = (expr,) + Tuple(*zip(variables, point))
obj = Expr.__new__(cls, *args)
return obj
def _eval_nseries(self, x, n):
from sympy import powsimp
arg = self.args[0]
k, l = Wild("k"), Wild("l")
r = arg.match(k * x**l)
if r is not None:
#k = r.get(r, S.One)
#l = r.get(l, S.Zero)
k, l = r[k], r[l]
if l != 0 and not l.has(x) and not k.has(x):
r = log(k) + l * log(x) # XXX true regardless of assumptions?
return r
order = C.Order(x**n, x)
arg = self.args[0]
use_lt = not C.Order(1, x).contains(arg)
if not use_lt:
arg0 = arg.limit(x, 0)
use_lt = (arg0 is S.Zero)
if use_lt: # singularity, #example: self = log(sin(x))
# arg = (arg / lt) * lt
lt = arg.as_leading_term(x) # arg = sin(x); lt = x
a = powsimp((arg / lt).expand(), deep=True,
combine='exp') # a = sin(x)/x
# the idea is to recursively call log(a).series(), but one needs to
# make sure that log(sin(x)/x) doesn't get "simplified" to
# -log(x)+log(sin(x)) and an infinite recursion occurs, see also the
# issue 252.
obj = log(lt) + log(a).nseries(x, n=n)
else:
# arg -> arg0 + (arg - arg0) -> arg0 * (1 + (arg/arg0 - 1))
z = (arg / arg0 - 1)
o = C.Order(z, x)
if o is S.Zero:
return log(1 + z) + log(arg0)
if o.expr.is_number:
e = log(order.expr * x) / log(x)
else:
e = log(order.expr) / log(o.expr)
n = e.limit(x, 0) + 1
if n.is_unbounded:
# requested accuracy gives infinite series,
# order is probably nonpolynomial e.g. O(exp(-1/x), x).
return log(1 + z) + log(arg0)
# XXX was int or floor intended? int used to behave like floor
try:
n = int(n)
except TypeError:
#well, the n is something more complicated (like 1+log(2))
n = int(n.evalf()) + 1 # XXX why is 1 being added?
assert n >= 0, ` n `
l = []
g = None
for i in xrange(n + 2):
g = log.taylor_term(i, z, g)
g = g.nseries(x, n=n)
l.append(g)
obj = Add(*l) + log(arg0)
obj2 = expand_log(powsimp(obj, deep=True, combine='exp'))
if obj2 != obj:
r = obj2.nseries(x, n=n)
else:
r = obj
if r == self:
return self
return r + order
def _solve_lambert(f, symbol, gens):
"""Return solution to ``f`` if it is a Lambert-type expression
else raise NotImplementedError.
For ``f(X, a..f) = a*log(b*X + c) + d*X - f = 0`` the solution
for ``X`` is ``X = -c/b + (a/d)*W(d/(a*b)*exp(c*d/a/b)*exp(f/a))``.
There are a variety of forms for `f(X, a..f)` as enumerated below:
1a1)
if B**B = R for R not in [0, 1] (since those cases would already
be solved before getting here) then log of both sides gives
log(B) + log(log(B)) = log(log(R)) and
X = log(B), a = 1, b = 1, c = 0, d = 1, f = log(log(R))
1a2)
if B*(b*log(B) + c)**a = R then log of both sides gives
log(B) + a*log(b*log(B) + c) = log(R) and
X = log(B), d=1, f=log(R)
1b)
if a*log(b*B + c) + d*B = R and
X = B, f = R
2a)
if (b*B + c)*exp(d*B + g) = R then log of both sides gives
log(b*B + c) + d*B + g = log(R) and
X = B, a = 1, f = log(R) - g
2b)
if g*exp(d*B + h) - b*B = c then the log form is
log(g) + d*B + h - log(b*B + c) = 0 and
X = B, a = -1, f = -h - log(g)
3)
if d*p**(a*B + g) - b*B = c then the log form is
log(d) + (a*B + g)*log(p) - log(b*B + c) = 0 and
X = B, a = -1, d = a*log(p), f = -log(d) - g*log(p)
"""
def _solve_even_degree_expr(expr, t, symbol):
"""Return the unique solutions of equations derived from
``expr`` by replacing ``t`` with ``+/- symbol``.
