#4413: (^^) is not correct for Double and Float -------------------------------------+------------------------------------- Reporter: | Owner: tcsavage daniel.is.fischer | Status: new Type: bug | Milestone: 7.6.2 Priority: low | Version: 7.1 Component: libraries/base | Keywords: Double, Float, Resolution: | exponentiation Operating System: Unknown/Multiple | Architecture: Unknown/Multiple Type of failure: Incorrect result | Difficulty: Unknown at runtime | Blocked By: Test Case: | Related Tickets: Blocking: | -------------------------------------+------------------------------------- Changes (by ekmett): * cc: ekmett@…, hvr (added) * difficulty: => Unknown Comment: When I first saw this, I was in favor of it, but then I chewed on it some more. The issue is unfortunately more subtle. While Daniel's proposal is sound for 2 and 0.5 which have exact representations in IEEE 754 floating point for both the original number and the reciprocal, this proposal yields worse numeric precision for any floating point number where the inverse isn't exactly representable. Consider 3. {{{

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recip 3^600 5.336385165377033e-287 recip (3^600) 5.336385165377108e-287 }}}

Now, lets compare these to answers that are more accurate, using my `rounded` library: {{{

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:m + Numeric.Rounded :set -XDataKinds recip (3^600) :: Rounded TowardZero 512 5.3363851653771076591089513443334745430695010187597757569958309540684063019552892464056775485633511783044398629355030763271677070438623409480523609685251156e-287 recip 3^600 :: Rounded TowardZero 512 5.3363851653771076591089513443334745430695010187597757569958309540684063019552892464056775485633511783044398629355030763271677070438623409480523609685247316e-287 }}}

The inexact version caused us 3 decimal places worth of accuracy. Those 10 bits of lost significand dwarf the error in the subnormal case. If we're stuck choosing, the existing implementation shoots an arrow that consistently winds up closer to the mark except in cases of underflow. -- Ticket URL: http://ghc.haskell.org/trac/ghc/ticket/4413#comment:7 GHC http://www.haskell.org/ghc/ The Glasgow Haskell Compiler