def field_isomorphism(a, b, **args):
"""Construct an isomorphism between two number fields. """
a, b = sympify(a), sympify(b)
if not a.is_AlgebraicNumber:
a = AlgebraicNumber(a)
if not b.is_AlgebraicNumber:
b = AlgebraicNumber(b)
if a == b:
return a.coeffs()
n = a.minpoly.degree()
m = b.minpoly.degree()
if n == 1:
return [a.root]
if m % n != 0:
return
if args.get('fast', True):
try:
result = field_isomorphism_pslq(a, b)
if result is not None:
return result
except NotImplementedError:
pass
return field_isomorphism_factor(a, b)
def __new__(cls, domain, density):
density = {sympify(key): sympify(val) for key, val in density.items()}
public_density = Dict(density)
obj = PSpace.__new__(cls, domain, public_density)
obj._density = density
return obj
def __init__(self, var, update_string, program_variables=None, is_random_var=False, random_var=None):
self.is_random_var = is_random_var
self.random_var = random_var
self.var = var
# check if this is a RV or expression update
rv = re.search(r"RV\((?P<params>.+)\)", update_string)
if rv is not None:
self.is_random_var = True
dist, *params = map(str.strip, rv.group('params').split(','))
params = list(map(sympify, params))
self.random_var = RandomVar(dist, params, var_name=str(self.var))
# here: if not is_random_var == else
if not self.is_random_var:
self.branches = []
branches = update_string.split(";")
for update in branches:
if '@' in update:
exp, prob = update.split("@")
else:
exp, prob = update, 1-sum([b[1] for b in self.branches])
prob = sympify(prob)
if prob.is_Float:
prob = Rational(str(prob))
if not prob.is_zero:
exp = make_symbols_positive(sympify(exp), program_variables)
prob = make_symbols_positive(prob, program_variables)
self.branches.append((sympify(exp), prob))
if sum([b[1] for b in self.branches]) != 1:
raise Exception(f"Branch probabilities for {self.var} update do not sum up to 1. Terminating.")
def test_sympyissue_6046():
assert str(sympify('Q & C', locals=_clash1)) == 'C & Q'
assert str(sympify('pi(x)', locals=_clash2)) == 'pi(x)'
assert str(sympify('pi(C, Q)', locals=_clash)) == 'pi(C, Q)'
locals = {}
exec('from diofant.abc import S, O', locals) # pylint: disable=exec-used
assert str(sympify('O&S', locals)) == 'O & S'
def test_product_basic():
H, T = 'H', 'T'
unit_line = Interval(0, 1)
d6 = FiniteSet(1, 2, 3, 4, 5, 6)
d4 = FiniteSet(1, 2, 3, 4)
coin = FiniteSet(H, T)
square = unit_line * unit_line
assert (0, 0) in square
assert 0 not in square
assert (H, T) in coin ** 2
assert (.5, .5, .5) in square * unit_line
assert (H, 3, 3) in coin*d6*d6
HH, TT = sympify(H), sympify(T)
assert set(coin**2) == {(HH, HH), (HH, TT), (TT, HH), (TT, TT)}
assert (d4*d4).is_subset(d6*d6)
assert (square.complement(Interval(-oo, oo)*Interval(-oo, oo)) ==
Union((Interval(-oo, 0, True, True) +
Interval(1, oo, True, True))*Interval(-oo, oo),
Interval(-oo, oo)*(Interval(-oo, 0, True, True) +
Interval(1, oo, True, True))))
assert (Interval(-5, 5)**3).is_subset(Interval(-10, 10)**3)
assert not (Interval(-10, 10)**3).is_subset(Interval(-5, 5)**3)
assert not (Interval(-5, 5)**2).is_subset(Interval(-10, 10)**3)
assert (Interval(.2, .5)*FiniteSet(.5)).is_subset(square) # segment in square
assert len(coin*coin*coin) == 8
