from sympy import (Symbol, S, exp, log, sqrt, oo, E, zoo, pi, tan, sin, cos, cot, sec, csc, Abs, symbols, I, re) from sympy.calculus.util import (function_range, continuous_domain, not_empty_in, periodicity, lcim, AccumBounds, is_convex) from sympy.core import Add, Mul, Pow from sympy.sets.sets import Interval, FiniteSet, Complement, Union from sympy.utilities.pytest import raises from sympy.abc import x a = Symbol('a', real=True) 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, S(2)/7), Interval(S(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) raises(NotImplementedError, lambda : function_range( exp(x)*(sin(x) - cos(x))/2 - x, x, S.Reals)) 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, 3*pi/2, True, True), Interval(3*pi/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(S(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)) def test_not_empty_in(): assert not_empty_in(FiniteSet(x, 2*x).intersect(Interval(1, 2, True, False)), x) == \ Interval(S(1)/2, 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(1)/2, -2), Interval(1, -S(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) == \ Union(Interval(S(3)/2, 2), FiniteSet(3)) 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)) 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) == S(4)/3 assert periodicity((sqrt(2)*(x+1)+x) % 3, x) == 3 / (sqrt(2)+1) assert periodicity((x**2+1) % x, x) == None assert periodicity(sin(re(x)), x) == 2*pi assert periodicity(sin(x)**2 + cos(x)**2, x) == S.Zero assert periodicity(tan(x), y) == 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) is 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 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(1)/2, S(2), S(3)]) == 6 assert lcim([pi/2, pi/4, pi]) == pi assert lcim([2*pi, pi/2]) == 2*pi assert lcim([S(1), 2*pi]) is None assert lcim([S(2) + 2*E, E/3 + S(1)/3, S(1) + 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 def test_AccumBounds(): assert AccumBounds(1, 2).args == (1, 2) assert AccumBounds(1, 2).delta == S(1) assert AccumBounds(1, 2).mid == S(3)/2 assert AccumBounds(1, 3).is_real == True assert AccumBounds(1, 1) == S(1) assert AccumBounds(1, 2) + 1 == AccumBounds(2, 3) assert 1 + AccumBounds(1, 2) == AccumBounds(2, 3) assert AccumBounds(1, 2) + AccumBounds(2, 3) == AccumBounds(3, 5) assert -AccumBounds(1, 2) == AccumBounds(-2, -1) assert AccumBounds(1, 2) - 1 == AccumBounds(0, 1) assert 1 - AccumBounds(1, 2) == AccumBounds(-1, 0) assert AccumBounds(2, 3) - AccumBounds(1, 2) == AccumBounds(0, 2) assert x + AccumBounds(1, 2) == Add(AccumBounds(1, 2), x) assert a + AccumBounds(1, 2) == AccumBounds(1 + a, 2 + a) assert AccumBounds(1, 2) - x == Add(AccumBounds(1, 2), -x) assert AccumBounds(-oo, 1) + oo == AccumBounds(-oo, oo) assert AccumBounds(1, oo) + oo == oo assert AccumBounds(1, oo) - oo == AccumBounds(-oo, oo) assert (-oo - AccumBounds(-1, oo)) == -oo assert AccumBounds(-oo, 1) - oo == -oo assert AccumBounds(1, oo) - oo == AccumBounds(-oo, oo) assert AccumBounds(-oo, 1) - (-oo) == AccumBounds(-oo, oo) assert (oo - AccumBounds(1, oo)) == AccumBounds(-oo, oo) assert (-oo - AccumBounds(1, oo)) == -oo assert AccumBounds(1, 2)/2 == AccumBounds(S(1)/2, 1) assert 2/AccumBounds(2, 3) == AccumBounds(S(2)/3, 1) assert 1/AccumBounds(-1, 1) == AccumBounds(-oo, oo) assert abs(AccumBounds(1, 2)) == AccumBounds(1, 2) assert abs(AccumBounds(-2, -1)) == AccumBounds(1, 2) assert abs(AccumBounds(-2, 1)) == AccumBounds(0, 2) assert