def _find_opts(expr):
if expr.is_Atom or expr.is_Order:
return
if iterable(expr):
list(map(_find_opts, expr))
return
if expr in seen_subexp:
return expr
seen_subexp.add(expr)
list(map(_find_opts, expr.args))
if _coeff_isneg(expr):
neg_expr = -expr
if not neg_expr.is_Atom:
opt_subs[expr] = Mul(S.NegativeOne, neg_expr, evaluate=False)
seen_subexp.add(neg_expr)
expr = neg_expr
if expr.is_Mul:
muls.add(expr)
elif expr.is_Add:
adds.add(expr)
elif expr.is_Pow:
if _coeff_isneg(expr.exp):
opt_subs[expr] = Pow(Pow(expr.base, -expr.exp), S.NegativeOne,
evaluate=False)
def as_numer_denom(self):
if not self.is_commutative:
return self, S.One
base, exp = self.as_base_exp()
n, d = base.as_numer_denom()
if d is not S.One:
if d.is_negative and n.is_negative:
n, d = -n, -d
if exp.is_Integer:
if exp.is_negative:
n, d = d, n
exp = -exp
return Pow(n, exp), Pow(d, exp)
elif exp.is_Rational or d.is_positive:
dneg = d.is_negative
if dneg is not None:
if dneg is True:
n = -n
d = -d
elif dneg is False:
n, d = d, n
exp = -exp
if _coeff_isneg(exp):
n, d = d, n
exp = -exp
return Pow(n, exp), Pow(d, exp)
# else we won't split up base but we check for neg expo below
if _coeff_isneg(exp):
return S.One, base**-exp
# unprocessed float or NumberSymbol exponent
# and Mul exp w/o negative sign
return self, S.One
def _find_opts(expr):
if not isinstance(expr, Basic):
return
if expr.is_Atom or expr.is_Order:
return
if iterable(expr):
list(map(_find_opts, expr))
return
if expr in seen_subexp:
return expr
seen_subexp.add(expr)
list(map(_find_opts, expr.args))
if _coeff_isneg(expr):
neg_expr = -expr
if not neg_expr.is_Atom:
opt_subs[expr] = Mul(S.NegativeOne, neg_expr, evaluate=False)
seen_subexp.add(neg_expr)
expr = neg_expr
if isinstance(expr, (Mul, MatMul)):
muls.add(expr)
elif isinstance(expr, (Add, MatAdd)):
adds.add(expr)
elif isinstance(expr, (Pow, MatPow)):
if _coeff_isneg(expr.exp):
opt_subs[expr] = Pow(Pow(expr.base, -expr.exp), S.NegativeOne,
evaluate=False)
def eval(cls, arg):
from sympy import acos
arg = sympify(arg)
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.Infinity
elif arg is S.NegativeInfinity:
return S.Infinity
elif arg is S.Zero:
return S.Pi*S.ImaginaryUnit / 2
elif arg is S.One:
return S.Zero
elif arg is S.NegativeOne:
return S.Pi*S.ImaginaryUnit
if arg.is_number:
cst_table = {
S.ImaginaryUnit: log(S.ImaginaryUnit*(1 + sqrt(2))),
-S.ImaginaryUnit: log(-S.ImaginaryUnit*(1 + sqrt(2))),
S.Half: S.Pi/3,
-S.Half: 2*S.Pi/3,
sqrt(2)/2: S.Pi/4,
-sqrt(2)/2: 3*S.Pi/4,
1/sqrt(2): S.Pi/4,
-1/sqrt(2): 3*S.Pi/4,
sqrt(3)/2: S.Pi/6,
-sqrt(3)/2: 5*S.Pi/6,
(sqrt(3) - 1)/sqrt(2**3): 5*S.Pi/12,
-(sqrt(3) - 1)/sqrt(2**3): 7*S.Pi/12,
sqrt(2 + sqrt(2))/2: S.Pi/8,
-sqrt(2 + sqrt(2))/2: 7*S.Pi/8,
sqrt(2 - sqrt(2))/2: 3*S.Pi/8,
-sqrt(2 - sqrt(2))/2: 5*S.Pi/8,
(1 + sqrt(3))/(2*sqrt(2)): S.Pi/12,
-(1 + sqrt(3))/(2*sqrt(2)): 11*S.Pi/12,
(sqrt(5) + 1)/4: S.Pi/5,
-(sqrt(5) + 1)/4: 4*S.Pi/5
}
if arg in cst_table:
if arg.is_real:
return cst_table[arg]*S.ImaginaryUnit
return cst_table[arg]
if arg is S.ComplexInfinity:
return S.Infinity
i_coeff = arg.as_coefficient(S.ImaginaryUnit)
if i_coeff is not None:
if _coeff_isneg(i_coeff):
return S.ImaginaryUnit * acos(i_coeff)
return S.ImaginaryUnit * acos(-i_coeff)
else:
if _coeff_isneg(arg):
return -cls(-arg)
def eval(cls, arg):
from sympy import cot
arg = sympify(arg)
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.One
elif arg is S.NegativeInfinity:
return S.NegativeOne
elif arg is S.Zero:
return S.ComplexInfinity
elif arg.is_negative:
return -cls(-arg)
else:
if arg is S.ComplexInfinity:
return S.NaN
i_coeff = arg.as_coefficient(S.ImaginaryUnit)
if i_coeff is not None:
if _coeff_isneg(i_coeff):
return S.ImaginaryUnit * cot(-i_coeff)
return -S.ImaginaryUnit * cot(i_coeff)
else:
if _coeff_isneg(arg):
return -cls(-arg)
if arg.is_Add:
x, m = _peeloff_ipi(arg)
if m:
cothm = coth(m)
if cothm is S.ComplexInfinity:
return coth(x)
else: # cothm == 0
return tanh(x)
if arg.func == asinh:
x = arg.args[0]
return sqrt(1 + x**2)/x
if arg.func == acosh:
x = arg.args[0]