assert len(S.EmptySet*S.EmptySet) == 0
assert len(S.EmptySet*coin) == 0
pytest.raises(TypeError, lambda: len(coin*Interval(0, 2)))
def test_sympyissue_7840():
# daveknippers' example
C393 = sympify(
'Piecewise((C391 - 1.65, C390 < 0.5), (Piecewise((C391 - 1.65, \
C391 > 2.35), (C392, True)), True))'
)
C391 = sympify(
'Piecewise((2.05*C390**(-1.03), C390 < 0.5), (2.5*C390**(-0.625), True))'
)
C393 = C393.subs({'C391': C391})
# simple substitution
sub = {}
sub['C390'] = 0.703451854
sub['C392'] = 1.01417794
ss_answer = C393.subs(sub)
# cse
substitutions, new_eqn = cse(C393)
for pair in substitutions:
sub[pair[0].name] = pair[1].subs(sub)
cse_answer = new_eqn[0].subs(sub)
# both methods should be the same
assert ss_answer == cse_answer
# GitRay's example
expr = Piecewise((Symbol('ON'), Eq(Symbol('mode'), Symbol('ON'))),
(Piecewise((Piecewise((Symbol('OFF'), Symbol('x') < Symbol('threshold')),
(Symbol('ON'), true)), Eq(Symbol('mode'), Symbol('AUTO'))),
(Symbol('OFF'), true)), true))
substitutions, new_eqn = cse(expr)
# this Piecewise should be exactly the same
assert new_eqn[0] == expr
# there should not be any replacements
assert len(substitutions) < 1
def test_sympyify_iterables():
ans = [Rational(3, 10), Rational(1, 5)]
assert sympify(['.3', '.2'], rational=True) == ans
assert sympify({'.3', '.2'}, rational=True) == set(ans)
assert sympify(('.3', '.2'), rational=True) == Tuple(*ans)
assert sympify({x: 0, y: 1}) == {x: 0, y: 1}
assert sympify(['1', '2', ['3', '4']]) == [Integer(1), Integer(2), [Integer(3), Integer(4)]]
def test_sympyissue_7840():
# daveknippers' example
C393 = sympify(
'Piecewise((C391 - 1.65, C390 < 0.5), (Piecewise((C391 - 1.65, \
C391 > 2.35), (C392, True)), True))')
C391 = sympify(
'Piecewise((2.05*C390**(-1.03), C390 < 0.5), (2.5*C390**(-0.625), True))'
)
C393 = C393.subs({'C391': C391})
# simple substitution
sub = {}
sub['C390'] = 0.703451854
sub['C392'] = 1.01417794
ss_answer = C393.subs(sub)
# cse
substitutions, new_eqn = cse(C393)
for pair in substitutions:
sub[pair[0].name] = pair[1].subs(sub)
cse_answer = new_eqn[0].subs(sub)
# both methods should be the same
assert ss_answer == cse_answer
# GitRay's example
expr = Piecewise(
(Symbol('ON'), Eq(Symbol('mode'), Symbol('ON'))), (Piecewise(
(Piecewise(
(Symbol('OFF'), Symbol('x') < Symbol('threshold')),
(Symbol('ON'), true)), Eq(Symbol('mode'), Symbol('AUTO'))),
(Symbol('OFF'), true)), true))
substitutions, new_eqn = cse(expr)
# this Piecewise should be exactly the same
assert new_eqn[0] == expr
# there should not be any replacements
assert len(substitutions) < 1
def test_product_basic():
H, T = 'H', 'T'
unit_line = Interval(0, 1)
d6 = FiniteSet(1, 2, 3, 4, 5, 6)
d4 = FiniteSet(1, 2, 3, 4)
coin = FiniteSet(H, T)
square = unit_line * unit_line
assert (0, 0) in square
assert 0 not in square
assert (H, T) in coin ** 2
assert (.5, .5, .5) in square * unit_line
assert (H, 3, 3) in coin*d6*d6
HH, TT = sympify(H), sympify(T)
assert set(coin**2) == {(HH, HH), (HH, TT), (TT, HH), (TT, TT)}
assert (d4*d4).is_subset(d6*d6)
assert (square.complement(Interval(-oo, oo)*Interval(-oo, oo)) ==
Union((Interval(-oo, 0, True, True) +
Interval(1, oo, True, True))*Interval(-oo, oo),
