def test_heurisch_hacking():
assert (heurisch(sqrt(1 + 7 * x**2), x,
hints=[]) == x * sqrt(1 + 7 * x**2) / 2 +
sqrt(7) * asinh(sqrt(7) * x) / 14)
assert (heurisch(sqrt(1 - 7 * x**2), x,
hints=[]) == x * sqrt(1 - 7 * x**2) / 2 +
sqrt(7) * asin(sqrt(7) * x) / 14)
assert heurisch(sqrt(y * x**2 - 1), x, hints=[]) is None
assert (heurisch(1 / sqrt(1 + 7 * x**2), x,
hints=[]) == sqrt(7) * asinh(sqrt(7) * x) / 7)
assert (heurisch(1 / sqrt(1 - 7 * x**2), x,
hints=[]) == sqrt(7) * asin(sqrt(7) * x) / 7)
assert heurisch(exp(-7 * x**2), x,
hints=[]) == sqrt(7 * pi) * erf(sqrt(7) * x) / 14
assert heurisch(exp(2 * x**2), x,
hints=[]) == sqrt(2) * sqrt(pi) * erfi(sqrt(2) * x) / 4
assert (heurisch(exp(2 * x**2 - 3 * x), x,
hints=[]) == sqrt(2) * sqrt(pi) *
erfi(sqrt(2) * x - 3 * sqrt(2) / 4) / (4 * E**Rational(9, 8)))
assert heurisch(1 / sqrt(9 - 4 * x**2), x, hints=[]) == asin(2 * x / 3) / 2
assert heurisch(1 / sqrt(9 + 4 * x**2), x,
hints=[]) == asinh(2 * x / 3) / 2
assert heurisch(li(x), x, hints=[]) == x * li(x) - Ei(2 * log(x))
assert heurisch(li(log(x)), x, hints=[]) is None
assert (heurisch(sqrt(1 + x), x,
hints=[x, sqrt(1 + x)]) == 2 * x * sqrt(x + 1) / 3 +
2 * sqrt(x + 1) / 3)
def test_heurisch_trigonometric():
assert heurisch(sin(x), x) == -cos(x)
assert heurisch(pi * sin(x) + 1, x) == x - pi * cos(x)
assert heurisch(cos(x), x) == sin(x)
assert heurisch(tan(x), x) in [
log(1 + tan(x)**2) / 2,
log(tan(x) + I) + I * x,
log(tan(x) - I) - I * x,
]
assert heurisch(sin(x) * sin(y), x) == -cos(x) * sin(y)
assert heurisch(sin(x) * sin(y), y) == -cos(y) * sin(x)
assert heurisch(sin(x) * cos(x), x) in [sin(x)**2 / 2, -cos(x)**2 / 2]
assert heurisch(cos(x) / sin(x), x) == log(sin(x))
assert heurisch(x * sin(7 * x), x) == sin(7 * x) / 49 - x * cos(7 * x) / 7
assert heurisch(
1 / pi / 4 * x**2 * cos(x),
x) == 1 / pi / 4 * (x**2 * sin(x) - 2 * sin(x) + 2 * x * cos(x))
assert heurisch(acos(x/4) * asin(x/4), x) == 2*x - (sqrt(16 - x**2))*asin(x/4) \
+ (sqrt(16 - x**2))*acos(x/4) + x*asin(x/4)*acos(x/4)
assert heurisch(1 / sin(1 / x) / x**2, x) == -log(tan(1 / x / 2))
def test_heurisch_trigonometric():
assert heurisch(sin(x), x) == -cos(x)
assert heurisch(pi * sin(x) + 1, x) == x - pi * cos(x)
assert heurisch(cos(x), x) == sin(x)
assert heurisch(tan(x), x) in [
log(1 + tan(x)**2) / 2,
log(tan(x) + I) + I * x,
log(tan(x) - I) - I * x,
]
assert heurisch(sin(x) * sin(y), x) == -cos(x) * sin(y)
assert heurisch(sin(x) * sin(y), y) == -cos(y) * sin(x)
# gives sin(x) in answer when run via setup.py and cos(x) when run via py.test
assert heurisch(sin(x) * cos(x), x) in [sin(x)**2 / 2, -cos(x)**2 / 2]
assert heurisch(cos(x) / sin(x), x) == log(sin(x))
assert heurisch(x * sin(7 * x), x) == sin(7 * x) / 49 - x * cos(7 * x) / 7
assert heurisch(
1 / pi / 4 * x**2 * cos(x),
x) == 1 / pi / 4 * (x**2 * sin(x) - 2 * sin(x) + 2 * x * cos(x))
assert heurisch(acos(x/4) * asin(x/4), x) == 2*x - (sqrt(16 - x**2))*asin(x/4) \
+ (sqrt(16 - x**2))*acos(x/4) + x*asin(x/4)*acos(x/4)
def test_basic6():
# pull sympy/sympy#22491
assert limit(1/asin(x), x, 0) == oo
assert limit(1/asin(x), x, 0, dir=1) == -oo
assert limit(1/sinh(x), x, 0) == oo
assert limit(1/sinh(x), x, 0, dir=1) == -oo
assert limit(log(1/x) + 1/sin(x), x, 0) == oo
assert limit(log(1/x) + 1/x, x, 0) == oo
def test_messy():
assert laplace_transform(Si(x), x, s) == ((-atan(s) + pi/2)/s, 0, True)
assert laplace_transform(Shi(x), x, s) == (acoth(s)/s, 1, True)
# where should the logs be simplified?
assert laplace_transform(Chi(x), x, s) == \
((log(s**(-2)) - log((s**2 - 1)/s**2))/(2*s), 1, True)
# TODO maybe simplify the inequalities?
assert laplace_transform(besselj(a, x), x, s)[1:] == \
(0, And(Integer(0) < re(a/2) + Rational(1, 2), Integer(0) < re(a/2) + 1))
# NOTE s < 0 can be done, but argument reduction is not good enough yet
assert fourier_transform(besselj(1, x)/x, x, s, noconds=False) == \
(Piecewise((0, 4*abs(pi**2*s**2) > 1),
(2*sqrt(-4*pi**2*s**2 + 1), True)), s > 0)
# TODO FT(besselj(0,x)) - conditions are messy (but for acceptable reasons)
# - folding could be better
assert integrate(E1(x)*besselj(0, x), (x, 0, oo), meijerg=True) == \
log(1 + sqrt(2))
assert integrate(E1(x)*besselj(1, x), (x, 0, oo), meijerg=True) == \
log(Rational(1, 2) + sqrt(2)/2)
assert integrate(1/x/sqrt(1 - x**2), x, meijerg=True) == \
Piecewise((-acosh(1/x), 1 < abs(x**(-2))), (I*asin(1/x), True))
def test_intrinsic_math1_codegen():
# not included: log10
name_expr = [
("test_fabs", abs(x)),
("test_acos", acos(x)),
("test_asin", asin(x)),
("test_atan", atan(x)),
("test_cos", cos(x)),
("test_cosh", cosh(x)),
("test_log", log(x)),
("test_ln", ln(x)),
("test_sin", sin(x)),
("test_sinh", sinh(x)),
("test_sqrt", sqrt(x)),
("test_tan", tan(x)),
("test_tanh", tanh(x)),
]
numerical_tests = []
for name, expr in name_expr:
for xval in 0.2, 0.5, 0.8:
expected = N(expr.subs({x: xval}), strict=False)
numerical_tests.append((name, (xval,), expected, 1e-14))
for lang, commands in valid_lang_commands:
if lang == "C":
name_expr_C = [("test_floor", floor(x)), ("test_ceil", ceiling(x))]
else:
name_expr_C = []
run_test("intrinsic_math1", name_expr + name_expr_C,
numerical_tests, lang, commands)
def test_messy():
assert laplace_transform(Si(x), x, s) == ((-atan(s) + pi / 2) / s, 0, True)
assert laplace_transform(Shi(x), x, s) == (acoth(s) / s, 1, True)