Parameters
==========
expr : Expr
The expression which includes a dummy variable t to be
replaced with +symbol and -symbol.
symbol : Symbol
The symbol for which a solution is being sought.
Returns
=======
List of unique solution of the two equations generated by
replacing ``t`` with positive and negative ``symbol``.
Notes
=====
If ``expr = 2*log(t) + x/2` then solutions for
``2*log(x) + x/2 = 0`` and ``2*log(-x) + x/2 = 0`` are
returned by this function. Though this may seem
counter-intuitive, one must note that the ``expr`` being
solved here has been derived from a different expression. For
an expression like ``eq = x**2*g(x) = 1``, if we take the
log of both sides we obtain ``log(x**2) + log(g(x)) = 0``. If
x is positive then this simplifies to
``2*log(x) + log(g(x)) = 0``; the Lambert-solving routines will
return solutions for this, but we must also consider the
solutions for ``2*log(-x) + log(g(x))`` since those must also
be a solution of ``eq`` which has the same value when the ``x``
in ``x**2`` is negated. If `g(x)` does not have even powers of
symbol then we don't want to replace the ``x`` there with
``-x``. So the role of the ``t`` in the expression received by
this function is to mark where ``+/-x`` should be inserted
before obtaining the Lambert solutions.
"""
nlhs, plhs = [expr.xreplace({t: sgn * symbol}) for sgn in (-1, 1)]
sols = _solve_lambert(nlhs, symbol, gens)
if plhs != nlhs:
sols.extend(_solve_lambert(plhs, symbol, gens))
# uniq is needed for a case like
# 2*log(t) - log(-z**2) + log(z + log(x) + log(z))
# where subtituting t with +/-x gives all the same solution;
# uniq, rather than list(set()), is used to maintain canonical
# order
return list(uniq(sols))
nrhs, lhs = f.as_independent(symbol, as_Add=True)
rhs = -nrhs
lamcheck = [
tmp for tmp in gens if (tmp.func in [exp, log] or (
tmp.is_Pow and symbol in tmp.exp.free_symbols))
]
if not lamcheck:
raise NotImplementedError()
if lhs.is_Add or lhs.is_Mul:
# replacing all even_degrees of symbol with dummy variable t
# since these will need special handling; non-Add/Mul do not
# need this handling
t = Dummy('t', **symbol.assumptions0)
lhs = lhs.replace(
lambda i: # find symbol**even
i.is_Pow and i.base == symbol and i.exp.is_even,
lambda i: # replace t**even
t**i.exp)
if lhs.is_Add and lhs.has(t):
t_indep = lhs.subs(t, 0)
t_term = lhs - t_indep
_rhs = rhs - t_indep
if not t_term.is_Add and _rhs and not (t_term.has(
S.ComplexInfinity, S.NaN)):
eq = expand_log(log(t_term) - log(_rhs))
return _solve_even_degree_expr(eq, t, symbol)
elif lhs.is_Mul and rhs:
# this needs to happen whether t is present or not
lhs = expand_log(log(lhs), force=True)
rhs = log(rhs)
if lhs.has(t) and lhs.is_Add:
# it expanded from Mul to Add
eq = lhs - rhs
return _solve_even_degree_expr(eq, t, symbol)
# restore symbol in lhs
lhs = lhs.xreplace({t: symbol})
lhs = powsimp(factor(lhs, deep=True))
# make sure we have inverted as completely as possible
r = Dummy()
i, lhs = _invert(lhs - r, symbol)
rhs = i.xreplace({r: rhs})
# For the first forms:
#
# 1a1) B**B = R will arrive here as B*log(B) = log(R)
# lhs is Mul so take log of both sides:
# log(B) + log(log(B)) = log(log(R))
# 1a2) B*(b*log(B) + c)**a = R will arrive unchanged so
# lhs is Mul, so take log of both sides:
# log(B) + a*log(b*log(B) + c) = log(R)
# 1b) d*log(a*B + b) + c*B = R will arrive unchanged so
# lhs is Add, so isolate c*B and expand log of both sides:
# log(c) + log(B) = log(R - d*log(a*B + b))
soln = []