Interval(-oo, oo)*(Interval(-oo, 0, True, True) +
Interval(1, oo, True, True))))
assert (Interval(-5, 5)**3).is_subset(Interval(-10, 10)**3)
assert not (Interval(-10, 10)**3).is_subset(Interval(-5, 5)**3)
assert not (Interval(-5, 5)**2).is_subset(Interval(-10, 10)**3)
assert (Interval(.2, .5)*FiniteSet(.5)).is_subset(square) # segment in square
assert len(coin*coin*coin) == 8
assert len(S.EmptySet*S.EmptySet) == 0
assert len(S.EmptySet*coin) == 0
pytest.raises(TypeError, lambda: len(coin*Interval(0, 2)))
def run(self, result: Result):
if result.AST.is_known():
return result
# Martingale expression has to be <= 0 eventually
if not is_invariant(self.martingale_expression, self.program):
return result
# Eventually one branch of LG_{i+1} - LG_i has to decrease more or equal than constant
branches = get_cases_for_expression(sympify(self.program.loop_guard),
self.program)
if self.program.contains_rvs:
branches = split_expressions_on_rvs(branches, self.program)
for branch, prob in branches:
bounds = bound_store.get_bounds_of_expr(
branch - sympify(self.program.loop_guard))
n = symbols("n", integer=True, positive=True)
if is_dominating_or_same(bounds.upper,
sympify(-1),
n,
direction=Direction.NegInf):
result.AST = Answer.TRUE
result.add_witness(
ASTWitness(self.program.loop_guard,
self.martingale_expression, branch,
bounds.upper, prob))
return result
return result
def test_sympyissue_10773():
with evaluate(False):
ans = Mul(Integer(-10), Pow(Integer(5), Integer(-1)))
assert sympify('-10/5', evaluate=False) == ans
with evaluate(False):
ans = Mul(Integer(-10), Pow(Integer(-5), Integer(-1)))
assert sympify('-10/-5', evaluate=False) == ans
def test_sympyissue_4988():
C = Symbol('C')
vars = {}
vars['C'] = C
exp1 = sympify('C')
assert exp1 == C # Make sure it did not get mixed up with diofant.C
exp2 = sympify('C', vars)
assert exp2 == C # Make sure it did not get mixed up with diofant.C
def test_sympify_gmpy():
if HAS_GMPY:
import gmpy2 as gmpy
value = sympify(gmpy.mpz(1000001))
assert value == Integer(1000001) and type(value) is Integer
value = sympify(gmpy.mpq(101, 127))
assert value == Rational(101, 127) and type(value) is Rational
def test_sympify5():
class A:
def __str__(self):
raise TypeError
with pytest.raises(SympifyError) as err:
sympify(A())
assert re.match(r"^Sympify of expression '<diofant\.tests\.core\.test_sympify"
r"\.test_sympify5\.<locals>\.A object at 0x[0-9a-f]+>' failed,"
' because of exception being raised:\nTypeError: $', str(err.value))
def test_lambda_raises():
pytest.raises(NotImplementedError,
lambda: sympify('lambda *args: args')) # args argument error
pytest.raises(
NotImplementedError,
lambda: sympify('lambda **kwargs: kwargs')) # kwargs argument error
pytest.raises(SympifyError,
lambda: sympify('lambda x = 1: x')) # Keyword argument error
with pytest.raises(SympifyError):
sympify('lambda: 1', strict=True)
def test_sympify2():
class A:
def _diofant_(self):
return Symbol('x')**3
a = A()
assert sympify(a, strict=True) == x**3
assert sympify(a) == x**3
assert a == x**3
def _minimal_polynomial_sq(p, n, x):
"""
Returns the minimal polynomial for the ``nth-root`` of a sum of surds
or ``None`` if it fails.
