from sympy import (Symbol, S, exp, log, sqrt, oo, E, zoo, pi, tan, sin, cos, cot, sec, csc, Abs, symbols, I, re, simplify, expint, Rational, Piecewise) from sympy.calculus.util import (function_range, continuous_domain, not_empty_in, periodicity, lcim, AccumBounds, is_convex, stationary_points, minimum, maximum) from sympy.core import Add, Mul, Pow from sympy.core.expr import unchanged from sympy.sets.sets import (Interval, FiniteSet, EmptySet, Complement, Union) from sympy.testing.pytest import raises, _both_exp_pow, XFAIL from sympy.abc import x a = Symbol('a', real=True) B = AccumBounds def test_function_range(): x, y, a, b = symbols('x y a b') assert function_range(sin(x), x, Interval(-pi/2, pi/2) ) == Interval(-1, 1) assert function_range(sin(x), x, Interval(0, pi) ) == Interval(0, 1) assert function_range(tan(x), x, Interval(0, pi) ) == Interval(-oo, oo) assert function_range(tan(x), x, Interval(pi/2, pi) ) == Interval(-oo, 0) assert function_range((x + 3)/(x - 2), x, Interval(-5, 5) ) == Union(Interval(-oo, Rational(2, 7)), Interval(Rational(8, 3), oo)) assert function_range(1/(x**2), x, Interval(-1, 1) ) == Interval(1, oo) assert function_range(exp(x), x, Interval(-1, 1) ) == Interval(exp(-1), exp(1)) assert function_range(log(x) - x, x, S.Reals ) == Interval(-oo, -1) assert function_range(sqrt(3*x - 1), x, Interval(0, 2) ) == Interval(0, sqrt(5)) assert function_range(x*(x - 1) - (x**2 - x), x, S.Reals ) == FiniteSet(0) assert function_range(x*(x - 1) - (x**2 - x) + y, x, S.Reals ) == FiniteSet(y) assert function_range(sin(x), x, Union(Interval(-5, -3), FiniteSet(4)) ) == Union(Interval(-sin(3), 1), FiniteSet(sin(4))) assert function_range(cos(x), x, Interval(-oo, -4) ) == Interval(-1, 1) assert function_range(cos(x), x, S.EmptySet) == S.EmptySet assert function_range(x/sqrt(x**2+1), x, S.Reals) == Interval.open(-1,1) raises(NotImplementedError, lambda : function_range( exp(x)*(sin(x) - cos(x))/2 - x, x, S.Reals)) raises(NotImplementedError, lambda : function_range( sin(x) + x, x, S.Reals)) # issue 13273 raises(NotImplementedError, lambda : function_range( log(x), x, S.Integers)) raises(NotImplementedError, lambda : function_range( sin(x)/2, x, S.Naturals)) def test_continuous_domain(): x = Symbol('x') assert continuous_domain(sin(x), x, Interval(0, 2*pi)) == Interval(0, 2*pi) assert continuous_domain(tan(x), x, Interval(0, 2*pi)) == \ Union(Interval(0, pi/2, False, True), Interval(pi/2, pi*Rational(3, 2), True, True), Interval(pi*Rational(3, 2), 2*pi, True, False)) assert continuous_domain((x - 1)/((x - 1)**2), x, S.Reals) == \ Union(Interval(-oo, 1, True, True), Interval(1, oo, True, True)) assert continuous_domain(log(x) + log(4*x - 1), x, S.Reals) == \ Interval(Rational(1, 4), oo, True, True) assert continuous_domain(1/sqrt(x - 3), x, S.Reals) == Interval(3, oo, True, True) assert continuous_domain(1/x - 2, x, S.Reals) == \ Union(Interval.open(-oo, 0), Interval.open(0, oo)) assert continuous_domain(1/(x**2 - 4) + 2, x, S.Reals) == \ Union(Interval.open(-oo, -2), Interval.open(-2, 2), Interval.open(2, oo)) domain = continuous_domain(log(tan(x)**2 + 1), x, S.Reals) assert not domain.contains(3*pi/2) assert domain.contains(5) d = Symbol('d', even=True, zero=False) assert continuous_domain(x**(1/d), x, S.Reals) == Interval(0, oo) def test_not_empty_in(): assert not_empty_in(FiniteSet(x, 2*x).intersect(Interval(1, 2, True, False)), x) == \ Interval(S.Half, 2, True, False) assert not_empty_in(FiniteSet(x, x**2).intersect(Interval(1, 2)), x) == \ Union(Interval(-sqrt(2), -1), Interval(1, 2)) assert not_empty_in(FiniteSet(x**2 + x, x).intersect(Interval(2, 4)), x) == \ Union(Interval(-sqrt(17)/2 - S.Half, -2), Interval(1, Rational(-1, 2) + sqrt(17)/2), Interval(2, 4)) assert not_empty_in(FiniteSet(x/(x - 1)).intersect(S.Reals), x) == \ Complement(S.Reals, FiniteSet(1)) assert not_empty_in(FiniteSet(a/(a - 1)).intersect(S.Reals), a) == \ Complement(S.Reals, FiniteSet(1)) assert not_empty_in(FiniteSet((x**2 - 3*x + 2)/(x - 1)).intersect(S.Reals), x) == \ Complement(S.Reals, FiniteSet(1)) assert not_empty_in(FiniteSet(3, 4, x/(x - 1)).intersect(Interval(2, 3)), x) == \ Interval(-oo, oo) assert not_empty_in(FiniteSet(4, x/(x - 1)).intersect(Interval(2, 3)), x) == \ Interval(S(3)/2, 2) assert not_empty_in(FiniteSet(x/(x**2 - 1)).intersect(S.Reals), x) == \ Complement(S.Reals, FiniteSet(-1, 1)) assert not_empty_in(FiniteSet(x, x**2).intersect(Union(Interval(1, 3, True, True), Interval(4, 5))), x) == \ Union(Interval(-sqrt(5), -2), Interval(-sqrt(3), -1, True, True), Interval(1, 3, True, True), Interval(4, 5)) assert not_empty_in(FiniteSet(1).intersect(Interval(3, 4)), x) == S.EmptySet assert not_empty_in(FiniteSet(x**2/(x + 2)).intersect(Interval(1, oo)), x) == \ Union(Interval(-2, -1, True, False), Interval(2, oo)) raises(ValueError, lambda: not_empty_in(x)) raises(ValueError, lambda: not_empty_in(Interval(0, 1), x)) raises(NotImplementedError, lambda: not_empty_in(FiniteSet(x).intersect(S.Reals), x, a)) @_both_exp_pow def test_periodicity(): x = Symbol('x') y = Symbol('y') z = Symbol('z', real=True) assert periodicity(sin(2*x), x) == pi assert periodicity((-2)*tan(4*x), x) == pi/4 assert periodicity(sin(x)**2, x) == 2*pi assert periodicity(3**tan(3*x), x) == pi/3 assert periodicity(tan(x)*cos(x), x) == 2*pi assert periodicity(sin(x)**(tan(x)), x) == 2*pi assert periodicity(tan(x)*sec(x), x) == 2*pi assert periodicity(sin(2*x)*cos(2*x) - y, x) == pi/2 assert periodicity(tan(x) + cot(x), x) == pi assert periodicity(sin(x) - cos(2*x), x) == 2*pi assert periodicity(sin(x) - 