def rs_tan(p, x, prec):
"""
Tangent of a series
Returns the series expansion of the tan of p, about 0.
Examples
========
>>> from sympy.polys.domains import QQ
>>> from sympy.polys.rings import ring
>>> from sympy.polys.ring_series import rs_tan
>>> R, x, y = ring('x, y', QQ)
>>> rs_tan(x + x*y, x, 4)
1/3*x**3*y**3 + x**3*y**2 + x**3*y + 1/3*x**3 + x*y + x
See Also
========
tan
"""
if rs_is_puiseux(p, x):
r = rs_puiseux(rs_tan, p, x, prec)
return r
R = p.ring
const = 0
c = _get_constant_term(p, x)
if c:
if R.domain is EX:
c_expr = c.as_expr()
const = tan(c_expr)
elif isinstance(c, PolyElement):
try:
c_expr = c.as_expr()
const = R(tan(c_expr))
except ValueError:
R = R.add_gens([tan(c_expr, )])
p = p.set_ring(R)
x = x.set_ring(R)
c = c.set_ring(R)
const = R(tan(c_expr))
else:
try:
const = R(tan(c))
except ValueError:
raise DomainError("The given series can't be expanded in "
"this domain.")
p1 = p - c
# Makes use of sympy fuctions to evaluate the values of the cos/sin
# of the constant term.
t2 = rs_tan(p1, x, prec)
t = rs_series_inversion(1 - const*t2, x, prec)
return rs_mul(const + t2, t, x, prec)
if R.ngens == 1:
return _tan1(p, x, prec)
else:
return rs_fun(p, rs_tan, x, prec)
def rs_tan(p, x, prec):
"""
Tangent of a series
Returns the series expansion of the tan of p, about 0.
Examples
========
>>> from sympy.polys.domains import QQ
>>> from sympy.polys.rings import ring
>>> from sympy.polys.ring_series import rs_tan
>>> R, x, y = ring('x, y', QQ)
>>> rs_tan(x + x*y, x, 4)
1/3*x**3*y**3 + x**3*y**2 + x**3*y + 1/3*x**3 + x*y + x
See Also
========
tan
"""
if rs_is_puiseux(p, x):
r = rs_puiseux(rs_tan, p, x, prec)
return r
R = p.ring
const = 0
if _has_constant_term(p, x):
zm = R.zero_monom
c = p[zm]
if R.domain is EX:
c_expr = c.as_expr()
const = tan(c_expr)
elif isinstance(c, PolyElement):
try:
c_expr = c.as_expr()
const = R(tan(c_expr))
except ValueError:
raise DomainError("The given series can't be expanded in "
"this domain.")
else:
try:
const = R(tan(c))
except ValueError:
raise DomainError("The given series can't be expanded in "
"this domain.")
p1 = p - c
# Makes use of sympy fuctions to evaluate the values of the cos/sin
# of the constant term.
t2 = rs_tan(p1, x, prec)
t = rs_series_inversion(1 - const * t2, x, prec)
return rs_mul(const + t2, t, x, prec)
if R.ngens == 1:
return _tan1(p, x, prec)
else:
return rs_fun(p, rs_tan, x, prec)
def test_rs_series():
x, a, b, c = symbols('x, a, b, c')
assert rs_series(a, a, 5).as_expr() == a
assert rs_series(sin(a), a, 5).as_expr() == (sin(a).series(a, 0,
5)).removeO()
assert rs_series(sin(a) + cos(a), a, 5).as_expr() == ((sin(a) +
cos(a)).series(a, 0, 5)).removeO()
assert rs_series(sin(a)*cos(a), a, 5).as_expr() == ((sin(a)*
cos(a)).series(a, 0, 5)).removeO()
p = (sin(a) - a)*(cos(a**2) + a**4/2)
assert expand(rs_series(p, a, 10).as_expr()) == expand(p.series(a, 0,
10).removeO())
p = sin(a**2/2 + a/3) + cos(a/5)*sin(a/2)**3
assert expand(rs_series(p, a, 5).as_expr()) == expand(p.series(a, 0,
5).removeO())
p = sin(x**2 + a)*(cos(x**3 - 1) - a - a**2)
assert expand(rs_series(p, a, 5).as_expr()) == expand(p.series(a, 0,
5).removeO())
p = sin(a**2 - a/3 + 2)**5*exp(a**3 - a/2)
assert expand(rs_series(p, a, 10).as_expr()) == expand(p.series(a, 0,
10).removeO())
p = sin(a + b + c)
assert expand(rs_series(p, a, 5).as_expr()) == expand(p.series(a, 0,
5).removeO())
p = tan(sin(a**2 + 4) + b + c)
assert expand(rs_series(p, a, 6).as_expr()) == expand(p.series(a, 0,
6).removeO())
p = a**QQ(2,5) + a**QQ(2,3) + a
r = rs_series(tan(p), a, 2)
assert r.as_expr() == a**QQ(9,5) + a**QQ(26,15) + a**QQ(22,15) + a**QQ(6,5)/3 + \
a + a**QQ(2,3) + a**QQ(2,5)
r = rs_series(exp(p), a, 1)
assert r.as_expr() == a**QQ(4,5)/2 + a**QQ(2,3) + a**QQ(2,5) + 1
r = rs_series(sin(p), a, 2)
assert r.as_expr() == -a**QQ(9,5)/2 - a**QQ(26,15)/2 - a**QQ(22,15)/2 - \
a**QQ(6,5)/6 + a + a**QQ(2,3) + a**QQ(2,5)
r = rs_series(cos(p), a, 2)
assert r.as_expr() == a**QQ(28,15)/6 - a**QQ(5,3) + a**QQ(8,5)/24 - a**QQ(7,5) - \
a**QQ(4,3)/2 - a**QQ(16,15) - a**QQ(4,5)/2 + 1
assert rs_series(sin(a)/7, a, 5).as_expr() == (sin(a)/7).series(a, 0,
5).removeO()
def rs_tan(p, x, prec):
"""
Tangent of a series
Returns the series expansion of the tan of p, about 0.
Examples
========
>>> from sympy.polys.domains import QQ
>>> from sympy.polys.rings import ring
>>> from sympy.polys.ring_series import rs_tan
>>> R, x, y = ring('x, y', QQ)
>>> rs_tan(x + x*y, x, 4)
1/3*x**3*y**3 + x**3*y**2 + x**3*y + 1/3*x**3 + x*y + x
See Also
========
tan
"""
R = p.ring
const = 0
if _has_constant_term(p, x):
zm = R.zero_monom
c = p[zm]
c_expr = c.as_expr()
if R.domain is EX:
const = tan(c_expr)
elif isinstance(c, PolyElement):
try:
const = R(tan(c_expr))
except ValueError:
raise DomainError("The given series can't be expanded in this " "domain.")
else:
raise DomainError("The given series can't be expanded in this " "domain")
