def test_nthroot1():
q = 1 + sqrt(2) + sqrt(3) + S(1) / 10**20
p = expand_multinomial(q**5)
assert nthroot(p, 5) == q
q = 1 + sqrt(2) + sqrt(3) + S(1) / 10**30
p = expand_multinomial(q**5)
assert nthroot(p, 5) == q
def test_nthroot1():
q = 1 + sqrt(2) + sqrt(3) + S(1)/10**20
p = expand_multinomial(q**5)
assert nthroot(p, 5) == q
q = 1 + sqrt(2) + sqrt(3) + S(1)/10**30
p = expand_multinomial(q**5)
assert nthroot(p, 5) == q
def test_minimal_polynomial_sq():
from sympy import Add, expand_multinomial
p = expand_multinomial((1 + 5 * sqrt(2) + 2 * sqrt(3))**3)
mp = minimal_polynomial(p**Rational(1, 3), x)
assert mp == x**4 - 4 * x**3 - 118 * x**2 + 244 * x + 1321
p = expand_multinomial((1 + sqrt(2) - 2 * sqrt(3) + sqrt(7))**3)
mp = minimal_polynomial(p**Rational(1, 3), x)
assert mp == x**8 - 8 * x**7 - 56 * x**6 + 448 * x**5 + 480 * x**4 - 5056 * x**3 + 1984 * x**2 + 7424 * x - 3008
p = Add(*[sqrt(i) for i in range(1, 12)])
mp = minimal_polynomial(p, x)
assert mp.subs({x: 0}) == -71965773323122507776
def test_minimal_polynomial_sq():
from sympy import Add, expand_multinomial
p = expand_multinomial((1 + 5*sqrt(2) + 2*sqrt(3))**3)
mp = minimal_polynomial(p**Rational(1, 3), x)
assert mp == x**4 - 4*x**3 - 118*x**2 + 244*x + 1321
p = expand_multinomial((1 + sqrt(2) - 2*sqrt(3) + sqrt(7))**3)
mp = minimal_polynomial(p**Rational(1, 3), x)
assert mp == x**8 - 8*x**7 - 56*x**6 + 448*x**5 + 480*x**4 - 5056*x**3 + 1984*x**2 + 7424*x - 3008
p = Add(*[sqrt(i) for i in range(1, 12)])
mp = minimal_polynomial(p, x)
assert mp.subs({x: 0}) == -71965773323122507776
def eval(self, e):
if e.ty == INTEGRAL:
raise NotImplementedError
if isinstance(e, str):
e = parser.parse_expr(e)
if expand_multinomial(expr.sympy_style(self.new_expr.normalize()).simplify()) != \
expand_multinomial(expr.sympy_style(e.normalize()).simplify()):
raise AssertionError("Rewriting by equation failed")
# if self.denom is None:
# if self.new_expr.normalize() != e.normalize():
# raise AssertionError("Rewriting by equation failed")
# else:
# if (self.new_expr * self.denom).normalize() != (e * self.denom).normalize():
# raise AssertionError("Rewriting by equation failed")
return self.new_expr
def _eval_nseries(self, x, n, logx):
if len(self.args) == 1:
from sympy import Order, ceiling, expand_multinomial
arg = self.args[0].nseries(x, n=n, logx=logx)
lt = arg.compute_leading_term(x, logx=logx)
lte = 1
if lt.is_Pow:
lte = lt.exp
if ceiling(n/lte) >= 1:
s = Add(*[(-S.One)**(k - 1)*Integer(k)**(k - 2)/
factorial(k - 1)*arg**k for k in range(1, ceiling(n/lte))])
s = expand_multinomial(s)
else:
s = S.Zero
return s + Order(x**n, x)
return super(LambertW, self)._eval_nseries(x, n, logx)
def test_nthroot():
assert nthroot(90 + 34 * sqrt(7), 3) == sqrt(7) + 3
q = 1 + sqrt(2) - 2 * sqrt(3) + sqrt(6) + sqrt(7)
assert nthroot(expand_multinomial(q**3), 3) == q
assert nthroot(41 + 29 * sqrt(2), 5) == 1 + sqrt(2)
assert nthroot(-41 - 29 * sqrt(2), 5) == -1 - sqrt(2)
expr = 1320 * sqrt(10) + 4216 + 2576 * sqrt(6) + 1640 * sqrt(15)
assert nthroot(expr, 5) == 1 + sqrt(6) + sqrt(15)