# where should the logs be simplified?
assert laplace_transform(Chi(x), x, s) == \
((log(s**(-2)) - log((s**2 - 1)/s**2))/(2*s), 1, True)
# TODO maybe simplify the inequalities?
assert laplace_transform(besselj(a, x), x, s)[1:] == \
(0, And(Integer(0) < re(a/2) + Rational(1, 2), Integer(0) < re(a/2) + 1))
# NOTE s < 0 can be done, but argument reduction is not good enough yet
assert fourier_transform(besselj(1, x)/x, x, s, noconds=False) == \
(Piecewise((0, 4*abs(pi**2*s**2) > 1),
(2*sqrt(-4*pi**2*s**2 + 1), True)), s > 0)
# TODO FT(besselj(0,x)) - conditions are messy (but for acceptable reasons)
# - folding could be better
assert integrate(E1(x)*besselj(0, x), (x, 0, oo), meijerg=True) == \
log(1 + sqrt(2))
assert integrate(E1(x)*besselj(1, x), (x, 0, oo), meijerg=True) == \
log(Rational(1, 2) + sqrt(2)/2)
assert integrate(1/x/sqrt(1 - x**2), x, meijerg=True) == \
Piecewise((-acosh(1/x), 1 < abs(x**(-2))), (I*asin(1/x), True))
def test_intrinsic_math1_codegen():
# not included: log10
name_expr = [
("test_fabs", abs(x)),
("test_acos", acos(x)),
("test_asin", asin(x)),
("test_atan", atan(x)),
("test_cos", cos(x)),
("test_cosh", cosh(x)),
("test_log", log(x)),
("test_ln", ln(x)),
("test_sin", sin(x)),
("test_sinh", sinh(x)),
("test_sqrt", sqrt(x)),
("test_tan", tan(x)),
("test_tanh", tanh(x)),
]
numerical_tests = []
for name, expr in name_expr:
for xval in 0.2, 0.5, 0.8:
expected = N(expr.subs(x, xval), strict=False)
numerical_tests.append((name, (xval, ), expected, 1e-14))
for lang, commands in valid_lang_commands:
if lang == "C":
name_expr_C = [("test_floor", floor(x)), ("test_ceil", ceiling(x))]
else:
name_expr_C = []
run_test("intrinsic_math1", name_expr + name_expr_C, numerical_tests,
lang, commands)
def test_acot_rewrite():
assert acot(x).rewrite(log) == I*log((x - I)/(x + I))/2
assert acot(x).rewrite(asin) == x*(-asin(sqrt(-x**2)/sqrt(-x**2 - 1)) + pi/2)*sqrt(x**(-2))
assert acot(x).rewrite(acos) == x*sqrt(x**(-2))*acos(sqrt(-x**2)/sqrt(-x**2 - 1))
assert acot(x).rewrite(atan) == atan(1/x)
assert acot(x).rewrite(asec) == x*sqrt(x**(-2))*asec(sqrt((x**2 + 1)/x**2))
assert acot(x).rewrite(acsc) == x*(-acsc(sqrt((x**2 + 1)/x**2)) + pi/2)*sqrt(x**(-2))
def test_atan_rewrite():
assert atan(x).rewrite(log) == I*log((1 - I*x)/(1 + I*x))/2
assert atan(x).rewrite(asin) == (-asin(1/sqrt(x**2 + 1)) + pi/2)*sqrt(x**2)/x
assert atan(x).rewrite(acos) == sqrt(x**2)*acos(1/sqrt(x**2 + 1))/x
assert atan(x).rewrite(acot) == acot(1/x)
assert atan(x).rewrite(asec) == sqrt(x**2)*asec(sqrt(x**2 + 1))/x
assert atan(x).rewrite(acsc) == (-acsc(sqrt(x**2 + 1)) + pi/2)*sqrt(x**2)/x
def test_intrinsic_math1_codegen():
# not included: log10
name_expr = [
('test_fabs', abs(x)),
('test_acos', acos(x)),
('test_asin', asin(x)),
('test_atan', atan(x)),
('test_cos', cos(x)),
('test_cosh', cosh(x)),
('test_log', log(x)),
('test_ln', ln(x)),
('test_sin', sin(x)),
('test_sinh', sinh(x)),
('test_sqrt', sqrt(x)),
('test_tan', tan(x)),
('test_tanh', tanh(x)),
]
numerical_tests = []
for name, expr in name_expr:
for xval in 0.2, 0.5, 0.8:
expected = N(expr.subs({x: xval}), strict=False)
numerical_tests.append((name, (xval, ), expected, 1e-14))
for lang, commands in valid_lang_commands:
if lang == 'C':
name_expr_C = [('test_floor', floor(x)), ('test_ceil', ceiling(x))]
else:
name_expr_C = []
run_test('intrinsic_math1', name_expr + name_expr_C, numerical_tests,
lang, commands)
def test_sympyissue_4492():
assert simplify(integrate(x**2 * sqrt(5 - x**2), x)) == Piecewise(
(I * (2 * x**5 - 15 * x**3 + 25 * x -
25 * sqrt(x**2 - 5) * acosh(sqrt(5) * x / 5)) /
(8 * sqrt(x**2 - 5)), 1 < Abs(x**2) / 5),
((-2 * x**5 + 15 * x**3 - 25 * x +
25 * sqrt(-x**2 + 5) * asin(sqrt(5) * x / 5)) /
(8 * sqrt(-x**2 + 5)), True))
def test_heurisch_hacking():
assert (heurisch(sqrt(1 + 7*x**2), x, hints=[]) ==
x*sqrt(1 + 7*x**2)/2 + sqrt(7)*asinh(sqrt(7)*x)/14)
assert (heurisch(sqrt(1 - 7*x**2), x, hints=[]) ==
x*sqrt(1 - 7*x**2)/2 + sqrt(7)*asin(sqrt(7)*x)/14)
assert (heurisch(1/sqrt(1 + 7*x**2), x, hints=[]) ==
sqrt(7)*asinh(sqrt(7)*x)/7)
assert (heurisch(1/sqrt(1 - 7*x**2), x, hints=[]) ==
sqrt(7)*asin(sqrt(7)*x)/7)
assert (heurisch(exp(-7*x**2), x, hints=[]) == sqrt(7*pi)*erf(sqrt(7)*x)/14)
assert heurisch(1/sqrt(9 - 4*x**2), x, hints=[]) == asin(2*x/3)/2
assert heurisch(1/sqrt(9 + 4*x**2), x, hints=[]) == asinh(2*x/3)/2
assert heurisch(li(x), x, hints=[]) == x*li(x) - Ei(2*log(x))
def test_acos_series():
assert acos(x).series(x, 0, 8) == \
pi/2 - x - x**3/6 - 3*x**5/40 - 5*x**7/112 + O(x**8)
assert acos(x).series(x, 0, 8) == pi/2 - asin(x).series(x, 0, 8)
t5 = acos(x).taylor_term(5, x)
assert t5 == -3*x**5/40
assert acos(x).taylor_term(7, x, t5, 0) == -5*x**7/112
assert acos(x).taylor_term(0, x) == pi/2
assert acos(x).taylor_term(2, x) == 0
def test_hyperexpand():
# Luke, Y. L. (1969), The Special Functions and Their Approximations,
# Volume 1, section 6.2
assert hyperexpand(hyper([], [], z)) == exp(z)
assert hyperexpand(hyper([1, 1], [2], -z)*z) == log(1 + z)
assert hyperexpand(hyper([], [S.Half], -z**2/4)) == cos(z)
assert hyperexpand(z*hyper([], [Rational(3, 2)], -z**2/4)) == sin(z)
assert hyperexpand(hyper([Rational(1, 2), Rational(1, 2)], [Rational(3, 2)], z**2)*z) \
== asin(z)
def test_heurisch_hacking():
assert heurisch(sqrt(1 + 7*x**2), x, hints=[]) == \
x*sqrt(1 + 7*x**2)/2 + sqrt(7)*asinh(sqrt(7)*x)/14
assert heurisch(sqrt(1 - 7*x**2), x, hints=[]) == \
x*sqrt(1 - 7*x**2)/2 + sqrt(7)*asin(sqrt(7)*x)/14
assert heurisch(1/sqrt(1 + 7*x**2), x, hints=[]) == \
sqrt(7)*asinh(sqrt(7)*x)/7
assert heurisch(1/sqrt(1 - 7*x**2), x, hints=[]) == \