if not soln:
mainlog = _mostfunc(lhs, log, symbol)
if mainlog:
if lhs.is_Mul and rhs != 0:
soln = _lambert(log(lhs) - log(rhs), symbol)
elif lhs.is_Add:
other = lhs.subs(mainlog, 0)
if other and not other.is_Add and [
tmp for tmp in other.atoms(Pow)
if symbol in tmp.free_symbols
]:
if not rhs:
diff = log(other) - log(other - lhs)
else:
diff = log(lhs - other) - log(rhs - other)
soln = _lambert(expand_log(diff), symbol)
else:
#it's ready to go
soln = _lambert(lhs - rhs, symbol)
# For the next forms,
#
# collect on main exp
# 2a) (b*B + c)*exp(d*B + g) = R
# lhs is mul, so take log of both sides:
# log(b*B + c) + d*B = log(R) - g
# 2b) g*exp(d*B + h) - b*B = R
# lhs is add, so add b*B to both sides,
# take the log of both sides and rearrange to give
# log(R + b*B) - d*B = log(g) + h
if not soln:
mainexp = _mostfunc(lhs, exp, symbol)
if mainexp:
lhs = collect(lhs, mainexp)
if lhs.is_Mul and rhs != 0:
soln = _lambert(expand_log(log(lhs) - log(rhs)), symbol)
elif lhs.is_Add:
# move all but mainexp-containing term to rhs
other = lhs.subs(mainexp, 0)
mainterm = lhs - other
rhs = rhs - other
if (mainterm.could_extract_minus_sign()
and rhs.could_extract_minus_sign()):
mainterm *= -1
rhs *= -1
diff = log(mainterm) - log(rhs)
soln = _lambert(expand_log(diff), symbol)
# For the last form:
#
# 3) d*p**(a*B + g) - b*B = c
# collect on main pow, add b*B to both sides,
# take log of both sides and rearrange to give
# a*B*log(p) - log(b*B + c) = -log(d) - g*log(p)
if not soln:
mainpow = _mostfunc(lhs, Pow, symbol)
if mainpow and symbol in mainpow.exp.free_symbols:
lhs = collect(lhs, mainpow)
if lhs.is_Mul and rhs != 0:
# b*B = 0
soln = _lambert(expand_log(log(lhs) - log(rhs)), symbol)
elif lhs.is_Add:
# move all but mainpow-containing term to rhs
other = lhs.subs(mainpow, 0)
mainterm = lhs - other
rhs = rhs - other
diff = log(mainterm) - log(rhs)
soln = _lambert(expand_log(diff), symbol)
if not soln:
raise NotImplementedError('%s does not appear to have a solution in '
'terms of LambertW' % f)
return list(ordered(soln))
def _lambert(eq, x):
"""
Given an expression assumed to be in the form
``F(X, a..f) = a*log(b*X + c) + d*X + f = 0``
where X = g(x) and x = g^-1(X), return the Lambert solution,
``x = g^-1(-c/b + (a/d)*W(d/(a*b)*exp(c*d/a/b)*exp(-f/a)))``.
"""
eq = _mexpand(expand_log(eq))
mainlog = _mostfunc(eq, log, x)
if not mainlog:
return [] # violated assumptions
other = eq.subs(mainlog, 0)
if isinstance(-other, log):
eq = (eq - other).subs(mainlog, mainlog.args[0])
mainlog = mainlog.args[0]
if not isinstance(mainlog, log):
return [] # violated assumptions
other = -(-other).args[0]
eq += other
if not x in other.free_symbols:
return [] # violated assumptions
d, f, X2 = _linab(other, x)
logterm = collect(eq - other, mainlog)
a = logterm.as_coefficient(mainlog)
if a is None or x in a.free_symbols:
return [] # violated assumptions
logarg = mainlog.args[0]
b, c, X1 = _linab(logarg, x)
if X1 != X2:
return [] # violated assumptions
# invert the generator X1 so we have x(u)
u = Dummy('rhs')
xusolns = solve(X1 - u, x)