1, x) == 2*pi assert periodicity(sin(4*x) + sin(x)*cos(x), x) == pi assert periodicity(exp(sin(x)), x) == 2*pi assert periodicity(log(cot(2*x)) - sin(cos(2*x)), x) == pi assert periodicity(sin(2*x)*exp(tan(x) - csc(2*x)), x) == pi assert periodicity(cos(sec(x) - csc(2*x)), x) == 2*pi assert periodicity(tan(sin(2*x)), x) == pi assert periodicity(2*tan(x)**2, x) == pi assert periodicity(sin(x%4), x) == 4 assert periodicity(sin(x)%4, x) == 2*pi assert periodicity(tan((3*x-2)%4), x) == Rational(4, 3) assert periodicity((sqrt(2)*(x+1)+x) % 3, x) == 3 / (sqrt(2)+1) assert periodicity((x**2+1) % x, x) is None assert periodicity(sin(re(x)), x) == 2*pi assert periodicity(sin(x)**2 + cos(x)**2, x) is S.Zero assert periodicity(tan(x), y) is S.Zero assert periodicity(sin(x) + I*cos(x), x) == 2*pi assert periodicity(x - sin(2*y), y) == pi assert periodicity(exp(x), x) is None assert periodicity(exp(I*x), x) == 2*pi assert periodicity(exp(I*z), z) == 2*pi assert periodicity(exp(z), z) is None assert periodicity(exp(log(sin(z) + I*cos(2*z)), evaluate=False), z) == 2*pi assert periodicity(exp(log(sin(2*z) + I*cos(z)), evaluate=False), z) == 2*pi assert periodicity(exp(sin(z)), z) == 2*pi assert periodicity(exp(2*I*z), z) == pi assert periodicity(exp(z + I*sin(z)), z) is None assert periodicity(exp(cos(z/2) + sin(z)), z) == 4*pi assert periodicity(log(x), x) is None assert periodicity(exp(x)**sin(x), x) is None assert periodicity(sin(x)**y, y) is None assert periodicity(Abs(sin(Abs(sin(x)))), x) == pi assert all(periodicity(Abs(f(x)), x) == pi for f in ( cos, sin, sec, csc, tan, cot)) assert periodicity(Abs(sin(tan(x))), x) == pi assert periodicity(Abs(sin(sin(x) + tan(x))), x) == 2*pi assert periodicity(sin(x) > S.Half, x) == 2*pi assert periodicity(x > 2, x) is None assert periodicity(x**3 - x**2 + 1, x) is None assert periodicity(Abs(x), x) is None assert periodicity(Abs(x**2 - 1), x) is None assert periodicity((x**2 + 4)%2, x) is None assert periodicity((E**x)%3, x) is None assert periodicity(sin(expint(1, x))/expint(1, x), x) is None # returning `None` for any Piecewise p = Piecewise((0, x < -1), (x**2, x <= 1), (log(x), True)) assert periodicity(p, x) is None def test_periodicity_check(): x = Symbol('x') y = Symbol('y') assert periodicity(tan(x), x, check=True) == pi assert periodicity(sin(x) + cos(x), x, check=True) == 2*pi assert periodicity(sec(x), x) == 2*pi assert periodicity(sin(x*y), x) == 2*pi/abs(y) assert periodicity(Abs(sec(sec(x))), x) == pi def test_lcim(): from sympy import pi assert lcim([S.Half, S(2), S(3)]) == 6 assert lcim([pi/2, pi/4, pi]) == pi assert lcim([2*pi, pi/2]) == 2*pi assert lcim([S.One, 2*pi]) is None assert lcim([S(2) + 2*E, E/3 + Rational(1, 3), S.One + E]) == S(2) + 2*E def test_is_convex(): assert