[GHC] #8539: Data.Complex shouldn't use default implementation of (**)

[GHC]

#8539: Data.Complex shouldn't use default implementation of (**)
-------------------------------------+-------------------------------------
        Reporter:  jjaredsimpson     |            Owner:
            Type:  bug               |           Status:  patch
        Priority:  low               |        Milestone:
       Component:  Prelude           |          Version:  7.6.3
      Resolution:                    |         Keywords:
Operating System:  Unknown/Multiple  |     Architecture:  Unknown/Multiple
 Type of failure:  Incorrect result  |       Difficulty:  Easy (less than 1
  at runtime                         |  hour)
       Test Case:                    |       Blocked By:
        Blocking:                    |  Related Tickets:
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Comment (by Scott Turner):

 I don't know that (!**) is more special. Maybe we are talking about it
 because it has the rule
 {{{
 x ** 0 == 1
 }}}
 which is peculiarly effective. It does battle with another rule
 {{{
 0 ** x == 0
 }}}
 and in the judgement of most numerical analysis, it wins the fight.

 I think the reason people aren't diving in to address how all the other
 operators handle complex infinities is that it's very ''complex''. Your
 patch handles a few cases well, but it doesn't come near to handling them
 all. For example, it doesn't take care of
 {{{
 (infinity:+infinity)*(infinity:+infinity) == (NaN:+infinity)
 }}}
 which is poor because the result should be a plain infinite value. The
 same kind of thing can happen, even if one of the operands is finite. Then
 there are cases in which there's an intermediate overflow but the best
 result is not infinite. That's because
 {{{
 (x:+y) * (x':+y')   =  (x*x'-y*y') :+ (x*y'+y*x')
 }}}
 subtracts after multiplying. The intermediate x*x' can be unrepresentable,
 while x*x'-y*y' is representable.

 It's all a lot of fun. It would be nice work if someone would pay us for
 doing it.

--
Ticket URL: <http://ghc.haskell.org/trac/ghc/ticket/8539#comment:18>
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