sqrt(7)*asin(sqrt(7)*x)/7
assert heurisch(exp(-7*x**2), x, hints=[]) == \
sqrt(7*pi)*erf(sqrt(7)*x)/14
assert heurisch(1/sqrt(9 - 4*x**2), x, hints=[]) == \
asin(2*x/3)/2
assert heurisch(1/sqrt(9 + 4*x**2), x, hints=[]) == \
asinh(2*x/3)/2
def test_acos_rewrite():
assert acos(x).rewrite(log) == pi/2 + I*log(I*x + sqrt(1 - x**2))
assert acos(x).rewrite(atan) == \
atan(sqrt(1 - x**2)/x) + (pi/2)*(1 - x*sqrt(1/x**2))
assert acos(0).rewrite(atan) == pi/2
assert acos(0.5).rewrite(atan) == acos(0.5).rewrite(log)
assert acos(x).rewrite(asin) == pi/2 - asin(x)
assert acos(x).rewrite(acot) == -2*acot((sqrt(-x**2 + 1) + 1)/x) + pi/2
assert acos(x).rewrite(asec) == asec(1/x)
assert acos(x).rewrite(acsc) == -acsc(1/x) + pi/2
def test_hyperexpand():
# Luke, Y. L. (1969), The Special Functions and Their Approximations,
# Volume 1, section 6.2
assert hyperexpand(hyper([], [], z)) == exp(z)
assert hyperexpand(hyper([1, 1], [2], -z)*z) == log(1 + z)
assert hyperexpand(hyper([], [Rational(1, 2)], -z**2/4)) == cos(z)
assert hyperexpand(z*hyper([], [Rational(3, 2)], -z**2/4)) == sin(z)
assert hyperexpand(hyper([Rational(1, 2), Rational(1, 2)], [Rational(3, 2)], z**2)*z) \
== asin(z)
def test_inverses():
pytest.raises(AttributeError, lambda: sin(x).inverse())
pytest.raises(AttributeError, lambda: cos(x).inverse())
assert tan(x).inverse() == atan
assert cot(x).inverse() == acot
pytest.raises(AttributeError, lambda: csc(x).inverse())
pytest.raises(AttributeError, lambda: sec(x).inverse())
assert asin(x).inverse() == sin
assert acos(x).inverse() == cos
assert atan(x).inverse() == tan
assert acot(x).inverse() == cot
def test_heurisch_hacking():
assert (heurisch(sqrt(1 + 7 * x**2), x,
hints=[]) == x * sqrt(1 + 7 * x**2) / 2 +
sqrt(7) * asinh(sqrt(7) * x) / 14)
assert (heurisch(sqrt(1 - 7 * x**2), x,
hints=[]) == x * sqrt(1 - 7 * x**2) / 2 +
sqrt(7) * asin(sqrt(7) * x) / 14)
assert (heurisch(1 / sqrt(1 + 7 * x**2), x,
hints=[]) == sqrt(7) * asinh(sqrt(7) * x) / 7)
assert (heurisch(1 / sqrt(1 - 7 * x**2), x,
hints=[]) == sqrt(7) * asin(sqrt(7) * x) / 7)
assert (heurisch(exp(-7 * x**2), x,
hints=[]) == sqrt(7 * pi) * erf(sqrt(7) * x) / 14)
assert heurisch(1 / sqrt(9 - 4 * x**2), x, hints=[]) == asin(2 * x / 3) / 2
assert heurisch(1 / sqrt(9 + 4 * x**2), x,
hints=[]) == asinh(2 * x / 3) / 2
assert heurisch(li(x), x, hints=[]) == x * li(x) - Ei(2 * log(x))
def test_asin_rewrite():
assert asin(x).rewrite(log) == -I*log(I*x + sqrt(1 - x**2))
assert asin(x).rewrite(atan) == 2*atan(x/(1 + sqrt(1 - x**2)))
assert asin(x).rewrite(acos) == pi/2 - acos(x)
assert asin(x).rewrite(acot) == 2*acot((sqrt(-x**2 + 1) + 1)/x)
assert asin(x).rewrite(asec) == -asec(1/x) + pi/2
assert asin(x).rewrite(acsc) == acsc(1/x)
def test_special_is_rational():
i = Symbol('i', integer=True)
i2 = Symbol('i2', integer=True)
ni = Symbol('ni', integer=True, nonzero=True)
r = Symbol('r', rational=True)
rn = Symbol('r', rational=True, nonzero=True)
nr = Symbol('nr', irrational=True)
x = Symbol('x')
assert sqrt(3).is_rational is False
assert (3 + sqrt(3)).is_rational is False
assert (3*sqrt(3)).is_rational is False
z = Symbol('z', zero=True)
assert exp(z).is_rational
assert exp(0, evaluate=False).is_rational
assert exp(3).is_rational is False
assert exp(ni).is_rational is False
assert exp(rn).is_rational is False
assert exp(x).is_rational is None
assert exp(log(3), evaluate=False).is_rational is True
assert log(exp(3), evaluate=False).is_rational is True
assert log(3).is_rational is False
assert log(ni + 1).is_rational is False
assert log(rn + 1).is_rational is False
assert log(x).is_rational is None
assert (sqrt(3) + sqrt(5)).is_rational is None
assert (sqrt(3) + pi).is_rational is False
assert (x**i).is_rational is None
assert (i**i).is_rational is True
assert (i**i2).is_rational is None
assert (r**i).is_rational is None
assert (r**r).is_rational is None
assert (r**x).is_rational is None
assert (nr**i).is_rational is None # issue sympy/sympy#8598
assert (nr**Symbol('z', zero=True)).is_rational
assert sin(1).is_rational is False
assert sin(ni).is_rational is False
assert sin(rn).is_rational is False
assert sin(x).is_rational is None
assert asin(rn).is_rational is False
assert sin(asin(3), evaluate=False).is_rational is True
def test_special_is_rational():
i = Symbol('i', integer=True)
i2 = Symbol('i2', integer=True)
ni = Symbol('ni', integer=True, nonzero=True)
r = Symbol('r', rational=True)
rn = Symbol('r', rational=True, nonzero=True)
nr = Symbol('nr', irrational=True)
x = Symbol('x')
assert sqrt(3).is_rational is False
assert (3 + sqrt(3)).is_rational is False
assert (3 * sqrt(3)).is_rational is False
z = Symbol('z', zero=True)
assert exp(z).is_rational
assert exp(0, evaluate=False).is_rational
assert exp(3).is_rational is False
assert exp(ni).is_rational is False
assert exp(rn).is_rational is False
assert exp(x).is_rational is None
assert exp(log(3), evaluate=False).is_rational is True
assert log(exp(3), evaluate=False).is_rational is True
assert log(3).is_rational is False
assert log(ni + 1).is_rational is False
assert log(rn + 1).is_rational is False
assert log(x).is_rational is None
assert (sqrt(3) + sqrt(5)).is_rational is None
assert (sqrt(3) + S.Pi).is_rational is False
assert (x**i).is_rational is None
assert (i**i).is_rational is True
assert (i**i2).is_rational is None
assert (r**i).is_rational is None
assert (r**r).is_rational is None
assert (r**x).is_rational is None
assert (nr**i).is_rational is None # issue sympy/sympy#8598
assert (nr**Symbol('z', zero=True)).is_rational
assert sin(1).is_rational is False
assert sin(ni).is_rational is False