is_convex(1/x, x, domain=Interval(0, oo)) == True assert is_convex(1/x, x, domain=Interval(-oo, 0)) == False assert is_convex(x**2, x, domain=Interval(0, oo)) == True assert is_convex(log(x), x) == False raises(NotImplementedError, lambda: is_convex(log(x), x, a)) def test_stationary_points(): x, y = symbols('x y') assert stationary_points(sin(x), x, Interval(-pi/2, pi/2) ) == {-pi/2, pi/2} assert stationary_points(sin(x), x, Interval.Ropen(0, pi/4) ) == EmptySet() assert stationary_points(tan(x), x, ) == EmptySet() assert stationary_points(sin(x)*cos(x), x, Interval(0, pi) ) == {pi/4, pi*Rational(3, 4)} assert stationary_points(sec(x), x, Interval(0, pi) ) == {0, pi} assert stationary_points((x+3)*(x-2), x ) == FiniteSet(Rational(-1, 2)) assert stationary_points((x + 3)/(x - 2), x, Interval(-5, 5) ) == EmptySet() assert stationary_points((x**2+3)/(x-2), x ) == {2 - sqrt(7), 2 + sqrt(7)} assert stationary_points((x**2+3)/(x-2), x, Interval(0, 5) ) == {2 + sqrt(7)} assert stationary_points(x**4 + x**3 - 5*x**2, x, S.Reals ) == FiniteSet(-2, 0, Rational(5, 4)) assert stationary_points(exp(x), x ) == EmptySet() assert stationary_points(log(x) - x, x, S.Reals ) == {1} assert stationary_points(cos(x), x, Union(Interval(0, 5), Interval(-6, -3)) ) == {0, -pi, pi} assert stationary_points(y, x, S.Reals ) == S.Reals assert stationary_points(y, x, S.EmptySet) == S.EmptySet def test_maximum(): x, y = symbols('x y') assert maximum(sin(x), x) is S.One assert maximum(sin(x), x, Interval(0, 1)) == sin(1) assert maximum(tan(x), x) is oo assert maximum(tan(x), x, Interval(-pi/4, pi/4)) is S.One assert maximum(sin(x)*cos(x), x, S.Reals) == S.Half assert simplify(maximum(sin(x)*cos(x), x, Interval(pi*Rational(3, 8), pi*Rational(5, 8))) ) == sqrt(2)/4 assert maximum((x+3)*(x-2), x) is oo assert maximum((x+3)*(x-2), x, Interval(-5, 0)) == S(14) assert maximum((x+3)/(x-2), x, Interval(-5, 0)) == Rational(2, 7) assert simplify(maximum(-x**4-x**3+x**2+10, x) ) == 41*sqrt(41)/512 + Rational(5419, 512) assert maximum(exp(x), x, Interval(-oo, 2)) == exp(2) assert maximum(log(x) - x, x, S.Reals) is S.NegativeOne assert maximum(cos(x), x, Union(Interval(0, 5), Interval(-6, -3)) ) is S.One assert maximum(cos(x)-sin(x), x, S.Reals) == sqrt(2) assert maximum(y, x, S.Reals) == y assert maximum(abs(a**3 + a), a, Interval(0, 2)) == 10 assert maximum(abs(60*a**3 + 24*a), a, Interval(0, 2)) == 528 assert maximum(abs(12*a*(5*a**2 + 2)), a, Interval(0, 2)) == 528 assert maximum(x/sqrt(x**2+1), x, S.Reals) == 1 raises(ValueError, lambda : maximum(sin(x), x, S.EmptySet)) raises(ValueError, lambda : maximum(log(cos(x)), x, S.EmptySet)) raises(ValueError, lambda : maximum(1/(x**2 + y**2 + 1), x, S.EmptySet)) raises(ValueError, lambda : maximum(sin(x), sin(x))) raises(ValueError, lambda : maximum(sin(x), x*y, S.EmptySet)) raises(ValueError, lambda : maximum(sin(x), S.One)) def test_minimum(): x, y = symbols('x y') assert minimum(sin(x), x) is S.NegativeOne assert minimum(sin(x), x, Interval(1, 4)) == sin(4) assert minimum(tan(x), x) is -oo assert minimum(tan(x), x, Interval(-pi/4, pi/4)) is S.NegativeOne assert minimum(sin(x)*cos(x), x, S.Reals) == Rational(-1, 2) assert simplify(minimum(sin(x)*cos(x), x, Interval(pi*Rational(3, 8), pi*Rational(5, 8))) ) == -sqrt(2)/4 assert minimum((x+3)*(x-2), x) == Rational(-25, 4) assert minimum((x+3)/(x-2), x, Interval(-5, 0)) == Rational(-3, 2) assert minimum(x**4-x**3+x**2+10, x) == S(10) assert minimum(exp(x), x, Interval(-2, oo)) == exp(-2) assert minimum(log(x) - x, x, S.Reals) is -oo assert minimum(cos(x), x, Union(Interval(0, 5), Interval(-6, -3)) ) is S.NegativeOne assert minimum(cos(x)-sin(x), x, S.Reals) == -sqrt(2) assert minimum(y, x, S.Reals) == y assert minimum(x/sqrt(x**2+1), x, S.Reals) == -1 raises(ValueError, lambda : minimum(sin(x), x, S.EmptySet)) raises(ValueError, lambda : minimum(log(cos(x)), x, S.EmptySet)) raises(ValueError, lambda : minimum(1/(x**2 + y**2 + 1), x, S.EmptySet)) raises(ValueError, lambda : minimum(sin(x), sin(x))) raises(ValueError, lambda : minimum(sin(x), x*y, S.EmptySet)) raises(ValueError, lambda : minimum(sin(x), S.One)) def test_issue_19869(): t = symbols('t') assert (maximum(sqrt(3)*(t - 1)/(3*sqrt(t**2 + 1)), t) ) == sqrt(3)/3 def test_AccumBounds(): assert B(1, 2).args == (1, 2) assert B(1, 2).delta is S.One assert B(1, 2).mid == Rational(3, 2) assert B(1, 3).is_real == True assert B(1, 1) is S.One assert B(1, 2) + 1 == B(2, 3) assert 1 + B(1, 2) == B(2, 3) assert B(1, 2) + B(2, 3) == B(3, 5) assert -B(1, 2) == B(-2, -1) assert B(1, 2) - 1 == B(0, 1) assert 1 - B(1, 2) == B(-1, 0) assert B(2, 3) - B(1, 2) == B(0, 2) assert x + B(1, 2) == Add(B(1, 2), x) assert a + B(1, 2) == B(1 + a, 2 + a) assert B(1, 2) - x == Add(B(1, 2), -x) assert B(-oo, 1) + oo == B(-oo, oo) assert B(1, oo) + oo is oo assert B(1, oo) - oo == B(-oo, oo) assert (-oo - B(-1, oo)) is -oo assert B(-oo, 1) - oo is -oo assert B(1, oo) - oo == B(-oo, oo) assert B(-oo, 1) - (-oo) == B(-oo, oo) assert (oo - B(1, oo)) == B(-oo, oo) assert (-oo - B(1, oo)) is -oo assert B(1, 2)/2 == B(S.Half, 1) assert 2/B(2, 3) == B(Rational(2, 3), 1) assert 1/B(-1, 1) == B(-oo, oo) assert abs(B(1, 2)) == B(1, 2) assert abs(B(-2, -1)) == B(1, 2) assert abs(B(-2, 1)) == B(0, 2) assert abs(B(-1, 2)) == B(0, 2) c = Symbol('c') raises(ValueError, lambda: B(0, c)) raises(ValueError, lambda: B(1, -1)) r = Symbol('r', real=True) raises(ValueError, lambda: B(r, r - 1)) def test_AccumBounds_mul(): assert B(1, 2)*2 == B(2, 4) assert 2*B(1, 2) == B(2, 4) assert B(1, 2)*B(2, 3) == B(2, 6) assert B(0, 2)*B(2, oo) == B(0, oo) l, r = B(-oo, oo), B(-a, a) assert l*r == B(-oo, oo) assert r*l == B(-oo, oo) l, r = B(1, oo), B(-3, -2) assert l*r == B(-oo, -2) assert r*l == B(-oo, -2) assert B(1, 2)*0 == 0 assert B(1, oo)*0 == B(0, oo) assert B(-oo, 1)*0 == B(-oo, 0) assert B(-oo, oo)*0 == B(-oo, oo) assert B(1, 2)*x == Mul(B(1, 2), x, evaluate=False) assert B(0, 2)*oo == B(0, oo) assert B(-2, 0)*oo == B(-oo, 0) assert B(0, 2)*(-oo) == B(-oo, 0) assert B(-2, 0)*(-oo) == B(0, oo) assert B(-1, 1)*oo == B(-oo, oo) assert B(-1, 1)*(-oo) == B(-oo, oo) assert B(-oo, oo)*oo == B(-oo, oo) def test_AccumBounds_div(): assert B(-1, 3)/B(3, 4) == B(Rational(-1, 3), 1) assert B(-2, 4)/B(-3, 4) == B(-oo, oo) assert B(-3, -2)/B(-4, 0) == B(S.Half, oo) # these two tests can have a better answer # after Union of B is improved assert B(-3, -2)/B(-2, 1) == B(-oo, oo) assert B(2, 3)/B(-2, 2) == B(-oo, oo) assert B(-3, -2)/B(0, 4) == B(-oo, Rational(-1, 2)) assert B(2, 4)/B(-3, 0) == B(-oo, Rational(-2, 3)) assert B(2, 4)/B(0, 3) == B(Rational(2, 3), oo) assert B(0, 1)/B(0, 1) == B(0, oo) assert B(-1, 0)/B(0, 1) == B(-oo, 0) assert B(-1, 2)/B(-2, 2) == B(-oo, oo) assert 1/B(-1, 2) == B(-oo, oo) assert 1/B(0, 2) == B(S.Half, oo) assert (-1)/B(0, 2) == B(-oo, Rational(-1, 2)) assert 1/B(-oo, 0) == B(-oo, 0) assert 1/B(-1, 0) == B(-oo, -1) assert (-2)/B(-oo, 0) == B(0, oo) assert 1/B(-oo, -1) == B(-1, 0) assert B(1, 2)/a == Mul(B(1, 2), 1/a, evaluate=False) assert B(1, 2)/0 == B(1, 2)*zoo assert B(1, oo)/oo == B(0, oo) assert B(1, oo)/(-oo) == B(-oo, 0) assert B(-oo, -1)/oo == B(-oo, 0) assert B(-oo, -1)/(-oo) == B(0, oo) assert B(-oo, oo)/oo == B(-oo, oo) assert B(-oo, oo)/(-oo) == B(-oo, oo) assert B(-1, oo)/oo == B(0, oo) assert B(-1, oo)/(-oo) == B(-oo, 0) assert B(-oo, 1)/oo == B(-oo, 0) assert B(-oo, 1)/(-oo) == B(0, oo) def test_issue_18795(): r = Symbol('r', real=True) a = B(-1,1) c = B(7, oo) b = B(-oo, oo) assert c - tan(r) == B(7-tan(r), oo) assert b + tan(r) == B(-oo, oo) assert (a + r)/a == B(-oo, oo)*B(r - 1, r + 1) assert (b + a)/a == B(-oo, oo) def test_AccumBounds_func(): assert (x**2 + 2*x + 1).subs(x, B(-1, 1)) == B(-1, 4) assert exp(B(0, 1)) == B(1, E) assert exp(B(-oo, oo)) == B(0, oo) assert log(B(3, 6)) == B(log(3), log(6)) @XFAIL def test_AccumBounds_powf(): nn = Symbol('nn', nonnegative=True) assert B(1 + nn, 2 + nn)**B(1, 2) == B(1 + nn, (2 + nn)**2) i = Symbol('i', integer=True, negative=True) assert B(1, 2)**i == B(2**i, 1) def test_AccumBounds_pow(): assert B(0, 2)**2 == B(0, 4) assert B(-1, 1)**2 == B(0, 1) assert B(1, 2)**2 == B(1, 4) assert B(-1, 2)**3 == B(-1, 8) assert B(-1, 1)**0 == 1 assert B(1, 2)**Rational(5, 2) == B(1, 4*sqrt(2)) assert B(0, 2)**S.Half == B(0, sqrt(2)) neg = Symbol('neg', negative=True) unchanged(Pow, B(neg, 1), S.Half) nn = Symbol('nn', nonnegative=True) assert B(nn, nn + 1)**S.Half == B(sqrt(nn), sqrt(nn + 1)) assert B(nn, nn + 1)**nn == B(nn**nn, (nn + 1)**nn) unchanged(Pow, B(nn, nn + 1), x) i = Symbol('i', integer=True) unchanged(Pow, B(1, 2), i) i = Symbol('i', integer=True, nonnegative=True) assert B(1, 2)**i == B(1, 2**i) assert B(0, 1)**i == B(0**i, 1) assert B(1, 5)**(-2) == B(Rational(1, 25), 1) assert B(-1, 3)**(-2) == B(0, oo) assert B(0, 2)**(-3) == B(Rational(1, 8), oo) assert B(-2, 0)**(-3) == B(-oo, -Rational(1, 8)) assert B(0, 2)**(-2) == B(Rational(1, 4), oo) assert B(-1, 2)**(-3) == B(-oo, oo) assert B(-3, -2)**(-3) == B(Rational(-1, 8), Rational(-1, 27)) assert B(-3, -2)**(-2) == B(Rational(1, 9), Rational(1, 4)) assert B(0, oo)**S.Half == B(0, oo) assert B(-oo, 0)**(-2) == B(0, oo) assert B(-2, 0)**(-2) == B(Rational(1, 4), oo) assert B(Rational(1, 3), S.Half)**oo is S.Zero assert B(0, S.Half)**oo is S.Zero assert B(S.Half, 1)**oo == B(0, oo) assert B(0, 1)**oo == B(0, oo) assert B(2, 3)**oo is oo assert B(1, 2)**oo == B(0, oo) assert B(S.Half, 3)**oo == B(0, oo) assert B(Rational(-1, 3), Rational(-1, 4))**oo is S.Zero assert B(-1, Rational(-1, 2))**oo is S.NaN assert B(-3, -2)**oo is zoo assert B(-2, -1)**oo is S.NaN assert B(-2, Rational(-1, 2))**oo is S.NaN assert B(Rational(-1, 2), S.Half)**oo is S.Zero assert B(Rational(-1, 2), 1)**oo == B(0, oo) assert B(Rational(-2, 3), 2)**oo == B(0, oo) assert B(-1, 1)**oo == B(-oo, oo) assert B(-1, S.Half)**oo == B(-oo, oo) assert B(-1, 2)**oo == B(-oo, oo) assert B(-2, S.Half)**oo == B(-oo, oo) assert B(1, 2)**x == Pow(B(1, 2), x, evaluate=False) assert B(2, 3)**(-oo) is S.Zero assert B(0, 2)**(-oo) == B(0, oo) assert B(-1, 2)**(-oo) == B(-oo, oo) assert (tan(x)**sin(2*x)).subs(x, B(0, pi/2)) == \ Pow(B(-oo, oo), B(0, 1)) def test_AccumBounds_exponent(): # base is 0 z = 0**B(a, a + S.Half) assert z.subs(a, 0) == B(0, 1) assert z.subs(a, 1) == 0 p = z.subs(a, -1) assert p.is_Pow and p.args == (0, B(-1, -S.Half)) # base > 0 # when base is 1 the type of bounds does not matter assert 1**B(a, a + 1) == 1 # otherwise we need to know if 0 is in the bounds assert S.Half**B(-2, 2) == B(S(1)/4, 4) assert 2**B(-2, 2) == B(S(1)/4, 4) # +eps may introduce +oo # if there is a negative integer exponent assert B(0, 1)**B(S(1)/2, 1) == B(0, 1) assert B(0, 1)**B(0, 1) == B(0, 1) # positive bases have positive bounds assert B(2, 3)**B(-3, -2) == B(S(1)/27, S(1)/4) assert B(2, 3)**B(-3, 2) == B(S(1)/27, 9) # bounds generating imaginary parts unevaluated unchanged(Pow, B(-1, 1), B(1, 2)) assert B(0, S(1)/2)**B(1, oo) == B(0, S(1)/2) assert B(0, 1)**B(1, oo) == B(0, oo) assert B(0, 2)**B(1, oo) == B(0, oo) assert B(0, oo)**B(1, oo) == B(0, oo) assert B(S(1)/2, 1)**B(1, oo) == B(0, oo) assert B(S(1)/2, 1)**B(-oo, -1) == B(0, oo) assert