def test_almosteq():
assert CC.almosteq(CC(2), 3) is False
assert CC.almosteq(2, CC(3)) is False
assert CC.almosteq(2, CC(2.5), 0.1) is False
assert CC.almosteq(2, CC(2.5), 1.0) is True
assert RR.almosteq(5, RR(2), 1) is True
assert RR._context.almosteq(RR(2), 1, None, 1) is True
def test_RR_Float():
f1 = Float('1.01', 15)
f2 = Float('1.0000000000000000000001')
assert f1._prec == 53
assert f2._prec == 80
assert RR(f1) - 1 > 1e-50
assert RR(f2) - 1 < 1e-50 # RR's precision is lower than f2's
RR2 = RealField(prec=f2._prec)
assert RR2(f1) - 1 > 1e-50
assert RR2(f2) - 1 > 1e-50 # RR's precision is equal to f2's
a = RR(2.1)
assert a.numerator == a and a.denominator == 1
def test_meijerg_with_Floats():
# see sympy/sympy#10681
f = meijerg(((3.0, 1), ()), ((Rational(3, 2),), (0,)), z)
a = -2.3632718012073
g = a*z**Rational(3, 2)*hyper((-0.5, Rational(3, 2)),
(Rational(5, 2),), z*exp_polar(I*pi))
assert RR.almosteq((hyperexpand(f)/g).n(), 1.0, 1e-12)
def test_Domain_interface():
pytest.raises(TypeError, lambda: DomainElement().parent)
assert RR(1).parent is RR
assert CC(1).parent is CC
assert RR.has_default_precision
assert CC.has_default_precision
RR3 = RealField(prec=53, dps=3)
assert str(RR3(1.7611107002)) == '1.76'
assert RealField(tol=3).tolerance == 3.0
assert RealField(tol=0.1).tolerance == 0.1
assert RealField(tol='0.1').tolerance == 0.1
pytest.raises(ValueError, lambda: RealField(tol=object()))
pytest.raises(AttributeError, lambda: CC.ring)
pytest.raises(DomainError, lambda: CC.get_exact())
assert str(EX(1)) == 'EX(1)'
assert EX(1).as_expr() == Integer(1)
assert bool(EX(1)) is True
assert bool(EX(0)) is False
def test_RR_double():
assert RR(3.14) > 1e-50
assert RR(1e-13) > 1e-50
assert RR(1e-14) > 1e-50
assert RR(1e-15) > 1e-50
assert RR(1e-20) > 1e-50
assert RR(1e-40) > 1e-50
def test_Domain_convert():
assert QQ.convert(10e-52) == QQ(
1684996666696915,
1684996666696914987166688442938726917102321526408785780068975640576)
R, x = ring('x', ZZ)
assert ZZ.convert(x - x) == 0
assert ZZ.convert(x - x, R) == 0
F3 = FF(3)
assert F3.convert(Float(2.0)) == F3.dtype(2)
assert F3.convert(PythonRational(2, 1)) == F3.dtype(2)
pytest.raises(CoercionFailed, lambda: F3.convert(PythonRational(1, 2)))
assert F3.convert(2.0) == F3.dtype(2)
pytest.raises(CoercionFailed, lambda: F3.convert(2.1))
assert RR.convert(CC(1)) == RR(1)
pytest.raises(CoercionFailed, lambda: RR.convert(CC(1, 2)))
assert QQ.convert(ALG(1), ALG) == QQ(1)
pytest.raises(CoercionFailed, lambda: QQ.convert(ALG([1, 1]), ALG))
assert ZZ.convert(ALG(1), ALG) == ZZ(1)
pytest.raises(CoercionFailed, lambda: ZZ.convert(ALG([1, 1]), ALG))
assert EX.convert(ALG([1, 1]), ALG) == sqrt(2) + sqrt(3) + 1
ALG2 = QQ.algebraic_field(sqrt(2))
a2 = ALG2.convert(sqrt(2))
a = ALG.convert(a2, ALG2)
assert a.rep.all_coeffs() == [0, -QQ(9, 2), 0, QQ(1, 2)]
assert RR.convert(a) == RR(1.4142135623730951)
assert CC.convert(a) == CC(1.4142135623730951)
assert ZZ_python.convert(3.0) == ZZ_python.dtype(3)
pytest.raises(CoercionFailed, lambda: ZZ_python.convert(3.2))
assert CC.convert(QQ_python(1, 2)) == CC(0.5)