assert sin(rn).is_rational is False
assert sin(x).is_rational is None
assert asin(rn).is_rational is False
assert sin(asin(3), evaluate=False).is_rational is True
def test_ansi_math1_codegen():
# not included: log10
name_expr = [
("test_fabs", Abs(x)),
("test_acos", acos(x)),
("test_asin", asin(x)),
("test_atan", atan(x)),
("test_ceil", ceiling(x)),
("test_cos", cos(x)),
("test_cosh", cosh(x)),
("test_floor", floor(x)),
("test_log", log(x)),
("test_ln", ln(x)),
("test_sin", sin(x)),
("test_sinh", sinh(x)),
("test_sqrt", sqrt(x)),
("test_tan", tan(x)),
("test_tanh", tanh(x)),
]
result = codegen(name_expr, "C", "file", header=False, empty=False)
assert result[0][0] == "file.c"
assert result[0][1] == (
'#include "file.h"\n#include <math.h>\n'
'double test_fabs(double x) {\n double test_fabs_result;\n test_fabs_result = fabs(x);\n return test_fabs_result;\n}\n'
'double test_acos(double x) {\n double test_acos_result;\n test_acos_result = acos(x);\n return test_acos_result;\n}\n'
'double test_asin(double x) {\n double test_asin_result;\n test_asin_result = asin(x);\n return test_asin_result;\n}\n'
'double test_atan(double x) {\n double test_atan_result;\n test_atan_result = atan(x);\n return test_atan_result;\n}\n'
'double test_ceil(double x) {\n double test_ceil_result;\n test_ceil_result = ceil(x);\n return test_ceil_result;\n}\n'
'double test_cos(double x) {\n double test_cos_result;\n test_cos_result = cos(x);\n return test_cos_result;\n}\n'
'double test_cosh(double x) {\n double test_cosh_result;\n test_cosh_result = cosh(x);\n return test_cosh_result;\n}\n'
'double test_floor(double x) {\n double test_floor_result;\n test_floor_result = floor(x);\n return test_floor_result;\n}\n'
'double test_log(double x) {\n double test_log_result;\n test_log_result = log(x);\n return test_log_result;\n}\n'
'double test_ln(double x) {\n double test_ln_result;\n test_ln_result = log(x);\n return test_ln_result;\n}\n'
'double test_sin(double x) {\n double test_sin_result;\n test_sin_result = sin(x);\n return test_sin_result;\n}\n'
'double test_sinh(double x) {\n double test_sinh_result;\n test_sinh_result = sinh(x);\n return test_sinh_result;\n}\n'
'double test_sqrt(double x) {\n double test_sqrt_result;\n test_sqrt_result = sqrt(x);\n return test_sqrt_result;\n}\n'
'double test_tan(double x) {\n double test_tan_result;\n test_tan_result = tan(x);\n return test_tan_result;\n}\n'
'double test_tanh(double x) {\n double test_tanh_result;\n test_tanh_result = tanh(x);\n return test_tanh_result;\n}\n'
)
assert result[1][0] == "file.h"
assert result[1][1] == (
'#ifndef PROJECT__FILE__H\n#define PROJECT__FILE__H\n'
'double test_fabs(double x);\ndouble test_acos(double x);\n'
'double test_asin(double x);\ndouble test_atan(double x);\n'
'double test_ceil(double x);\ndouble test_cos(double x);\n'
'double test_cosh(double x);\ndouble test_floor(double x);\n'
'double test_log(double x);\ndouble test_ln(double x);\n'
'double test_sin(double x);\ndouble test_sinh(double x);\n'
'double test_sqrt(double x);\ndouble test_tan(double x);\n'
'double test_tanh(double x);\n#endif\n'
)
def test_hyperexpand_bases():
assert hyperexpand(hyper([2], [a], z)) == \
a + z**(-a + 1)*(-a**2 + 3*a + z*(a - 1) - 2)*exp(z) * \
lowergamma(a - 1, z) - 1
# TODO [a+1, a+Rational(-1, 2)], [2*a]
assert hyperexpand(hyper([1, 2], [3], z)) == -2/z - 2*log(-z + 1)/z**2
assert hyperexpand(hyper([Rational(1, 2), 2], [Rational(3, 2)], z)) == \
-1/(2*z - 2) + atanh(sqrt(z))/sqrt(z)/2
assert hyperexpand(hyper([Rational(1, 2), Rational(1, 2)], [Rational(5, 2)], z)) == \
(-3*z + 3)/4/(z*sqrt(-z + 1)) \
+ (6*z - 3)*asin(sqrt(z))/(4*z**Rational(3, 2))
assert hyperexpand(hyper([1, 2], [Rational(3, 2)], z)) == -1/(2*z - 2) \
- asin(sqrt(z))/(sqrt(z)*(2*z - 2)*sqrt(-z + 1))
assert hyperexpand(hyper([Rational(-1, 2) - 1, 1, 2], [Rational(1, 2), 3], z)) == \
sqrt(z)*(6*z/7 - Rational(6, 5))*atanh(sqrt(z)) \
+ (-30*z**2 + 32*z - 6)/35/z - 6*log(-z + 1)/(35*z**2)
assert hyperexpand(hyper([1 + Rational(1, 2), 1, 1], [2, 2], z)) == \
-4*log(sqrt(-z + 1)/2 + Rational(1, 2))/z
# TODO hyperexpand(hyper([a], [2*a + 1], z))
# TODO [Rational(1, 2), a], [Rational(3, 2), a+1]
assert hyperexpand(hyper([2], [b, 1], z)) == \
z**(-b/2 + Rational(1, 2))*besseli(b - 1, 2*sqrt(z))*gamma(b) \
+ z**(-b/2 + 1)*besseli(b, 2*sqrt(z))*gamma(b)
def test_hyperexpand_bases():
assert hyperexpand(hyper([2], [a], z)) == \
a + z**(-a + 1)*(-a**2 + 3*a + z*(a - 1) - 2)*exp(z) * \
lowergamma(a - 1, z) - 1
# TODO [a+1, a-S.Half], [2*a]
assert hyperexpand(hyper([1, 2], [3], z)) == -2/z - 2*log(-z + 1)/z**2
assert hyperexpand(hyper([S.Half, 2], [Rational(3, 2)], z)) == \
-1/(2*z - 2) + atanh(sqrt(z))/sqrt(z)/2
assert hyperexpand(hyper([Rational(1, 2), Rational(1, 2)], [Rational(5, 2)], z)) == \
(-3*z + 3)/4/(z*sqrt(-z + 1)) \
+ (6*z - 3)*asin(sqrt(z))/(4*z**Rational(3, 2))
assert hyperexpand(hyper([1, 2], [Rational(3, 2)], z)) == -1/(2*z - 2) \
- asin(sqrt(z))/(sqrt(z)*(2*z - 2)*sqrt(-z + 1))
assert hyperexpand(hyper([-S.Half - 1, 1, 2], [S.Half, 3], z)) == \
sqrt(z)*(6*z/7 - Rational(6, 5))*atanh(sqrt(z)) \
+ (-30*z**2 + 32*z - 6)/35/z - 6*log(-z + 1)/(35*z**2)
assert hyperexpand(hyper([1 + S.Half, 1, 1], [2, 2], z)) == \
-4*log(sqrt(-z + 1)/2 + Rational(1, 2))/z
# TODO hyperexpand(hyper([a], [2*a + 1], z))
# TODO [S.Half, a], [Rational(3, 2), a+1]
assert hyperexpand(hyper([2], [b, 1], z)) == \
z**(-b/2 + Rational(1, 2))*besseli(b - 1, 2*sqrt(z))*gamma(b) \
+ z**(-b/2 + 1)*besseli(b, 2*sqrt(z))*gamma(b)
def test_leading_terms():
for func in [sin, cos, tan, cot, asin, acos, atan, acot]:
for arg in (1/x, Rational(1, 2)):