Parameters
==========
p : sum of surds
n : positive integer
x : variable of the returned polynomial
Examples
========
>>> from diofant import sqrt
>>> from diofant.abc import x
>>> q = 1 + sqrt(2) + sqrt(3)
>>> _minimal_polynomial_sq(q, 3, x)
x**12 - 4*x**9 - 4*x**6 + 16*x**3 - 8
"""
from diofant.simplify.simplify import _is_sum_surds
p = sympify(p)
n = sympify(n)
r = _is_sum_surds(p)
if not n.is_Integer or not n > 0 or not _is_sum_surds(p):
return
pn = p**Rational(1, n)
# eliminate the square roots
p -= x
while 1:
p1 = _separate_sq(p)
if p1 is p:
p = p1.subs({x: x**n})
break
else:
p = p1
# _separate_sq eliminates field extensions in a minimal way, so that
# if n = 1 then `p = constant*(minimal_polynomial(p))`
# if n > 1 it contains the minimal polynomial as a factor.
if n == 1:
p1 = Poly(p)
if p.coeff(x**p1.degree(x)) < 0:
p = -p
p = p.primitive()[1]
return p
# by construction `p` has root `pn`
# the minimal polynomial is the factor vanishing in x = pn
factors = factor_list(p)[1]
result = _choose_factor(factors, x, pn)
return result
def __init__(self, ranking_martingale, martingale_expression, bound):
super(PASTWitness, self).__init__("PAST")
ranking_martingale = sympify(ranking_martingale).as_expr()
martingale_expression = sympify(martingale_expression).as_expr()
bound = sympify(bound).as_expr()
self.data = {
"Ranking SM": ranking_martingale,
"SM expression": martingale_expression,
"SM expression bound": bound,
}
self.explanation = f"Eventually, '{ranking_martingale}' is a ranking supermartingale. That's because eventually\n" \
f"the bound of the supermartingale expression is '{bound}'."
def test_nsimplify():
assert nsimplify(0) == 0
assert nsimplify(-1) == -1
assert nsimplify(1) == 1
assert nsimplify(1 + x) == 1 + x
assert nsimplify(2.7) == Rational(27, 10)
assert nsimplify(1 - GoldenRatio) == (1 - sqrt(5))/2
assert nsimplify((1 + sqrt(5))/4, [GoldenRatio]) == GoldenRatio/2
assert nsimplify(2/GoldenRatio, [GoldenRatio]) == 2*GoldenRatio - 2
assert nsimplify(exp(5*pi*I/3, evaluate=False)) == \
sympify('1/2 - sqrt(3)*I/2')
assert nsimplify(sin(3*pi/5, evaluate=False)) == \
sympify('sqrt(sqrt(5)/8 + 5/8)')
assert nsimplify(sqrt(atan('1', evaluate=False))*(2 + I), [pi]) == \
sqrt(pi) + sqrt(pi)/2*I
assert nsimplify(2 + exp(2*atan('1/4')*I)) == sympify('49/17 + 8*I/17')
assert nsimplify(pi, tolerance=0.01) == Rational(22, 7)
assert nsimplify(pi, tolerance=0.001) == Rational(355, 113)
assert nsimplify(0.33333, tolerance=1e-4) == Rational(1, 3)
assert nsimplify(2.0**(1/3.), tolerance=0.001) == Rational(635, 504)
assert nsimplify(2.0**(1/3.), tolerance=0.001, full=True) == \
2**Rational(1, 3)
assert nsimplify(x + .5, rational=True) == Rational(1, 2) + x
assert nsimplify(1/.3 + x, rational=True) == Rational(10, 3) + x
assert nsimplify(log(3).n(), rational=True) == \
sympify('109861228866811/100000000000000')
assert nsimplify(Float(0.272198261287950), [pi, log(2)]) == pi*log(2)/8
assert nsimplify(Float(0.272198261287950).n(3), [pi, log(2)]) == \
-pi/4 - log(2) + Rational(7, 4)
assert nsimplify(x/7.0) == x/7