raise NotImplementedError
p1 = p - c
# Makes use of sympy fuctions to evaluate the values of the cos/sin
# of the constant term.
t2 = rs_tan(p1, x, prec)
t = rs_series_inversion(1 - const * t2, x, prec)
return rs_mul(const + t2, t, x, prec)
if R.ngens == 1:
return _tan1(p, x, prec)
else:
return fun(p, _tan1, x, prec)
def test_rs_series():
x, a, b, c = symbols('x, a, b, c')
assert rs_series(a, a, 5).as_expr() == a
assert rs_series(sin(1/a), a, 5).as_expr() == sin(1/a)
assert rs_series(sin(a), a, 5).as_expr() == (sin(a).series(a, 0,
5)).removeO()
assert rs_series(sin(a) + cos(a), a, 5).as_expr() == ((sin(a) +
cos(a)).series(a, 0, 5)).removeO()
assert rs_series(sin(a)*cos(a), a, 5).as_expr() == ((sin(a)*
cos(a)).series(a, 0, 5)).removeO()
p = (sin(a) - a)*(cos(a**2) + a**4/2)
assert expand(rs_series(p, a, 10).as_expr()) == expand(p.series(a, 0,
10).removeO())
p = sin(a**2/2 + a/3) + cos(a/5)*sin(a/2)**3
assert expand(rs_series(p, a, 5).as_expr()) == expand(p.series(a, 0,
5).removeO())
p = sin(x**2 + a)*(cos(x**3 - 1) - a - a**2)
assert expand(rs_series(p, a, 5).as_expr()) == expand(p.series(a, 0,
5).removeO())
p = sin(a**2 - a/3 + 2)**5*exp(a**3 - a/2)
assert expand(rs_series(p, a, 10).as_expr()) == expand(p.series(a, 0,
10).removeO())
p = sin(a + b + c)
assert expand(rs_series(p, a, 5).as_expr()) == expand(p.series(a, 0,
5).removeO())
p = tan(sin(a**2 + 4) + b + c)
assert expand(rs_series(p, a, 6).as_expr()) == expand(p.series(a, 0,
6).removeO())
def _get_general_solution(self, *, simplify: bool = True):
a, b, c, d = self.wilds_match()
fx = self.ode_problem.func
x = self.ode_problem.sym
(C1,) = self.ode_problem.get_numbered_constants(num=1)
mu = sqrt(4*d*b - (a - c)**2)
gensol = Eq(fx, (a - c - mu*tan(mu/(2*a)*log(x) + C1))/(2*b*x))
return [gensol]
def test_C99CodePrinter__precision():
n = symbols('n', integer=True)
f32_printer = C99CodePrinter(dict(type_aliases={real: float32}))
f64_printer = C99CodePrinter(dict(type_aliases={real: float64}))
f80_printer = C99CodePrinter(dict(type_aliases={real: float80}))
assert f32_printer.doprint(sin(x+2.1)) == 'sinf(x + 2.1F)'
assert f64_printer.doprint(sin(x+2.1)) == 'sin(x + 2.1000000000000001)'
assert f80_printer.doprint(sin(x+Float('2.0'))) == 'sinl(x + 2.0L)'
for printer, suffix in zip([f32_printer, f64_printer, f80_printer], ['f', '', 'l']):
def check(expr, ref):
assert printer.doprint(expr) == ref.format(s=suffix, S=suffix.upper())
check(Abs(n), 'abs(n)')
check(Abs(x + 2.0), 'fabs{s}(x + 2.0{S})')
check(sin(x + 4.0)**cos(x - 2.0), 'pow{s}(sin{s}(x + 4.0{S}), cos{s}(x - 2.0{S}))')
check(exp(x*8.0), 'exp{s}(8.0{S}*x)')
check(exp2(x), 'exp2{s}(x)')
check(expm1(x*4.0), 'expm1{s}(4.0{S}*x)')
check(Mod(n, 2), '((n) % (2))')
check(Mod(2*n + 3, 3*n + 5), '((2*n + 3) % (3*n + 5))')
check(Mod(x + 2.0, 3.0), 'fmod{s}(1.0{S}*x + 2.0{S}, 3.0{S})')
check(Mod(x, 2.0*x + 3.0), 'fmod{s}(1.0{S}*x, 2.0{S}*x + 3.0{S})')
check(log(x/2), 'log{s}((1.0{S}/2.0{S})*x)')
check(log10(3*x/2), 'log10{s}((3.0{S}/2.0{S})*x)')
check(log2(x*8.0), 'log2{s}(8.0{S}*x)')
check(log1p(x), 'log1p{s}(x)')
check(2**x, 'pow{s}(2, x)')
check(2.0**x, 'pow{s}(2.0{S}, x)')
check(x**3, 'pow{s}(x, 3)')
check(x**4.0, 'pow{s}(x, 4.0{S})')
check(sqrt(3+x), 'sqrt{s}(x + 3)')
check(Cbrt(x-2.0), 'cbrt{s}(x - 2.0{S})')
check(hypot(x, y), 'hypot{s}(x, y)')
check(sin(3.*x + 2.), 'sin{s}(3.0{S}*x + 2.0{S})')
check(cos(3.*x - 1.), 'cos{s}(3.0{S}*x - 1.0{S})')
check(tan(4.*y + 2.), 'tan{s}(4.0{S}*y + 2.0{S})')
check(asin(3.*x + 2.), 'asin{s}(3.0{S}*x + 2.0{S})')
check(acos(3.*x + 2.), 'acos{s}(3.0{S}*x + 2.0{S})')
check(atan(3.*x + 2.), 'atan{s}(3.0{S}*x + 2.0{S})')
check(atan2(3.*x, 2.*y), 'atan2{s}(3.0{S}*x, 2.0{S}*y)')
check(sinh(3.*x + 2.), 'sinh{s}(3.0{S}*x + 2.0{S})')
check(cosh(3.*x - 1.), 'cosh{s}(3.0{S}*x - 1.0{S})')
check(tanh(4.0*y + 2.), 'tanh{s}(4.0{S}*y + 2.0{S})')
check(asinh(3.*x + 2.), 'asinh{s}(3.0{S}*x + 2.0{S})')
check(acosh(3.*x + 2.), 'acosh{s}(3.0{S}*x + 2.0{S})')
check(atanh(3.*x + 2.), 'atanh{s}(3.0{S}*x + 2.0{S})')
check(erf(42.*x), 'erf{s}(42.0{S}*x)')
check(erfc(42.*x), 'erfc{s}(42.0{S}*x)')
check(gamma(x), 'tgamma{s}(x)')
check(loggamma(x), 'lgamma{s}(x)')
check(ceiling(x + 2.), "ceil{s}(x + 2.0{S})")
check(floor(x + 2.), "floor{s}(x + 2.0{S})")
check(fma(x, y, -z), 'fma{s}(x, y, -z)')
check(Max(x, 8.0, x**4.0), 'fmax{s}(8.0{S}, fmax{s}(x, pow{s}(x, 4.0{S})))')
check(Min(x, 2.0), 'fmin{s}(2.0{S}, x)')
def test_C99CodePrinter__precision():
n = symbols('n', integer=True)
f32_printer = C99CodePrinter(dict(type_aliases={real: float32}))
f64_printer = C99CodePrinter(dict(type_aliases={real: float64}))
f80_printer = C99CodePrinter(dict(type_aliases={real: float80}))
assert f32_printer.doprint(sin(x+2.1)) == 'sinf(x + 2.1F)'
assert f64_printer.doprint(sin(x+2.1)) == 'sin(x + 2.1000000000000001)'
assert f80_printer.doprint(sin(x+Float('2.0'))) == 'sinl(x + 2.0L)'
for printer, suffix in zip([f32_printer, f64_printer, f80_printer], ['f', '', 'l']):
def check(expr, ref):
assert printer.doprint(expr) == ref.format(s=suffix, S=suffix.upper())
check(Abs(n), 'abs(n)')
check(Abs(x + 2.0), 'fabs{s}(x + 2.0{S})')
check(sin(x + 4.0)**cos(x - 2.0), 'pow{s}(sin{s}(x + 4.0{S}), cos{s}(x - 2.0{S}))')
check(exp(x*8.0), 'exp{s}(8.0{S}*x)')
check(exp2(x), 'exp2{s}(x)')
check(expm1(x*4.0), 'expm1{s}(4.0{S}*x)')
check(Mod(n, 2), '((n) % (2))')
check(Mod(2*n + 3, 3*n + 5), '((2*n + 3) % (3*n + 5))')
check(Mod(x + 2.0, 3.0), 'fmod{s}(1.0{S}*x + 2.0{S}, 3.0{S})')
check(Mod(x, 2.0*x + 3.0), 'fmod{s}(1.0{S}*x, 2.0{S}*x + 3.0{S})')
check(log(x/2), 'log{s}((1.0{S}/2.0{S})*x)')
check(log10(3*x/2), 'log10{s}((3.0{S}/2.0{S})*x)')
check(log2(x*8.0), 'log2{s}(8.0{S}*x)')
check(log1p(x), 'log1p{s}(x)')
check(2**x, 'pow{s}(2, x)')
check(2.0**x, 'pow{s}(2.0{S}, x)')
check(x**3, 'pow{s}(x, 3)')
check(x**4.0, 'pow{s}(x, 4.0{S})')
check(sqrt(3+x), 'sqrt{s}(x + 3)')
check(Cbrt(x-2.0), 'cbrt{s}(x - 2.0{S})')
check(hypot(x, y), 'hypot{s}(x, y)')
check(sin(3.*x + 2.), 'sin{s}(3.0{S}*x + 2.0{S})')
check(cos(3.*x - 1.), 'cos{s}(3.0{S}*x - 1.0{S})')
check(tan(4.*y + 2.), 'tan{s}(4.0{S}*y + 2.0{S})')
check(asin(3.*x + 2.), 'asin{s}(3.0{S}*x + 2.0{S})')
check(acos(3.*x + 2.), 'acos{s}(3.0{S}*x + 2.0{S})')