eq = func(arg)
assert eq.as_leading_term(x) == eq
# issue sympy/sympy#5272
assert sin(x).as_leading_term(x) == x
assert cos(x).as_leading_term(x) == 1
assert tan(x).as_leading_term(x) == x
assert cot(x).as_leading_term(x) == 1/x
assert asin(x).as_leading_term(x) == x
assert acos(x).as_leading_term(x) == x
assert atan(x).as_leading_term(x) == x
assert acot(x).as_leading_term(x) == x
def test_heurisch_trigonometric():
assert heurisch(sin(x), x) == -cos(x)
assert heurisch(pi*sin(x) + 1, x) == x - pi*cos(x)
assert heurisch(cos(x), x) == sin(x)
assert heurisch(tan(x), x) in [
log(1 + tan(x)**2)/2,
log(tan(x) + I) + I*x,
log(tan(x) - I) - I*x,
]
assert heurisch(sin(x)*sin(y), x) == -cos(x)*sin(y)
assert heurisch(sin(x)*sin(y), y) == -cos(y)*sin(x)
# gives sin(x) in answer when run via setup.py and cos(x) when run via py.test
assert heurisch(sin(x)*cos(x), x) in [sin(x)**2 / 2, -cos(x)**2 / 2]
assert heurisch(cos(x)/sin(x), x) == log(sin(x))
assert heurisch(x*sin(7*x), x) == sin(7*x) / 49 - x*cos(7*x) / 7
assert heurisch(1/pi/4 * x**2*cos(x), x) == 1/pi/4*(x**2*sin(x) -
2*sin(x) + 2*x*cos(x))
assert heurisch(acos(x/4) * asin(x/4), x) == 2*x - (sqrt(16 - x**2))*asin(x/4) \
+ (sqrt(16 - x**2))*acos(x/4) + x*asin(x/4)*acos(x/4)
def test_hyperexpand():
# Luke, Y. L. (1969), The Special Functions and Their Approximations,
# Volume 1, section 6.2
assert hyperexpand(hyper([], [], z)) == exp(z)
assert hyperexpand(hyper([1, 1], [2], -z)*z) == log(1 + z)
assert hyperexpand(hyper([], [Rational(1, 2)], -z**2/4)) == cos(z)
assert hyperexpand(z*hyper([], [Rational(3, 2)], -z**2/4)) == sin(z)
assert hyperexpand(hyper([Rational(1, 2), Rational(1, 2)], [Rational(3, 2)], z**2)*z) \
== asin(z)
# Test place option
f = meijerg(((0, 1), ()), ((Rational(1, 2),), (0,)), z**2)
assert hyperexpand(f) == sqrt(pi)/sqrt(1 + z**(-2))
assert hyperexpand(f, place=0) == sqrt(pi)*z/sqrt(z**2 + 1)
assert hyperexpand(f, place=zoo) == sqrt(pi)/sqrt(1 + z**(-2))
def test_acsc():
assert acsc(nan) == nan
assert acsc(1) == pi/2
assert acsc(-1) == -pi/2
assert acsc(oo) == 0
assert acsc(-oo) == 0
assert acsc(zoo) == 0
assert acsc(x).diff(x) == -1/(x**2*sqrt(1 - 1/x**2))
assert acsc(x).as_leading_term(x) == log(x)
assert acsc(x).rewrite(log) == -I*log(sqrt(1 - 1/x**2) + I/x)
assert acsc(x).rewrite(asin) == asin(1/x)
assert acsc(x).rewrite(acos) == -acos(1/x) + pi/2
assert acsc(x).rewrite(atan) == (-atan(sqrt(x**2 - 1)) + pi/2)*sqrt(x**2)/x
assert acsc(x).rewrite(acot) == (-acot(1/sqrt(x**2 - 1)) + pi/2)*sqrt(x**2)/x
assert acsc(x).rewrite(asec) == -asec(x) + pi/2
pytest.raises(ArgumentIndexError, lambda: acsc(x).fdiff(2))
assert acsc(x).as_leading_term(x) == log(x)
assert acsc(1/x).as_leading_term(x) == acsc(1/x)
def test_mathml_trig():
mml = mp._print(sin(x))
assert mml.childNodes[0].nodeName == 'sin'
mml = mp._print(cos(x))
assert mml.childNodes[0].nodeName == 'cos'
mml = mp._print(tan(x))
assert mml.childNodes[0].nodeName == 'tan'
mml = mp._print(asin(x))
assert mml.childNodes[0].nodeName == 'arcsin'
mml = mp._print(acos(x))
assert mml.childNodes[0].nodeName == 'arccos'
mml = mp._print(atan(x))
assert mml.childNodes[0].nodeName == 'arctan'
mml = mp._print(sinh(x))
assert mml.childNodes[0].nodeName == 'sinh'
mml = mp._print(cosh(x))
assert mml.childNodes[0].nodeName == 'cosh'
mml = mp._print(tanh(x))
assert mml.childNodes[0].nodeName == 'tanh'
mml = mp._print(asinh(x))
assert mml.childNodes[0].nodeName == 'arcsinh'
mml = mp._print(atanh(x))
assert mml.childNodes[0].nodeName == 'arctanh'
mml = mp._print(acosh(x))
assert mml.childNodes[0].nodeName == 'arccosh'
def test_asec():
z = Symbol('z', zero=True)
assert asec(z) == zoo
assert asec(nan) == nan
assert asec(1) == 0
assert asec(-1) == pi
assert asec(oo) == pi/2
assert asec(-oo) == pi/2
assert asec(zoo) == pi/2
assert asec(x).diff(x) == 1/(x**2*sqrt(1 - 1/x**2))
assert asec(x).as_leading_term(x) == log(x)
assert asec(x).rewrite(log) == I*log(sqrt(1 - 1/x**2) + I/x) + pi/2
assert asec(x).rewrite(asin) == -asin(1/x) + pi/2
assert asec(x).rewrite(acos) == acos(1/x)
assert asec(x).rewrite(atan) == (2*atan(x + sqrt(x**2 - 1)) - pi/2)*sqrt(x**2)/x
assert asec(x).rewrite(acot) == (2*acot(x - sqrt(x**2 - 1)) - pi/2)*sqrt(x**2)/x
assert asec(x).rewrite(acsc) == -acsc(x) + pi/2
pytest.raises(ArgumentIndexError, lambda: asec(x).fdiff(2))
assert asec(x).as_leading_term(x) == log(x)
assert asec(1/x).as_leading_term(x) == asec(1/x)
def test_mathml_trig():
mml = mp._print(sin(x))
assert mml.childNodes[0].nodeName == 'sin'
mml = mp._print(cos(x))
assert mml.childNodes[0].nodeName == 'cos'
mml = mp._print(tan(x))
assert mml.childNodes[0].nodeName == 'tan'
mml = mp._print(asin(x))
assert mml.childNodes[0].nodeName == 'arcsin'
mml = mp._print(acos(x))
assert mml.childNodes[0].nodeName == 'arccos'
mml = mp._print(atan(x))
assert mml.childNodes[0].nodeName == 'arctan'
mml = mp._print(sinh(x))
assert mml.childNodes[0].nodeName == 'sinh'
mml = mp._print(cosh(x))
assert mml.childNodes[0].nodeName == 'cosh'
mml = mp._print(tanh(x))
assert mml.childNodes[0].nodeName == 'tanh'
mml = mp._print(asinh(x))
assert mml.childNodes[0].nodeName == 'arcsinh'
mml = mp._print(atanh(x))
assert mml.childNodes[0].nodeName == 'arctanh'
mml = mp._print(acosh(x))
assert mml.childNodes[0].nodeName == 'arccosh'
def test_sympyissue_3504():
e = asin(a * x) / x
assert e.series(x, 4, n=2).removeO() == \
(x - 4)*(a/(4*sqrt(-16*a**2 + 1)) - asin(4*a)/16) + asin(4*a)/4
def test_sympyissue_4492():
assert simplify(integrate(x**2 * sqrt(5 - x**2), x)) == Piecewise(
(I*(2*x**5 - 15*x**3 + 25*x - 25*sqrt(x**2 - 5)*acosh(sqrt(5)*x/5)) /