assert nsimplify(pi/1e2) == pi/100
assert nsimplify(pi/1e2, rational=False) == pi/100.0
assert nsimplify(pi/1e-7) == 10000000*pi
assert not nsimplify(
factor(-3.0*z**2*(z**2)**(-2.5) + 3*(z**2)**(-1.5))).atoms(Float)
e = x**0.0
assert e.is_Pow and nsimplify(x**0.0) == 1
assert nsimplify(3.333333, tolerance=0.1, rational=True) == Rational(10, 3)
assert nsimplify(3.333333, tolerance=0.01, rational=True) == Rational(10, 3)
assert nsimplify(3.666666, tolerance=0.1, rational=True) == Rational(11, 3)
assert nsimplify(3.666666, tolerance=0.01, rational=True) == Rational(11, 3)
assert nsimplify(33, tolerance=10, rational=True) == Rational(33)
assert nsimplify(33.33, tolerance=10, rational=True) == Rational(30)
assert nsimplify(37.76, tolerance=10, rational=True) == Rational(40)
assert nsimplify(-203.1) == -Rational(2031, 10)
assert nsimplify(.2, tolerance=0) == S.One/5
assert nsimplify(-.2, tolerance=0) == -S.One/5
assert nsimplify(.2222, tolerance=0) == Rational(1111, 5000)
assert nsimplify(-.2222, tolerance=0) == -Rational(1111, 5000)
# issue 7211, PR 4112
assert nsimplify(Float(2e-8)) == Rational(1, 50000000)
# issue 7322 direct test
assert nsimplify(1e-42, rational=True) != 0
def __init__(self, repulsing_martingale, martingale_expression):
super(NONPASTWitness, self).__init__("Not PAST")
repulsing_martingale = sympify(repulsing_martingale).as_expr()
martingale_expression = sympify(martingale_expression).as_expr()
self.data = {
"Repulsing SM": repulsing_martingale,
"SM expression": martingale_expression,
}
self.explanation = f"There is always a positive probability of having a next iteration.\n" \
f"Moreover, '{repulsing_martingale}' eventually is a repulsing supermartingale\n" \
f"decreasing with epsilons '0'. Also, the repulsing SM has differences bounded\n" \
f"by a constant."
def test_Wild_properties():
# these tests only include Atoms
x = Symbol("x")
y = Symbol("y")
p = Symbol("p", positive=True)
k = Symbol("k", integer=True)
n = Symbol("n", integer=True, positive=True)
given_patterns = [
x, y, p, k, -k, n, -n,
sympify(-3),
sympify(3), pi,
Rational(3, 2), I
]
def integerp(k):
return k.is_integer
def positivep(k):
return k.is_positive
def symbolp(k):
return k.is_Symbol
def realp(k):
return k.is_extended_real
S = Wild("S", properties=[symbolp])
R = Wild("R", properties=[realp])
Y = Wild("Y", exclude=[x, p, k, n])
P = Wild("P", properties=[positivep])
K = Wild("K", properties=[integerp])
N = Wild("N", properties=[positivep, integerp])
given_wildcards = [S, R, Y, P, K, N]
goodmatch = {
S: (x, y, p, k, n),
R: (p, k, -k, n, -n, -3, 3, pi, Rational(3, 2)),
Y: (y, -3, 3, pi, Rational(3, 2), I),
P: (p, n, 3, pi, Rational(3, 2)),
K: (k, -k, n, -n, -3, 3),
N: (n, 3)
}
for A in given_wildcards:
for pat in given_patterns:
d = pat.match(A)
if pat in goodmatch[A]:
assert d[A] in goodmatch[A]
else:
assert d is None
def __new__(cls, *args):
n = args[BinomialDistribution._argnames.index('n')]
p = args[BinomialDistribution._argnames.index('p')]
n_sym = sympify(n)
p_sym = sympify(p)
if fuzzy_not(fuzzy_and((n_sym.is_integer, n_sym.is_nonnegative))):