check(atan(3.*x + 2.), 'atan{s}(3.0{S}*x + 2.0{S})')
check(atan2(3.*x, 2.*y), 'atan2{s}(3.0{S}*x, 2.0{S}*y)')
check(sinh(3.*x + 2.), 'sinh{s}(3.0{S}*x + 2.0{S})')
check(cosh(3.*x - 1.), 'cosh{s}(3.0{S}*x - 1.0{S})')
check(tanh(4.0*y + 2.), 'tanh{s}(4.0{S}*y + 2.0{S})')
check(asinh(3.*x + 2.), 'asinh{s}(3.0{S}*x + 2.0{S})')
check(acosh(3.*x + 2.), 'acosh{s}(3.0{S}*x + 2.0{S})')
check(atanh(3.*x + 2.), 'atanh{s}(3.0{S}*x + 2.0{S})')
check(erf(42.*x), 'erf{s}(42.0{S}*x)')
check(erfc(42.*x), 'erfc{s}(42.0{S}*x)')
check(gamma(x), 'tgamma{s}(x)')
check(loggamma(x), 'lgamma{s}(x)')
check(ceiling(x + 2.), "ceil{s}(x + 2.0{S})")
check(floor(x + 2.), "floor{s}(x + 2.0{S})")
check(fma(x, y, -z), 'fma{s}(x, y, -z)')
check(Max(x, 8.0, x**4.0), 'fmax{s}(8.0{S}, fmax{s}(x, pow{s}(x, 4.0{S})))')
check(Min(x, 2.0), 'fmin{s}(2.0{S}, x)')
def test_issue_21195():
t = symbols('t')
x = Function('x')(t)
dx = x.diff(t)
exp1 = cos(x) + cos(x) * dx
exp2 = sin(x) + tan(x) * (dx.diff(t))
exp3 = sin(x) * sin(t) * (dx.diff(t)).diff(t)
A = Matrix([[exp1], [exp2], [exp3]])
B = Matrix([[exp1.diff(x)], [exp2.diff(x)], [exp3.diff(x)]])
assert A.diff(x) == B
def test_trigintegrate_mixed():
assert trigintegrate(sin(x)*sec(x), x) == -log(sin(x)**2 - 1)/2
assert trigintegrate(sin(x)*csc(x), x) == x
assert trigintegrate(sin(x)*cot(x), x) == sin(x)
assert trigintegrate(cos(x)*sec(x), x) == x
assert trigintegrate(cos(x)*csc(x), x) == log(cos(x)**2 - 1)/2
assert trigintegrate(cos(x)*tan(x), x) == -cos(x)
assert trigintegrate(cos(x)*cot(x), x) == log(cos(x) - 1)/2 \
- log(cos(x) + 1)/2 + cos(x)
def test_trigintegrate_mixed():
assert trigintegrate(sin(x) * sec(x), x) == -log(sin(x)**2 - 1) / 2
assert trigintegrate(sin(x) * csc(x), x) == x
assert trigintegrate(sin(x) * cot(x), x) == sin(x)
assert trigintegrate(cos(x) * sec(x), x) == x
assert trigintegrate(cos(x) * csc(x), x) == log(cos(x)**2 - 1) / 2
assert trigintegrate(cos(x) * tan(x), x) == -cos(x)
assert trigintegrate(cos(x)*cot(x), x) == log(cos(x) - 1)/2 \
- log(cos(x) + 1)/2 + cos(x)
def test_rs_series():
x, a, b, c = symbols('x, a, b, c')
assert rs_series(a, a, 5).as_expr() == a
assert rs_series(sin(1 / a), a, 5).as_expr() == sin(1 / a)
assert rs_series(sin(a), a, 5).as_expr() == (sin(a).series(a, 0,
5)).removeO()
assert rs_series(sin(a) + cos(a), a,
5).as_expr() == ((sin(a) + cos(a)).series(a, 0,
5)).removeO()
assert rs_series(sin(a) * cos(a), a,
5).as_expr() == ((sin(a) * cos(a)).series(a, 0,
5)).removeO()
p = (sin(a) - a) * (cos(a**2) + a**4 / 2)
assert expand(rs_series(p, a, 10).as_expr()) == expand(
p.series(a, 0, 10).removeO())
p = sin(a**2 / 2 + a / 3) + cos(a / 5) * sin(a / 2)**3
assert expand(rs_series(p, a, 5).as_expr()) == expand(
p.series(a, 0, 5).removeO())
p = sin(x**2 + a) * (cos(x**3 - 1) - a - a**2)
assert expand(rs_series(p, a, 5).as_expr()) == expand(
p.series(a, 0, 5).removeO())
p = sin(a**2 - a / 3 + 2)**5 * exp(a**3 - a / 2)
assert expand(rs_series(p, a, 10).as_expr()) == expand(
p.series(a, 0, 10).removeO())
p = sin(a + b + c)
assert expand(rs_series(p, a, 5).as_expr()) == expand(
p.series(a, 0, 5).removeO())
p = tan(sin(a**2 + 4) + b + c)
assert expand(rs_series(p, a, 6).as_expr()) == expand(
p.series(a, 0, 6).removeO())
def Trig_Check(s):
if sin(s.args[0])/s is S.One or cos(s.args[0])/s is S.One \
or csc(s.args[0])/s is S.One or sec(s.args[0])/s is S.One \
or tan(s.args[0])/s is S.One or cot(s.args[0])/s is S.One:
return True
def test_tan():
R, x, y = ring('x, y', QQ)
assert rs_tan(x, x, 9) == \
x + x**3/3 + 2*x**5/15 + 17*x**7/315
assert rs_tan(x*y + x**2*y**3, x, 9) == 4/3*x**8*y**11 + 17/45*x**8*y**9 + \
4/3*x**7*y**9 + 17/315*x**7*y**7 + 1/3*x**6*y**9 + 2/3*x**6*y**7 + \
x**5*y**7 + 2/15*x**5*y**5 + x**4*y**5 + 1/3*x**3*y**3 + x**2*y**3 + x*y
# Constant term in series
a = symbols('a')
R, x, y = ring('x, y', QQ[tan(a), a])
assert rs_tan(x + a, x, 5) == (tan(a)**5 + 5*tan(a)**3/3 + \
2*tan(a)/3)*x**4 + (tan(a)**4 + 4*tan(a)**2/3 + 1/3)*x**3 + \
(tan(a)**3 + tan(a))*x**2 + (tan(a)**2 + 1)*x + tan(a)
assert rs_tan(x + x**2*y + a, x, 4) == (2*tan(a)**3 + 2*tan(a))*x**3*y + \
(tan(a)**4 + 4/3*tan(a)**2 + 1/3)*x**3 + (tan(a)**2 + 1)*x**2*y + \
(tan(a)**3 + tan(a))*x**2 + (tan(a)**2 + 1)*x + tan(a)
R, x, y = ring('x, y', EX)
assert rs_tan(x + a, x, 5) == EX(tan(a)**5 + 5*tan(a)**3/3 + \
2*tan(a)/3)*x**4 + EX(tan(a)**4 + 4*tan(a)**2/3 + EX(1)/3)*x**3 + \
EX(tan(a)**3 + tan(a))*x**2 + EX(tan(a)**2 + 1)*x + EX(tan(a))
assert rs_tan(x + x**2*y + a, x, 4) == EX(2*tan(a)**3 + \
2*tan(a))*x**3*y + EX(tan(a)**4 + 4*tan(a)**2/3 + EX(1)/3)*x**3 + \
EX(tan(a)**2 + 1)*x**2*y + EX(tan(a)**3 + tan(a))*x**2 + \
EX(tan(a)**2 + 1)*x + EX(tan(a))
p = x + x**2 + 5
assert rs_atan(p, x, 10).compose(x, 10) == EX(atan(5) + 67701870330562640/ \
668083460499)
def test_tan():
R, x, y = ring('x, y', QQ)
assert rs_tan(x, x, 9) == \
x + x**3/3 + 2*x**5/15 + 17*x**7/315
assert rs_tan(x*y + x**2*y**3, x, 9) == 4/3*x**8*y**11 + 17/45*x**8*y**9 + \
4/3*x**7*y**9 + 17/315*x**7*y**7 + 1/3*x**6*y**9 + 2/3*x**6*y**7 + \
x**5*y**7 + 2/15*x**5*y**5 + x**4*y**5 + 1/3*x**3*y**3 + x**2*y**3 + x*y
# Constant term in series
a = symbols('a')
R, x, y = ring('x, y', QQ[tan(a), a])
assert rs_tan(x + a, x, 5) == (tan(a)**5 + 5*tan(a)**3/3 + \
2*tan(a)/3)*x**4 + (tan(a)**4 + 4*tan(a)**2/3 + 1/3)*x**3 + \
(tan(a)**3 + tan(a))*x**2 + (tan(a)**2 + 1)*x + tan(a)
assert rs_tan(x + x**2*y + a, x, 4) == (2*tan(a)**3 + 2*tan(a))*x**3*y + \
(tan(a)**4 + 4/3*tan(a)**2 + 1/3)*x**3 + (tan(a)**2 + 1)*x**2*y + \
(tan(a)**3 + tan(a))*x**2 + (tan(a)**2 + 1)*x + tan(a)
R, x, y = ring('x, y', EX)
assert rs_tan(x + a, x, 5) == EX(tan(a)**5 + 5*tan(a)**3/3 + \
2*tan(a)/3)*x**4 + EX(tan(a)**4 + 4*tan(a)**2/3 + EX(1)/3)*x**3 + \
EX(tan(a)**3 + tan(a))*x**2 + EX(tan(a)**2 + 1)*x + EX(tan(a))
assert rs_tan(x + x**2*y + a, x, 4) == EX(2*tan(a)**3 + \
2*tan(a))*x**3*y + EX(tan(a)**4 + 4*tan(a)**2/3 + EX(1)/3)*x**3 + \
EX(tan(a)**2 + 1)*x**2*y + EX(tan(a)**3 + tan(a))*x**2 + \
EX(tan(a)**2 + 1)*x + EX(tan(a))
p = x + x**2 + 5
assert rs_atan(p, x, 10).compose(x, 10) == EX(atan(5) + 67701870330562640/ \
668083460499)
def exptrigsimp(expr, simplify=True):
"""
Simplifies exponential / trigonometric / hyperbolic functions.