(8*sqrt(x**2 - 5)), 1 < Abs(x**2)/5),
((-2*x**5 + 15*x**3 - 25*x + 25*sqrt(-x**2 + 5)*asin(sqrt(5)*x/5)) /
(8*sqrt(-x**2 + 5)), True))
def test_intrinsic_math_codegen():
# not included: log10
name_expr = [
("test_abs", Abs(x)),
("test_acos", acos(x)),
("test_asin", asin(x)),
("test_atan", atan(x)),
("test_cos", cos(x)),
("test_cosh", cosh(x)),
("test_log", log(x)),
("test_ln", ln(x)),
("test_sin", sin(x)),
("test_sinh", sinh(x)),
("test_sqrt", sqrt(x)),
("test_tan", tan(x)),
("test_tanh", tanh(x)),
]
result = codegen(name_expr, "F95", "file", header=False, empty=False)
assert result[0][0] == "file.f90"
expected = (
'REAL*8 function test_abs(x)\n'
'implicit none\n'
'REAL*8, intent(in) :: x\n'
'test_abs = Abs(x)\n'
'end function\n'
'REAL*8 function test_acos(x)\n'
'implicit none\n'
'REAL*8, intent(in) :: x\n'
'test_acos = acos(x)\n'
'end function\n'
'REAL*8 function test_asin(x)\n'
'implicit none\n'
'REAL*8, intent(in) :: x\n'
'test_asin = asin(x)\n'
'end function\n'
'REAL*8 function test_atan(x)\n'
'implicit none\n'
'REAL*8, intent(in) :: x\n'
'test_atan = atan(x)\n'
'end function\n'
'REAL*8 function test_cos(x)\n'
'implicit none\n'
'REAL*8, intent(in) :: x\n'
'test_cos = cos(x)\n'
'end function\n'
'REAL*8 function test_cosh(x)\n'
'implicit none\n'
'REAL*8, intent(in) :: x\n'
'test_cosh = cosh(x)\n'
'end function\n'
'REAL*8 function test_log(x)\n'
'implicit none\n'
'REAL*8, intent(in) :: x\n'
'test_log = log(x)\n'
'end function\n'
'REAL*8 function test_ln(x)\n'
'implicit none\n'
'REAL*8, intent(in) :: x\n'
'test_ln = log(x)\n'
'end function\n'
'REAL*8 function test_sin(x)\n'
'implicit none\n'
'REAL*8, intent(in) :: x\n'
'test_sin = sin(x)\n'
'end function\n'
'REAL*8 function test_sinh(x)\n'
'implicit none\n'
'REAL*8, intent(in) :: x\n'
'test_sinh = sinh(x)\n'
'end function\n'
'REAL*8 function test_sqrt(x)\n'
'implicit none\n'
'REAL*8, intent(in) :: x\n'
'test_sqrt = sqrt(x)\n'
'end function\n'
'REAL*8 function test_tan(x)\n'
'implicit none\n'
'REAL*8, intent(in) :: x\n'
'test_tan = tan(x)\n'
'end function\n'
'REAL*8 function test_tanh(x)\n'
'implicit none\n'
'REAL*8, intent(in) :: x\n'
'test_tanh = tanh(x)\n'
'end function\n'
)
assert result[0][1] == expected
assert result[1][0] == "file.h"
expected = (
'interface\n'
'REAL*8 function test_abs(x)\n'
'implicit none\n'
'REAL*8, intent(in) :: x\n'
'end function\n'
'end interface\n'
'interface\n'
'REAL*8 function test_acos(x)\n'
'implicit none\n'
'REAL*8, intent(in) :: x\n'
'end function\n'
'end interface\n'
'interface\n'
'REAL*8 function test_asin(x)\n'
'implicit none\n'
'REAL*8, intent(in) :: x\n'
'end function\n'
'end interface\n'
'interface\n'
'REAL*8 function test_atan(x)\n'
'implicit none\n'
'REAL*8, intent(in) :: x\n'
'end function\n'
'end interface\n'
'interface\n'
'REAL*8 function test_cos(x)\n'
'implicit none\n'
'REAL*8, intent(in) :: x\n'
'end function\n'
'end interface\n'
'interface\n'
'REAL*8 function test_cosh(x)\n'
'implicit none\n'
'REAL*8, intent(in) :: x\n'
'end function\n'
'end interface\n'
'interface\n'
'REAL*8 function test_log(x)\n'
'implicit none\n'
'REAL*8, intent(in) :: x\n'
'end function\n'
'end interface\n'
'interface\n'
'REAL*8 function test_ln(x)\n'
'implicit none\n'
'REAL*8, intent(in) :: x\n'
'end function\n'
'end interface\n'
'interface\n'
'REAL*8 function test_sin(x)\n'
'implicit none\n'
'REAL*8, intent(in) :: x\n'
'end function\n'
'end interface\n'
'interface\n'
'REAL*8 function test_sinh(x)\n'
'implicit none\n'
'REAL*8, intent(in) :: x\n'
'end function\n'
'end interface\n'
'interface\n'
'REAL*8 function test_sqrt(x)\n'
'implicit none\n'
'REAL*8, intent(in) :: x\n'
'end function\n'
'end interface\n'
'interface\n'
'REAL*8 function test_tan(x)\n'
'implicit none\n'
'REAL*8, intent(in) :: x\n'
'end function\n'
'end interface\n'
'interface\n'
'REAL*8 function test_tanh(x)\n'
'implicit none\n'
'REAL*8, intent(in) :: x\n'
'end function\n'
'end interface\n'
)
assert result[1][1] == expected
def test_sympyissue_12221():
e = sqrt(1 - x)/x
r = 2*I*(-sqrt(2) - asin(sqrt(3)/3) + asin(sqrt(5)/5) + 2)
assert integrate(e, (x, 3, 5)).simplify() == r
def test_sympyissue_4551():
assert (integrate(1/(x*sqrt(1 - x**2)), x) ==
Piecewise((-acosh(1/x), abs(x**-2) > 1), (I*asin(1/x), True)))
def test_sympyissue_3504():
e = asin(a*x)/x
assert e.series(x, 4, n=2).removeO() == \
(x - 4)*(a/(4*sqrt(-16*a**2 + 1)) - asin(4*a)/16) + asin(4*a)/4
def test_f3():
# issue sympy/sympy#3504
assert limit(asin(a * x) / x, x, 0) == a
def test_sympyissue_4551():
assert (integrate(1/(x*sqrt(1 - x**2)), x) ==
Piecewise((-acosh(1/x), Abs(x**(-2)) > 1), (I*asin(1/x), True)))
def test_plane():
p1 = Point3D(0, 0, 0)
p2 = Point3D(1, 1, 1)
p3 = Point3D(1, 2, 3)
p4 = Point3D(x, x, x)
p5 = Point3D(y, y, y)
pl3 = Plane(p1, p2, p3)
pl4 = Plane(p1, normal_vector=(1, 1, 1))
pl4b = Plane(p1, p2)
pl5 = Plane(p3, normal_vector=(1, 2, 3))
pl6 = Plane(Point3D(2, 3, 7), normal_vector=(2, 2, 2))
pl7 = Plane(Point3D(1, -5, -6), normal_vector=(1, -2, 1))
l1 = Line3D(Point3D(5, 0, 0), Point3D(1, -1, 1))
l2 = Line3D(Point3D(0, -2, 0), Point3D(3, 1, 1))
l3 = Line3D(Point3D(0, -1, 0), Point3D(5, -1, 9))
assert Plane(p1, p2, p3) != Plane(p1, p3, p2)
assert Plane(p1, p2, p3).is_coplanar(Plane(p1, p3, p2))
assert pl3 == Plane(Point3D(0, 0, 0), normal_vector=(1, -2, 1))
assert pl3 != pl4
assert pl4 == pl4b
assert pl5 == Plane(Point3D(1, 2, 3), normal_vector=(1, 2, 3))
assert pl5.equation(x, y, z) == x + 2 * y + 3 * z - 14
assert pl3.equation(x, y, z) == x - 2 * y + z
assert pl3.p1 == p1
assert pl4.p1 == p1
assert pl5.p1 == p3
assert pl4.normal_vector == (1, 1, 1)
assert pl5.normal_vector == (1, 2, 3)
