def test_sets():
x = Integer(2)
y = Integer(3)
x1 = sympy.Integer(2)
y1 = sympy.Integer(3)
assert Interval(x, y) == Interval(x1, y1)
assert Interval(x1, y) == Interval(x1, y1)
assert Interval(x, y)._sympy_() == sympy.Interval(x1, y1)
assert sympify(sympy.Interval(x1, y1)) == Interval(x, y)
assert sympify(sympy.EmptySet()) == EmptySet()
assert sympy.EmptySet() == EmptySet()._sympy_()
assert FiniteSet(x, y) == FiniteSet(x1, y1)
assert FiniteSet(x1, y) == FiniteSet(x1, y1)
assert FiniteSet(x, y)._sympy_() == sympy.FiniteSet(x1, y1)
assert sympify(sympy.FiniteSet(x1, y1)) == FiniteSet(x, y)
x = Interval(1, 2)
y = Interval(2, 3)
x1 = sympy.Interval(1, 2)
y1 = sympy.Interval(2, 3)
assert Union(x, y) == Union(x1, y1)
assert Union(x1, y) == Union(x1, y1)
assert Union(x, y)._sympy_() == sympy.Union(x1, y1)
assert sympify(sympy.Union(x1, y1)) == Union(x, y)
assert Complement(x, y) == Complement(x1, y1)
assert Complement(x1, y) == Complement(x1, y1)
assert Complement(x, y)._sympy_() == sympy.Complement(x1, y1)
assert sympify(sympy.Complement(x1, y1)) == Complement(x, y)
def __init__(self, difficulty, var=None):
if var is None:
var = x
if difficulty == 1:
c = 0
m = random.randint(-coefficients_bound + 2, coefficients_bound - 4)
if m >= 0: # make sure m is not 0 or 1
m += 2
elif difficulty == 2:
c = not_named_yet.randint_no_zero(-coefficients_bound,
coefficients_bound)
m = 1
elif difficulty == 3:
c = not_named_yet.randint_no_zero(-coefficients_bound,
coefficients_bound)
m = random.randint(-coefficients_bound + 2, coefficients_bound - 4)
if m >= 0: # make sure m is not 0 or 1
m += 2
else:
raise ValueError(
'You have supplied an invalid difficulty level! Choose between 1, 2 or 3.'
)
self.equation = m * var + c
self.domain = sympy.Interval(-sympy.oo, sympy.oo, True, True)
self.range = sympy.Interval(-sympy.oo, sympy.oo, True, True)
def __init__(self, difficulty):
if difficulty not in [1, 2, 3]:
raise ValueError('You gave an invalid difficulty of %d!' %
difficulty)
while True:
a = not_named_yet.randint_no_zero(-coefficients_bound + 2,
coefficients_bound - 2)
b = random.randint(-coefficients_bound * 4, coefficients_bound * 4)
c = random.randint(-coefficients_bound * 4, coefficients_bound * 4)
discriminant = b**2 - 4 * a * c
if difficulty == 1 and discriminant == 0 and (b == 0 or c == 0):
break
elif difficulty == 2 and discriminant > 0 and gmpy.is_square(
discriminant):
break
elif difficulty == 3 and discriminant > 0 and not gmpy.is_square(
discriminant):
break
self.equation = a * x**2 + b * x + c
self.discriminant = b**2 - 4 * a * c
self.domain = sympy.Interval(-sympy.oo, sympy.oo, True, True)
vertex_x = -sympy.Rational(b, 2 * a)
vertex_y = self.equation.subs({x: vertex_x})
if a > 0:
self.range = sympy.Interval(vertex_y, sympy.oo, False, True)
else:
self.range = sympy.Interval(-sympy.oo, vertex_y, True, False)
def __init__(self, difficulty):
a = not_named_yet.randint_no_zero(-3, 2)
k = not_named_yet.randint_no_zero(-2, 2)
c = not_named_yet.randint_no_zero(-5, 5)
if a == 1:
a += 1
if difficulty == 1:
self.equation = sympy.exp(k * x) + c
elif difficulty == 2:
self.equation = a * sympy.exp(k * x) + c
elif difficulty == 3:
inner_function = request_linear(difficulty=3).equation
self.equation = a * sympy.exp(inner_function) + c
else:
raise ValueError(
'You have given an invalid difficulty level! Please use difficulty levels 1-3'
)
self.domain = sympy.Interval(-sympy.oo, sympy.oo, True, True)
if a < 0:
self.range = sympy.Interval(-sympy.oo, c, True, True)
else:
self.range = sympy.Interval(c, sympy.oo, True, True)
def __init__(self, part):
self.num_lines, self.num_marks = 2, 1
self._qp = copy.deepcopy(part._qp)
self._qp['interval'] = random.choice([
sympy.Interval(-sympy.oo, self._qp['mean']),
sympy.Interval(self._qp['mean'], sympy.oo)
])
def to_port_annotation(parsed):
# convert both single port and ranges into sympy.Interval
# (open intervals)
if isinstance(parsed, int):
rv = sm.Interval(parsed - 1, parsed + 1, True, True)
else:
rv = sm.Interval(parsed[0] - 1, parsed[1] + 1, True, True)
return rv
def Inflection_Interval(self):
I=self.inflection_points()
if len(I)>1:
a=min(I)
b=max(I)
return sp.Interval(a,b)
else:
return sp.Interval(-10,10)
def __init__(self, part):
self.num_lines, self.num_marks = 12, 3
self._qp = copy.deepcopy(part._qp)
domain = sympy.Interval(-2 * sympy.pi, 2 * sympy.pi)
bounds = sensible_values.integral_domain(self._qp['equation'], domain)
self._qp['domain'] = sympy.Interval(bounds[0], bounds[1])
self._qp['answer'] = self._qp['hidden_objective'].integrate(
(x, self._qp['domain'].left, self._qp['domain'].right))
def solution_statement(self):
lines = solutions.Lines()
left_pr = sympy.Rational(1, 2)
if self._qp['event_provided_to_student'].left == -sympy.oo:
symmetrically_opposite_event = sympy.Interval(
self._qp['mean'], self._qp['point'])
opposite_half_of_normal_distribution = sympy.Interval(
self._qp['mean'], sympy.oo)
event_provided_to_student = sympy.Interval(
self._qp['event_provided_to_student'].right, sympy.oo)
right_pr = self._qi['unknown_variable']
elif self._qp['event_provided_to_student'].right == sympy.oo:
symmetrically_opposite_event = sympy.Interval(
self._qp['point'], self._qp['mean'])
opposite_half_of_normal_distribution = sympy.Interval(
-sympy.oo, self._qp['mean'])
event_provided_to_student = sympy.Interval(
-sympy.oo, self._qp['event_provided_to_student'].left)
right_pr = 1 - self._qi['unknown_variable']
elif self._qp['point'] < self._qp['mean']:
symmetrically_opposite_event = sympy.Interval(
-sympy.oo, self._qp['point'])
opposite_half_of_normal_distribution = sympy.Interval(
-sympy.oo, self._qp['mean'])
event_provided_to_student = self._qp['event_provided_to_student']
right_pr = self._qi['unknown_variable']
else:
symmetrically_opposite_event = sympy.Interval(
self._qp['point'], sympy.oo)
opposite_half_of_normal_distribution = sympy.Interval(
self._qp['mean'], sympy.oo)
event_provided_to_student = self._qp['event_provided_to_student']
right_pr = 1 - self._qi['unknown_variable']
# e.g. Pr(56 <= X <= 60) = Pr(60 <= X <= 64) = Pr(X >= 60) - Pr(X >= 64)
lines += r'''$Pr({desired_event}) = Pr({symmetrically_opposite_event}) =
Pr({opposite_half_of_normal_distribution}) - Pr({event_provided_to_student})$'''.format(
desired_event=expressions.relation(self._qp['desired_event'],
var=X),
symmetrically_opposite_event=expressions.relation(
symmetrically_opposite_event, var=X),
opposite_half_of_normal_distribution=expressions.relation(
opposite_half_of_normal_distribution, var=X),
event_provided_to_student=expressions.relation(
event_provided_to_student, var=X),
)
# e.g. = (1/2) - (-c + 1) = c - 1/2
lines += r'$= ({half_interval_probability}) - ({probability_derived_from_question}) = {answer}$'.format(
half_interval_probability=sympy.latex(left_pr),
probability_derived_from_question=sympy.latex(right_pr),
answer=sympy.latex(self._qp['answer']))
return lines.write()