CC01 = ComplexField(tol=0.1)
assert CC.convert(CC01(0.3)) == CC(0.3)
assert RR.convert(complex(2 + 0j)) == RR(2)
pytest.raises(CoercionFailed, lambda: RR.convert(complex(2 + 3j)))
assert ALG.convert(EX(sqrt(2)), EX) == ALG.from_expr(sqrt(2))
pytest.raises(CoercionFailed, lambda: ALG.convert(EX(sqrt(5)), EX))
pytest.raises(CoercionFailed, lambda: ALG2.convert(ALG.unit))
def test_Domain_get_exact():
assert EX.get_exact() == EX
assert ZZ.get_exact() == ZZ
assert QQ.get_exact() == QQ
assert RR.get_exact() == QQ
assert ALG.get_exact() == ALG
assert ZZ.inject(x).get_exact() == ZZ.inject(x)
assert QQ.inject(x).get_exact() == QQ.inject(x)
assert ZZ.inject(x, y).get_exact() == ZZ.inject(x, y)
assert QQ.inject(x, y).get_exact() == QQ.inject(x, y)
assert ZZ.inject(x).field.get_exact() == ZZ.inject(x).field
assert QQ.inject(x).field.get_exact() == QQ.inject(x).field
assert ZZ.inject(x, y).field.get_exact() == ZZ.inject(x, y).field
assert QQ.inject(x, y).field.get_exact() == QQ.inject(x, y).field
def test_sympyissue_10681():
f = integrate(r**2 * (R**2 - r**2)**0.5, r, meijerg=True)
g = (1.0 / 3) * R**1.0 * r**3 * hyper(
(-0.5, Rational(3, 2)),
(Rational(5, 2), ), r**2 * exp_polar(2 * I * pi) / R**2)
assert RR.almosteq((f / g).n(), 1.0, 1e-12)
def test_construct_domain():
assert construct_domain([1, 2, 3]) == (ZZ, [ZZ(1), ZZ(2), ZZ(3)])
assert construct_domain([1, 2, 3],
field=True) == (QQ, [QQ(1), QQ(2),
QQ(3)])
assert construct_domain([Integer(1), Integer(2),
Integer(3)]) == (ZZ, [ZZ(1), ZZ(2),
ZZ(3)])
assert construct_domain(
[Integer(1), Integer(2), Integer(3)],
field=True) == (QQ, [QQ(1), QQ(2), QQ(3)])
assert construct_domain([Rational(1, 2),
Integer(2)]) == (QQ, [QQ(1, 2), QQ(2)])
assert construct_domain([3.14, 1, Rational(1, 2)
]) == (RR, [RR(3.14), RR(1.0),
RR(0.5)])
assert construct_domain([3.14, sqrt(2)],
extension=False) == (EX, [EX(3.14),
EX(sqrt(2))])
assert construct_domain([3.14, sqrt(2)]) == (EX, [EX(3.14), EX(sqrt(2))])
assert construct_domain([sqrt(2), 3.14]) == (EX, [EX(sqrt(2)), EX(3.14)])
assert construct_domain([1, sqrt(2)],
extension=False) == (EX, [EX(1),
EX(sqrt(2))])
assert construct_domain([x, sqrt(x)]) == (EX, [EX(x), EX(sqrt(x))])
assert construct_domain([x, sqrt(x), sqrt(y)
]) == (EX, [EX(x),
EX(sqrt(x)),
EX(sqrt(y))])
alg = QQ.algebraic_field(sqrt(2))
assert (construct_domain(
[7, Rational(1, 2),
sqrt(2)]) == (alg, [alg([7]),
alg([Rational(1, 2)]),
alg([1, 0])]))
alg = QQ.algebraic_field(sqrt(2) + sqrt(3))
assert (construct_domain([7, sqrt(2), sqrt(3)]) == (alg, [
alg([7]), alg.from_expr(sqrt(2)),
alg.from_expr(sqrt(3))
]))
dom = ZZ.inject(x)
assert construct_domain([2 * x, 3]) == (dom, [dom(2 * x), dom(3)])
dom = ZZ.inject(x, y)
assert construct_domain([2 * x, 3 * y]) == (dom, [dom(2 * x), dom(3 * y)])
dom = QQ.inject(x)
assert construct_domain([x / 2, 3]) == (dom, [dom(x / 2), dom(3)])
dom = QQ.inject(x, y)
assert construct_domain([x / 2, 3 * y]) == (dom, [dom(x / 2), dom(3 * y)])
dom = RR.inject(x)
assert construct_domain([x / 2, 3.5]) == (dom, [dom(x / 2), dom(3.5)])