# There are infinitely many branches for LambertW
# but only branches for k = -1 and 0 might be real. The k = 0
# branch is real and the k = -1 branch is real if the LambertW argumen
# in in range [-1/e, 0]. Since `solve` does not return infinite
# solutions we will only include the -1 branch if it tests as real.
# Otherwise, inclusion of any LambertW in the solution indicates to
# the user that there are imaginary solutions corresponding to
# different k values.
lambert_real_branches = [-1, 0]
sol = []
# solution of the given Lambert equation is like
# sol = -c/b + (a/d)*LambertW(arg, k),
# where arg = d/(a*b)*exp((c*d-b*f)/a/b) and k in lambert_real_branches.
# Instead of considering the single arg, `d/(a*b)*exp((c*d-b*f)/a/b)`,
# the individual `p` roots obtained when writing `exp((c*d-b*f)/a/b)`
# as `exp(A/p) = exp(A)**(1/p)`, where `p` is an Integer, are used.
# calculating args for LambertW
num, den = ((c * d - b * f) / a / b).as_numer_denom()
p, den = den.as_coeff_Mul()
e = exp(num / den)
t = Dummy('t')
args = [d / (a * b) * t for t in roots(t**p - e, t).keys()]
# calculating solutions from args
for arg in args:
for k in lambert_real_branches:
w = LambertW(arg, k)
if k and not w.is_real:
continue
rhs = -c / b + (a / d) * w
for xu in xusolns:
sol.append(xu.subs(u, rhs))
return sol
untouched.append(with_func)
return e.func(numsum, *substituted, *untouched, *other_non_num_terms)
def __call__(self, expr):
alt1 = super().__call__(expr)
alt2 = super().__call__(expr.factor())
return self.cheapest(alt1, alt2)
expm1_opt = FuncMinusOneOptim(exp, expm1)
cosm1_opt = FuncMinusOneOptim(cos, cosm1)
log1p_opt = ReplaceOptim(
lambda e: isinstance(e, log),
lambda l: expand_log(l.replace(log, lambda arg: log(arg.factor()))
).replace(log(_u + 1), log1p(_u)))
def create_expand_pow_optimization(limit, *, base_req=lambda b: b.is_symbol):
""" Creates an instance of :class:`ReplaceOptim` for expanding ``Pow``.
Explanation
===========
The requirements for expansions are that the base needs to be a symbol
and the exponent needs to be an Integer (and be less than or equal to
``limit``).
Parameters
==========
def _lambert(eq, x, domain=S.Complexes):
"""
Given an expression assumed to be in the form
``F(X, a..f) = a*log(b*X + c) + d*X + f = 0``
where X = g(x) and x = g^-1(X), return the Lambert solution,
``x = g^-1(-c/b + (a/d)*W(d/(a*b)*exp(c*d/a/b)*exp(-f/a)))``.
"""
eq = _mexpand(expand_log(eq))
mainlog = _mostfunc(eq, log, x)
if not mainlog:
return [] # violated assumptions
other = eq.subs(mainlog, 0)
if isinstance(-other, log):
eq = (eq - other).subs(mainlog, mainlog.args[0])
mainlog = mainlog.args[0]
if not isinstance(mainlog, log):
return [] # violated assumptions
other = -(-other).args[0]
eq += other
if not x in other.free_symbols:
return [] # violated assumptions
d, f, X2 = _linab(other, x)
logterm = collect(eq - other, mainlog)
a = logterm.as_coefficient(mainlog)
if a is None or x in a.free_symbols:
return [] # violated assumptions
logarg = mainlog.args[0]
b, c, X1 = _linab(logarg, x)
if X1 != X2:
return [] # violated assumptions
# invert the generator X1 so we have x(u)
u = Dummy('rhs')
xusolns = solve(X1 - u, x, domain=domain)
# There are infinitely many branches for LambertW
# but only branches for k = -1 and 0 might be real. The k = 0
# branch is real and the k = -1 branch is real if the LambertW argumen
# in in range [-1/e, 0]. Since `solve` does not return infinite
# solutions we will only include the -1 branch if it tests as real.
# Otherwise, inclusion of any LambertW in the solution indicates to
# the user that there are imaginary solutions corresponding to
# different k values.
lambert_real_branches = [-1, 0]
sol = []