def test_is_monotone_increasing():
assert functions.is_monotone_increasing(sympy.exp(x))
assert functions.is_monotone_increasing(x**3)
assert functions.is_monotone_increasing(x**2,
sympy.Interval(0, oo, False, True))
assert functions.is_monotone_increasing(
sympy.sin(x), sympy.Interval(0, sympy.pi / 2, False, False))
assert functions.is_monotone_increasing(-1 / x)
assert not functions.is_monotone_increasing(x**4)
def detect_multi_stability(cls, chnk_stable, chnk_unstable, bi_data_np):
if cls.is_list_empty(chnk_stable) or cls.is_list_empty(chnk_unstable):
return False
chnk_stable_pcp_ranges = []
for i in range(len(chnk_stable)):
end_ind = len(chnk_stable[i]) - 1
chnk_stable_pcp_ranges.append([
bi_data_np[chnk_stable[i][0], 0],
bi_data_np[chnk_stable[i][end_ind], 0]
])
[i.sort() for i in chnk_stable_pcp_ranges]
chnk_unstable_pcp_ranges = []
for i in range(len(chnk_unstable)):
end_ind = len(chnk_unstable[i]) - 1
chnk_unstable_pcp_ranges.append([
bi_data_np[chnk_unstable[i][0], 0],
bi_data_np[chnk_unstable[i][end_ind], 0]
])
[i.sort() for i in chnk_unstable_pcp_ranges]
chnk_unstable_pcp_intervals = []
chnk_stable_pcp_intervals = []
for i in range(len(chnk_unstable)):
chnk_unstable_pcp_intervals.append(
sympy.Interval(chnk_unstable_pcp_ranges[i][0],
chnk_unstable_pcp_ranges[i][1]))
for i in range(len(chnk_stable)):
chnk_stable_pcp_intervals.append(
sympy.Interval(chnk_stable_pcp_ranges[i][0],
chnk_stable_pcp_ranges[i][1]))
# building intersections of unstable branch with stable branches
unstable_intersections = []
for i in chnk_unstable_pcp_intervals:
temp_list = []
for j in chnk_stable_pcp_intervals:
temp = i.intersect(j)
if temp != sympy.EmptySet():
if not temp.is_FiniteSet:
temp_list.append(1)
elif temp.is_FiniteSet and len(list(temp)) > 1:
temp_list.append(1)
unstable_intersections.append(temp_list)
return any([sum(i) >= 2 for i in unstable_intersections])
def relation_to_interval(relation):
if isinstance(relation, sympy.Or):
return functools.reduce(operator.or_, [relation_to_interval(i) for i in relation.args])
elif isinstance(relation, sympy.And):
return functools.reduce(operator.and_, [relation_to_interval(i) for i in relation.args])
if relation.rel_op == '>':
if isinstance(relation.lhs, sympy.Symbol):
return sympy.Interval(relation.rhs, sympy.oo, True, True)
else:
return sympy.Interval(-sympy.oo, relation.lhs, True, True)
elif relation.rel_op == '>=':
if isinstance(relation.lhs, sympy.Symbol):
return sympy.Interval(relation.rhs, sympy.oo)
else:
return sympy.Interval(-sympy.oo, relation.lhs)
elif relation.rel_op == '<':
if isinstance(relation.lhs, sympy.Symbol):
return sympy.Interval(-sympy.oo, relation.rhs, True, True)
else:
return sympy.Interval(relation.lhs, sympy.oo, True, True)
elif relation.rel_op == '<=':
if isinstance(relation.lhs, sympy.Symbol):
return sympy.Interval(-sympy.oo, relation.rhs)
else:
return sympy.Interval(relation.lhs, sympy.oo)
def construct_interval(boundary1, boundary2):
"""Construct a sympy interval from two boundaries, when we don't know which boundary is bigger.
>>> SimpleNormalDistribution.construct_interval(0, 1)
[0, 1]
>>> SimpleNormalDistribution.construct_interval(sympy.oo, 1)
[1, oo)
"""
if boundary1 < boundary2:
return sympy.Interval(boundary1, boundary2)
else:
return sympy.Interval(boundary2, boundary1)
def integer_domain(low=-5, high=5, minimum_distance=3):
""" Return a sympy.Interval domain with min/max bounds of input variables low and high.