When ``simplify`` is True (default) the expression obtained after the
simplification step will be then be passed through simplify to
precondition it so the final transformations will be applied.
Examples
========
>>> from sympy import exptrigsimp, exp, cosh, sinh
>>> from sympy.abc import z
>>> exptrigsimp(exp(z) + exp(-z))
2*cosh(z)
>>> exptrigsimp(cosh(z) - sinh(z))
exp(-z)
"""
from sympy.simplify.fu import hyper_as_trig, TR2i
from sympy.simplify.simplify import bottom_up
def exp_trig(e):
# select the better of e, and e rewritten in terms of exp or trig
# functions
choices = [e]
if e.has(*_trigs):
choices.append(e.rewrite(exp))
choices.append(e.rewrite(cos))
return min(*choices, key=count_ops)
newexpr = bottom_up(expr, exp_trig)
if simplify:
newexpr = newexpr.simplify()
# conversion from exp to hyperbolic
ex = newexpr.atoms(exp, S.Exp1)
ex = [ei for ei in ex if 1/ei not in ex]
## sinh and cosh
for ei in ex:
e2 = ei**-2
if e2 in ex:
a = e2.args[0]/2 if not e2 is S.Exp1 else S.Half
newexpr = newexpr.subs((e2 + 1)*ei, 2*cosh(a))
newexpr = newexpr.subs((e2 - 1)*ei, 2*sinh(a))
## exp ratios to tan and tanh
for ei in ex:
n, d = ei - 1, ei + 1
et = n/d
etinv = d/n # not 1/et or else recursion errors arise
a = ei.args[0] if ei.func is exp else S.One
if a.is_Mul or a is S.ImaginaryUnit:
c = a.as_coefficient(I)
if c:
t = S.ImaginaryUnit*tan(c/2)
newexpr = newexpr.subs(etinv, 1/t)
newexpr = newexpr.subs(et, t)
continue
t = tanh(a/2)
newexpr = newexpr.subs(etinv, 1/t)
newexpr = newexpr.subs(et, t)
# sin/cos and sinh/cosh ratios to tan and tanh, respectively
if newexpr.has(HyperbolicFunction):
e, f = hyper_as_trig(newexpr)
newexpr = f(TR2i(e))
if newexpr.has(TrigonometricFunction):
newexpr = TR2i(newexpr)
# can we ever generate an I where there was none previously?
if not (newexpr.has(I) and not expr.has(I)):
expr = newexpr
return expr
def test_rs_series():
x, a, b, c = symbols('x, a, b, c')
assert rs_series(a, a, 5).as_expr() == a
assert rs_series(sin(a), a, 5).as_expr() == (sin(a).series(a, 0,
5)).removeO()
assert rs_series(sin(a) + cos(a), a,
5).as_expr() == ((sin(a) + cos(a)).series(a, 0,
5)).removeO()
assert rs_series(sin(a) * cos(a), a,
5).as_expr() == ((sin(a) * cos(a)).series(a, 0,
5)).removeO()
p = (sin(a) - a) * (cos(a**2) + a**4 / 2)
assert expand(rs_series(p, a, 10).as_expr()) == expand(
p.series(a, 0, 10).removeO())
p = sin(a**2 / 2 + a / 3) + cos(a / 5) * sin(a / 2)**3
assert expand(rs_series(p, a, 5).as_expr()) == expand(
p.series(a, 0, 5).removeO())
p = sin(x**2 + a) * (cos(x**3 - 1) - a - a**2)
assert expand(rs_series(p, a, 5).as_expr()) == expand(
p.series(a, 0, 5).removeO())
p = sin(a**2 - a / 3 + 2)**5 * exp(a**3 - a / 2)
assert expand(rs_series(p, a, 10).as_expr()) == expand(
p.series(a, 0, 10).removeO())
p = sin(a + b + c)
assert expand(rs_series(p, a, 5).as_expr()) == expand(
p.series(a, 0, 5).removeO())
p = tan(sin(a**2 + 4) + b + c)
assert expand(rs_series(p, a, 6).as_expr()) == expand(
p.series(a, 0, 6).removeO())
p = a**QQ(2, 5) + a**QQ(2, 3) + a
r = rs_series(tan(p), a, 2)
assert r.as_expr() == a**QQ(9,5) + a**QQ(26,15) + a**QQ(22,15) + a**QQ(6,5)/3 + \
a + a**QQ(2,3) + a**QQ(2,5)
r = rs_series(exp(p), a, 1)
assert r.as_expr() == a**QQ(4, 5) / 2 + a**QQ(2, 3) + a**QQ(2, 5) + 1
r = rs_series(sin(p), a, 2)
assert r.as_expr() == -a**QQ(9,5)/2 - a**QQ(26,15)/2 - a**QQ(22,15)/2 - \
a**QQ(6,5)/6 + a + a**QQ(2,3) + a**QQ(2,5)
r = rs_series(cos(p), a, 2)
assert r.as_expr() == a**QQ(28,15)/6 - a**QQ(5,3) + a**QQ(8,5)/24 - a**QQ(7,5) - \
a**QQ(4,3)/2 - a**QQ(16,15) - a**QQ(4,5)/2 + 1
assert rs_series(sin(a) / 7, a,
5).as_expr() == (sin(a) / 7).series(a, 0, 5).removeO()
assert rs_series(log(1 + x), x, 5).as_expr() == -x**4/4 + x**3/3 - \
x**2/2 + x
assert rs_series(log(1 + 4*x), x, 5).as_expr() == -64*x**4 + 64*x**3/3 - \
8*x**2 + 4*x
assert rs_series(log(1 + x + x**2), x, 10).as_expr() == -2*x**9/9 + \
x**8/8 + x**7/7 - x**6/3 + x**5/5 + x**4/4 - 2*x**3/3 + \
x**2/2 + x
assert rs_series(log(1 + x*a**2), x, 7).as_expr() == -x**6*a**12/6 + \
x**5*a**10/5 - x**4*a**8/4 + x**3*a**6/3 - \
x**2*a**4/2 + x*a**2
def __init__(self):
self.s, self.t, self.x, self.y, self.z = symbols('s,t,x,y,z')
self.stack = []
self.defs = {}
self.mode = 0
self.hist = [('', [])] # Innehåller en lista med (kommandorad, stack)
self.lastx = ''
self.clear = True
self.op0 = {
's': lambda: self.s,
't': lambda: self.t,
'x': lambda: self.x,
'y': lambda: self.y,
'z': lambda: self.z,
'oo': lambda: S('oo'),
'inf': lambda: S('oo'),
'infinity': lambda: S('oo'),
'?': lambda: self.help(),
'help': lambda: self.help(),
'hist': lambda: self.history(),
'history': lambda: self.history(),
'sketch': lambda: self.sketch(),
}
self.op1 = {
'radians':
lambda x: pi / 180 * x,
'sin':
lambda x: sin(x),
'cos':
lambda x: cos(x),
'tan':
lambda x: tan(x),
'sq':
lambda x: x**2,
'sqrt':
lambda x: sqrt(x),
'ln':
lambda x: ln(x),
'exp':
lambda x: exp(x),
'log':
lambda x: log(x),
'simplify':
lambda x: simplify(x),
'polynom':
lambda x: self.polynom(x),
'inv':
lambda x: 1 / x,
'chs':
lambda x: -x,
'center':
lambda x: x.center,
'radius':
lambda x: x.radius,
'expand':
lambda x: x.expand(),
'factor':
lambda x: x.factor(),
'incircle':
lambda x: x.incircle,
'circumcircle':
lambda x: x.circumcircle,
'xdiff':
lambda x: x.diff(self.x),
'ydiff':
lambda x: x.diff(self.y),
'xint':
lambda x: x.integrate(self.x),
'xsolve':
lambda x: solve(x, self.x),
'xapart':
lambda x: apart(x, self.x),
'xtogether':
lambda x: together(x, self.x),
'N':
lambda x: N(x),
'info':
lambda x:
[x.__class__.__name__, [m for m in dir(x) if m[0] != '_']],
}
self.op2 = {
'+': lambda x, y: y + x,
'-': lambda x, y: y - x,
'*': lambda x, y: y * x,
'/': lambda x, y: y / x,
'**': lambda x, y: y**x,
'item': lambda x, y: y[x],
'point': lambda x, y: Point(y, x),
'line': lambda x, y: Line(y, x),
'circle': lambda x, y: Circle(y, x),
'tangent_lines': lambda x, y: y.tangent_lines(x),
'intersection': lambda x, y: intersection(x, y),
'perpendicular_line': lambda x, y: y.perpendicular_line(x),
'diff': lambda x, y: y.diff(x),
'int': lambda x, y: y.integrate(x),
'solve': lambda x, y: solve(y, x),
'apart': lambda x, y: apart(y, x),
'together': lambda x, y: together(y, x),
'xeval': lambda x, y: y.subs(self.x, x),
}
self.op3 = {
'triangle': lambda x, y, z: Triangle(x, y, z),
'limit': lambda x, y, z: limit(
z, y, x), # limit(sin(x)/x,x,0) <=> x sin x / x 0 limit
'eval': lambda x, y, z: z.subs(y, x),
}
self.op4 = {
'sum': lambda x, y, z, t: Sum(t, (z, y, x)).doit(
) # Sum(1/x**2,(x,1,oo)).doit() <=> 1 x x * / x 1 oo sum
}
self.lastx = ''
def _trigpats():
global _trigpat
a, b, c = symbols('a b c', cls=Wild)
d = Wild('d', commutative=False)