dom = RR.inject(x, y)
assert construct_domain([x / 2,
3.5 * y]) == (dom, [dom(x / 2),
dom(3.5 * y)])
dom = ZZ.inject(x).field
assert construct_domain([2 / x, 3]) == (dom, [dom(2 / x), dom(3)])
dom = ZZ.inject(x, y).field
assert construct_domain([2 / x, 3 * y]) == (dom, [dom(2 / x), dom(3 * y)])
dom = RR.inject(x).field
assert construct_domain([2 / x, 3.5]) == (dom, [dom(2 / x), dom(3.5)])
dom = RR.inject(x, y).field
assert construct_domain([2 / x,
3.5 * y]) == (dom, [dom(2 / x),
dom(3.5 * y)])
assert construct_domain(2) == (ZZ, ZZ(2))
assert construct_domain(Rational(2, 3)) == (QQ, QQ(2, 3))
assert construct_domain({}) == (ZZ, {})
assert construct_domain([-x * y + x * (y + 42) - 42 * x
]) == (EX, [EX(-x * y + x * (y + 42) - 42 * x)])
def test_RealField_from_expr():
assert RR.convert(Integer(0)) == RR.dtype(0)
assert RR.convert(Float(0.0)) == RR.dtype(0.0)
assert RR.convert(Integer(1)) == RR.dtype(1)
assert RR.convert(Float(1.0)) == RR.dtype(1.0)
assert RR.convert(sin(1)) == RR.dtype(sin(1).evalf())
assert RR.convert(oo) == RR('+inf')
assert RR.convert(-oo) == RR('-inf')
pytest.raises(CoercionFailed, lambda: RR.convert(x))
def test_FractionField_convert():
F, X, Y = field('x y', QQ)
assert F.convert(QQ_python(1, 3)) == F.one / 3
assert F.convert(RR(1)) == F.one
def test_Domain_unify():
F3 = GF(3)
assert unify(F3, F3) == F3
assert unify(F3, ZZ) == F3
assert unify(F3, QQ) == QQ
assert unify(F3, ALG) == ALG
assert unify(F3, RR) == RR
assert unify(F3, CC) == CC
assert unify(F3, ZZ.inject(x)) == F3.inject(x)
assert unify(F3, ZZ.inject(x).field) == F3.inject(x).field
assert unify(F3, EX) == EX
assert unify(ZZ, F3) == F3
assert unify(ZZ, ZZ) == ZZ
assert unify(ZZ, QQ) == QQ
assert unify(ZZ, ALG) == ALG
assert unify(ZZ, RR) == RR
assert unify(ZZ, CC) == CC
assert unify(ZZ, ZZ.inject(x)) == ZZ.inject(x)
assert unify(ZZ, ZZ.inject(x).field) == ZZ.inject(x).field
assert unify(ZZ, EX) == EX
assert unify(QQ, F3) == QQ
assert unify(QQ, ZZ) == QQ
assert unify(QQ, QQ) == QQ
assert unify(QQ, ALG) == ALG
assert unify(QQ, RR) == RR
assert unify(QQ, CC) == CC
assert unify(QQ, ZZ.inject(x)) == QQ.inject(x)
assert unify(QQ, ZZ.inject(x).field) == QQ.inject(x).field
assert unify(QQ, EX) == EX
assert unify(RR, F3) == RR
assert unify(RR, ZZ) == RR
assert unify(RR, QQ) == RR
assert unify(RR, ALG) == RR
assert unify(RR, RR) == RR
assert unify(RR, CC) == CC
assert unify(RR, ZZ.inject(x)) == RR.inject(x)
assert unify(RR, ZZ.inject(x).field) == RR.inject(x).field
assert unify(RR, EX) == EX
assert unify(CC, F3) == CC
assert unify(CC, ZZ) == CC
assert unify(CC, QQ) == CC
assert unify(CC, ALG) == CC
assert unify(CC, RR) == CC
assert unify(CC, CC) == CC
assert unify(CC, ZZ.inject(x)) == CC.inject(x)
assert unify(CC, ZZ.inject(x).field) == CC.inject(x).field
assert unify(CC, EX) == EX
CC2 = ComplexField(prec=20)
assert unify(CC, CC2) == unify(CC2, CC) == ComplexField(prec=CC.precision,
tol=CC2.tolerance)
RR2 = RealField(prec=20)
assert unify(RR, RR2) == unify(RR2, RR) == RealField(prec=RR.precision,
tol=RR2.tolerance)
assert unify(ZZ.inject(x), F3) == F3.inject(x)