The interval is chosen by selecting integers randomly: random.randint(low, high)
"""
while True:
b1 = random.randint(low, high)
b2 = random.randint(low, high)
if b1 + minimum_distance <= b2:
return sympy.Interval(b1, b2, False, False)
elif b2 + minimum_distance <= b1:
return sympy.Interval(b2, b1, False, False)
def rand_interval_type(a, b):
"""
Generate an interval with random type of bounds.
"""
bl = rd.choice([True, False])
br = rd.choice([True, False])
return sp.Interval(a, b, left_open=bl, right_open=br)
def remove_nonzero_factors(eqn, reduce_monomials=False, atom_bounds=None):
if eqn == 0:
return eqn
result = 1
domains = {}
for atom in [a for a in eqn.atoms() if a.is_symbol]:
if atom_bounds and atom in atom_bounds:
domains[atom] = sp.Interval(*atom_bounds[atom])
else:
domains[atom] = sp.Reals
for factor in sp.factor(sp.cancel(eqn)).args:
if isinstance(factor, sp.Pow) and factor.args[1] < 0:
continue
if reduce_monomials:
atoms = [a for a in factor.atoms() if a.is_symbol]
if len(atoms) == 1:
a, = atoms
solve_set = sp.solveset(factor, domain=domains[a])
if not solve_set:
continue
result *= factor
return result
def axvspan(self, axes=None, phase_offsets=[0], **kwargs):
""" Overlay range on matplotlib axes.
N.B. set phase_offsets=[0,1] to overlay on
phaseogram that varies from 0 to 2 the phase
range both on the 0-1 and the 1-2 part of the plot. """
import pylab as P
if axes is None: axes=P.gca()
label=kwargs.pop('label',None)
if phase_offsets != [0] and phase_offsets !=0:
# kind of ugly, but create a larger PhaseRange object
# temporarily with the offsets. This allows for
# merging needed offsets.
# (a) create a giant list of all phases
all_phases = reduce(add,[[[a+o,b+o] for a,b in self.tolist(dense=False)] for o in phase_offsets])
# (b) turn the list of ranges into a sympy object
interval = sympy.Union([sympy.Interval(a,b) for a,b in all_phases])
# (c) cretae a temporary phase range object which spans all intervals
temp = PhaseRange()
temp.range = interval
else:
temp=self
ret = []
for a,b in temp.tolist(dense=False):
ret.append(axes.axvspan(a, b, label=label, **kwargs))
label=None
return ret
def transform_set(x, expr, sympy_set):
""" Transform a sympy_set by an expression
>>> x = sympy.Symbol('x')
>>> domain = sympy.Interval(-sympy.pi / 4, -sympy.pi / 6, False, True) | sympy.Interval(sympy.pi / 6, sympy.pi / 4, True, False)
>>> transform_set(x, -2 * x, domain)
[-pi/2, -pi/3) U (pi/3, pi/2]
"""
if isinstance(sympy_set, sympy.Union):
return sympy.Union(
transform_set(x, expr, arg) for arg in sympy_set.args)
if isinstance(sympy_set, sympy.Intersection):
return sympy.Intersection(
transform_set(x, expr, arg) for arg in sympy_set.args)
f = sympy.Lambda(x, expr)
if isinstance(sympy_set, sympy.Interval):
left, right = f(sympy_set.left), f(sympy_set.right)
if left < right:
new_left_open = sympy_set.left_open
new_right_open = sympy_set.right_open
else:
new_left_open = sympy_set.right_open
new_right_open = sympy_set.left_open
return sympy.Interval(sympy.Min(left, right), sympy.Max(left, right),
new_left_open, new_right_open)
if isinstance(sympy_set, sympy.FiniteSet):
return sympy.FiniteSet(list(map(f, sympy_set)))
def conditional_integral(expr, domain):
''' Return two points within the domain (excluding the end-points) on which to perform a conditional integral from one side of the domain.
