def _eval_expand_complex(self, *args):
if self.args[0].is_real:
return self
re, im = self.args[0].as_real_imag()
denom = sin(re)**2 + C.sinh(im)**2
return (sin(re)*cos(re) - \
S.ImaginaryUnit*C.sinh(im)*C.cosh(im))/denom
def _eval_expand_complex(self, *args):
if self.args[0].is_real:
return self
re, im = self.args[0].as_real_imag()
denom = sin(re)**2 + C.sinh(im)**2
return (sin(re)*cos(re) - \
S.ImaginaryUnit*C.sinh(im)*C.cosh(im))/denom
def _eval_expand_complex(self, deep=True, **hints):
if self.args[0].is_real:
if deep:
hints['complex'] = False
return self.expand(deep, **hints)
else:
return self
if deep:
re, im = self.args[0].expand(deep, **hints).as_real_imag()
else:
re, im = self.args[0].as_real_imag()
return sin(re) * C.cosh(im) + S.ImaginaryUnit * cos(re) * C.sinh(im)
def _eval_expand_complex(self, deep=True, **hints):
if self.args[0].is_real:
if deep:
hints['complex'] = False
return self.expand(deep, **hints)
else:
return self
if deep:
re, im = self.args[0].expand(deep, **hints).as_real_imag()
else:
re, im = self.args[0].as_real_imag()
return sin(re)*C.cosh(im) + S.ImaginaryUnit*cos(re)*C.sinh(im)
def as_real_imag(self, deep=True, **hints):
if self.args[0].is_real:
if deep:
hints['complex'] = False
return (self.expand(deep, **hints), S.Zero)
else:
return (self, S.Zero)
if deep:
re, im = self.args[0].expand(deep, **hints).as_real_imag()
else:
re, im = self.args[0].as_real_imag()
return (cos(re) * C.cosh(im), -sin(re) * C.sinh(im))
def as_real_imag(self, deep=True, **hints):
if self.args[0].is_real:
if deep:
hints['complex'] = False
return (self.expand(deep, **hints), S.Zero)
else:
return (self, S.Zero)
if deep:
re, im = self.args[0].expand(deep, **hints).as_real_imag()
else:
re, im = self.args[0].as_real_imag()
return (cos(re)*C.cosh(im), -sin(re)*C.sinh(im))
def as_real_imag(self, deep=True, **hints):
if self.args[0].is_real:
if deep:
hints["complex"] = False
return (self.expand(deep, **hints), S.Zero)
else:
return (self, S.Zero)
if deep:
re, im = self.args[0].expand(deep, **hints).as_real_imag()
else:
re, im = self.args[0].as_real_imag()
denom = cos(re) ** 2 + C.sinh(im) ** 2
return (sin(re) * cos(re) / denom, C.sinh(im) * C.cosh(im) / denom)
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_negative:
return cls(-arg)
else:
i_coeff = arg.as_coefficient(S.ImaginaryUnit)
if i_coeff is not None:
return C.cosh(i_coeff)
else:
pi_coeff = arg.as_coefficient(S.Pi)
if pi_coeff is not None:
if pi_coeff.is_Rational:
cst_table_some = {
1: S.One,
2: S.Zero,
3: S.Half,
4: S.Half * sqrt(2),
6: S.Half * sqrt(3),
}
cst_table_more = {
(1, 5): (sqrt(5) + 1) / 4,
(2, 5): (sqrt(5) - 1) / 4
}
p = pi_coeff.p
q = pi_coeff.q
Q, P = 2 * p // q, p % q
try:
result = cst_table_some[q]
except KeyError:
if abs(P) > q // 2:
P = q - P
try:
result = cst_table_more[(P, q)]
except KeyError:
if P != p:
result = cls(C.Rational(P, q) * S.Pi)
else:
return None
if Q % 4 in (1, 2):
return -result
else:
return result
if arg.is_Mul and arg.args[0].is_negative:
return cls(-arg)
if arg.is_Add:
x, m = arg.as_independent(S.Pi)
if m in [S.Pi / 2, S.Pi]:
return cos(m) * cos(x) - sin(m) * sin(x)
# normalize cos(-x-y) to cos(x+y)
if arg.args[0].is_Mul:
if arg.args[0].args[0].is_negative:
# e.g. arg = -x - y
if (-arg).args[0].is_Mul:
if (-arg).args[0].args[0].is_negative:
# This is to prevent infinite recursion in
# the case cos(-x+y), for which
# -arg = -y + x. See also #838 for the
# root of the problem here.
return
# convert cos(-x-y) to cos(x+y)
return cls(-arg)
if arg.args[0].is_negative:
if (-arg).args[0].is_negative:
# This is to avoid infinite recursion in the case
# sin(-x-1)
return
return cls(-arg)
if isinstance(arg, acos):
return arg.args[0]
if isinstance(arg, atan):
x = arg.args[0]
return 1 / sqrt(1 + x**2)
if isinstance(arg, asin):
x = arg.args[0]
return sqrt(1 - x**2)
if isinstance(arg, acot):
x = arg.args[0]
return 1 / sqrt(1 + 1 / x**2)
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.Infinity or arg is S.NegativeInfinity:
# In this cases, it is unclear if we should
# return S.NaN or leave un-evaluated. One
# useful test case is how "limit(sin(x)/x,x,oo)"
# is handled.
# See test_sin_cos_with_infinity() an
# Test for issue 209
# http://code.google.com/p/sympy/issues/detail?id=2097
# For now, we return un-evaluated.
return
if arg.could_extract_minus_sign():
return cls(-arg)
i_coeff = arg.as_coefficient(S.ImaginaryUnit)
if i_coeff is not None:
return C.cosh(i_coeff)
pi_coeff = _pi_coeff(arg)
if pi_coeff is not None:
if pi_coeff.is_integer:
return (S.NegativeOne)**pi_coeff
if not pi_coeff.is_Rational:
narg = pi_coeff*S.Pi
if narg != arg:
return cls(narg)
return None
# cosine formula #####################
# http://code.google.com/p/sympy/issues/detail?id=2949
# explicit calculations are preformed for
# cos(k pi / 8), cos(k pi /10), and cos(k pi / 12)
# Some other exact values like cos(k pi/15) can be
# calculated using a partial-fraction decomposition
# by calling cos( X ).rewrite(sqrt)
cst_table_some = {
3: S.Half,
5: (sqrt(5) + 1)/4,
}
if pi_coeff.is_Rational:
q = pi_coeff.q
p = pi_coeff.p % (2*q)
if p > q:
narg = (pi_coeff - 1)*S.Pi
return -cls(narg)
if 2*p > q:
narg = (1 - pi_coeff)*S.Pi
return -cls(narg)
# If nested sqrt's are worse than un-evaluation
# you can require q in (1, 2, 3, 4, 6)
# q <= 12 returns expressions with 2 or fewer nestings.
if q > 12:
return None
if q in cst_table_some:
cts = cst_table_some[pi_coeff.q]
return C.chebyshevt(pi_coeff.p, cts).expand()
if 0 == q % 2:
narg = (pi_coeff*2)*S.Pi
nval = cls(narg)
if None == nval:
return None
x = (2*pi_coeff + 1)/2
sign_cos = (-1)**((-1 if x < 0 else 1)*int(abs(x)))
return sign_cos*sqrt( (1 + nval)/2 )
return None
if arg.is_Add:
x, m = _peeloff_pi(arg)
if m:
return cos(m)*cos(x) - sin(m)*sin(x)
if arg.func is acos:
return arg.args[0]
if arg.func is atan:
x = arg.args[0]
return 1 / sqrt(1 + x**2)
if arg.func is atan2:
y, x = arg.args
return x / sqrt(x**2 + y**2)
if arg.func is asin:
x = arg.args[0]
return sqrt(1 - x ** 2)
if arg.func is acot:
x = arg.args[0]
return 1 / sqrt(1 + 1 / x**2)
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.Infinity:
return
if arg.could_extract_minus_sign():
return cls(-arg)
i_coeff = arg.as_coefficient(S.ImaginaryUnit)
if i_coeff is not None:
return C.cosh(i_coeff)
pi_coeff = _pi_coeff(arg)
if pi_coeff is not None:
if not pi_coeff.is_Rational:
if pi_coeff.is_integer:
return (S.NegativeOne)**pi_coeff
narg = pi_coeff*S.Pi
if narg != arg:
return cls(narg)
return None
cst_table_some = {
1 : S.One,
2 : S.Zero,
3 : S.Half,
4 : S.Half*sqrt(2),
6 : S.Half*sqrt(3),
}
cst_table_more = {
(1, 5) : (sqrt(5) + 1)/4,
(2, 5) : (sqrt(5) - 1)/4
}