assert p1 in pl3
assert p1 in pl4
assert p3 in pl5
assert pl3.projection(Point(0, 0)) == p1
p = pl3.projection(Point3D(1, 1, 0))
assert p == Point3D(7 / 6, 2 / 3, 1 / 6)
assert p in pl3
l = pl3.projection_line(Line(Point(0, 0), Point(1, 1)))
assert l == Line3D(Point3D(0, 0, 0), Point3D(7 / 6, 2 / 3, 1 / 6))
assert l in pl3
# get a segment that does not intersect the plane which is also
# parallel to pl3's normal veector
t = Dummy()
r = pl3.random_point()
a = pl3.perpendicular_line(r).arbitrary_point(t)
s = Segment3D(a.subs(t, 1), a.subs(t, 2))
assert s.p1 not in pl3 and s.p2 not in pl3
assert pl3.projection_line(s).equals(r)
assert pl3.projection_line(Segment(Point(1, 0), Point(1, 1))) == \
Segment3D(Point3D(5/6, 1/3, -1/6), Point3D(7/6, 2/3, 1/6))
assert pl6.projection_line(Ray(Point(1, 0), Point(1, 1))) == \
Ray3D(Point3D(14/3, 11/3, 11/3), Point3D(13/3, 13/3, 10/3))
assert pl3.perpendicular_line(r.args) == pl3.perpendicular_line(r)
assert pl3.is_parallel(pl6) is False
assert pl4.is_parallel(pl6)
assert pl6.is_parallel(l1) is False
assert pl3.is_perpendicular(pl6)
assert pl4.is_perpendicular(pl7)
assert pl6.is_perpendicular(pl7)
assert pl6.is_perpendicular(l1) is False
assert pl7.distance(Point3D(1, 3, 5)) == 5 * sqrt(6) / 6
assert pl6.distance(Point3D(0, 0, 0)) == 4 * sqrt(3)
assert pl6.distance(pl6.p1) == 0
assert pl7.distance(pl6) == 0
assert pl7.distance(l1) == 0
assert pl6.distance(Segment3D(Point3D(2, 3, 1), Point3D(1, 3, 4))) == 0
pl6.distance(Plane(Point3D(5, 5, 5), normal_vector=(8, 8, 8))) == sqrt(3)
assert pl6.angle_between(pl3) == pi / 2
assert pl6.angle_between(pl6) == 0
assert pl6.angle_between(pl4) == 0
assert pl7.angle_between(Line3D(Point3D(2, 3, 5), Point3D(2, 4, 6))) == \
-asin(sqrt(3)/6)
assert pl6.angle_between(Ray3D(Point3D(2, 4, 1), Point3D(6, 5, 3))) == \
asin(sqrt(7)/3)
assert pl7.angle_between(Segment3D(Point3D(5, 6, 1), Point3D(1, 2, 4))) == \
-asin(7*sqrt(246)/246)
assert are_coplanar(l1, l2, l3) is False
assert are_coplanar(l1) is False
assert are_coplanar(Point3D(2, 7, 2), Point3D(0, 0, 2), Point3D(1, 1, 2),
Point3D(1, 2, 2))
assert are_coplanar(Plane(p1, p2, p3), Plane(p1, p3, p2))
assert Plane.are_concurrent(pl3, pl4, pl5) is False
assert Plane.are_concurrent(pl6) is False
pytest.raises(ValueError, lambda: Plane.are_concurrent(Point3D(0, 0, 0)))
assert pl3.parallel_plane(Point3D(1, 2,
5)) == Plane(Point3D(1, 2, 5),
normal_vector=(1, -2, 1))
# perpendicular_plane
p = Plane((0, 0, 0), (1, 0, 0))
# default
assert p.perpendicular_plane() == Plane(Point3D(0, 0, 0), (0, 1, 0))
# 1 pt
assert p.perpendicular_plane(Point3D(1, 0, 1)) == \
Plane(Point3D(1, 0, 1), (0, 1, 0))
# pts as tuples
assert p.perpendicular_plane((1, 0, 1), (1, 1, 1)) == \
Plane(Point3D(1, 0, 1), (0, 0, -1))
a, b = Point3D(0, 0, 0), Point3D(0, 1, 0)
Z = (0, 0, 1)
p = Plane(a, normal_vector=Z)
# case 4
assert p.perpendicular_plane(a, b) == Plane(a, (1, 0, 0))
n = Point3D(*Z)
# case 1
assert p.perpendicular_plane(a, n) == Plane(a, (-1, 0, 0))
# case 2
assert Plane(a, normal_vector=b.args).perpendicular_plane(a, a + b) == \
Plane(Point3D(0, 0, 0), (1, 0, 0))
# case 1&3
assert Plane(b, normal_vector=Z).perpendicular_plane(b, b + n) == \
Plane(Point3D(0, 1, 0), (-1, 0, 0))
# case 2&3
assert Plane(b, normal_vector=b.args).perpendicular_plane(n, n + b) == \
Plane(Point3D(0, 0, 1), (1, 0, 0))
assert pl6.intersection(pl6) == [pl6]
assert pl4.intersection(pl4.p1) == [pl4.p1]
assert pl3.intersection(pl6) == [
Line3D(Point3D(8, 4, 0), Point3D(2, 4, 6))
]
assert pl3.intersection(Line3D(Point3D(1, 2, 4),
Point3D(4, 4,
2))) == [Point3D(2, 8 / 3, 10 / 3)]
assert pl3.intersection(Plane(Point3D(6, 0, 0),
normal_vector=(2, -5, 3))) == [
Line3D(Point3D(-24, -12, 0),
Point3D(-25, -13, -1))
]
assert pl6.intersection(Ray3D(Point3D(2, 3, 1),
Point3D(1, 3, 4))) == [Point3D(-1, 3, 10)]
assert pl6.intersection(Segment3D(Point3D(2, 3, 1),
Point3D(1, 3,
4))) == [Point3D(-1, 3, 10)]
assert pl7.intersection(Line(Point(2, 3),
Point(4, 2))) == [Point3D(13 / 2, 3 / 4, 0)]
r = Ray(Point(2, 3), Point(4, 2))
assert Plane((1, 2, 0), normal_vector=(0, 0, 1)).intersection(r) == [
Ray3D(Point(2, 3), Point(4, 2))
]
assert pl3.random_point() in pl3
# issue 8570
l2 = Line3D(
Point3D(Rational(50000004459633, 5000000000000),
Rational(-891926590718643, 1000000000000000),
Rational(231800966893633, 100000000000000)),
Point3D(Rational(50000004459633, 50000000000000),
Rational(-222981647679771, 250000000000000),
Rational(231800966893633, 100000000000000)))
p2 = Plane(
Point3D(Rational(402775636372767, 100000000000000),
Rational(-97224357654973, 100000000000000),
Rational(216793600814789, 100000000000000)),
(-Float(9.00000087501922), Float(-4.81170658872543e-13), Float(0.0)))
assert sstr([i.n(2) for i in p2.intersection(l2)]) == \
'[Point3D(4.0, -0.89, 2.3)]'
def test_plane():
p1 = Point3D(0, 0, 0)
p2 = Point3D(1, 1, 1)
p3 = Point3D(1, 2, 3)
pl3 = Plane(p1, p2, p3)
pl4 = Plane(p1, normal_vector=(1, 1, 1))
pl4b = Plane(p1, p2)
pl5 = Plane(p3, normal_vector=(1, 2, 3))
pl6 = Plane(Point3D(2, 3, 7), normal_vector=(2, 2, 2))
pl7 = Plane(Point3D(1, -5, -6), normal_vector=(1, -2, 1))
l1 = Line3D(Point3D(5, 0, 0), Point3D(1, -1, 1))
l2 = Line3D(Point3D(0, -2, 0), Point3D(3, 1, 1))
l3 = Line3D(Point3D(0, -1, 0), Point3D(5, -1, 9))
pytest.raises(ValueError, lambda: Plane(p1, normal_vector=(1, 1)))
assert Plane(p1, p2, p3) != Plane(p1, p3, p2)
assert Plane(p1, p2, p3).is_coplanar(Plane(p1, p3, p2))
assert pl3 == Plane(Point3D(0, 0, 0), normal_vector=(1, -2, 1))
assert pl3 != pl4
assert pl4 == pl4b
assert pl5 == Plane(Point3D(1, 2, 3), normal_vector=(1, 2, 3))