abs(AccumBounds(-1, 2)) == AccumBounds(0, 2) def test_AccumBounds_mul(): assert AccumBounds(1, 2)*2 == AccumBounds(2, 4) assert 2*AccumBounds(1, 2) == AccumBounds(2, 4) assert AccumBounds(1, 2)*AccumBounds(2, 3) == AccumBounds(2, 6) assert AccumBounds(1, 2)*0 == 0 assert AccumBounds(1, oo)*0 == AccumBounds(0, oo) assert AccumBounds(-oo, 1)*0 == AccumBounds(-oo, 0) assert AccumBounds(-oo, oo)*0 == AccumBounds(-oo, oo) assert AccumBounds(1, 2)*x == Mul(AccumBounds(1, 2), x, evaluate=False) assert AccumBounds(0, 2)*oo == AccumBounds(0, oo) assert AccumBounds(-2, 0)*oo == AccumBounds(-oo, 0) assert AccumBounds(0, 2)*(-oo) == AccumBounds(-oo, 0) assert AccumBounds(-2, 0)*(-oo) == AccumBounds(0, oo) assert AccumBounds(-1, 1)*oo == AccumBounds(-oo, oo) assert AccumBounds(-1, 1)*(-oo) == AccumBounds(-oo, oo) assert AccumBounds(-oo, oo)*oo == AccumBounds(-oo, oo) def test_AccumBounds_div(): assert AccumBounds(-1, 3)/AccumBounds(3, 4) == AccumBounds(-S(1)/3, 1) assert AccumBounds(-2, 4)/AccumBounds(-3, 4) == AccumBounds(-oo, oo) assert AccumBounds(-3, -2)/AccumBounds(-4, 0) == AccumBounds(S(1)/2, oo) # these two tests can have a better answer # after Union of AccumBounds is improved assert AccumBounds(-3, -2)/AccumBounds(-2, 1) == AccumBounds(-oo, oo) assert AccumBounds(2, 3)/AccumBounds(-2, 2) == AccumBounds(-oo, oo) assert AccumBounds(-3, -2)/AccumBounds(0, 4) == AccumBounds(-oo, -S(1)/2) assert AccumBounds(2, 4)/AccumBounds(-3, 0) == AccumBounds(-oo, -S(2)/3) assert AccumBounds(2, 4)/AccumBounds(0, 3) == AccumBounds(S(2)/3, oo) assert AccumBounds(0, 1)/AccumBounds(0, 1) == AccumBounds(0, oo) assert AccumBounds(-1, 0)/AccumBounds(0, 1) == AccumBounds(-oo, 0) assert AccumBounds(-1, 2)/AccumBounds(-2, 2) == AccumBounds(-oo, oo) assert 1/AccumBounds(-1, 2) == AccumBounds(-oo, oo) assert 1/AccumBounds(0, 2) == AccumBounds(S(1)/2, oo) assert (-1)/AccumBounds(0, 2) == AccumBounds(-oo, -S(1)/2) assert 1/AccumBounds(-oo, 0) == AccumBounds(-oo, 0) assert 1/AccumBounds(-1, 0) == AccumBounds(-oo, -1) assert (-2)/AccumBounds(-oo, 0) == AccumBounds(0, oo) assert 1/AccumBounds(-oo, -1) == AccumBounds(-1, 0) assert AccumBounds(1, 2)/a == Mul(AccumBounds(1, 2), 1/a, evaluate=False) assert AccumBounds(1, 2)/0 == AccumBounds(1, 2)*zoo assert AccumBounds(1, oo)/oo == AccumBounds(0, oo) assert AccumBounds(1, oo)/(-oo) == AccumBounds(-oo, 0) assert AccumBounds(-oo, -1)/oo == AccumBounds(-oo, 0) assert AccumBounds(-oo, -1)/(-oo) == AccumBounds(0, oo) assert AccumBounds(-oo, oo)/oo == AccumBounds(-oo, oo) assert AccumBounds(-oo, oo)/(-oo) == AccumBounds(-oo, oo) assert AccumBounds(-1, oo)/oo == AccumBounds(0, oo) assert AccumBounds(-1, oo)/(-oo) == AccumBounds(-oo, 0) assert AccumBounds(-oo, 1)/oo == AccumBounds(-oo, 0) assert AccumBounds(-oo, 1)/(-oo) == AccumBounds(0, oo) def test_AccumBounds_func(): assert (x**2 + 2*x + 1).subs(x, AccumBounds(-1, 1)) == AccumBounds(-1, 4) assert exp(AccumBounds(0, 1)) == AccumBounds(1, E) assert exp(AccumBounds(-oo, oo)) == AccumBounds(0, oo) assert log(AccumBounds(3, 6)) == AccumBounds(log(3), log(6)) def test_AccumBounds_pow(): assert AccumBounds(0, 2)**2 == AccumBounds(0, 4) assert AccumBounds(-1, 1)**2 == AccumBounds(0, 1) assert AccumBounds(1, 2)**2 == AccumBounds(1, 4) assert AccumBounds(-1, 2)**3 == AccumBounds(-1, 8) assert AccumBounds(-1, 1)**0 == 1 assert AccumBounds(1, 2)**(S(5)/2) == AccumBounds(1, 4*sqrt(2)) assert AccumBounds(-1, 2)**(S(1)/3) == AccumBounds(-1, 2**(S(1)/3)) assert