raise ValueError("'n' must be positive integer. n = %s." % str(n))
elif fuzzy_not(
fuzzy_and((p_sym.is_nonnegative, (p_sym - 1).is_nonpositive))):
raise ValueError("'p' must be: 0 <= p <= 1 . p = %s" % str(p))
else:
return super(BinomialDistribution, cls).__new__(cls, *args)
def compute_moment(self, k):
if self.distribution == 'finite':
return sum([p * (b ** k) for b, p in self.parameters])
if self.distribution == 'uniform':
l, u = self.parameters
return (u**(k+1)-l**(k+1))/((k+1)*(u-l))
if self.distribution == 'gauss' or self.distribution == 'normal':
mu, sigma_squared = self.parameters
# For low moments avoid scipy.stats.moments as it does not support
# parametric parameters. In the future get all moments directly,
# using the following properties:
# https://math.stackexchange.com/questions/1945448/methods-for-finding-raw-moments-of-the-normal-distribution
if k == 0:
return 1
elif k == 1:
return mu
elif k == 2:
return mu**2 + sigma_squared
elif k == 3:
return mu*(mu**2 + 3*sigma_squared)
elif k == 4:
return mu**4 + 6*mu**2*sigma_squared + 3*sigma_squared**2
moment = norm(loc=mu, scale=sqrt(sigma_squared)).moment(k)
return Rational(moment)
if self.distribution == 'unknown':
return sympify(f"{self.var_name}(0)^{k}")
def _minpoly_cos(ex, x):
"""
Returns the minimal polynomial of ``cos(ex)``
see http://mathworld.wolfram.com/TrigonometryAngles.html
"""
from diofant import sqrt
c, a = ex.args[0].as_coeff_Mul()
if a is pi:
if c.is_rational:
if c.p == 1:
if c.q == 7:
return 8 * x**3 - 4 * x**2 - 4 * x + 1
if c.q == 9:
return 8 * x**3 - 6 * x + 1
elif c.p == 2:
q = sympify(c.q)
if q.is_prime:
s = _minpoly_sin(ex, x)
return _mexpand(s.subs({x: sqrt((1 - x) / 2)}))
# for a = pi*p/q, cos(q*a) =T_q(cos(a)) = (-1)**p
n = int(c.q)
a = dup_chebyshevt(n, ZZ)
a = [x**(n - i) * a[i] for i in range(n + 1)]
r = Add(*a) - (-1)**c.p
_, factors = factor_list(r)
res = _choose_factor(factors, x, ex)
return res
raise NotAlgebraic("%s doesn't seem to be an algebraic element" % ex)
def pspace(expr):
"""
Returns the underlying Probability Space of a random expression.
For internal use.
Examples
========
>>> from diofant.stats import pspace, Normal
>>> from diofant.stats.rv import ProductPSpace
>>> X = Normal('X', 0, 1)
>>> pspace(2*X + 1) == X.pspace
True
"""
expr = sympify(expr)
rvs = random_symbols(expr)
if not rvs:
raise ValueError("Expression containing Random Variable expected, not %s" % (expr))
# If only one space present
if all(rv.pspace == rvs[0].pspace for rv in rvs):
return rvs[0].pspace
# Otherwise make a product space
return ProductPSpace(*[rv.pspace for rv in rvs])
def to_number_field(extension, theta=None, **args):
"""Express `extension` in the field generated by `theta`. """
gen = args.get('gen')
if hasattr(extension, '__iter__'):
extension = list(extension)
else:
extension = [extension]
if len(extension) == 1 and type(extension[0]) is tuple:
return AlgebraicNumber(extension[0])
minpoly, coeffs = primitive_element(extension, gen, polys=True)
root = sum(coeff * ext for coeff, ext in zip(coeffs, extension))