return x/(sqrt(x - 1) * sqrt(x + 1))
if arg.func == atanh:
return 1/arg.args[0]
if arg.func == acoth:
return arg.args[0]
def eval(cls, arg):
arg = sympify(arg)
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.Zero
elif arg is S.NegativeInfinity:
return S.Zero
elif arg is S.Zero:
return S.Pi*S.ImaginaryUnit / 2
elif arg is S.One:
return S.Infinity
elif arg is S.NegativeOne:
return S.NegativeInfinity
elif arg.is_negative:
return -cls(-arg)
else:
if arg is S.ComplexInfinity:
return 0
i_coeff = arg.as_coefficient(S.ImaginaryUnit)
if i_coeff is not None:
return -S.ImaginaryUnit * C.acot(i_coeff)
else:
if _coeff_isneg(arg):
return -cls(-arg)
def _set_function(f, self):
expr = f.expr
if not isinstance(expr, Expr):
return
if len(f.variables) > 1:
return
n = f.variables[0]
# f(x) + c and f(-x) + c cover the same integers
# so choose the form that has the fewest negatives
c = f(0)
fx = f(n) - c
f_x = f(-n) - c
neg_count = lambda e: sum(_coeff_isneg(_) for _ in Add.make_args(e))
if neg_count(f_x) < neg_count(fx):
expr = f_x + c
a = Wild('a', exclude=[n])
b = Wild('b', exclude=[n])
match = expr.match(a*n + b)
if match and match[a]:
# canonical shift
expr = match[a]*n + match[b] % match[a]
if expr != f.expr:
return ImageSet(Lambda(n, expr), S.Integers)
def as_numer_denom(self):
if not self.is_commutative:
return self, S.One
base, exp = self.as_base_exp()
n, d = base.as_numer_denom()
# this should be the same as ExpBase.as_numer_denom wrt
# exponent handling
neg_exp = exp.is_negative
if not neg_exp and not (-exp).is_negative:
neg_exp = _coeff_isneg(exp)
int_exp = exp.is_integer
# the denominator cannot be separated from the numerator if
# its sign is unknown unless the exponent is an integer, e.g.
# sqrt(a/b) != sqrt(a)/sqrt(b) when a=1 and b=-1. But if the
# denominator is negative the numerator and denominator can
# be negated and the denominator (now positive) separated.
if not (d.is_real or int_exp):
n = base
d = S.One
dnonpos = d.is_nonpositive
if dnonpos:
n, d = -n, -d
elif dnonpos is None and not int_exp:
n = base
d = S.One
if neg_exp:
n, d = d, n
exp = -exp
return self.func(n, exp), self.func(d, exp)
def eval(cls, arg):
from sympy import atan
arg = sympify(arg)
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Zero:
return S.Zero
elif arg is S.One:
return S.Infinity
elif arg is S.NegativeOne:
return S.NegativeInfinity
elif arg is S.Infinity:
return -S.ImaginaryUnit * atan(arg)
elif arg is S.NegativeInfinity:
return S.ImaginaryUnit * atan(-arg)
elif arg.is_negative:
return -cls(-arg)
else:
if arg is S.ComplexInfinity:
from sympy.calculus.util import AccumBounds
return S.ImaginaryUnit*AccumBounds(-S.Pi/2, S.Pi/2)
i_coeff = arg.as_coefficient(S.ImaginaryUnit)
if i_coeff is not None:
return S.ImaginaryUnit * atan(i_coeff)
else:
if _coeff_isneg(arg):
return -cls(-arg)
def _print_Mul(self, expr):
if _coeff_isneg(expr):
x = self.dom.createElement('apply')
x.appendChild(self.dom.createElement('minus'))
x.appendChild(self._print_Mul(-expr))
return x
from sympy.simplify import fraction
numer, denom = fraction(expr)
if denom is not S.One:
x = self.dom.createElement('apply')
x.appendChild(self.dom.createElement('divide'))
x.appendChild(self._print(numer))
x.appendChild(self._print(denom))
return x
coeff, terms = expr.as_coeff_mul()
if coeff is S.One and len(terms) == 1:
# XXX since the negative coefficient has been handled, I don't
# think a coeff of 1 can remain
return self._print(terms[0])
if self.order != 'old':
terms = Mul._from_args(terms).as_ordered_factors()
x = self.dom.createElement('apply')
x.appendChild(self.dom.createElement('times'))
if(coeff != 1):
x.appendChild(self._print(coeff))
for term in terms:
x.appendChild(self._print(term))
return x
def _print_Add(self, expr, order=None):
args = self._as_ordered_terms(expr, order=order)
lastProcessed = self._print(args[0])
plusNodes = []
for arg in args[1:]:
if _coeff_isneg(arg):
# use minus
x = self.dom.createElement('apply')
x.appendChild(self.dom.createElement('minus'))
x.appendChild(lastProcessed)