B(S(1)/2, 1)**B(-oo, oo) == B(0, oo) assert B(S(1)/2, 2)**B(1, oo) == B(0, oo) assert B(S(1)/2, 2)**B(-oo, -1) == B(0, oo) assert B(S(1)/2, 2)**B(-oo, oo) == B(0, oo) assert B(S(1)/2, oo)**B(1, oo) == B(0, oo) assert B(S(1)/2, oo)**B(-oo, -1) == B(0, oo) assert B(S(1)/2, oo)**B(-oo, oo) == B(0, oo) assert B(1, 2)**B(1, oo) == B(0, oo) assert B(1, 2)**B(-oo, -1) == B(0, oo) assert B(1, 2)**B(-oo, oo) == B(0, oo) assert B(1, oo)**B(1, oo) == B(0, oo) assert B(1, oo)**B(-oo, -1) == B(0, oo) assert B(1, oo)**B(-oo, oo) == B(0, oo) assert B(2, oo)**B(1, oo) == B(2, oo) assert B(2, oo)**B(-oo, -1) == B(0, S(1)/2) assert B(2, oo)**B(-oo, oo) == B(0, oo) def test_comparison_AccumBounds(): assert (B(1, 3) < 4) == S.true assert (B(1, 3) < -1) == S.false assert (B(1, 3) < 2).rel_op == '<' assert (B(1, 3) <= 2).rel_op == '<=' assert (B(1, 3) > 4) == S.false assert (B(1, 3) > -1) == S.true assert (B(1, 3) > 2).rel_op == '>' assert (B(1, 3) >= 2).rel_op == '>=' assert (B(1, 3) < B(4, 6)) == S.true assert (B(1, 3) < B(2, 4)).rel_op == '<' assert (B(1, 3) < B(-2, 0)) == S.false assert (B(1, 3) <= B(4, 6)) == S.true assert (B(1, 3) <= B(-2, 0)) == S.false assert (B(1, 3) > B(4, 6)) == S.false assert (B(1, 3) > B(-2, 0)) == S.true assert (B(1, 3) >= B(4, 6)) == S.false assert (B(1, 3) >= B(-2, 0)) == S.true # issue 13499 assert (cos(x) > 0).subs(x, oo) == (B(-1, 1) > 0) c = Symbol('c') raises(TypeError, lambda: (B(0, 1) < c)) raises(TypeError, lambda: (B(0, 1) <= c)) raises(TypeError, lambda: (B(0, 1) > c)) raises(TypeError, lambda: (B(0, 1) >= c)) def test_contains_AccumBounds(): assert (1 in B(1, 2)) == S.true raises(TypeError, lambda: a in B(1, 2)) assert 0 in B(-1, 0) raises(TypeError, lambda: (cos(1)**2 + sin(1)**2 - 1) in B(-1, 0)) assert (-oo in B(1, oo)) == S.true assert (oo in B(-oo, 0)) == S.true # issue 13159 assert Mul(0, B(-1, 1)) == Mul(B(-1, 1), 0) == 0 import itertools for perm in itertools.permutations([0, B(-1, 1), x]): assert Mul(*perm) == 0 def test_intersection_AccumBounds(): assert B(0, 3).intersection(B(1, 2)) == B(1, 2) assert B(0, 3).intersection(B(1, 4)) == B(1, 3) assert B(0, 3).intersection(B(-1, 2)) == B(0, 2) assert B(0, 3).intersection(B(-1, 4)) == B(0, 3) assert B(0, 1).intersection(B(2, 3)) == S.EmptySet raises(TypeError, lambda: B(0, 3).intersection(1)) def test_union_AccumBounds(): assert B(0, 3).union(B(1, 2)) == B(0, 3) assert B(0, 3).union(B(1, 4)) == B(0, 4) assert B(0, 3).union(B(-1, 2)) == B(-1, 3) assert B(0, 3).union(B(-1, 4)) == B(-1, 4) raises(TypeError, lambda: B(0, 3).union(1)) def test_issue_16469(): x = Symbol("x", real=True) f = abs(x) assert function_range(f, x, S.Reals) == Interval(0, oo, False, True) @_both_exp_pow def test_issue_18747(): assert periodicity(exp(pi*I*(x/4+S.Half/2)), x) == 8