if theta is None:
return AlgebraicNumber((minpoly, root))
else:
theta = sympify(theta)
if not theta.is_AlgebraicNumber:
theta = AlgebraicNumber(theta, gen=gen)
coeffs = field_isomorphism(root, theta)
if coeffs is not None:
return AlgebraicNumber(theta, coeffs)
else:
raise IsomorphismFailed("%s is not in a subfield of %s" %
(root, theta.root))
def isolate(alg, eps=None, fast=False):
"""Give a rational isolating interval for an algebraic number. """
alg = sympify(alg)
if alg.is_Rational:
return alg, alg
elif not alg.is_extended_real:
raise NotImplementedError(
"complex algebraic numbers are not supported")
func = lambdify((), alg, modules="mpmath", printer=IntervalPrinter())
poly = minpoly(alg, polys=True)
intervals = poly.intervals(sqf=True)
dps, done = mp.dps, False
try:
while not done:
alg = func()
for a, b in intervals:
if a <= alg.a and alg.b <= b:
done = True
break
else:
mp.dps *= 2
finally:
mp.dps = dps
if eps is not None:
a, b = poly.refine_root(a, b, eps=eps, fast=fast)
return a, b
def __init__(self, repulsing_martingale, martingale_expression, epsilons,
cs):
super(NONASTWitness, self).__init__("Not AST")
repulsing_martingale = sympify(repulsing_martingale).as_expr()
martingale_expression = sympify(martingale_expression).as_expr()
epsilons = sympify(epsilons).as_expr()
cs = sympify(cs).as_expr()
self.data = {
"Repulsing SM": repulsing_martingale,
"SM expression": martingale_expression,
"Epsilons": epsilons,
"Cs": cs
}
self.explanation = f"There is always a positive probability of having a next iteration.\n" \
f"Moreover, '{repulsing_martingale}' eventually is a repulsing supermartingale\n" \
f"decreasing with epsilons '{epsilons}'. Also, the repulsing SM has differences bound\n" \
f"by '{cs}' which is O(epsilons)."
def run(self, result: Result):
if result.PAST.is_known() and result.AST.is_known():
return result
# Martingale expression has to be <= 0 eventually
if not is_invariant(self.martingale_expression, self.program):
return result
branches = get_cases_for_expression(sympify(self.program.loop_guard),
self.program)
if self.program.contains_rvs:
branches = split_expressions_on_rvs(branches, self.program)
branches = [
simplify(branch - sympify(self.program.loop_guard))
for branch, _ in branches
]
bounds = [bound_store.get_bounds_of_expr(case) for case in branches]
def __init__(self, martingale, martingale_expression, decreasing_branch,
bound, prob):
super(ASTWitness, self).__init__("AST")
martingale = sympify(martingale).as_expr()
martingale_expression = sympify(martingale_expression).as_expr()
decreasing_branch = sympify(decreasing_branch).as_expr()
bound = sympify(bound).as_expr()
self.data = {
"SM": martingale,
"SM expression": martingale_expression,
"Decreasing branch": decreasing_branch,
"Branch change bound": bound,
"Probability": prob
}
self.explanation = f"Eventually, '{martingale}' is a supermartingale. Also eventually, taking the branch\n" \
f"'{decreasing_branch}' (which happens with probability {prob}) " \
f"changes the supermartingale by at least {bound}."
