[Bug libstdc++/83140] New: assoc_legendre returns negated value when m is odd

["kwsm.gcc at gmail dot com" <gcc-bugzilla at gcc dot gnu dot org>]

https://gcc.gnu.org/bugzilla/show_bug.cgi?id=83140

            Bug ID: 83140
           Summary: assoc_legendre returns negated value when m is odd
           Product: gcc
           Version: 7.1.0
            Status: UNCONFIRMED
          Severity: normal
          Priority: P3
         Component: libstdc++
          Assignee: unassigned at gcc dot gnu.org
          Reporter: kwsm.gcc at gmail dot com
  Target Milestone: ---

In the standard, Legendre polynomials and associated Legendre functions are
defined as
  P_l = (1 / 2^l l!) d^l/dx^l (x^2 - 1)^l,
and
  P_l^m = (1 - x^2)^(m/2) d^m/dx^m P_l,
respectively.

Then, for example, P_1^1 should be √(1 - x^2), but GCC returns -√(1 - x^2): the
return value of std::assoc_legendre(1, 1, 0.5) is -0.866.

GCC's implementation uses recurrence of associated Legendre function
  P_m^m = (-1)^m (2m - 1)!! (1 - x^2)^(m/2),
but this corresponds to
  P_l^m = (-1)^m (1 - x^2)^(m/2) d^m/dx^m  P_l.

Namely, std::assoc_legendre returns negated value when m is odd.