assert unify(ZZ.inject(x), ZZ) == ZZ.inject(x)
assert unify(ZZ.inject(x), QQ) == QQ.inject(x)
assert unify(ZZ.inject(x), ALG) == ALG.inject(x)
assert unify(ZZ.inject(x), RR) == RR.inject(x)
assert unify(ZZ.inject(x), CC) == CC.inject(x)
assert unify(ZZ.inject(x), ZZ.inject(x)) == ZZ.inject(x)
assert unify(ZZ.inject(x), ZZ.inject(x).field) == ZZ.inject(x).field
assert unify(ZZ.inject(x), EX) == EX
assert unify(ZZ.inject(x).field, F3) == F3.inject(x).field
assert unify(ZZ.inject(x).field, ZZ) == ZZ.inject(x).field
assert unify(ZZ.inject(x).field, QQ) == QQ.inject(x).field
assert unify(ZZ.inject(x).field, ALG) == ALG.inject(x).field
assert unify(ZZ.inject(x).field, RR) == RR.inject(x).field
assert unify(ZZ.inject(x).field, CC) == CC.inject(x).field
assert unify(ZZ.inject(x).field, ZZ.inject(x)) == ZZ.inject(x).field
assert unify(ZZ.inject(x).field, ZZ.inject(x).field) == ZZ.inject(x).field
assert unify(ZZ.inject(x).field, EX) == EX
assert unify(EX, F3) == EX
assert unify(EX, ZZ) == EX
assert unify(EX, QQ) == EX
assert unify(EX, ALG) == EX
assert unify(EX, RR) == EX
assert unify(EX, CC) == EX
assert unify(EX, ZZ.inject(x)) == EX
assert unify(EX, ZZ.inject(x).field) == EX
assert unify(EX, EX) == EX
def test_Domain__contains__():
assert (0 in EX) is True
assert (0 in ZZ) is True
assert (0 in QQ) is True
assert (0 in RR) is True
assert (0 in CC) is True
assert (0 in ALG) is True
assert (0 in ZZ.inject(x, y)) is True
assert (0 in QQ.inject(x, y)) is True
assert (0 in RR.inject(x, y)) is True
assert (-7 in EX) is True
assert (-7 in ZZ) is True
assert (-7 in QQ) is True
assert (-7 in RR) is True
assert (-7 in CC) is True
assert (-7 in ALG) is True
assert (-7 in ZZ.inject(x, y)) is True
assert (-7 in QQ.inject(x, y)) is True
assert (-7 in RR.inject(x, y)) is True
assert (17 in EX) is True
assert (17 in ZZ) is True
assert (17 in QQ) is True
assert (17 in RR) is True
assert (17 in CC) is True
assert (17 in ALG) is True
assert (17 in ZZ.inject(x, y)) is True
assert (17 in QQ.inject(x, y)) is True
assert (17 in RR.inject(x, y)) is True
assert (-Rational(1, 7) in EX) is True
assert (-Rational(1, 7) in ZZ) is False
assert (-Rational(1, 7) in QQ) is True
assert (-Rational(1, 7) in RR) is True
assert (-Rational(1, 7) in CC) is True
assert (-Rational(1, 7) in ALG) is True
assert (-Rational(1, 7) in ZZ.inject(x, y)) is False
assert (-Rational(1, 7) in QQ.inject(x, y)) is True
assert (-Rational(1, 7) in RR.inject(x, y)) is True
assert (Rational(3, 5) in EX) is True
assert (Rational(3, 5) in ZZ) is False
assert (Rational(3, 5) in QQ) is True
assert (Rational(3, 5) in RR) is True
assert (Rational(3, 5) in CC) is True
assert (Rational(3, 5) in ALG) is True
assert (Rational(3, 5) in ZZ.inject(x, y)) is False
assert (Rational(3, 5) in QQ.inject(x, y)) is True
assert (Rational(3, 5) in RR.inject(x, y)) is True
assert (3.0 in EX) is True
assert (3.0 in ZZ) is True
assert (3.0 in QQ) is True
assert (3.0 in RR) is True
assert (3.0 in CC) is True
assert (3.0 in ALG) is True
assert (3.0 in ZZ.inject(x, y)) is True