# for the simplifications like sinh/cosh -> tanh:
# DO NOT REORDER THE FIRST 14 since these are assumed to be in this
# order in _match_div_rewrite.
matchers_division = (
(a * sin(b)**c / cos(b)**c, a * tan(b)**c, sin(b), cos(b)),
(a * tan(b)**c * cos(b)**c, a * sin(b)**c, sin(b), cos(b)),
(a * cot(b)**c * sin(b)**c, a * cos(b)**c, sin(b), cos(b)),
(a * tan(b)**c / sin(b)**c, a / cos(b)**c, sin(b), cos(b)),
(a * cot(b)**c / cos(b)**c, a / sin(b)**c, sin(b), cos(b)),
(a * cot(b)**c * tan(b)**c, a, sin(b), cos(b)),
(a * (cos(b) + 1)**c * (cos(b) - 1)**c, a * (-sin(b)**2)**c,
cos(b) + 1, cos(b) - 1),
(a * (sin(b) + 1)**c * (sin(b) - 1)**c, a * (-cos(b)**2)**c,
sin(b) + 1, sin(b) - 1),
(a * sinh(b)**c / cosh(b)**c, a * tanh(b)**c, S.One, S.One),
(a * tanh(b)**c * cosh(b)**c, a * sinh(b)**c, S.One, S.One),
(a * coth(b)**c * sinh(b)**c, a * cosh(b)**c, S.One, S.One),
(a * tanh(b)**c / sinh(b)**c, a / cosh(b)**c, S.One, S.One),
(a * coth(b)**c / cosh(b)**c, a / sinh(b)**c, S.One, S.One),
(a * coth(b)**c * tanh(b)**c, a, S.One, S.One),
(c * (tanh(a) + tanh(b)) / (1 + tanh(a) * tanh(b)), tanh(a + b) * c,
S.One, S.One),
)
matchers_add = (
(c * sin(a) * cos(b) + c * cos(a) * sin(b) + d, sin(a + b) * c + d),
(c * cos(a) * cos(b) - c * sin(a) * sin(b) + d, cos(a + b) * c + d),
(c * sin(a) * cos(b) - c * cos(a) * sin(b) + d, sin(a - b) * c + d),
(c * cos(a) * cos(b) + c * sin(a) * sin(b) + d, cos(a - b) * c + d),
(c * sinh(a) * cosh(b) + c * sinh(b) * cosh(a) + d,
sinh(a + b) * c + d),
(c * cosh(a) * cosh(b) + c * sinh(a) * sinh(b) + d,
cosh(a + b) * c + d),
)
# for cos(x)**2 + sin(x)**2 -> 1
matchers_identity = (
(a * sin(b)**2, a - a * cos(b)**2),
(a * tan(b)**2, a * (1 / cos(b))**2 - a),
(a * cot(b)**2, a * (1 / sin(b))**2 - a),
(a * sin(b + c), a * (sin(b) * cos(c) + sin(c) * cos(b))),
(a * cos(b + c), a * (cos(b) * cos(c) - sin(b) * sin(c))),
(a * tan(b + c), a * ((tan(b) + tan(c)) / (1 - tan(b) * tan(c)))),
(a * sinh(b)**2, a * cosh(b)**2 - a),
(a * tanh(b)**2, a - a * (1 / cosh(b))**2),
(a * coth(b)**2, a + a * (1 / sinh(b))**2),
(a * sinh(b + c), a * (sinh(b) * cosh(c) + sinh(c) * cosh(b))),
(a * cosh(b + c), a * (cosh(b) * cosh(c) + sinh(b) * sinh(c))),
(a * tanh(b + c), a * ((tanh(b) + tanh(c)) / (1 + tanh(b) * tanh(c)))),
)
# Reduce any lingering artifacts, such as sin(x)**2 changing
# to 1-cos(x)**2 when sin(x)**2 was "simpler"
artifacts = (
(a - a * cos(b)**2 + c, a * sin(b)**2 + c, cos),
(a - a * (1 / cos(b))**2 + c, -a * tan(b)**2 + c, cos),
(a - a * (1 / sin(b))**2 + c, -a * cot(b)**2 + c, sin),
(a - a * cosh(b)**2 + c, -a * sinh(b)**2 + c, cosh),
(a - a * (1 / cosh(b))**2 + c, a * tanh(b)**2 + c, cosh),
(a + a * (1 / sinh(b))**2 + c, a * coth(b)**2 + c, sinh),
# same as above but with noncommutative prefactor
(a * d - a * d * cos(b)**2 + c, a * d * sin(b)**2 + c, cos),
(a * d - a * d * (1 / cos(b))**2 + c, -a * d * tan(b)**2 + c, cos),
(a * d - a * d * (1 / sin(b))**2 + c, -a * d * cot(b)**2 + c, sin),
(a * d - a * d * cosh(b)**2 + c, -a * d * sinh(b)**2 + c, cosh),
(a * d - a * d * (1 / cosh(b))**2 + c, a * d * tanh(b)**2 + c, cosh),
(a * d + a * d * (1 / sinh(b))**2 + c, a * d * coth(b)**2 + c, sinh),
)
_trigpat = (a, b, c, d, matchers_division, matchers_add, matchers_identity,
artifacts)
return _trigpat
def test_tan():
R, x, y = ring('x, y', QQ)
assert rs_tan(x, x, 9)/x**5 == \
S(17)/315*x**2 + S(2)/15 + S(1)/3*x**(-2) + x**(-4)
assert rs_tan(x*y + x**2*y**3, x, 9) == 4*x**8*y**11/3 + 17*x**8*y**9/45 + \
4*x**7*y**9/3 + 17*x**7*y**7/315 + x**6*y**9/3 + 2*x**6*y**7/3 + \
x**5*y**7 + 2*x**5*y**5/15 + x**4*y**5 + x**3*y**3/3 + x**2*y**3 + x*y
# Constant term in series
a = symbols('a')
R, x, y = ring('x, y', QQ[tan(a), a])
assert rs_tan(x + a, x, 5) == (tan(a)**5 + 5*tan(a)**S(3)/3 +
2*tan(a)/3)*x**4 + (tan(a)**4 + 4*tan(a)**2/3 + S(1)/3)*x**3 + \
(tan(a)**3 + tan(a))*x**2 + (tan(a)**2 + 1)*x + tan(a)
assert rs_tan(x + x**2*y + a, x, 4) == (2*tan(a)**3 + 2*tan(a))*x**3*y + \
(tan(a)**4 + S(4)/3*tan(a)**2 + S(1)/3)*x**3 + (tan(a)**2 + 1)*x**2*y + \
(tan(a)**3 + tan(a))*x**2 + (tan(a)**2 + 1)*x + tan(a)
R, x, y = ring('x, y', EX)
assert rs_tan(x + a, x, 5) == EX(tan(a)**5 + 5*tan(a)**3/3 +
2*tan(a)/3)*x**4 + EX(tan(a)**4 + 4*tan(a)**2/3 + EX(1)/3)*x**3 + \
EX(tan(a)**3 + tan(a))*x**2 + EX(tan(a)**2 + 1)*x + EX(tan(a))
assert rs_tan(x + x**2*y + a, x, 4) == EX(2*tan(a)**3 +
2*tan(a))*x**3*y + EX(tan(a)**4 + 4*tan(a)**2/3 + EX(1)/3)*x**3 + \
EX(tan(a)**2 + 1)*x**2*y + EX(tan(a)**3 + tan(a))*x**2 + \
EX(tan(a)**2 + 1)*x + EX(tan(a))
p = x + x**2 + 5
assert rs_atan(p, x, 10).compose(x, 10) == EX(atan(5) + S(67701870330562640) / \
668083460499)
def test_rs_series():
x, a, b, c = symbols('x, a, b, c')
assert rs_series(a, a, 5).as_expr() == a
assert rs_series(sin(a), a, 5).as_expr() == (sin(a).series(a, 0,
5)).removeO()
assert rs_series(sin(a) + cos(a), a, 5).as_expr() == ((sin(a) +
cos(a)).series(a, 0, 5)).removeO()
assert rs_series(sin(a)*cos(a), a, 5).as_expr() == ((sin(a)*
cos(a)).series(a, 0, 5)).removeO()
p = (sin(a) - a)*(cos(a**2) + a**4/2)
assert expand(rs_series(p, a, 10).as_expr()) == expand(p.series(a, 0,
10).removeO())
p = sin(a**2/2 + a/3) + cos(a/5)*sin(a/2)**3
assert expand(rs_series(p, a, 5).as_expr()) == expand(p.series(a, 0,
5).removeO())
p = sin(x**2 + a)*(cos(x**3 - 1) - a - a**2)
assert expand(rs_series(p, a, 5).as_expr()) == expand(p.series(a, 0,
5).removeO())
