def METHOD_NAME(self, x, y):
    """
    Return the Kazhdan-Lusztig `P` polynomial.
    If the rank is large, this runs slowly at first but speeds up
    as you do repeated calculations due to the caching.
    INPUT:
    - ``x``, ``y`` -- elements of the underlying Coxeter group
    .. SEEALSO::
        :mod:`~sage.libs.coxeter3.coxeter_group.CoxeterGroup.kazhdan_lusztig_polynomial`
        for a faster implementation using Fokko Ducloux's Coxeter3 C++ library.
    EXAMPLES::
        sage: R.<q> = QQ[]
        sage: W = WeylGroup("A3", prefix="s")
        sage: [s1,s2,s3] = W.simple_reflections()
        sage: KL = KazhdanLusztigPolynomial(W, q)
        sage: KL.P(s2,s2*s1*s3*s2)
        q + 1
    """
    if x == 1:
        x = self._one
    if y == 1:
        y = self._one
    if x == y:
        return self._base_ring.one()
    if not x.bruhat_le(y):
        return self._base_ring.zero()
    if y.length() == 0:
        if x.length() == 0:
            return self._base_ring.one()
        else:
            return self._base_ring.zero()
    p = sum(-self.R(x, t) * self.METHOD_NAME(t, y)
            for t in self._coxeter_group.bruhat_interval(x, y) if t != x)
    tr = (y.length() - x.length() + 1) // 2
    ret = p.truncate(tr)
    if self._trace:
        print("    P({},{})={}".format(x, y, ret))
    return ret