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Why sympy cannot correctly determine identity of simple sqrt expressions?

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Even if it correctly determine the inequalities of both sides, it cannot determine the identity.

from sympy import *
assert not 2 * sqrt(2 - sqrt(3)) == -sqrt(2) + sqrt(6)
assert 2 * sqrt(2 - sqrt(3)) <= -sqrt(2) + sqrt(6)
assert 2 * sqrt(2 - sqrt(3)) >= -sqrt(2) + sqrt(6)

assert not 2 * sqrt(2 - sqrt(3)) == sqrt(8 - 4 * sqrt(3))
assert 2 * sqrt(2 - sqrt(3)) <= sqrt(8 - 4 * sqrt(3))
assert 2 * sqrt(2 - sqrt(3)) >= sqrt(8 - 4 * sqrt(3))
Solution

  • The reason is, as commented by @OscarBenjamin, "a==b tests structural equality not mathematical equality."

    The equality operator (==) tests whether expressions have identical form, not whether they are mathematically equivalent.

    https://github.com/sympy/sympy/wiki/Faq#why-does-sympy-say-that-two-equal-expressions-are-unequal

    Double equals signs (==) are used to test equality. However, this tests expressions exactly, not symbolically.

    https://docs.sympy.org/latest/explanation/gotchas.html#double-equals-signs

    Except to use simplify(lhs - rhs) and check if the expression reduces to 0, Eq(lhs, rhs) can be used.

    from sympy import *
assert Eq(2 * sqrt(2 - sqrt(3)), -sqrt(2) + sqrt(6))

    This class is not the same as the == operator. The == operator tests for exact structural equality between two expressions; this class compares expressions mathematically.

    https://docs.sympy.org/latest/modules/core.html#sympy.core.relational.Equality

    See also: https://docs.sympy.org/latest/tutorials/intro-tutorial/gotchas.html#equals-signs

    P.S. .equals tests if two expressions are equal by evaluating them numerically at random points.