x.appendChild(self._print(-arg))
# invert expression since this is now minused
lastProcessed = x
if(arg == args[-1]):
plusNodes.append(lastProcessed)
else:
plusNodes.append(lastProcessed)
lastProcessed = self._print(arg)
if(arg == args[-1]):
plusNodes.append(self._print(arg))
if len(plusNodes) == 1:
return lastProcessed
x = self.dom.createElement('apply')
x.appendChild(self.dom.createElement('plus'))
while len(plusNodes) > 0:
x.appendChild(plusNodes.pop(0))
return x
def eval(cls, arg):
from sympy import asin
arg = sympify(arg)
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.Infinity
elif arg is S.NegativeInfinity:
return S.NegativeInfinity
elif arg is S.Zero:
return S.Zero
elif arg is S.One:
return log(sqrt(2) + 1)
elif arg is S.NegativeOne:
return log(sqrt(2) - 1)
elif arg.is_negative:
return -cls(-arg)
else:
if arg is S.ComplexInfinity:
return S.ComplexInfinity
i_coeff = arg.as_coefficient(S.ImaginaryUnit)
if i_coeff is not None:
return S.ImaginaryUnit * asin(i_coeff)
else:
if _coeff_isneg(arg):
return -cls(-arg)
def _print_Add(self, expr, order=None):
if self.order == 'none':
terms = list(expr.args)
else:
terms = self._as_ordered_terms(expr, order=order)
pforms, indices = [], []
def pretty_negative(pform, index):
"""Prepend a minus sign to a pretty form. """
if index == 0:
if pform.height() > 1:
pform_neg = '- '
else:
pform_neg = '-'
else:
pform_neg = ' - '
pform = stringPict.next(pform_neg, pform)
return prettyForm(binding=prettyForm.NEG, *pform)
for i, term in enumerate(terms):
if term.is_Mul and _coeff_isneg(term):
pform = self._print(-term)
pforms.append(pretty_negative(pform, i))
elif term.is_Rational and term.q > 1:
pforms.append(None)
indices.append(i)
elif term.is_Number and term < 0:
pform = self._print(-term)
pforms.append(pretty_negative(pform, i))
else:
pforms.append(self._print(term))
if indices:
large = True
for pform in pforms:
if pform is not None and pform.height() > 1:
break
else:
large = False
for i in indices:
term, negative = terms[i], False
if term < 0:
term, negative = -term, True
if large:
pform = prettyForm(str(term.p))/prettyForm(str(term.q))
else:
pform = self._print(term)
if negative:
pform = pretty_negative(pform, i)
pforms[i] = pform
return prettyForm.__add__(*pforms)
def _print_Add(self, expr):
tex = str(self._print(expr.args[0]))
for term in expr.args[1:]:
if _coeff_isneg(term):
tex += r" %s" % self._print(term)
else:
tex += r" + %s" % self._print(term)
return tex
def eval(cls, arg):
from sympy import tan
arg = sympify(arg)
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.One
elif arg is S.NegativeInfinity:
return S.NegativeOne
elif arg is S.Zero:
return S.Zero
elif arg.is_negative:
return -cls(-arg)
else:
if arg is S.ComplexInfinity:
return S.NaN
i_coeff = arg.as_coefficient(S.ImaginaryUnit)
if i_coeff is not None:
if _coeff_isneg(i_coeff):
return -S.ImaginaryUnit * tan(-i_coeff)
return S.ImaginaryUnit * tan(i_coeff)
else:
if _coeff_isneg(arg):
return -cls(-arg)
if arg.func == asinh:
x = arg.args[0]
return x / sqrt(1 + x ** 2)
if arg.func == acosh:
x = arg.args[0]
return sqrt(x - 1) * sqrt(x + 1) / x
if arg.func == atanh:
return arg.args[0]
if arg.func == acoth:
return 1 / arg.args[0]
def _print_Add(self, expr, order=None):
if self.order == "none":
terms = list(expr.args)
else:
terms = self._as_ordered_terms(expr, order=order)
tex = self._print(terms[0])
for term in terms[1:]:
if not _coeff_isneg(term):
tex += " +"
tex += " " + self._print(term)
return tex
def _print_Mul(self, expr):
prec = precedence(expr)
if _coeff_isneg(expr):
expr = -expr
sign = "-"
else:
sign = ""
a = [] # items in the numerator
b = [] # items that are in the denominator (if any)
if self.order not in ('old', 'none'):
args = expr.as_ordered_factors()
else:
# use make_args in case expr was something like -x -> x
args = Mul.make_args(expr)
# Gather args for numerator/denominator
for item in args:
if item.is_commutative and item.is_Pow and item.exp.is_Rational and item.exp.is_negative:
if item.exp != -1:
b.append(Pow(item.base, -item.exp, evaluate=False))