p = pi_coeff.p
q = pi_coeff.q
Q, P = 2*p // q, p % q
try:
result = cst_table_some[q]
except KeyError:
if abs(P) > q // 2:
P = q - P
try:
result = cst_table_more[(P, q)]
except KeyError:
if P != p:
result = cls(C.Rational(P, q)*S.Pi)
else:
return None
if Q % 4 in (1, 2):
return -result
else:
return result
if arg.is_Add:
x, m = _peeloff_pi(arg)
if m:
return cos(m)*cos(x)-sin(m)*sin(x)
if arg.func is acos:
return arg.args[0]
if arg.func is atan:
x = arg.args[0]
return 1 / sqrt(1 + x**2)
if arg.func is asin:
x = arg.args[0]
return sqrt(1 - x ** 2)
if arg.func is acot:
x = arg.args[0]
return 1 / sqrt(1 + 1 / x**2)
def canonize(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_negative:
return cls(-arg)
else:
i_coeff = arg.as_coefficient(S.ImaginaryUnit)
if i_coeff is not None:
return C.cosh(i_coeff)
else:
pi_coeff = arg.as_coefficient(S.Pi)
if pi_coeff is not None:
if pi_coeff.is_Rational:
cst_table = {
1 : S.One,
2 : S.Zero,
3 : S.Half,
4 : S.Half*sqrt(2),
6 : S.Half*sqrt(3),
}
try:
result = cst_table[pi_coeff.q]
if (2*pi_coeff.p // pi_coeff.q) % 4 in (1, 2):
return -result
else:
return result
except KeyError:
pass
if arg.is_Mul and arg.args[0].is_negative:
return cls(-arg)
if arg.is_Add:
x, m = arg.as_independent(S.Pi)
if m in [S.Pi/2, S.Pi]:
return cos(m)*cos(x)-sin(m)*sin(x)
# normalize cos(-x-y) to cos(x+y)
if arg.args[0].is_Mul:
if arg.args[0].args[0].is_negative:
# e.g. arg = -x - y
if (-arg).args[0].is_Mul:
if (-arg).args[0].args[0].is_negative:
# This is to prevent infinite recursion in
# the case cos(-x+y), for which
# -arg = -y + x. See also #838 for the
# root of the problem here.
return
# convert cos(-x-y) to cos(x+y)
return cls(-arg)
if arg.args[0].is_negative:
if (-arg).args[0].is_negative:
# This is to avoid infinite recursion in the case
# sin(-x-1)
return
return cls(-arg)
if isinstance(arg, acos):
return arg.args[0]
if isinstance(arg, atan):
x = arg.args[0]
return 1 / sqrt(1 + x**2)
if isinstance(arg, asin):
x = arg.args[0]
return sqrt(1 - x ** 2)
if isinstance(arg, acot):
x = arg.args[0]
return 1 / sqrt(1 + 1 / x**2)
def _eval_expand_complex(self, *args):
if self.args[0].is_real:
return self
re, im = self.args[0].as_real_imag()
return cos(re)*C.cosh(im) - \
S.ImaginaryUnit*sin(re)*C.sinh(im)
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_negative:
return cls(-arg)
else:
i_coeff = arg.as_coefficient(S.ImaginaryUnit)
if i_coeff is not None:
return C.cosh(i_coeff)
else:
pi_coeff = arg.as_coefficient(S.Pi)
if pi_coeff is not None:
if pi_coeff.is_Rational:
cst_table_some = {
1 : S.One,
2 : S.Zero,
3 : S.Half,
4 : S.Half*sqrt(2),
6 : S.Half*sqrt(3),
}
cst_table_more = {
(1, 5) : (sqrt(5) + 1)/4,
(2, 5) : (sqrt(5) - 1)/4
}
p = pi_coeff.p
q = pi_coeff.q
Q, P = 2*p // q, p % q
try:
result = cst_table_some[q]
except KeyError:
if abs(P) > q // 2:
P = q - P
try:
result = cst_table_more[(P, q)]
except KeyError:
if P != p:
result = cls(C.Rational(P, q)*S.Pi)
else:
return None
if Q % 4 in (1, 2):
return -result
else:
return result
if arg.is_Mul and arg.args[0].is_negative:
return cls(-arg)
if arg.is_Add:
x, m = arg.as_independent(S.Pi)
if m in [S.Pi/2, S.Pi]:
return cos(m)*cos(x)-sin(m)*sin(x)
# normalize cos(-x-y) to cos(x+y)
if arg.args[0].is_Mul:
if arg.args[0].args[0].is_negative:
# e.g. arg = -x - y
if (-arg).args[0].is_Mul:
if (-arg).args[0].args[0].is_negative:
# This is to prevent infinite recursion in
# the case cos(-x+y), for which
# -arg = -y + x. See also #838 for the
# root of the problem here.
return
# convert cos(-x-y) to cos(x+y)
return cls(-arg)
if arg.args[0].is_negative:
if (-arg).args[0].is_negative:
# This is to avoid infinite recursion in the case
# sin(-x-1)
return
return cls(-arg)
if isinstance(arg, acos):
return arg.args[0]
if isinstance(arg, atan):
x = arg.args[0]
return 1 / sqrt(1 + x**2)
if isinstance(arg, asin):
x = arg.args[0]
return sqrt(1 - x ** 2)
if isinstance(arg, acot):
x = arg.args[0]
return 1 / sqrt(1 + 1 / x**2)
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.Infinity or arg is S.NegativeInfinity:
# In this cases, it is unclear if we should
# return S.NaN or leave un-evaluated. One
# useful test case is how "limit(sin(x)/x,x,oo)"
# is handled.
# See test_sin_cos_with_infinity() an
# Test for issue 209
# http://code.google.com/p/sympy/issues/detail?id=2097
# For now, we return un-evaluated.
return
if arg.could_extract_minus_sign():
return cls(-arg)
i_coeff = arg.as_coefficient(S.ImaginaryUnit)
if i_coeff is not None:
return C.cosh(i_coeff)
pi_coeff = _pi_coeff(arg)
if pi_coeff is not None:
if pi_coeff.is_integer:
return (S.NegativeOne)**pi_coeff
if not pi_coeff.is_Rational:
narg = pi_coeff * S.Pi
if narg != arg:
return cls(narg)
return None
# cosine formula #####################
# http://code.google.com/p/sympy/issues/detail?id=2949
# explicit calculations are preformed for
# cos(k pi / 8), cos(k pi /10), and cos(k pi / 12)
# Some other exact values like cos(k pi/15) can be
# calculated using a partial-fraction decomposition
# by calling cos( X ).rewrite(sqrt)
cst_table_some = {
3: S.Half,
5: (sqrt(5) + 1) / 4,
}
if pi_coeff.is_Rational:
q = pi_coeff.q
p = pi_coeff.p % (2 * q)
if p > q:
narg = (pi_coeff - 1) * S.Pi
return -cls(narg)
if 2 * p > q:
narg = (1 - pi_coeff) * S.Pi
return -cls(narg)