p = sin(a**2 - a/3 + 2)**5*exp(a**3 - a/2)
assert expand(rs_series(p, a, 10).as_expr()) == expand(p.series(a, 0,
10).removeO())
p = sin(a + b + c)
assert expand(rs_series(p, a, 5).as_expr()) == expand(p.series(a, 0,
5).removeO())
p = tan(sin(a**2 + 4) + b + c)
assert expand(rs_series(p, a, 6).as_expr()) == expand(p.series(a, 0,
6).removeO())
p = a**QQ(2,5) + a**QQ(2,3) + a
r = rs_series(tan(p), a, 2)
assert r.as_expr() == a**QQ(9,5) + a**QQ(26,15) + a**QQ(22,15) + a**QQ(6,5)/3 + \
a + a**QQ(2,3) + a**QQ(2,5)
r = rs_series(exp(p), a, 1)
assert r.as_expr() == a**QQ(4,5)/2 + a**QQ(2,3) + a**QQ(2,5) + 1
r = rs_series(sin(p), a, 2)
assert r.as_expr() == -a**QQ(9,5)/2 - a**QQ(26,15)/2 - a**QQ(22,15)/2 - \
a**QQ(6,5)/6 + a + a**QQ(2,3) + a**QQ(2,5)
r = rs_series(cos(p), a, 2)
assert r.as_expr() == a**QQ(28,15)/6 - a**QQ(5,3) + a**QQ(8,5)/24 - a**QQ(7,5) - \
a**QQ(4,3)/2 - a**QQ(16,15) - a**QQ(4,5)/2 + 1
assert rs_series(sin(a)/7, a, 5).as_expr() == (sin(a)/7).series(a, 0,
5).removeO()
assert rs_series(log(1 + x), x, 5).as_expr() == -x**4/4 + x**3/3 - \
x**2/2 + x
assert rs_series(log(1 + 4*x), x, 5).as_expr() == -64*x**4 + 64*x**3/3 - \
8*x**2 + 4*x
assert rs_series(log(1 + x + x**2), x, 10).as_expr() == -2*x**9/9 + \
x**8/8 + x**7/7 - x**6/3 + x**5/5 + x**4/4 - 2*x**3/3 + \
x**2/2 + x
assert rs_series(log(1 + x*a**2), x, 7).as_expr() == -x**6*a**12/6 + \
x**5*a**10/5 - x**4*a**8/4 + x**3*a**6/3 - \
x**2*a**4/2 + x*a**2
plt.style.use("ggplot")
# Define the variable and the function to approximate and point to approximate around.
x = sy.Symbol('x')
p0 = float(
input("What point would you like to approximate your function around?: "))
"Analytic functions"
#f = np.e**(x)
#f = np.log(x)
"The Trigonometric functions"
#f = sin(x)
#f = cos(x)
f = tan(x)
"The Hyperbolic functions"
#f = sinh(x)
#f = cosh(x)
#f = tanh(x)
# Factorial function
def factorial(n):
if n <= 0:
return 1
else:
return n * factorial(n - 1)
def test_tensorflow_math():
if not tf:
skip("TensorFlow not installed")
expr = Abs(x)
assert tensorflow_code(expr) == "tensorflow.math.abs(x)"
_compare_tensorflow_scalar((x, ), expr)
expr = sign(x)
assert tensorflow_code(expr) == "tensorflow.math.sign(x)"
_compare_tensorflow_scalar((x, ), expr)
expr = ceiling(x)
assert tensorflow_code(expr) == "tensorflow.math.ceil(x)"
_compare_tensorflow_scalar((x, ), expr, rng=lambda: random.random())
expr = floor(x)
assert tensorflow_code(expr) == "tensorflow.math.floor(x)"
_compare_tensorflow_scalar((x, ), expr, rng=lambda: random.random())
expr = exp(x)
assert tensorflow_code(expr) == "tensorflow.math.exp(x)"
_compare_tensorflow_scalar((x, ), expr, rng=lambda: random.random())
expr = sqrt(x)
assert tensorflow_code(expr) == "tensorflow.math.sqrt(x)"
_compare_tensorflow_scalar((x, ), expr, rng=lambda: random.random())
expr = x**4
assert tensorflow_code(expr) == "tensorflow.math.pow(x, 4)"
_compare_tensorflow_scalar((x, ), expr, rng=lambda: random.random())
expr = cos(x)
assert tensorflow_code(expr) == "tensorflow.math.cos(x)"
_compare_tensorflow_scalar((x, ), expr, rng=lambda: random.random())
expr = acos(x)
assert tensorflow_code(expr) == "tensorflow.math.acos(x)"
_compare_tensorflow_scalar((x, ),
expr,
rng=lambda: random.uniform(0, 0.95))
expr = sin(x)
assert tensorflow_code(expr) == "tensorflow.math.sin(x)"
_compare_tensorflow_scalar((x, ), expr, rng=lambda: random.random())
expr = asin(x)
assert tensorflow_code(expr) == "tensorflow.math.asin(x)"
_compare_tensorflow_scalar((x, ), expr, rng=lambda: random.random())
expr = tan(x)
assert tensorflow_code(expr) == "tensorflow.math.tan(x)"
_compare_tensorflow_scalar((x, ), expr, rng=lambda: random.random())
expr = atan(x)
assert tensorflow_code(expr) == "tensorflow.math.atan(x)"
_compare_tensorflow_scalar((x, ), expr, rng=lambda: random.random())
expr = atan2(y, x)
assert tensorflow_code(expr) == "tensorflow.math.atan2(y, x)"
_compare_tensorflow_scalar((y, x), expr, rng=lambda: random.random())
expr = cosh(x)
assert tensorflow_code(expr) == "tensorflow.math.cosh(x)"
_compare_tensorflow_scalar((x, ), expr, rng=lambda: random.random())
expr = acosh(x)
assert tensorflow_code(expr) == "tensorflow.math.acosh(x)"
_compare_tensorflow_scalar((x, ), expr, rng=lambda: random.uniform(1, 2))
expr = sinh(x)
assert tensorflow_code(expr) == "tensorflow.math.sinh(x)"
_compare_tensorflow_scalar((x, ), expr, rng=lambda: random.uniform(1, 2))
expr = asinh(x)
assert tensorflow_code(expr) == "tensorflow.math.asinh(x)"
_compare_tensorflow_scalar((x, ), expr, rng=lambda: random.uniform(1, 2))
expr = tanh(x)
assert tensorflow_code(expr) == "tensorflow.math.tanh(x)"
_compare_tensorflow_scalar((x, ), expr, rng=lambda: random.uniform(1, 2))
expr = atanh(x)
assert tensorflow_code(expr) == "tensorflow.math.atanh(x)"
_compare_tensorflow_scalar((x, ),
expr,
rng=lambda: random.uniform(-.5, .5))
expr = erf(x)
assert tensorflow_code(expr) == "tensorflow.math.erf(x)"
_compare_tensorflow_scalar((x, ), expr, rng=lambda: random.random())
expr = loggamma(x)
assert tensorflow_code(expr) == "tensorflow.math.lgamma(x)"
_compare_tensorflow_scalar((x, ), expr, rng=lambda: random.random())
def exptrigsimp(expr, simplify=True):
"""
Simplifies exponential / trigonometric / hyperbolic functions.
When ``simplify`` is True (default) the expression obtained after the
simplification step will be then be passed through simplify to
precondition it so the final transformations will be applied.