AccumBounds(0, 2)**(S(1)/2) == AccumBounds(0, sqrt(2)) assert AccumBounds(-4, 2)**(S(2)/3) == AccumBounds(0, 2*2**(S(1)/3)) assert AccumBounds(-1, 5)**(S(1)/2) == AccumBounds(0, sqrt(5)) assert AccumBounds(-oo, 2)**(S(1)/2) == AccumBounds(0, sqrt(2)) assert AccumBounds(-2, 3)**(S(-1)/4) == AccumBounds(0, oo) assert AccumBounds(1, 5)**(-2) == AccumBounds(S(1)/25, 1) assert AccumBounds(-1, 3)**(-2) == AccumBounds(0, oo) assert AccumBounds(0, 2)**(-2) == AccumBounds(S(1)/4, oo) assert AccumBounds(-1, 2)**(-3) == AccumBounds(-oo, oo) assert AccumBounds(-3, -2)**(-3) == AccumBounds(S(-1)/8, -S(1)/27) assert AccumBounds(-3, -2)**(-2) == AccumBounds(S(1)/9, S(1)/4) assert AccumBounds(0, oo)**(S(1)/2) == AccumBounds(0, oo) assert AccumBounds(-oo, -1)**(S(1)/3) == AccumBounds(-oo, -1) assert AccumBounds(-2, 3)**(-S(1)/3) == AccumBounds(-oo, oo) assert AccumBounds(-oo, 0)**(-2) == AccumBounds(0, oo) assert AccumBounds(-2, 0)**(-2) == AccumBounds(S(1)/4, oo) assert AccumBounds(S(1)/3, S(1)/2)**oo == S(0) assert AccumBounds(0, S(1)/2)**oo == S(0) assert AccumBounds(S(1)/2, 1)**oo == AccumBounds(0, oo) assert AccumBounds(0, 1)**oo == AccumBounds(0, oo) assert AccumBounds(2, 3)**oo == oo assert AccumBounds(1, 2)**oo == AccumBounds(0, oo) assert AccumBounds(S(1)/2, 3)**oo == AccumBounds(0, oo) assert AccumBounds(-S(1)/3, -S(1)/4)**oo == S(0) assert AccumBounds(-1, -S(1)/2)**oo == AccumBounds(-oo, oo) assert AccumBounds(-3, -2)**oo == FiniteSet(-oo, oo) assert AccumBounds(-2, -1)**oo == AccumBounds(-oo, oo) assert AccumBounds(-2, -S(1)/2)**oo == AccumBounds(-oo, oo) assert AccumBounds(-S(1)/2, S(1)/2)**oo == S(0) assert AccumBounds(-S(1)/2, 1)**oo == AccumBounds(0, oo) assert AccumBounds(-S(2)/3, 2)**oo == AccumBounds(0, oo) assert AccumBounds(-1, 1)**oo == AccumBounds(-oo, oo) assert AccumBounds(-1, S(1)/2)**oo == AccumBounds(-oo, oo) assert AccumBounds(-1, 2)**oo == AccumBounds(-oo, oo) assert AccumBounds(-2, S(1)/2)**oo == AccumBounds(-oo, oo) assert AccumBounds(1, 2)**x == Pow(AccumBounds(1, 2), x, evaluate=False) assert AccumBounds(2, 3)**(-oo) == S(0) assert AccumBounds(0, 2)**(-oo) == AccumBounds(0, oo) assert AccumBounds(-1, 2)**(-oo) == AccumBounds(-oo, oo) assert (tan(x)**sin(2*x)).subs(x, AccumBounds(0, pi/2)) == \ Pow(AccumBounds(-oo, oo), AccumBounds(0, 1), evaluate=False) def test_comparison_AccumBounds(): assert (AccumBounds(1, 3) < 4) == S.true assert (AccumBounds(1, 3) < -1) == S.false assert (AccumBounds(1, 3) < 2).rel_op == '<' assert (AccumBounds(1, 3) <= 2).rel_op == '<=' assert (AccumBounds(1, 3) > 4) == S.false assert (AccumBounds(1, 3) > -1) == S.true assert (AccumBounds(1, 3) > 2).rel_op == '>' assert (AccumBounds(1, 3) >= 2).rel_op == '>=' assert (AccumBounds(1, 3) < AccumBounds(4, 6)) == S.true assert (AccumBounds(1, 3) < AccumBounds(2, 4)).rel_op == '<' assert (AccumBounds(1, 3) < AccumBounds(-2, 0)) == S.false # issue 13499 assert (cos(x) > 0).subs(x, oo) == (AccumBounds(-1, 1) > 0) def test_contains_AccumBounds(): assert (1 in AccumBounds(1, 2)) == S.true raises(TypeError, lambda: a in AccumBounds(1, 2)) assert 0 in AccumBounds(-1, 0) raises(TypeError, lambda: (cos(1)**2 + sin(1)**2 - 1) in AccumBounds(-1, 0)) assert (-oo in AccumBounds(1, oo)) == S.true assert (oo in AccumBounds(-oo, 0)) == S.true # issue 13159 assert Mul(0, AccumBounds(-1, 1)) == Mul(AccumBounds(-1, 1), 0) == 0 import itertools for perm in itertools.permutations([0, AccumBounds(-1, 1), x]): assert Mul(*perm) == 0