else:
b.append(Pow(item.base, -item.exp))
elif item.is_Rational and item is not S.Infinity:
if item.p != 1:
a.append(Rational(item.p))
if item.q != 1:
b.append(Rational(item.q))
else:
a.append(item)
a = a or [S.One]
a_str = map(lambda x:self.parenthesize(x, prec), a)
b_str = map(lambda x:self.parenthesize(x, prec), b)
if len(b) == 0:
return sign + '*'.join(a_str)
elif len(b) == 1:
if len(a) == 1 and not (a[0].is_Atom or a[0].is_Add):
return sign + "%s/"%a_str[0] + '*'.join(b_str)
else:
return sign + '*'.join(a_str) + "/%s"%b_str[0]
else:
return sign + '*'.join(a_str) + "/(%s)"%'*'.join(b_str)
def eval(cls, arg):
from sympy import sin
arg = sympify(arg)
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.Infinity
elif arg is S.NegativeInfinity:
return S.NegativeInfinity
elif arg is S.Zero:
return S.Zero
elif arg.is_negative:
return -cls(-arg)
else:
if arg is S.ComplexInfinity:
return S.NaN
i_coeff = arg.as_coefficient(S.ImaginaryUnit)
if i_coeff is not None:
return S.ImaginaryUnit * sin(i_coeff)
else:
if _coeff_isneg(arg):
return -cls(-arg)
if arg.is_Add:
x, m = _peeloff_ipi(arg)
if m:
return sinh(m)*cosh(x) + cosh(m)*sinh(x)
if arg.func == asinh:
return arg.args[0]
if arg.func == acosh:
x = arg.args[0]
return sqrt(x - 1) * sqrt(x + 1)
if arg.func == atanh:
x = arg.args[0]
return x/sqrt(1 - x**2)
if arg.func == acoth:
x = arg.args[0]
return 1/(sqrt(x - 1) * sqrt(x + 1))
def _print_Mul(self, expr):
def multiply(expr, mrow):
from sympy.simplify import fraction
numer, denom = fraction(expr)
if denom is not S.One:
frac = self.dom.createElement('mfrac')
if self._settings["fold_short_frac"] and len(str(expr)) < 7:
frac.setAttribute('bevelled', 'true')
xnum = self._print(numer)
xden = self._print(denom)
frac.appendChild(xnum)
frac.appendChild(xden)
mrow.appendChild(frac)
return mrow
coeff, terms = expr.as_coeff_mul()
if coeff is S.One and len(terms) == 1:
mrow.appendChild(self._print(terms[0]))
return mrow
if self.order != 'old':
terms = Mul._from_args(terms).as_ordered_factors()
if coeff != 1:
x = self._print(coeff)
y = self.dom.createElement('mo')
y.appendChild(self.dom.createTextNode(self.mathml_tag(expr)))
mrow.appendChild(x)
mrow.appendChild(y)
for term in terms:
x = self._print(term)
mrow.appendChild(x)
if not term == terms[-1]:
y = self.dom.createElement('mo')
y.appendChild(self.dom.createTextNode(self.mathml_tag(expr)))
mrow.appendChild(y)
return mrow
mrow = self.dom.createElement('mrow')
if _coeff_isneg(expr):
x = self.dom.createElement('mo')
x.appendChild(self.dom.createTextNode('-'))
mrow.appendChild(x)
mrow = multiply(-expr, mrow)
else:
mrow = multiply(expr, mrow)
return mrow
def clear_coefficients(expr, rhs=S.Zero):
"""Return `p, r` where `p` is the expression obtained when Rational
additive and multiplicative coefficients of `expr` have been stripped
away in a naive fashion (i.e. without simplification). The operations
needed to remove the coefficients will be applied to `rhs` and returned
as `r`.
Examples
========
>>> from sympy.simplify.simplify import clear_coefficients
>>> from sympy.abc import x, y
>>> from sympy import Dummy
>>> expr = 4*y*(6*x + 3)
>>> clear_coefficients(expr - 2)
(y*(2*x + 1), 1/6)
When solving 2 or more expressions like `expr = a`,
`expr = b`, etc..., it is advantageous to provide a Dummy symbol
for `rhs` and simply replace it with `a`, `b`, etc... in `r`.
>>> rhs = Dummy('rhs')
>>> clear_coefficients(expr, rhs)
(y*(2*x + 1), _rhs/12)
>>> _[1].subs(rhs, 2)
1/6
"""
was = None
free = expr.free_symbols
if expr.is_Rational:
return (S.Zero, rhs - expr)
while expr and was != expr:
was = expr
m, expr = (
expr.as_content_primitive()
if free else
factor_terms(expr).as_coeff_Mul(rational=True))
rhs /= m
c, expr = expr.as_coeff_Add(rational=True)
rhs -= c
expr = signsimp(expr, evaluate = False)
if _coeff_isneg(expr):
expr = -expr
rhs = -rhs
return expr, rhs
def _print_Add(self, expr, order=None):
mrow = self.dom.createElement('mrow')
args = self._as_ordered_terms(expr, order=order)
mrow.appendChild(self._print(args[0]))
for arg in args[1:]:
if _coeff_isneg(arg):
# use minus
x = self.dom.createElement('mo')
x.appendChild(self.dom.createTextNode('-'))
y = self._print(-arg)
# invert expression since this is now minused
else:
x = self.dom.createElement('mo')
x.appendChild(self.dom.createTextNode('+'))
y = self._print(arg)
mrow.appendChild(x)
mrow.appendChild(y)
return mrow
def __new__(cls, b, e, evaluate=None):
if evaluate is None:
evaluate = global_evaluate[0]
from sympy.functions.elementary.exponential import exp_polar
b = _sympify(b)
e = _sympify(e)
if evaluate:
if e is S.Zero:
return S.One
elif e is S.One:
return b
elif e.is_integer and _coeff_isneg(b):
if e.is_even:
b = -b
elif e.is_odd:
return -Pow(-b, e)
if b is S.One:
if e in (S.NaN, S.Infinity, -S.Infinity):
return S.NaN
return S.One
elif S.NaN in (b, e): # XXX S.NaN**x -> S.NaN under assumption that x != 0
return S.NaN
else:
# recognize base as E
if not e.is_Atom and b is not S.Exp1 and b.func is not exp_polar:
from sympy import numer, denom, log, sign, im, factor_terms
c, ex = factor_terms(e, sign=False).as_coeff_Mul()
den = denom(ex)
if den.func is log and den.args[0] == b:
return S.Exp1**(c*numer(ex))
elif den.is_Add:
s = sign(im(b))
if s.is_Number and s and den == \
log(-factor_terms(b, sign=False)) + s*S.ImaginaryUnit*S.Pi:
return S.Exp1**(c*numer(ex))
obj = b._eval_power(e)
if obj is not None:
return obj
obj = Expr.__new__(cls, b, e)
obj.is_commutative = (b.is_commutative and e.is_commutative)
return obj
def eval(cls, arg):
arg = sympify(arg)
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.Zero
elif arg is S.NegativeInfinity:
return S.Zero
elif arg is S.Zero:
return S.ComplexInfinity
elif arg is S.One:
return log(1 + sqrt(2))
elif arg is S.NegativeOne:
return - log(1 + sqrt(2))
if arg.is_number:
cst_table = {
S.ImaginaryUnit: -S.Pi / 2,
S.ImaginaryUnit*(sqrt(2) + sqrt(6)): -S.Pi / 12,
S.ImaginaryUnit*(1 + sqrt(5)): -S.Pi / 10,
S.ImaginaryUnit*2 / sqrt(2 - sqrt(2)): -S.Pi / 8,
S.ImaginaryUnit*2: -S.Pi / 6,
S.ImaginaryUnit*sqrt(2 + 2/sqrt(5)): -S.Pi / 5,
S.ImaginaryUnit*sqrt(2): -S.Pi / 4,
S.ImaginaryUnit*(sqrt(5)-1): -3*S.Pi / 10,
S.ImaginaryUnit*2 / sqrt(3): -S.Pi / 3,
S.ImaginaryUnit*2 / sqrt(2 + sqrt(2)): -3*S.Pi / 8,
S.ImaginaryUnit*sqrt(2 - 2/sqrt(5)): -2*S.Pi / 5,
S.ImaginaryUnit*(sqrt(6) - sqrt(2)): -5*S.Pi / 12,
S(2): -S.ImaginaryUnit*log((1+sqrt(5))/2),
}
if arg in cst_table:
return cst_table[arg]*S.ImaginaryUnit
if arg is S.ComplexInfinity:
return S.Zero
if _coeff_isneg(arg):
return -cls(-arg)
def _print_Add(self, expr, order=None):
if(self._settings['use_operators']):
return CodePrinter._print_Add(self, expr, order=order)
terms = expr.as_ordered_terms()
def partition(p,l):
return reduce(lambda x, y: (x[0]+[y], x[1]) if p(y) else (x[0], x[1]+[y]), l, ([], []))
def add(a,b):
return self._print_Function_with_args('add', (a, b))
# return self.known_functions['add']+'(%s, %s)' % (a,b)
neg, pos = partition(lambda arg: _coeff_isneg(arg), terms)
s = pos = reduce(lambda a,b: add(a,b), map(lambda t: self._print(t),pos))
if(len(neg) > 0):
# sum the absolute values of the negative terms
neg = reduce(lambda a,b: add(a,b), map(lambda n: self._print(-n),neg))
# then subtract them from the positive terms
s = self._print_Function_with_args('sub', (pos,neg))
# s = self.known_functions['sub']+'(%s, %s)' % (pos,neg)
return s
def eval(cls, arg):
arg = sympify(arg)
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.Infinity
elif arg is S.NegativeInfinity:
return S.Infinity
elif arg is S.Zero:
return S.One
elif arg.is_negative:
return cls(-arg)
else:
if arg is S.ComplexInfinity:
return S.NaN
i_coeff = arg.as_coefficient(S.ImaginaryUnit)
if i_coeff is not None:
return C.cos(i_coeff)
else:
if _coeff_isneg(arg):
return cls(-arg)
if arg.func == asinh:
return sqrt(1+arg.args[0]**2)
if arg.func == acosh:
return arg.args[0]
if arg.func == atanh:
return 1/sqrt(1-arg.args[0]**2)
if arg.func == acoth:
x = arg.args[0]
return x/(sqrt(x-1) * sqrt(x+1))
def as_numer_denom(self):
"""
Returns this with a positive exponent as a 2-tuple (a fraction).