'''
modified_domain = sympy.Interval(domain.left, domain.right, True, True)
return _delegate(expr, modified_domain, num=2)
def solution_statement(self):
lines = solutions.Lines()
X = sympy.Symbol('X')
lines += r'Let $X = {exp_of_x}$'.format(
exp_of_x=sympy.latex(sympy.exp(x))
)
hidden_quadratic = self._qp['equation'].replace(sympy.exp(x), X). replace(sympy.exp(2 * x), X ** 2)
lines += r'$\therefore {equation} = {hidden_quadratic}$'.format(
equation=sympy.latex(self._qp['equation']),
hidden_quadratic=sympy.latex(hidden_quadratic)
)
lines += r'$= {factorised_hidden_quadratic}$'.format(
factorised_hidden_quadratic=sympy.latex(hidden_quadratic.factor())
)
lines += expressions.shrink_solution_set(expr=hidden_quadratic, domain=sympy.Interval(0, sympy.oo), var=X)
valid_solution = [i for i in sympy.solve(hidden_quadratic) if i > 0][0]
lines += r'${exp_of_x} = {valid_solution}$'.format(
exp_of_x=sympy.latex(sympy.exp(x)),
valid_solution=valid_solution
)
answer = sympy.log(valid_solution)
lines += r'$x = {answer}$'.format(
answer=sympy.latex(answer)
)
return lines.write()
def str2interval(s, local_dict={}):
"""
Convert a latex string into an interval.
"""
s = s.strip()
s = s.replace("\emptyset", "EmptySet()")
s = s.replace("\R", "S.Reals")
s = s.replace("\infty", "oo")
pattern = re.compile(r'\\{(.*)\\}')
if pattern.search(s) is not None:
return sp.FiniteSet(str2expr(pattern.search(s).group(1)))
pattern = re.compile(r'\\lbrack(.*),(.*)\\rbrack')
if pattern.search(s) is not None:
return sp.Interval(str2expr(pattern.search(s).group(1)),
str2expr(pattern.search(s).group(2)))
pattern = re.compile(r'\\lbrack(.*),(.*)\\lbrack')
if pattern.search(s) is not None:
return sp.Interval.Ropen(str2expr(pattern.search(s).group(1)),
str2expr(pattern.search(s).group(2)))
pattern = re.compile(r'\\rbrack(.*),(.*)\\rbrack')
if pattern.search(s) is not None:
return sp.Interval.Lopen(str2expr(pattern.search(s).group(1)),
str2expr(pattern.search(s).group(2)))
pattern = re.compile(r'\\rbrack(.*),(.*)\\lbrack')
if pattern.search(s) is not None:
return sp.Interval.open(str2expr(pattern.search(s).group(1)),
str2expr(pattern.search(s).group(2)))
def __init__(self):
self.num_lines = 7
self.num_marks = 3
self._qp = {}
function = all_functions.request_linear(difficulty=3).equation
self._qp['equation'] = 1 / function
# we can't integrate over a range of a hyperbola that has the vertical asymptote in the middle
# we will only integrate on one side of the hyperbola
vertical_asymptote = sympy.solve(function)[0]
if random.randint(0, 1): # use the side to the left of the asymptote
bound = vertical_asymptote.p // vertical_asymptote.q
possible_x_values = list(range(bound - 5, bound))
else:
bound = vertical_asymptote.p // vertical_asymptote.q + 1
possible_x_values = list(range(bound, bound + 5))
boundary = all_functions.choose_bounds(possible_x_values)
self._qp['boundary'] = sympy.Interval(*boundary)
integral_result = not_named_yet.soft_logcombine(
self._qp['equation'].integrate(
(x, self._qp['boundary'].left, self._qp['boundary'].right)))
self._qp['answer'] = integral_result.match(sympy.log(x0) / x1)[x0]
p = sympy.Symbol('p')
self._qp['variable_expression'] = integral_result.replace(
sympy.log(x0), sympy.log(p))
def piecewise_one(self, expr):
if self.hs._dimension is None:
to_val = sympy.oo
else:
to_val = self.hs.dimension
return Piecewise(
(1, sympy.Interval(0, to_val).as_relational(expr)), (0, True)
)
def round_interval(interval):
if interval.is_empty:
return interval