assert pl5.equation(x, y, z) == x + 2*y + 3*z - 14
assert pl3.equation(x, y, z) == x - 2*y + z
assert pl3.p1 == p1
assert pl4.p1 == p1
assert pl5.p1 == p3
assert pl4.normal_vector == (1, 1, 1)
assert pl5.normal_vector == (1, 2, 3)
assert p1 in pl3
assert p1 in pl4
assert p3 in pl5
assert pl3.projection(Point(0, 0)) == p1
p = pl3.projection(Point3D(1, 1, 0))
assert p == Point3D(7/6, 2/3, 1/6)
assert p in pl3
l = pl3.projection_line(Line(Point(0, 0), Point(1, 1)))
assert l == Line3D(Point3D(0, 0, 0), Point3D(7/6, 2/3, 1/6))
assert l in pl3
# get a segment that does not intersect the plane which is also
# parallel to pl3's normal veector
t = Dummy()
r = pl3.random_point()
a = pl3.perpendicular_line(r).arbitrary_point(t)
s = Segment3D(a.subs({t: 1}), a.subs({t: 2}))
assert s.p1 not in pl3 and s.p2 not in pl3
assert pl3.projection_line(s).equals(r)
assert pl3.projection_line(Segment(Point(1, 0), Point(1, 1))) == \
Segment3D(Point3D(5/6, 1/3, -1/6), Point3D(7/6, 2/3, 1/6))
assert pl6.projection_line(Ray(Point(1, 0), Point(1, 1))) == \
Ray3D(Point3D(14/3, 11/3, 11/3), Point3D(13/3, 13/3, 10/3))
assert pl3.perpendicular_line(r.args) == pl3.perpendicular_line(r)
assert pl3.is_parallel(pl6) is False
assert pl4.is_parallel(pl6)
assert pl6.is_parallel(l1) is False
assert pl3.is_perpendicular(pl6)
assert pl4.is_perpendicular(pl7)
assert pl6.is_perpendicular(pl7)
assert pl6.is_perpendicular(l1) is False
assert pl7.distance(Point3D(1, 3, 5)) == 5*sqrt(6)/6
assert pl6.distance(Point3D(0, 0, 0)) == 4*sqrt(3)
assert pl6.distance(pl6.p1) == 0
assert pl7.distance(pl6) == 0
assert pl7.distance(l1) == 0
assert pl6.distance(Segment3D(Point3D(2, 3, 1), Point3D(1, 3, 4))) == 0
pl6.distance(Plane(Point3D(5, 5, 5), normal_vector=(8, 8, 8))) == sqrt(3)
assert pl6.angle_between(pl3) == pi/2
assert pl6.angle_between(pl6) == 0
assert pl6.angle_between(pl4) == 0
assert pl7.angle_between(Line3D(Point3D(2, 3, 5), Point3D(2, 4, 6))) == \
-asin(sqrt(3)/6)
assert pl6.angle_between(Ray3D(Point3D(2, 4, 1), Point3D(6, 5, 3))) == \
asin(sqrt(7)/3)
assert pl7.angle_between(Segment3D(Point3D(5, 6, 1), Point3D(1, 2, 4))) == \
-asin(7*sqrt(246)/246)
assert are_coplanar(l1, l2, l3) is False
assert are_coplanar(l1) is False
assert are_coplanar(Point3D(2, 7, 2), Point3D(0, 0, 2),
Point3D(1, 1, 2), Point3D(1, 2, 2))
assert are_coplanar(Plane(p1, p2, p3), Plane(p1, p3, p2))
assert Plane.are_concurrent(pl3, pl4, pl5) is False
assert Plane.are_concurrent(pl6) is False
pytest.raises(ValueError, lambda: Plane.are_concurrent(Point3D(0, 0, 0)))
assert pl3.parallel_plane(Point3D(1, 2, 5)) == Plane(Point3D(1, 2, 5),
normal_vector=(1, -2, 1))
# perpendicular_plane
p = Plane((0, 0, 0), (1, 0, 0))
# default
assert p.perpendicular_plane() == Plane(Point3D(0, 0, 0), (0, 1, 0))
# 1 pt
assert p.perpendicular_plane(Point3D(1, 0, 1)) == \
Plane(Point3D(1, 0, 1), (0, 1, 0))
# pts as tuples
assert p.perpendicular_plane((1, 0, 1), (1, 1, 1)) == \
Plane(Point3D(1, 0, 1), (0, 0, -1))
pytest.raises(ValueError, lambda: p.perpendicular_plane(Point3D(1, 0, 1),
Point3D(1, 0, 2),
Point3D(1, 0, 3)))
a, b = Point3D(0, 0, 0), Point3D(0, 1, 0)
Z = (0, 0, 1)
p = Plane(a, normal_vector=Z)
# case 4
assert p.perpendicular_plane(a, b) == Plane(a, (1, 0, 0))
n = Point3D(*Z)
# case 1
assert p.perpendicular_plane(a, n) == Plane(a, (-1, 0, 0))
# case 2
assert Plane(a, normal_vector=b.args).perpendicular_plane(a, a + b) == \
Plane(Point3D(0, 0, 0), (1, 0, 0))
# case 1&3
assert Plane(b, normal_vector=Z).perpendicular_plane(b, b + n) == \
Plane(Point3D(0, 1, 0), (-1, 0, 0))
# case 2&3
assert Plane(b, normal_vector=b.args).perpendicular_plane(n, n + b) == \
Plane(Point3D(0, 0, 1), (1, 0, 0))
assert pl6.intersection(pl6) == [pl6]
assert pl4.intersection(pl4.p1) == [pl4.p1]
assert pl3.intersection(pl6) == [
Line3D(Point3D(8, 4, 0), Point3D(2, 4, 6))]
assert pl3.intersection(Line3D(Point3D(1, 2, 4), Point3D(4, 4, 2))) == [
Point3D(2, 8/3, 10/3)]
assert (pl3.intersection(Plane(Point3D(6, 0, 0),
normal_vector=(2, -5, 3))) ==
[Line3D(Point3D(-24, -12, 0), Point3D(-25, -13, -1))])
assert pl6.intersection(Ray3D(Point3D(2, 3, 1), Point3D(1, 3, 4))) == [
Point3D(-1, 3, 10)]
assert pl6.intersection(Segment3D(Point3D(2, 3, 1), Point3D(1, 3, 4))) == [
Point3D(-1, 3, 10)]
assert pl7.intersection(Line(Point(2, 3), Point(4, 2))) == [
Point3D(13/2, 3/4, 0)]
r = Ray(Point(2, 3), Point(4, 2))
assert Plane((1, 2, 0), normal_vector=(0, 0, 1)).intersection(r) == [
Ray3D(Point(2, 3), Point(4, 2))]
assert pl3.random_point() in pl3
# issue sympy/sympy#8570
l2 = Line3D(Point3D(Rational(50000004459633, 5000000000000),
Rational(-891926590718643, 1000000000000000),
Rational(231800966893633, 100000000000000)),
Point3D(Rational(50000004459633, 50000000000000),
Rational(-222981647679771, 250000000000000),
Rational(231800966893633, 100000000000000)))
p2 = Plane(Point3D(Rational(402775636372767, 100000000000000),
Rational(-97224357654973, 100000000000000),
Rational(216793600814789, 100000000000000)),
(-Float(9.00000087501922), Float(-4.81170658872543e-13),
Float(0.0)))
assert sstr([i.evalf(2) for i in p2.intersection(l2)]) == \
'[Point3D(4.0, -0.89, 2.3)]'
def test_sympyissue_12221():
e = sqrt(1 - x) / x
r = 2 * I * (-sqrt(2) - asin(sqrt(3) / 3) + asin(sqrt(5) / 5) + 2)
assert integrate(e, (x, 3, 5)).simplify() == r
def test_sympyissue_4052():
f = Rational(1, 2)*asin(x) + x*sqrt(1 - x**2)/2
assert integrate(cos(asin(x)), x) == f
assert integrate(sin(acos(x)), x) == f
def test_sympyissue_4052():
f = Rational(1, 2) * asin(x) + x * sqrt(1 - x**2) / 2
assert integrate(cos(asin(x)), x) == f
assert integrate(sin(acos(x)), x) == f