Examples
========
>>> from sympy.functions import exp
>>> from sympy.abc import x
>>> exp(-x).as_numer_denom()
(1, exp(x))
>>> exp(x).as_numer_denom()
(exp(x), 1)
"""
# this should be the same as Pow.as_numer_denom wrt
# exponent handling
exp = self.exp
neg_exp = exp.is_negative
if not neg_exp and not (-exp).is_negative:
neg_exp = _coeff_isneg(exp)
if neg_exp:
return S.One, self.func(-exp)
return self, S.One
def _print_Mul(self, expr):
prec = precedence(expr)
# Check for unevaluated Mul. In this case we need to make sure the
# identities are visible, multiple Rational factors are not combined
# etc so we display in a straight-forward form that fully preserves all
# args and their order.
args = expr.args
if args[0] is S.One or any(
isinstance(a, Number) or a.is_Pow and all(ai.is_Integer
for ai in a.args)
for a in args[1:]):
d, n = sift(args,
lambda x: isinstance(x, Pow) and bool(
x.exp.as_coeff_Mul()[0] < 0),
binary=True)
for i, di in enumerate(d):
if di.exp.is_Number:
e = -di.exp
else:
dargs = list(di.exp.args)
dargs[0] = -dargs[0]
e = Mul._from_args(dargs)
d[i] = Pow(di.base, e, evaluate=False) if e - 1 else di.base
# don't parenthesize first factor if negative
if _coeff_isneg(n[0]):
pre = [str(n.pop(0))]
else:
pre = []
nfactors = pre + [
self.parenthesize(a, prec, strict=False) for a in n
]
# don't parenthesize first of denominator unless singleton
if len(d) > 1 and _coeff_isneg(d[0]):
pre = [str(d.pop(0))]
else:
pre = []
dfactors = pre + [
self.parenthesize(a, prec, strict=False) for a in d
]
n = '*'.join(nfactors)
d = '*'.join(dfactors)
if len(dfactors) > 1:
return '%s/(%s)' % (n, d)
elif dfactors:
return '%s/%s' % (n, d)
return n
c, e = expr.as_coeff_Mul()
if c < 0:
expr = _keep_coeff(-c, e)
sign = "-"
else:
sign = ""
a = [] # items in the numerator
b = [] # items that are in the denominator (if any)
pow_paren = [
] # Will collect all pow with more than one base element and exp = -1
if self.order not in ('old', 'none'):
args = expr.as_ordered_factors()
else:
# use make_args in case expr was something like -x -> x
args = Mul.make_args(expr)
# Gather args for numerator/denominator
def apow(i):
b, e = i.as_base_exp()
eargs = list(Mul.make_args(e))
if eargs[0] is S.NegativeOne:
eargs = eargs[1:]
else:
eargs[0] = -eargs[0]
e = Mul._from_args(eargs)
if isinstance(i, Pow):
return i.func(b, e, evaluate=False)
return i.func(e, evaluate=False)
for item in args:
if (item.is_commutative and isinstance(item, Pow)
and bool(item.exp.as_coeff_Mul()[0] < 0)):
if item.exp is not S.NegativeOne:
b.append(apow(item))
else:
if (len(item.args[0].args) != 1
and isinstance(item.base, (Mul, Pow))):
# To avoid situations like #14160
pow_paren.append(item)
b.append(item.base)
elif item.is_Rational and item is not S.Infinity:
if item.p != 1:
a.append(Rational(item.p))
if item.q != 1:
b.append(Rational(item.q))
else:
a.append(item)
a = a or [S.One]
a_str = [self.parenthesize(x, prec, strict=False) for x in a]
b_str = [self.parenthesize(x, prec, strict=False) for x in b]
# To parenthesize Pow with exp = -1 and having more than one Symbol
for item in pow_paren:
if item.base in b:
b_str[b.index(item.base)] = "(%s)" % b_str[b.index(item.base)]
if not b:
return sign + '*'.join(a_str)
elif len(b) == 1:
return sign + '*'.join(a_str) + "/" + b_str[0]
else:
return sign + '*'.join(a_str) + "/(%s)" % '*'.join(b_str)
def cse(exprs, symbols=None, optimizations=None, postprocess=None):
""" Perform common subexpression elimination on an expression.
Parameters
==========
exprs : list of sympy expressions, or a single sympy expression
The expressions to reduce.
symbols : infinite iterator yielding unique Symbols
The symbols used to label the common subexpressions which are pulled
out. The ``numbered_symbols`` generator is useful. The default is a
stream of symbols of the form "x0", "x1", etc. This must be an infinite
iterator.
optimizations : list of (callable, callable) pairs, optional
The (preprocessor, postprocessor) pairs. If not provided,
``sympy.simplify.cse.cse_optimizations`` is used.
postprocess : a function which accepts the two return values of cse and
returns the desired form of output from cse, e.g. if you want the
replacements reversed the function might be the following lambda:
lambda r, e: return reversed(r), e
Returns
=======
replacements : list of (Symbol, expression) pairs
All of the common subexpressions that were replaced. Subexpressions
earlier in this list might show up in subexpressions later in this list.
reduced_exprs : list of sympy expressions
The reduced expressions with all of the replacements above.