left = sp.floor(interval.left) if interval.left > 0 else sp.ceiling(interval.left)
right = sp.floor(interval.right) if interval.right > 0 else sp.ceiling(interval.right)
left_open = not interval.contains(left)
right_open = not interval.contains(right)
return sp.Interval(left, right, left_open=left_open, right_open=right_open)
def __init__(self, part):
self.num_lines, self.num_marks = 6, 2
self._qp = copy.deepcopy(part._qp)
self._qi = {}
self._qi['unknown_variable'] = a
self._qp['point'] = not_named_yet.randint(self._qp['mean'] - 10,
self._qp['mean'] + 10,
exclude=[self._qp['mean']])
self._qp['symmetrically_opposite_point'] = 2 * self._qp[
'mean'] - self._qp['point']
# there are 4 connected intervals:
# where "2*mean - point" is the symmetric opposite of "point" in terms of the mean
# (-sympy.oo, point) pr = q
# (2*mean - point, sympy.oo) pr = q
# (mean, point) pr = 0.5 - q
# (2*mean - point, mean) pr = 0.5 - q
if random.choice([True, False
]): # we do not have infinity as one of the bounds
self._qp[
'event_provided_to_student'] = SimpleNormalDistribution.construct_interval(
self._qp['mean'], self._qp['point'])
if self._qp['point'] > self._qp['mean']:
self._qp['desired_event'] = sympy.Interval(
-sympy.oo, self._qp['symmetrically_opposite_point'])
else:
self._qp['desired_event'] = sympy.Interval(
self._qp['symmetrically_opposite_point'], sympy.oo)
self._qp['answer'] = sympy.Rational(
1, 2) - self._qi['unknown_variable']
else: # we do have infinity as one of the bounds
self._qp[
'desired_event'] = SimpleNormalDistribution.construct_interval(
self._qp['symmetrically_opposite_point'], self._qp['mean'])
if self._qp['point'] > self._qp['mean']:
self._qp['event_provided_to_student'] = sympy.Interval(
-sympy.oo, self._qp['point'])
else:
self._qp['event_provided_to_student'] = sympy.Interval(
self._qp['point'], sympy.oo)
self._qp['answer'] = -sympy.Rational(
1, 2) + self._qi['unknown_variable']
def test_sets():
x = Integer(2)
y = Integer(3)
x1 = sympy.Integer(2)
y1 = sympy.Integer(3)
assert Interval(x, y) == Interval(x1, y1)
assert Interval(x1, y) == Interval(x1, y1)
assert Interval(x, y)._sympy_() == sympy.Interval(x1, y1)
assert sympify(sympy.Interval(x1, y1)) == Interval(x, y)
assert sympify(sympy.S.EmptySet) == EmptySet()
assert sympy.S.EmptySet == EmptySet()._sympy_()
assert sympify(sympy.S.UniversalSet) == UniversalSet()
assert sympy.S.UniversalSet == UniversalSet()._sympy_()
assert sympify(sympy.S.Reals) == Reals()
assert sympy.S.Reals == Reals()._sympy_()
assert sympify(sympy.S.Rationals) == Rationals()
assert sympy.S.Rationals == Rationals()._sympy_()
assert sympify(sympy.S.Integers) == Integers()
assert sympy.S.Integers == Integers()._sympy_()
assert FiniteSet(x, y) == FiniteSet(x1, y1)
assert FiniteSet(x1, y) == FiniteSet(x1, y1)
assert FiniteSet(x, y)._sympy_() == sympy.FiniteSet(x1, y1)
assert sympify(sympy.FiniteSet(x1, y1)) == FiniteSet(x, y)
x = Interval(1, 2)
y = Interval(2, 3)
x1 = sympy.Interval(1, 2)
y1 = sympy.Interval(2, 3)
assert Union(x, y) == Union(x1, y1)
assert Union(x1, y) == Union(x1, y1)
assert Union(x, y)._sympy_() == sympy.Union(x1, y1)
assert sympify(sympy.Union(x1, y1)) == Union(x, y)
assert Complement(x, y) == Complement(x1, y1)
assert Complement(x1, y) == Complement(x1, y1)
assert Complement(x, y)._sympy_() == sympy.Complement(x1, y1)
assert sympify(sympy.Complement(x1, y1)) == Complement(x, y)
def maximal_domain(expr, domain=sympy.Interval(-oo, oo)):
''' Return the maximal domain of an expression.