Examples
========
>>> from sympy import exptrigsimp, exp, cosh, sinh
>>> from sympy.abc import z
>>> exptrigsimp(exp(z) + exp(-z))
2*cosh(z)
>>> exptrigsimp(cosh(z) - sinh(z))
exp(-z)
"""
from sympy.simplify.fu import hyper_as_trig, TR2i
from sympy.simplify.simplify import bottom_up
def exp_trig(e):
# select the better of e, and e rewritten in terms of exp or trig
# functions
choices = [e]
if e.has(*_trigs):
choices.append(e.rewrite(exp))
choices.append(e.rewrite(cos))
return min(*choices, key=count_ops)
newexpr = bottom_up(expr, exp_trig)
if simplify:
newexpr = newexpr.simplify()
# conversion from exp to hyperbolic
ex = newexpr.atoms(exp, S.Exp1)
ex = [ei for ei in ex if 1 / ei not in ex]
## sinh and cosh
for ei in ex:
e2 = ei**-2
if e2 in ex:
a = e2.args[0] / 2 if not e2 is S.Exp1 else S.Half
newexpr = newexpr.subs((e2 + 1) * ei, 2 * cosh(a))
newexpr = newexpr.subs((e2 - 1) * ei, 2 * sinh(a))
## exp ratios to tan and tanh
for ei in ex:
n, d = ei - 1, ei + 1
et = n / d
etinv = d / n # not 1/et or else recursion errors arise
a = ei.args[0] if ei.func is exp else S.One
if a.is_Mul or a is S.ImaginaryUnit:
c = a.as_coefficient(I)
if c:
t = S.ImaginaryUnit * tan(c / 2)
newexpr = newexpr.subs(etinv, 1 / t)
newexpr = newexpr.subs(et, t)
continue
t = tanh(a / 2)
newexpr = newexpr.subs(etinv, 1 / t)
newexpr = newexpr.subs(et, t)
# sin/cos and sinh/cosh ratios to tan and tanh, respectively
if newexpr.has(HyperbolicFunction):
e, f = hyper_as_trig(newexpr)
newexpr = f(TR2i(e))
if newexpr.has(TrigonometricFunction):
newexpr = TR2i(newexpr)
# can we ever generate an I where there was none previously?
if not (newexpr.has(I) and not expr.has(I)):
expr = newexpr
return expr
def test_tan():
R, x, y = ring('x, y', QQ)
assert rs_tan(x, x, 9)/x**5 == \
Rational(17, 315)*x**2 + Rational(2, 15) + Rational(1, 3)*x**(-2) + x**(-4)
assert rs_tan(x*y + x**2*y**3, x, 9) == 4*x**8*y**11/3 + 17*x**8*y**9/45 + \
4*x**7*y**9/3 + 17*x**7*y**7/315 + x**6*y**9/3 + 2*x**6*y**7/3 + \
x**5*y**7 + 2*x**5*y**5/15 + x**4*y**5 + x**3*y**3/3 + x**2*y**3 + x*y
# Constant term in series
a = symbols('a')
R, x, y = ring('x, y', QQ[tan(a), a])
assert rs_tan(x + a, x, 5) == (tan(a)**5 + 5*tan(a)**3/3 +
2*tan(a)/3)*x**4 + (tan(a)**4 + 4*tan(a)**2/3 + Rational(1, 3))*x**3 + \
(tan(a)**3 + tan(a))*x**2 + (tan(a)**2 + 1)*x + tan(a)
assert rs_tan(x + x**2*y + a, x, 4) == (2*tan(a)**3 + 2*tan(a))*x**3*y + \
(tan(a)**4 + Rational(4, 3)*tan(a)**2 + Rational(1, 3))*x**3 + (tan(a)**2 + 1)*x**2*y + \
(tan(a)**3 + tan(a))*x**2 + (tan(a)**2 + 1)*x + tan(a)
R, x, y = ring('x, y', EX)
assert rs_tan(x + a, x, 5) == EX(tan(a)**5 + 5*tan(a)**3/3 +
2*tan(a)/3)*x**4 + EX(tan(a)**4 + 4*tan(a)**2/3 + EX(1)/3)*x**3 + \
EX(tan(a)**3 + tan(a))*x**2 + EX(tan(a)**2 + 1)*x + EX(tan(a))
assert rs_tan(x + x**2*y + a, x, 4) == EX(2*tan(a)**3 +
2*tan(a))*x**3*y + EX(tan(a)**4 + 4*tan(a)**2/3 + EX(1)/3)*x**3 + \
EX(tan(a)**2 + 1)*x**2*y + EX(tan(a)**3 + tan(a))*x**2 + \
EX(tan(a)**2 + 1)*x + EX(tan(a))
p = x + x**2 + 5
assert rs_atan(p, x, 10).compose(x, 10) == EX(atan(5) + S(67701870330562640) / \
668083460499)
def test_C99CodePrinter__precision():
n = symbols("n", integer=True)
f32_printer = C99CodePrinter(dict(type_aliases={real: float32}))
f64_printer = C99CodePrinter(dict(type_aliases={real: float64}))
f80_printer = C99CodePrinter(dict(type_aliases={real: float80}))
assert f32_printer.doprint(sin(x + 2.1)) == "sinf(x + 2.1F)"
assert f64_printer.doprint(sin(x + 2.1)) == "sin(x + 2.1000000000000001)"
assert f80_printer.doprint(sin(x + Float("2.0"))) == "sinl(x + 2.0L)"
for printer, suffix in zip([f32_printer, f64_printer, f80_printer], ["f", "", "l"]):
def check(expr, ref):
assert printer.doprint(expr) == ref.format(s=suffix, S=suffix.upper())
check(Abs(n), "abs(n)")
check(Abs(x + 2.0), "fabs{s}(x + 2.0{S})")
check(
sin(x + 4.0) ** cos(x - 2.0),
"pow{s}(sin{s}(x + 4.0{S}), cos{s}(x - 2.0{S}))",
)
check(exp(x * 8.0), "exp{s}(8.0{S}*x)")
check(exp2(x), "exp2{s}(x)")
check(expm1(x * 4.0), "expm1{s}(4.0{S}*x)")
check(Mod(n, 2), "((n) % (2))")
check(Mod(2 * n + 3, 3 * n + 5), "((2*n + 3) % (3*n + 5))")
check(Mod(x + 2.0, 3.0), "fmod{s}(1.0{S}*x + 2.0{S}, 3.0{S})")
check(Mod(x, 2.0 * x + 3.0), "fmod{s}(1.0{S}*x, 2.0{S}*x + 3.0{S})")
check(log(x / 2), "log{s}((1.0{S}/2.0{S})*x)")
check(log10(3 * x / 2), "log10{s}((3.0{S}/2.0{S})*x)")
check(log2(x * 8.0), "log2{s}(8.0{S}*x)")
check(log1p(x), "log1p{s}(x)")
check(2 ** x, "pow{s}(2, x)")
check(2.0 ** x, "pow{s}(2.0{S}, x)")
check(x ** 3, "pow{s}(x, 3)")
check(x ** 4.0, "pow{s}(x, 4.0{S})")
check(sqrt(3 + x), "sqrt{s}(x + 3)")
check(Cbrt(x - 2.0), "cbrt{s}(x - 2.0{S})")
check(hypot(x, y), "hypot{s}(x, y)")
check(sin(3.0 * x + 2.0), "sin{s}(3.0{S}*x + 2.0{S})")
check(cos(3.0 * x - 1.0), "cos{s}(3.0{S}*x - 1.0{S})")
check(tan(4.0 * y + 2.0), "tan{s}(4.0{S}*y + 2.0{S})")
check(asin(3.0 * x + 2.0), "asin{s}(3.0{S}*x + 2.0{S})")
check(acos(3.0 * x + 2.0), "acos{s}(3.0{S}*x + 2.0{S})")
check(atan(3.0 * x + 2.0), "atan{s}(3.0{S}*x + 2.0{S})")
check(atan2(3.0 * x, 2.0 * y), "atan2{s}(3.0{S}*x, 2.0{S}*y)")
check(sinh(3.0 * x + 2.0), "sinh{s}(3.0{S}*x + 2.0{S})")
check(cosh(3.0 * x - 1.0), "cosh{s}(3.0{S}*x - 1.0{S})")
check(tanh(4.0 * y + 2.0), "tanh{s}(4.0{S}*y + 2.0{S})")
check(asinh(3.0 * x + 2.0), "asinh{s}(3.0{S}*x + 2.0{S})")
check(acosh(3.0 * x + 2.0), "acosh{s}(3.0{S}*x + 2.0{S})")
check(atanh(3.0 * x + 2.0), "atanh{s}(3.0{S}*x + 2.0{S})")
check(erf(42.0 * x), "erf{s}(42.0{S}*x)")
check(erfc(42.0 * x), "erfc{s}(42.0{S}*x)")
check(gamma(x), "tgamma{s}(x)")
check(loggamma(x), "lgamma{s}(x)")
check(ceiling(x + 2.0), "ceil{s}(x + 2.0{S})")
check(floor(x + 2.0), "floor{s}(x + 2.0{S})")
check(fma(x, y, -z), "fma{s}(x, y, -z)")
check(Max(x, 8.0, x ** 4.0), "fmax{s}(8.0{S}, fmax{s}(x, pow{s}(x, 4.0{S})))")
check(Min(x, 2.0), "fmin{s}(2.0{S}, x)")
def _trigpats():
global _trigpat
a, b, c = symbols('a b c', cls=Wild)
d = Wild('d', commutative=False)