# If nested sqrt's are worse than un-evaluation
# you can require q in (1, 2, 3, 4, 6)
# q <= 12 returns expressions with 2 or fewer nestings.
if q > 12:
return None
if q in cst_table_some:
cts = cst_table_some[pi_coeff.q]
return C.chebyshevt(pi_coeff.p, cts).expand()
if 0 == q % 2:
narg = (pi_coeff * 2) * S.Pi
nval = cls(narg)
if None == nval:
return None
x = (2 * pi_coeff + 1) / 2
sign_cos = (-1)**((-1 if x < 0 else 1) * int(abs(x)))
return sign_cos * sqrt((1 + nval) / 2)
return None
if arg.is_Add:
x, m = _peeloff_pi(arg)
if m:
return cos(m) * cos(x) - sin(m) * sin(x)
if arg.func is acos:
return arg.args[0]
if arg.func is atan:
x = arg.args[0]
return 1 / sqrt(1 + x**2)
if arg.func is atan2:
y, x = arg.args
return x / sqrt(x**2 + y**2)
if arg.func is asin:
x = arg.args[0]
return sqrt(1 - x**2)
if arg.func is acot:
x = arg.args[0]
return 1 / sqrt(1 + 1 / x**2)
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.Infinity:
return
if arg.could_extract_minus_sign():
return cls(-arg)
i_coeff = arg.as_coefficient(S.ImaginaryUnit)
if i_coeff is not None:
return C.cosh(i_coeff)
pi_coeff = _pi_coeff(arg)
if pi_coeff is not None:
if not pi_coeff.is_Rational:
if pi_coeff.is_integer:
even = pi_coeff.is_even
if even:
return S.One
elif even is False:
return S.NegativeOne
narg = pi_coeff * S.Pi
if narg != arg:
return cls(narg)
return None
cst_table_some = {
1: S.One,
2: S.Zero,
3: S.Half,
4: S.Half * sqrt(2),
6: S.Half * sqrt(3),
}
cst_table_more = {
(1, 5): (sqrt(5) + 1) / 4,
(2, 5): (sqrt(5) - 1) / 4
}
p = pi_coeff.p
q = pi_coeff.q
Q, P = 2 * p // q, p % q
try:
result = cst_table_some[q]
except KeyError:
if abs(P) > q // 2:
P = q - P
try:
result = cst_table_more[(P, q)]
except KeyError:
if P != p:
result = cls(C.Rational(P, q) * S.Pi)
else:
return None
if Q % 4 in (1, 2):
return -result
else:
return result
if arg.is_Add:
x, m = _peeloff_pi(arg)
if m:
return cos(m) * cos(x) - sin(m) * sin(x)
if arg.func is acos:
return arg.args[0]
if arg.func is atan:
x = arg.args[0]
return 1 / sqrt(1 + x**2)
if arg.func is asin:
x = arg.args[0]
return sqrt(1 - x**2)
if arg.func is acot:
x = arg.args[0]
return 1 / sqrt(1 + 1 / x**2)
def canonize(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_negative:
return cls(-arg)
else:
i_coeff = arg.as_coefficient(S.ImaginaryUnit)
if i_coeff is not None:
return C.cosh(i_coeff)
else:
pi_coeff = arg.as_coefficient(S.Pi)
if pi_coeff is not None:
if pi_coeff.is_Rational:
cst_table = {
1: S.One,
2: S.Zero,
3: S.Half,
4: S.Half * sqrt(2),
6: S.Half * sqrt(3),
}
try:
result = cst_table[pi_coeff.q]
if (2 * pi_coeff.p // pi_coeff.q) % 4 in (1, 2):
return -result
else:
return result
except KeyError:
pass
if arg.is_Mul and arg.args[0].is_negative:
return cls(-arg)
if arg.is_Add:
x, m = arg.as_independent(S.Pi)
if m in [S.Pi / 2, S.Pi]:
return cos(m) * cos(x) - sin(m) * sin(x)
# normalize cos(-x-y) to cos(x+y)
if arg.args[0].is_Mul:
if arg.args[0].args[0].is_negative:
# e.g. arg = -x - y
if (-arg).args[0].is_Mul:
if (-arg).args[0].args[0].is_negative:
# This is to prevent infinite recursion in
# the case cos(-x+y), for which
# -arg = -y + x. See also #838 for the
# root of the problem here.
return
# convert cos(-x-y) to cos(x+y)
return cls(-arg)
if arg.args[0].is_negative:
if (-arg).args[0].is_negative:
# This is to avoid infinite recursion in the case
# sin(-x-1)
return
return cls(-arg)
if isinstance(arg, acos):
return arg.args[0]
if isinstance(arg, atan):
x = arg.args[0]
return 1 / sqrt(1 + x**2)
if isinstance(arg, asin):
x = arg.args[0]
return sqrt(1 - x**2)
if isinstance(arg, acot):
x = arg.args[0]
return 1 / sqrt(1 + 1 / x**2)
def _eval_expand_complex(self, *args):
if self.args[0].is_real:
return self
re, im = self.args[0].as_real_imag()
return cos(re)*C.cosh(im) - \
S.ImaginaryUnit*sin(re)*C.sinh(im)