assert (3.0 in QQ.inject(x, y)) is True
assert (3.0 in RR.inject(x, y)) is True
assert (3.14 in EX) is True
assert (3.14 in ZZ) is False
assert (3.14 in QQ) is True
assert (3.14 in RR) is True
assert (3.14 in CC) is True
assert (3.14 in ALG) is True
assert (3.14 in ZZ.inject(x, y)) is False
assert (3.14 in QQ.inject(x, y)) is True
assert (3.14 in RR.inject(x, y)) is True
assert (oo in EX) is True
assert (oo in ZZ) is False
assert (oo in QQ) is False
assert (oo in RR) is True
assert (oo in CC) is True
assert (oo in ALG) is False
assert (oo in ZZ.inject(x, y)) is False
assert (oo in QQ.inject(x, y)) is False
assert (oo in RR.inject(x, y)) is True
assert (-oo in EX) is True
assert (-oo in ZZ) is False
assert (-oo in QQ) is False
assert (-oo in RR) is True
assert (-oo in CC) is True
assert (-oo in ALG) is False
assert (-oo in ZZ.inject(x, y)) is False
assert (-oo in QQ.inject(x, y)) is False
assert (-oo in RR.inject(x, y)) is True
assert (sqrt(7) in EX) is True
assert (sqrt(7) in ZZ) is False
assert (sqrt(7) in QQ) is False
assert (sqrt(7) in RR) is True
assert (sqrt(7) in CC) is True
assert (sqrt(7) in ALG) is False
assert (sqrt(7) in ZZ.inject(x, y)) is False
assert (sqrt(7) in QQ.inject(x, y)) is False
assert (sqrt(7) in RR.inject(x, y)) is True
assert (2 * sqrt(3) + 1 in EX) is True
assert (2 * sqrt(3) + 1 in ZZ) is False
assert (2 * sqrt(3) + 1 in QQ) is False
assert (2 * sqrt(3) + 1 in RR) is True
assert (2 * sqrt(3) + 1 in CC) is True
assert (2 * sqrt(3) + 1 in ALG) is True
assert (2 * sqrt(3) + 1 in ZZ.inject(x, y)) is False
assert (2 * sqrt(3) + 1 in QQ.inject(x, y)) is False
assert (2 * sqrt(3) + 1 in RR.inject(x, y)) is True
assert (sin(1) in EX) is True
assert (sin(1) in ZZ) is False
assert (sin(1) in QQ) is False
assert (sin(1) in RR) is True
assert (sin(1) in CC) is True
assert (sin(1) in ALG) is False
assert (sin(1) in ZZ.inject(x, y)) is False
assert (sin(1) in QQ.inject(x, y)) is False
assert (sin(1) in RR.inject(x, y)) is True
assert (x**2 + 1 in EX) is True
assert (x**2 + 1 in ZZ) is False
assert (x**2 + 1 in QQ) is False
assert (x**2 + 1 in RR) is False
assert (x**2 + 1 in CC) is False
assert (x**2 + 1 in ALG) is False
assert (x**2 + 1 in ZZ.inject(x)) is True
assert (x**2 + 1 in QQ.inject(x)) is True
assert (x**2 + 1 in RR.inject(x)) is True
assert (x**2 + 1 in ZZ.inject(x, y)) is True
assert (x**2 + 1 in QQ.inject(x, y)) is True
assert (x**2 + 1 in RR.inject(x, y)) is True
assert (x**2 + y**2 in EX) is True
assert (x**2 + y**2 in ZZ) is False
assert (x**2 + y**2 in QQ) is False
assert (x**2 + y**2 in RR) is False
assert (x**2 + y**2 in CC) is False
assert (x**2 + y**2 in ALG) is False
assert (x**2 + y**2 in ZZ.inject(x)) is False
assert (x**2 + y**2 in QQ.inject(x)) is False
assert (x**2 + y**2 in RR.inject(x)) is False
assert (x**2 + y**2 in ZZ.inject(x, y)) is True
assert (x**2 + y**2 in QQ.inject(x, y)) is True
assert (x**2 + y**2 in RR.inject(x, y)) is True
assert (Rational(3, 2) * x / (y + 1) - z in QQ.inject(x, y, z)) is False
R = QQ.inject(x)
assert R(1) in ZZ
F = R.field
assert F(1) in ZZ