# for the simplifications like sinh/cosh -> tanh:
# DO NOT REORDER THE FIRST 14 since these are assumed to be in this
# order in _match_div_rewrite.
matchers_division = (
(a*sin(b)**c/cos(b)**c, a*tan(b)**c, sin(b), cos(b)),
(a*tan(b)**c*cos(b)**c, a*sin(b)**c, sin(b), cos(b)),
(a*cot(b)**c*sin(b)**c, a*cos(b)**c, sin(b), cos(b)),
(a*tan(b)**c/sin(b)**c, a/cos(b)**c, sin(b), cos(b)),
(a*cot(b)**c/cos(b)**c, a/sin(b)**c, sin(b), cos(b)),
(a*cot(b)**c*tan(b)**c, a, sin(b), cos(b)),
(a*(cos(b) + 1)**c*(cos(b) - 1)**c,
a*(-sin(b)**2)**c, cos(b) + 1, cos(b) - 1),
(a*(sin(b) + 1)**c*(sin(b) - 1)**c,
a*(-cos(b)**2)**c, sin(b) + 1, sin(b) - 1),
(a*sinh(b)**c/cosh(b)**c, a*tanh(b)**c, S.One, S.One),
(a*tanh(b)**c*cosh(b)**c, a*sinh(b)**c, S.One, S.One),
(a*coth(b)**c*sinh(b)**c, a*cosh(b)**c, S.One, S.One),
(a*tanh(b)**c/sinh(b)**c, a/cosh(b)**c, S.One, S.One),
(a*coth(b)**c/cosh(b)**c, a/sinh(b)**c, S.One, S.One),
(a*coth(b)**c*tanh(b)**c, a, S.One, S.One),
(c*(tanh(a) + tanh(b))/(1 + tanh(a)*tanh(b)),
tanh(a + b)*c, S.One, S.One),
)
matchers_add = (
(c*sin(a)*cos(b) + c*cos(a)*sin(b) + d, sin(a + b)*c + d),
(c*cos(a)*cos(b) - c*sin(a)*sin(b) + d, cos(a + b)*c + d),
(c*sin(a)*cos(b) - c*cos(a)*sin(b) + d, sin(a - b)*c + d),
(c*cos(a)*cos(b) + c*sin(a)*sin(b) + d, cos(a - b)*c + d),
(c*sinh(a)*cosh(b) + c*sinh(b)*cosh(a) + d, sinh(a + b)*c + d),
(c*cosh(a)*cosh(b) + c*sinh(a)*sinh(b) + d, cosh(a + b)*c + d),
)
# for cos(x)**2 + sin(x)**2 -> 1
matchers_identity = (
(a*sin(b)**2, a - a*cos(b)**2),
(a*tan(b)**2, a*(1/cos(b))**2 - a),
(a*cot(b)**2, a*(1/sin(b))**2 - a),
(a*sin(b + c), a*(sin(b)*cos(c) + sin(c)*cos(b))),
(a*cos(b + c), a*(cos(b)*cos(c) - sin(b)*sin(c))),
(a*tan(b + c), a*((tan(b) + tan(c))/(1 - tan(b)*tan(c)))),
(a*sinh(b)**2, a*cosh(b)**2 - a),
(a*tanh(b)**2, a - a*(1/cosh(b))**2),
(a*coth(b)**2, a + a*(1/sinh(b))**2),
(a*sinh(b + c), a*(sinh(b)*cosh(c) + sinh(c)*cosh(b))),
(a*cosh(b + c), a*(cosh(b)*cosh(c) + sinh(b)*sinh(c))),
(a*tanh(b + c), a*((tanh(b) + tanh(c))/(1 + tanh(b)*tanh(c)))),
)
# Reduce any lingering artifacts, such as sin(x)**2 changing
# to 1-cos(x)**2 when sin(x)**2 was "simpler"
artifacts = (
(a - a*cos(b)**2 + c, a*sin(b)**2 + c, cos),
(a - a*(1/cos(b))**2 + c, -a*tan(b)**2 + c, cos),
(a - a*(1/sin(b))**2 + c, -a*cot(b)**2 + c, sin),
(a - a*cosh(b)**2 + c, -a*sinh(b)**2 + c, cosh),
(a - a*(1/cosh(b))**2 + c, a*tanh(b)**2 + c, cosh),
(a + a*(1/sinh(b))**2 + c, a*coth(b)**2 + c, sinh),
# same as above but with noncommutative prefactor
(a*d - a*d*cos(b)**2 + c, a*d*sin(b)**2 + c, cos),
(a*d - a*d*(1/cos(b))**2 + c, -a*d*tan(b)**2 + c, cos),
(a*d - a*d*(1/sin(b))**2 + c, -a*d*cot(b)**2 + c, sin),
(a*d - a*d*cosh(b)**2 + c, -a*d*sinh(b)**2 + c, cosh),
(a*d - a*d*(1/cosh(b))**2 + c, a*d*tanh(b)**2 + c, cosh),
(a*d + a*d*(1/sinh(b))**2 + c, a*d*coth(b)**2 + c, sinh),
)
_trigpat = (a, b, c, d, matchers_division, matchers_add,
matchers_identity, artifacts)
return _trigpat
def test_torch_math():
if not torch:
skip("Torch not installed")
ma = torch.tensor([[1, 2, -3, -4]])
expr = Abs(x)
assert torch_code(expr) == "torch.abs(x)"
f = lambdify(x, expr, 'torch')
y = f(ma)
c = torch.abs(ma)
assert (y == c).all()
expr = sign(x)
assert torch_code(expr) == "torch.sign(x)"
_compare_torch_scalar((x, ), expr, rng=lambda: random.randint(0, 10))
expr = ceiling(x)
assert torch_code(expr) == "torch.ceil(x)"
_compare_torch_scalar((x, ), expr, rng=lambda: random.random())
expr = floor(x)
assert torch_code(expr) == "torch.floor(x)"
_compare_torch_scalar((x, ), expr, rng=lambda: random.random())
expr = exp(x)
assert torch_code(expr) == "torch.exp(x)"
_compare_torch_scalar((x, ), expr, rng=lambda: random.random())
# expr = sqrt(x)
# assert torch_code(expr) == "torch.sqrt(x)"
# _compare_torch_scalar((x,), expr, rng=lambda: random.random())
# expr = x ** 4
# assert torch_code(expr) == "torch.pow(x, 4)"
# _compare_torch_scalar((x,), expr, rng=lambda: random.random())
expr = cos(x)
assert torch_code(expr) == "torch.cos(x)"
_compare_torch_scalar((x, ), expr, rng=lambda: random.random())
expr = acos(x)
assert torch_code(expr) == "torch.acos(x)"
_compare_torch_scalar((x, ), expr, rng=lambda: random.uniform(0, 0.95))
expr = sin(x)
assert torch_code(expr) == "torch.sin(x)"
_compare_torch_scalar((x, ), expr, rng=lambda: random.random())
expr = asin(x)
assert torch_code(expr) == "torch.asin(x)"
_compare_torch_scalar((x, ), expr, rng=lambda: random.random())
expr = tan(x)
assert torch_code(expr) == "torch.tan(x)"
_compare_torch_scalar((x, ), expr, rng=lambda: random.random())
expr = atan(x)
assert torch_code(expr) == "torch.atan(x)"
_compare_torch_scalar((x, ), expr, rng=lambda: random.random())
# expr = atan2(y, x)
# assert torch_code(expr) == "torch.atan2(y, x)"
# _compare_torch_scalar((y, x), expr, rng=lambda: random.random())
expr = cosh(x)
assert torch_code(expr) == "torch.cosh(x)"
_compare_torch_scalar((x, ), expr, rng=lambda: random.random())
expr = acosh(x)
assert torch_code(expr) == "torch.acosh(x)"
_compare_torch_scalar((x, ), expr, rng=lambda: random.uniform(1, 2))
expr = sinh(x)
assert torch_code(expr) == "torch.sinh(x)"
_compare_torch_scalar((x, ), expr, rng=lambda: random.uniform(1, 2))
expr = asinh(x)
assert torch_code(expr) == "torch.asinh(x)"
_compare_torch_scalar((x, ), expr, rng=lambda: random.uniform(1, 2))
expr = tanh(x)
assert torch_code(expr) == "torch.tanh(x)"
_compare_torch_scalar((x, ), expr, rng=lambda: random.uniform(1, 2))
expr = atanh(x)
assert torch_code(expr) == "torch.atanh(x)"
_compare_torch_scalar((x, ), expr, rng=lambda: random.uniform(-.5, .5))
expr = erf(x)
assert torch_code(expr) == "torch.erf(x)"
_compare_torch_scalar((x, ), expr, rng=lambda: random.random())
expr = loggamma(x)
assert torch_code(expr) == "torch.lgamma(x)"
_compare_torch_scalar((x, ), expr, rng=lambda: random.random())