# solution of the given Lambert equation is like
# sol = -c/b + (a/d)*LambertW(arg, k),
# where arg = d/(a*b)*exp((c*d-b*f)/a/b) and k in lambert_real_branches.
# Instead of considering the single arg, `d/(a*b)*exp((c*d-b*f)/a/b)`,
# the individual `p` roots obtained when writing `exp((c*d-b*f)/a/b)`
# as `exp(A/p) = exp(A)**(1/p)`, where `p` is an Integer, are used.
# calculating args for LambertW
num, den = ((c * d - b * f) / a / b).as_numer_denom()
p, den = den.as_coeff_Mul()
e = exp(num / den)
t = Dummy('t')
real = None
if p == 1:
t = e
args = [d / (a * b) * t]
elif domain.is_real:
ind_ls = [d / (a * b) * t for t in roots(t**p - e, t).keys()]
args = []
j = -1
for i in ind_ls:
j += 1
if not isinstance(i, int):
if not i.has(I):
args.append(ind_ls[j])
real = True
elif isinstance(i, int):
args.append(ind_ls[j])
else:
args = [d / (a * b) * t for t in roots(t**p - e, t).keys()]
if len(args) == 0:
return S.EmptySet
# calculating solutions from args
for arg in args:
if not len(list(eq.atoms(Symbol, Dummy))) > 1:
if not domain.is_subset(S.Reals) and (
re(arg) < -1 / E or
arg.has(I)) and (not LambertW(arg).is_real) and (not real):
integer = Symbol('integer', integer=True)
rhs = -c / b + (a / d) * LambertW(arg, integer)
for xu in xusolns:
sol.append(xu.subs(u, rhs))
else:
if -1 / E <= re(arg) < 0 and LambertW(arg).is_real:
for k in lambert_real_branches:
w = LambertW(arg, k)
rhs = -c / b + (a / d) * w
for xu in xusolns:
sol.append(xu.subs(u, rhs))
elif re(arg) >= 0 and LambertW(arg).is_real:
w = LambertW(arg)
rhs = -c / b + (a / d) * w
for xu in xusolns:
sol.append(xu.subs(u, rhs))
elif re(arg) < -1 / E:
return S.EmptySet
elif (arg.has(Symbol)) and not arg.has(Dummy):
if (list(arg.atoms(Symbol))[0]).is_negative and (list(
arg.atoms(Symbol))[0]).is_real and domain.is_subset(
S.Reals):
for k in lambert_real_branches:
w = LambertW(arg, k)
rhs = -c / b + (a / d) * w
if rhs.has(I) or w.has(I):
continue
for xu in xusolns:
sol.append(xu.subs(u, rhs))
if (list(arg.atoms(Symbol))[0]).is_positive and (list(
arg.atoms(Symbol))[0]).is_real and domain.is_subset(
S.Reals):
w = LambertW(arg)
rhs = -c / b + (a / d) * w
if rhs.has(I) or w.has(I):
continue
for xu in xusolns:
sol.append(xu.subs(u, rhs))
if (list(arg.atoms(Symbol))[0]).is_negative and (list(
arg.atoms(Symbol))[0]).is_real and not domain.is_subset(
S.Reals):
integer = Symbol('integer', integer=True)
rhs = -c / b + (a / d) * LambertW(arg, integer)
for xu in xusolns:
sol.append(xu.subs(u, rhs))
elif domain.is_subset(S.Reals):
for k in lambert_real_branches:
w = LambertW(arg, k)
rhs = -c / b + (a / d) * w
if rhs.has(I) or w.has(I):
continue
for xu in xusolns:
sol.append(xu.subs(u, rhs))
elif domain.is_subset(S.Complexes):
integer = Symbol('integer', integer=True)
rhs = -c / b + (a / d) * LambertW(arg, integer)
for xu in xusolns:
sol.append(xu.subs(u, rhs))
else:
if not domain.is_subset(
S.Reals) and (not LambertW(arg).is_real) and (not real):
integer = Symbol('integer', integer=True)
rhs = Lambda(integer,
-c / b + (a / d) * LambertW(arg, integer))
for xu in xusolns:
sol.append(xu.subs(u, rhs))
else:
w = LambertW(arg)
rhs = -c / b + (a / d) * w
for xu in xusolns:
sol.append(xu.subs(u, rhs))
return list(set(sol))