"""
from sympy.matrices import Matrix
from sympy.simplify.simplify import fraction
if symbols is None:
symbols = numbered_symbols()
else:
# In case we get passed an iterable with an __iter__ method instead of
# an actual iterator.
symbols = iter(symbols)
seen_subexp = set()
muls = set()
adds = set()
to_eliminate = []
to_eliminate_ops_count = []
if optimizations is None:
# Pull out the default here just in case there are some weird
# manipulations of the module-level list in some other thread.
optimizations = list(cse_optimizations)
# Handle the case if just one expression was passed.
if isinstance(exprs, Basic):
exprs = [exprs]
# Preprocess the expressions to give us better optimization opportunities.
reduced_exprs = [preprocess_for_cse(e, optimizations) for e in exprs]
# Find all of the repeated subexpressions.
def insert(subtree):
'''This helper will insert the subtree into to_eliminate while
maintaining the ordering by op count and will skip the insertion
if subtree is already present.'''
ops_count = (subtree.count_ops(), subtree.is_Mul) # prefer non-Mul to Mul
index_to_insert = bisect.bisect(to_eliminate_ops_count, ops_count)
# all i up to this index have op count <= the current op count
# so check that subtree is not yet present from this index down
# (if necessary) to zero.
for i in xrange(index_to_insert - 1, -1, -1):
if to_eliminate_ops_count[i] == ops_count and \
subtree == to_eliminate[i]:
return # already have it
to_eliminate_ops_count.insert(index_to_insert, ops_count)
to_eliminate.insert(index_to_insert, subtree)
for expr in reduced_exprs:
if not isinstance(expr, Basic):
continue
pt = preorder_traversal(expr, key=default_sort_key)
for subtree in pt:
inv = 1/subtree if subtree.is_Pow else None
if subtree.is_Atom or iterable(subtree) or inv and inv.is_Atom:
# Exclude atoms, since there is no point in renaming them.
continue
if subtree in seen_subexp:
if inv and _coeff_isneg(subtree.exp):
# save the form with positive exponent
subtree = inv
insert(subtree)
pt.skip()
continue
if inv and inv in seen_subexp:
if _coeff_isneg(subtree.exp):
# save the form with positive exponent
subtree = inv
insert(subtree)
pt.skip()
continue
elif subtree.is_Mul:
muls.add(subtree)
elif subtree.is_Add:
adds.add(subtree)
seen_subexp.add(subtree)
# process adds - any adds that weren't repeated might contain
# subpatterns that are repeated, e.g. x+y+z and x+y have x+y in common
adds = [set(a.args) for a in adds]
for i in xrange(len(adds)):
for j in xrange(i + 1, len(adds)):
com = adds[i].intersection(adds[j])
if len(com) > 1:
insert(Add(*com))
# remove this set of symbols so it doesn't appear again
adds[i] = adds[i].difference(com)
adds[j] = adds[j].difference(com)
for k in xrange(j + 1, len(adds)):
if not com.difference(adds[k]):
adds[k] = adds[k].difference(com)
# process muls - any muls that weren't repeated might contain
# subpatterns that are repeated, e.g. x*y*z and x*y have x*y in common
# use SequenceMatcher on the nc part to find the longest common expression
# in common between the two nc parts
sm = difflib.SequenceMatcher()
muls = [a.args_cnc(cset=True) for a in muls]
for i in xrange(len(muls)):
if muls[i][1]:
sm.set_seq1(muls[i][1])
for j in xrange(i + 1, len(muls)):
# the commutative part in common
ccom = muls[i][0].intersection(muls[j][0])
# the non-commutative part in common
if muls[i][1] and muls[j][1]:
# see if there is any chance of an nc match
ncom = set(muls[i][1]).intersection(set(muls[j][1]))
if len(ccom) + len(ncom) < 2:
continue
# now work harder to find the match
sm.set_seq2(muls[j][1])
i1, _, n = sm.find_longest_match(0, len(muls[i][1]),
0, len(muls[j][1]))
ncom = muls[i][1][i1:i1 + n]
else:
ncom = []
com = list(ccom) + ncom
if len(com) < 2:
continue
insert(Mul(*com))
# remove ccom from all if there was no ncom; to update the nc part
# would require finding the subexpr and then replacing it with a
# dummy to keep bounding nc symbols from being identified as a
# subexpr, e.g. removing B*C from A*B*C*D might allow A*D to be
# identified as a subexpr which would not be right.
if not ncom:
muls[i][0] = muls[i][0].difference(ccom)
for k in xrange(j, len(muls)):
if not ccom.difference(muls[k][0]):
muls[k][0] = muls[k][0].difference(ccom)
# Substitute symbols for all of the repeated subexpressions.
replacements = []
reduced_exprs = list(reduced_exprs)
hit = True
for i, subtree in enumerate(to_eliminate):
if hit:
sym = symbols.next()
hit = False
if subtree.is_Pow and subtree.exp.is_Rational:
update = lambda x: x.xreplace({subtree: sym, 1/subtree: 1/sym})
else:
update = lambda x: x.subs(subtree, sym)
# Make the substitution in all of the target expressions.
for j, expr in enumerate(reduced_exprs):
old = reduced_exprs[j]
reduced_exprs[j] = update(expr)
hit = hit or (old != reduced_exprs[j])
# Make the substitution in all of the subsequent substitutions.
for j in range(i+1, len(to_eliminate)):
old = to_eliminate[j]
to_eliminate[j] = update(to_eliminate[j])
hit = hit or (old != to_eliminate[j])
if hit:
replacements.append((sym, subtree))
# Postprocess the expressions to return the expressions to canonical form.
for i, (sym, subtree) in enumerate(replacements):
subtree = postprocess_for_cse(subtree, optimizations)
replacements[i] = (sym, subtree)
reduced_exprs = [postprocess_for_cse(e, optimizations)
for e in reduced_exprs]
if isinstance(exprs, Matrix):
reduced_exprs = [Matrix(exprs.rows, exprs.cols, reduced_exprs)]
if postprocess is None:
return replacements, reduced_exprs
return postprocess(replacements, reduced_exprs)
def precedence_Mul(item):
if _coeff_isneg(item):
return PRECEDENCE["Add"]
return PRECEDENCE["Mul"]
def _print_Add(self, expr, order=None):
"""Print a Add object.