q = 1 + sqrt(2) + sqrt(3) + sqrt(5)
assert expand_multinomial(nthroot(expand_multinomial(q**5), 5)) == q
q = 1 + sqrt(2) + 7 * sqrt(6) + 2 * sqrt(10)
assert nthroot(expand_multinomial(q**5), 5, 8) == q
q = 1 + sqrt(2) - 2 * sqrt(3) + 1171 * sqrt(6)
assert nthroot(expand_multinomial(q**3), 3) == q
assert nthroot(expand_multinomial(q**6), 6) == q
def test_nthroot():
assert nthroot(90 + 34*sqrt(7), 3) == sqrt(7) + 3
q = 1 + sqrt(2) - 2*sqrt(3) + sqrt(6) + sqrt(7)
assert nthroot(expand_multinomial(q**3), 3) == q
assert nthroot(41 + 29*sqrt(2), 5) == 1 + sqrt(2)
assert nthroot(-41 - 29*sqrt(2), 5) == -1 - sqrt(2)
expr = 1320*sqrt(10) + 4216 + 2576*sqrt(6) + 1640*sqrt(15)
assert nthroot(expr, 5) == 1 + sqrt(6) + sqrt(15)
q = 1 + sqrt(2) + sqrt(3) + sqrt(5)
assert expand_multinomial(nthroot(expand_multinomial(q**5), 5)) == q
q = 1 + sqrt(2) + 7*sqrt(6) + 2*sqrt(10)
assert nthroot(expand_multinomial(q**5), 5, 8) == q
q = 1 + sqrt(2) - 2*sqrt(3) + 1171*sqrt(6)
assert nthroot(expand_multinomial(q**3), 3) == q
assert nthroot(expand_multinomial(q**6), 6) == q
def __new__(cls, expr, *args, **kwargs):
expr = sympify(expr)
if not args:
if expr.is_Order:
variables = expr.variables
point = expr.point
else:
variables = list(expr.free_symbols)
point = [S.Zero] * len(variables)
else:
args = list(args if is_sequence(args) else [args])
variables, point = [], []
if is_sequence(args[0]):
for a in args:
v, p = list(map(sympify, a))
variables.append(v)
point.append(p)
else:
variables = list(map(sympify, args))
point = [S.Zero] * len(variables)
if not all(v.is_symbol for v in variables):
raise TypeError('Variables are not symbols, got %s' % variables)
if len(list(uniq(variables))) != len(variables):
raise ValueError(
'Variables are supposed to be unique symbols, got %s' %
variables)
if expr.is_Order:
expr_vp = dict(expr.args[1:])
new_vp = dict(expr_vp)
vp = dict(zip(variables, point))
for v, p in vp.items():
if v in new_vp.keys():
if p != new_vp[v]:
raise NotImplementedError(
"Mixing Order at different points is not supported."
)
else:
new_vp[v] = p
if set(expr_vp.keys()) == set(new_vp.keys()):
return expr
else:
variables = list(new_vp.keys())
point = [new_vp[v] for v in variables]
if expr is S.NaN:
return S.NaN
if any(x in p.free_symbols for x in variables for p in point):
raise ValueError('Got %s as a point.' % point)
if variables:
if any(p != point[0] for p in point):
raise NotImplementedError(
"Multivariable orders at different points are not supported."
)
if point[0] is S.Infinity:
s = {k: 1 / Dummy() for k in variables}
rs = {1 / v: 1 / k for k, v in s.items()}
elif point[0] is S.NegativeInfinity:
s = {k: -1 / Dummy() for k in variables}
rs = {-1 / v: -1 / k for k, v in s.items()}
elif point[0] is not S.Zero:
s = dict((k, Dummy() + point[0]) for k in variables)
rs = dict((v - point[0], k - point[0]) for k, v in s.items())
else:
s = ()
rs = ()
expr = expr.subs(s)
if expr.is_Add:
from sympy import expand_multinomial
expr = expand_multinomial(expr)
if s:
args = tuple([r[0] for r in rs.items()])
else:
args = tuple(variables)
if len(variables) > 1:
# XXX: better way? We need this expand() to
# workaround e.g: expr = x*(x + y).
# (x*(x + y)).as_leading_term(x, y) currently returns
# x*y (wrong order term!). That's why we want to deal with
# expand()'ed expr (handled in "if expr.is_Add" branch below).
expr = expr.expand()
if expr.is_Add:
lst = expr.extract_leading_order(args)
expr = Add(*[f.expr for (e, f) in lst])
elif expr:
expr = expr.as_leading_term(*args)
expr = expr.as_independent(*args, as_Add=False)[1]
expr = expand_power_base(expr)
expr = expand_log(expr)
if len(args) == 1:
# The definition of O(f(x)) symbol explicitly stated that
# the argument of f(x) is irrelevant. That's why we can
# combine some power exponents (only "on top" of the
# expression tree for f(x)), e.g.:
# x**p * (-x)**q -> x**(p+q) for real p, q.
x = args[0]
margs = list(
Mul.make_args(expr.as_independent(x, as_Add=False)[1]))
for i, t in enumerate(margs):
if t.is_Power:
b, q = t.args
if b in (x, -x) and q.is_real and not q.has(x):
margs[i] = x**q
elif b.is_Power and not b.exp.has(x):
b, r = b.args
if b in (x, -x) and r.is_real:
margs[i] = x**(r * q)
elif b.is_Mul and b.args[0] is S.NegativeOne:
b = -b
if b.is_Power and not b.exp.has(x):
b, r = b.args
if b in (x, -x) and r.is_real:
margs[i] = x**(r * q)
expr = Mul(*margs)
expr = expr.subs(rs)
if expr is S.Zero:
return expr
if expr.is_Order:
expr = expr.expr
if not expr.has(*variables):
expr = S.One
# create Order instance:
vp = dict(zip(variables, point))
variables.sort(key=default_sort_key)
point = [vp[v] for v in variables]
args = (expr, ) + Tuple(*zip(variables, point))
obj = Expr.__new__(cls, *args)
return obj
def __new__(cls, expr, *args, **kwargs):
expr = sympify(expr)
if not args:
if expr.is_Order:
variables = expr.variables
point = expr.point
else:
variables = list(expr.free_symbols)
point = [S.Zero]*len(variables)
else:
args = list(args if is_sequence(args) else [args])
variables, point = [], []
if is_sequence(args[0]):
for a in args:
v, p = list(map(sympify, a))
variables.append(v)
point.append(p)
else:
variables = list(map(sympify, args))
point = [S.Zero]*len(variables)
if not all(isinstance(v, Symbol) for v in variables):
raise TypeError('Variables are not symbols, got %s' % variables)
if len(list(uniq(variables))) != len(variables):
raise ValueError('Variables are supposed to be unique symbols, got %s' % variables)
if expr.is_Order:
expr_vp = dict(expr.args[1:])
new_vp = dict(expr_vp)
vp = dict(zip(variables, point))
for v, p in vp.items():
if v in new_vp.keys():
if p != new_vp[v]:
raise NotImplementedError(
"Mixing Order at different points is not supported.")
else:
new_vp[v] = p
if set(expr_vp.keys()) == set(new_vp.keys()):
return expr
else:
variables = list(new_vp.keys())
point = [new_vp[v] for v in variables]
if expr is S.NaN:
return S.NaN
if any(x in p.free_symbols for x in variables for p in point):
raise ValueError('Got %s as a point.' % point)
if variables:
if any(p != point[0] for p in point):
raise NotImplementedError
if point[0] is S.Infinity:
s = dict([(k, 1/Dummy()) for k in variables])
rs = dict([(1/v, 1/k) for k, v in s.items()])
elif point[0] is not S.Zero:
s = dict((k, Dummy() + point[0]) for k in variables)
rs = dict((v - point[0], k - point[0]) for k, v in s.items())
else:
s = ()
rs = ()
expr = expr.subs(s)
if expr.is_Add:
from sympy import expand_multinomial
expr = expand_multinomial(expr)
if s:
args = tuple([r[0] for r in rs.items()])
else:
args = tuple(variables)
if len(variables) > 1:
# XXX: better way? We need this expand() to
# workaround e.g: expr = x*(x + y).
# (x*(x + y)).as_leading_term(x, y) currently returns
# x*y (wrong order term!). That's why we want to deal with
# expand()'ed expr (handled in "if expr.is_Add" branch below).
expr = expr.expand()
if expr.is_Add:
lst = expr.extract_leading_order(args)
expr = Add(*[f.expr for (e, f) in lst])
elif expr:
expr = expr.as_leading_term(*args)
expr = expr.as_independent(*args, as_Add=False)[1]
expr = expand_power_base(expr)
expr = expand_log(expr)
if len(args) == 1:
# The definition of O(f(x)) symbol explicitly stated that
# the argument of f(x) is irrelevant. That's why we can
# combine some power exponents (only "on top" of the
# expression tree for f(x)), e.g.:
# x**p * (-x)**q -> x**(p+q) for real p, q.
x = args[0]
margs = list(Mul.make_args(
expr.as_independent(x, as_Add=False)[1]))
for i, t in enumerate(margs):
if t.is_Pow:
b, q = t.args
if b in (x, -x) and q.is_real and not q.has(x):
margs[i] = x**q
elif b.is_Pow and not b.exp.has(x):
b, r = b.args
if b in (x, -x) and r.is_real:
margs[i] = x**(r*q)
elif b.is_Mul and b.args[0] is S.NegativeOne:
b = -b
if b.is_Pow and not b.exp.has(x):
b, r = b.args
if b in (x, -x) and r.is_real:
margs[i] = x**(r*q)
expr = Mul(*margs)
expr = expr.subs(rs)
if expr is S.Zero:
return expr
if expr.is_Order:
expr = expr.expr
if not expr.has(*variables):
expr = S.One
# create Order instance:
vp = dict(zip(variables, point))
variables.sort(key=default_sort_key)
point = [vp[v] for v in variables]
args = (expr,) + Tuple(*zip(variables, point))
obj = Expr.__new__(cls, *args)
return obj
def integral_rewrite_expr():
data = json.loads(request.get_data().decode('utf-8'))
problem = integral.parser.parse_expr(data['problem'])
old_expr = integral.parser.parse_expr(data['old_expr'])
try:
new_expr = integral.parser.parse_expr(data['new_expr'])
location = data['relative_location']
if expand_multinomial(integral.expr.sympy_style(new_expr.normalize()).simplify()) != expand_multinomial(integral.expr.sympy_style(old_expr.normalize()).simplify()) or new_expr.findVar()[0].name != problem.var:
return jsonify({
'flag': False
})
new_problem = integral.expr.Integral(problem.var, problem.lower, problem.upper, problem.body.replace_expr(location, new_expr))
if location == "":
rel_loc = "0"
else:
rel_loc = "0." + location
if old_expr.ty == integral.expr.OP and old_expr.op == "/" or\
old_expr.ty == integral.expr.OP and old_expr.op == "*" and\
old_expr.args[1].ty == integral.expr.OP and old_expr.args[1].op == "^" and\
old_expr.args[1].args[1] == integral.expr.Const(-1):
denom = old_expr.args[1]
step = {
'flag': True,
'text': str(new_problem),
'latex': integral.latex.convert_expr(new_problem),
'reason': "Rewrite",
'_latex_reason': "Rewrite \\(%s\\) to \\(%s\\)"%(integral.latex.convert_expr(old_expr),
integral.latex.convert_expr(new_expr)),
'params': {
'rhs': data['new_expr'],
'denom': str(denom)
},
"location": data['absolute_location']
}
return jsonify(step)
else:
step = {
'flag': True,
'text': str(new_problem),
'latex': integral.latex.convert_expr(new_problem),
'reason': "Rewrite",
'_latex_reason': "Rewrite \\(%s\\) to \\(%s\\)"%(integral.latex.convert_expr(old_expr),
integral.latex.convert_expr(new_expr)),
'params': {
'rhs': data['new_expr']
},
"location": data['absolute_location']
}
return jsonify(step)
except (exceptions.UnexpectedCharacters, exceptions.UnexpectedToken) as e:
return jsonify({
'flag': False
})