def get_loop_guard_change(program: Program):
"""
Returns E[LG_{n+1} - LG_{n}]
"""
n = symbols("n", integer=True, positive=True)
lg = sympify(LOOP_GUARD_VAR).as_poly(program.variables)
expected_lg = get_expected(program, lg)
expected_lg_plus = expected_lg.xreplace({n: n + 1})
return expand(expected_lg_plus - expected_lg)
def test_sympyissue_8821():
s = str(pi.evalf(128))
p = N(s)
assert abs(sin(p)) < 1e-15
p = N(s, 64)
assert abs(sin(p)) < 1e-64
s = str(pi.evalf(128))
p = sympify(s)
assert abs(sin(p)) < 1e-127
def test_Tuple():
t = (1, 2, 3, 4)
st = Tuple(*t)
assert set(sympify(t)) == set(st)
assert len(t) == len(st)
assert set(sympify(t[:2])) == set(st[:2])
assert isinstance(st[:], Tuple)
assert st == Tuple(1, 2, 3, 4)
assert st.func(*st.args) == st
t2 = (p, q, r, s)
st2 = Tuple(*t2)
assert st2.atoms() == set(t2)
assert st == st2.subs({p: 1, q: 2, r: 3, s: 4})
# issue sympy/sympy#5505
assert all(isinstance(arg, Basic) for arg in st.args)
assert Tuple(p, 1).subs({p: 0}) == Tuple(0, 1)
assert Tuple(p, Tuple(p, 1)).subs({p: 0}) == Tuple(0, Tuple(0, 1))
assert Tuple(t2) == Tuple(Tuple(*t2))
def t(m, a, b):
a, b = sympify([a, b])
m_ = m
m = hyperexpand(m)
if not m == Piecewise((a, abs(z) < 1), (b, abs(1/z) < 1), (m_, True)):
return False
if not (m.args[0].args[0] == a and m.args[1].args[0] == b):
return False
z0 = randcplx()/10
if abs(m.subs({z: z0}) - a.subs({z: z0})).evalf(strict=False) > 1e-10:
return False
if abs(m.subs({z: 1/z0}) - b.subs({z: 1/z0})).evalf(strict=False) > 1e-10:
return False
return True
def test_numexpr_printer():
# if translation/printing is done incorrectly then evaluating
# a lambdified numexpr expression will throw an exception
blacklist = ('where', 'complex', 'contains')
arg_tuple = (x, y, z) # some functions take more than one argument
for sym in NumExprPrinter._numexpr_functions:
if sym in blacklist:
continue
ssym = sympify(sym)
if hasattr(ssym, '_nargs'):
nargs = ssym._nargs[0]
else:
nargs = 1
args = arg_tuple[:nargs]
f = lambdify(args, ssym(*args), modules='numexpr')
assert f(*(1, )*nargs) is not None
def test_mathematica():
d = {
'- 6x': '-6*x',
'Sin[x]^2': 'sin(x)**2',
'2(x-1)': '2*(x-1)',
'(x-1)2': '(x-1)*2',
'(x-y)': '(x-y)',
'3y+8': '3*y+8',
'Arcsin[2x+9(4-x)^2]/x': 'asin(2*x+9*(4-x)**2)/x',
'x+y': 'x+y',
'355/113': '355/113',
'2.718281828': '2.718281828',
'Sin[12]': 'sin(12)',
'Exp[Log[4]]': 'exp(log(4))',
'(x+1)(x+3)': '(x+1)*(x+3)',
'Cos[Arccos[3.6]]': 'cos(acos(3.6))',
'Cos[x]==Sin[y]': 'cos(x)==sin(y)',
'2*Sin[x+y]': '2*sin(x+y)',
'Sin[x]+Cos[y]': 'sin(x)+cos(y)',
'Sin[Cos[x]]': 'sin(cos(x))',
'2*Sqrt[x+y]': '2*sqrt(x+y)'} # Test case from the issue sympy/sympy#4259
for e in d:
assert mathematica(e) == sympify(d[e])
def NS(e, n=15, **options):
return sstr(sympify(e).evalf(n, **options), full_prec=True)
def test_S_repr_is_identity():
p = Piecewise((10, Eq(x, 0)), (12, True))
q = sympify(repr(p))
assert p == q
def test_as_immutable():
X = Matrix([[1, 2], [3, 4]])
assert sympify(X) == X.as_immutable() == ImmutableMatrix([[1, 2], [3, 4]])
X = SparseMatrix(5, 5, {})
assert sympify(X) == X.as_immutable() == ImmutableSparseMatrix(
[[0 for i in range(5)] for i in range(5)])
def test_cdf():
D = Die('D', 6)
o = Integer(1)
assert cdf(
D) == sympify({1: o/6, 2: o/3, 3: o/2, 4: 2*o/3, 5: 5*o/6, 6: o})
def eval(cls, arg):
arg = sympify(arg)
if arg == 0:
return sympify(0)
def NS(e, n=15, **options):
return str(sympify(e).evalf(n, **options))