:param expr: The expression.
:rtype : bce.dom.mathml.all.Base
:return: The printed MathML object.
"""
assert isinstance(expr, _sympy.Add)
args = self._as_ordered_terms(expr, order=order)
PREC = _sympy_precedence.precedence(expr)
dt = _mathml.RowComponent()
args_len = len(args)
# Iterator each part.
for arg_id in range(0, args_len):
cur_arg = args[arg_id]
if cur_arg.is_negative:
# Get the negative number.
neg_arg = -cur_arg
# Get the precedence.
CUR_PREC = _sympy_precedence.precedence(neg_arg)
# Add a '-' operator.
dt.append_object(
_mathml.OperatorComponent(_mathml.OPERATOR_MINUS))
# noinspection PyProtectedMember
if CUR_PREC < PREC or (_coeff_isneg(neg_arg) and arg_id != 0):
# Insert the argument with parentheses around.
dt.append_object(
_mathml.OperatorComponent(
_mathml.OPERATOR_LEFT_PARENTHESIS))
dt.append_object(self._print(neg_arg))
dt.append_object(
_mathml.OperatorComponent(
_mathml.OPERATOR_RIGHT_PARENTHESIS))
else:
# Insert the argument only.
dt.append_object(self._print(neg_arg))
else:
# Add a '+' operator if the argument is not the first one.
if arg_id != 0:
dt.append_object(
_mathml.OperatorComponent(_mathml.OPERATOR_PLUS))
# Get the precedence.
CUR_PREC = _sympy_precedence.precedence(cur_arg)
# noinspection PyProtectedMember
if CUR_PREC < PREC or (_coeff_isneg(cur_arg) and arg_id != 0):
# Insert the argument with parentheses around.
dt.append_object(
_mathml.OperatorComponent(
_mathml.OPERATOR_LEFT_PARENTHESIS))
dt.append_object(self._print(cur_arg))
dt.append_object(
_mathml.OperatorComponent(
_mathml.OPERATOR_RIGHT_PARENTHESIS))
else:
# Insert the argument only.
dt.append_object(self._print(cur_arg))
return dt
def precedence_Mul(item):
if _coeff_isneg(item):
return PRECEDENCE["Add"]
return PRECEDENCE["Mul"]
def _print_Mul(self, expr):
"""Print a Mul object.
:param expr: The expression.
:rtype : bce.dom.mathml.all.Base
:return: The printed MathML object.
"""
assert isinstance(expr, _sympy.Mul)
# noinspection PyProtectedMember
if _coeff_isneg(expr):
x = _mathml.RowComponent()
x.append_object(_mathml.OperatorComponent(_mathml.OPERATOR_MINUS))
x.append_object(self._print(-expr))
return x
PREC = _sympy_precedence.precedence(expr)
from sympy.simplify import fraction
numer, denom = fraction(expr)
if denom is not _sympy.S.One:
return _mathml.FractionComponent(self._print(numer),
self._print(denom))
coeff, terms = expr.as_coeff_mul()
if coeff is _sympy.S.One and len(terms) == 1:
# Since the negative coefficient has been handled, I don't
# thing a coeff of 1 can remain
if _sympy_precedence.precedence(terms[0]) < PREC:
# Return the argument with parentheses around.
tmp_node = _mathml.RowComponent()
tmp_node.append_object(
_mathml.OperatorComponent(
_mathml.OPERATOR_LEFT_PARENTHESIS))
tmp_node.append_object(self._print(terms[0]))
tmp_node.append_object(
_mathml.OperatorComponent(
_mathml.OPERATOR_RIGHT_PARENTHESIS))
return tmp_node
else:
# Return the argument only.
return self._print(terms[0])
if self.order != "old":
# noinspection PyProtectedMember
terms = _sympy.Mul._from_args(terms).as_ordered_factors()
# Build result row element(node).
x = _mathml.RowComponent()
if coeff != 1:
if _sympy_precedence.precedence(coeff) < PREC:
# Insert the coefficient number with parentheses around.
x.append_object(
_mathml.OperatorComponent(
_mathml.OPERATOR_LEFT_PARENTHESIS))
x.append_object(self._print(coeff))
x.append_object(
_mathml.OperatorComponent(
_mathml.OPERATOR_RIGHT_PARENTHESIS))
else:
# Insert the coefficient number only.
x.append_object(self._print(coeff))
# Insert a multiply operator.
if not terms[0].is_Symbol:
x.append_object(
_mathml.OperatorComponent(_mathml.OPERATOR_MULTIPLY))
terms_len = len(terms)
for term_id in range(0, terms_len):
cur_term = terms[term_id]
if _sympy_precedence.precedence(cur_term) < PREC:
x.append_object(
_mathml.OperatorComponent(
_mathml.OPERATOR_LEFT_PARENTHESIS))
x.append_object(self._print(cur_term))
x.append_object(
_mathml.OperatorComponent(
_mathml.OPERATOR_RIGHT_PARENTHESIS))
else:
x.append_object(self._print(cur_term))
if term_id + 1 != terms_len and not cur_term.is_Symbol:
x.append_object(
_mathml.OperatorComponent(_mathml.OPERATOR_MULTIPLY))
return x