>>> maximal_domain(sympy.log(x**2 - 1))
(-oo, -1) U (1, oo)
>>> maximal_domain(1 / (x**2 - 4))
(-oo, -2) U (-2, 2) U (2, oo)
>>> maximal_domain((x - 2) ** (sympy.Rational(3, 2)))
(2, oo)
'''
# 4 possible scenarios:
# 1. 1/(a) -- a != 0
# 2. sqrt(b) -- b > 0
# 3. log(c) -- c > 0
# 4. tan(d) -- d != the solutions of tan
for symbol in expr.free_symbols:
if symbol == x:
expr = expr.replace(x, sympy.Symbol('x', real=True))
else:
raise NotImplementedError('only x is currently supported - but that can be easily changed')
powers = expr.find(sympy.Pow)
# case 1
for power in powers:
if power.args[1] < 0:
denominator = power.args[0]
solution = sympy.solve(denominator)
domain &= -sympy.FiniteSet(solution)
# case 2
if isinstance(power.args[1], sympy.Rational):
if power.args[1].q % 2 != 0:
continue
sqrt_interior = power.args[0]
solution = sympy.solve(sqrt_interior > 0)
domain &= relation_to_interval(solution)
# case 3
logs = expr.find(sympy.log)
for log in logs:
interior = log.args[0]
solution = sympy.solve(interior > 0)
domain &= relation_to_interval(solution)
# case 4
if expr.find(sympy.tan) or expr.find(sympy.cot) or expr.find(sympy.sec) or expr.find(sympy.csc):
raise NotImplementedError('tan/cot/sec/csc are not supported')
return domain
def __init__(self, *args):
if len(args) == 1 and isinstance(args[0], PhaseRange):
self.range = copy.deepcopy(args[0].range)
return
if len(args) == 1 and np.alltrue(len(i) == 2 for i in args):
args = args[0]
if len(args) == 2 and \
isinstance(args[0],numbers.Real) and \
isinstance(args[1],numbers.Real):
args = [list(args)]
ranges = []
for range in args:
if range[0] not in PhaseRange.allowed_phase_input or \
range[1] not in PhaseRange.allowed_phase_input:
raise Exception(
"Error, phase range %s is outside allowed range." %
str(range))
if np.allclose(range[0]-range[1],1) or \
np.allclose(range[1]-range[0],1):
range = [0, 1]
else:
for i in [0, 1]:
if range[i] != 1: range[i] %= 1
if range[0] > range[1]:
# worry about wraping phase ranges, for example [0.6, 0.2] -> [0,0.2]U[0.6,1]
if range[1] == 0:
ranges.append(sympy.Interval(range[0], 1))
else:
ranges += [
sympy.Interval(0, range[1]),
sympy.Interval(range[0], 1)
]
else:
ranges.append(sympy.Interval(*range))
self.range = sympy.Union(*ranges)
def is_reasonable_tangent(tangent):
"""Check if the tangent has student-friendly coefficients.
"""
gradient = tangent.coeff(x, 1)
if not isinstance(gradient, sympy.Integer):
return False
elif gradient not in sympy.Interval(-5, 5):
return False
elif gradient == 0:
return False
y_intercept = tangent.coeff(x, 0)
if not isinstance(y_intercept, sympy.Integer):
return False
elif y_intercept not in sympy.Interval(-10, 10):
return False
return True
def solution_statement(self):
inverse = all_functions.inverse(self._qp['equation'])
maximal_domain_original = functions.maximal_domain(self._qp['equation'])
maximal_domain_inverse = functions.maximal_domain(inverse)
maximal_domain = maximal_domain_original & maximal_domain_inverse
expr_domain = maximal_domain & sympy.Interval(-self._qp['MAX_PLOT_RANGE'], self._qp['MAX_PLOT_RANGE'])
# an inverse of an inverse is always just "y = x"
path = plot.plot(x,
plot_domain=sympy.Interval(-self._qp['MAX_PLOT_RANGE'], self._qp['MAX_PLOT_RANGE']),
plot_range=sympy.Interval(-self._qp['MAX_PLOT_RANGE'], self._qp['MAX_PLOT_RANGE']),
expr_domain=expr_domain
)
return r'''{0}'''.format(plot.latex(path))