{-# OPTIONS_GHC -Wall -fwarn-tabs #-} -- Unfortunately we need -fglasgow-exts in order to actually pick -- up on the rules (see -ddump-rules). The -frewrite-rules flag -- doesn't do what you want. -- <http://hackage.haskell.org/trac/ghc/ticket/2213> -- <http://www.mail-archive.com/glasgow-haskell-users@haskell.org/msg14313.html> {-# OPTIONS_GHC -fglasgow-exts #-} ---------------------------------------------------------------- -- ~ 2009.03.09 -- | -- Module : Data.Number.Transfinite -- Copyright : Copyright (c) 2007--2010 wren ng thornton -- License : BSD3 -- Maintainer : wren@community.haskell.org -- Stability : stable -- Portability : portable -- -- This module presents a type class for numbers which have -- representations for transfinite values. The idea originated from -- the IEEE-754 floating-point special values, used by -- "Data.Number.LogFloat". However not all 'Fractional' types -- necessarily support transfinite values. In particular, @Ratio@ -- types including 'Rational' do not have portable representations. -- -- For the Glasgow compiler (GHC 6.8.2), "GHC.Real" defines @1%0@ -- and @0%0@ as representations for 'infinity' and 'notANumber', -- but most operations on them will raise exceptions. If 'toRational' -- is used on an infinite floating value, the result is a rational -- with a numerator sufficiently large that it will overflow when -- converted back to a @Double@. If used on NaN, the result would -- buggily convert back as 'negativeInfinity'. For more discussion -- on why this approach is problematic, see: -- -- * <http://www.haskell.org/pipermail/haskell-prime/2006-February/000791.html> -- -- * <http://www.haskell.org/pipermail/haskell-prime/2006-February/000821.html> -- -- Hugs (September 2006) stays closer to the haskell98 spec and -- offers no way of constructing those values, raising arithmetic -- overflow errors if attempted. ---------------------------------------------------------------- module Data.Number.Transfinite ( Transfinite(..) , log ) where import Prelude hiding (log, isInfinite, isNaN) import qualified Prelude (log) import qualified Hugs.RealFloat as Prelude (isInfinite, isNaN) import Data.Number.PartialOrd ---------------------------------------------------------------- -- | Many numbers are not 'Bounded' yet, even though they can -- represent arbitrarily large values, they are not necessarily -- able to represent transfinite values such as infinity itself. -- This class is for types which are capable of representing such -- values. Notably, this class does not require the type to be -- 'Fractional' nor 'Floating' since integral types could also have -- representations for transfinite values. By popular demand the -- 'Num' restriction has been lifted as well, due to complications -- of defining 'Show' or 'Eq' for some types. -- -- In particular, this class extends the ordered projection to have -- a maximum value 'infinity' and a minimum value 'negativeInfinity', -- as well as an exceptional value 'notANumber'. All the natural -- laws regarding @infinity@ and @negativeInfinity@ should pertain. -- (Some of these are discussed below.) -- -- Hugs (September 2006) has buggy Prelude definitions for -- 'Prelude.isNaN' and 'Prelude.isInfinite' on Float and Double. -- This module provides correct definitions, so long as "Hugs.RealFloat" -- is compiled correctly. class (PartialOrd a) => Transfinite a where -- | A transfinite value which is greater than all finite values. -- Adding or subtracting any finite value is a no-op. As is -- multiplying by any non-zero positive value (including -- @infinity@), and dividing by any positive finite value. Also -- obeys the law @negate infinity = negativeInfinity@ with all -- appropriate ramifications. infinity :: a -- | A transfinite value which is less than all finite values. -- Obeys all the same laws as @infinity@ with the appropriate -- changes for the sign difference. negativeInfinity :: a -- | An exceptional transfinite value for dealing with undefined -- results when manipulating infinite values. The following -- operations must return @notANumber@, where @inf@ is any value -- which @isInfinite@: -- -- * @infinity + negativeInfinity@ -- -- * @negativeInfinity + infinity@ -- -- * @infinity - infinity@ -- -- * @negativeInfinity - negativeInfinity@ -- -- * @inf * 0@ -- -- * @0 * inf@ -- -- * @inf \/ inf@ -- -- * @inf `div` inf@ -- -- * @0 \/ 0@ -- -- * @0 `div` 0@ -- -- Additionally, any mathematical operations on @notANumber@ -- must also return @notANumber@, and any equality or ordering -- comparison on @notANumber@ must return @False@ (violating -- the law of the excluded middle, often assumed but not required -- for 'Eq'; thus, 'eq' and 'ne' are preferred over ('==') and -- ('/=')). Since it returns false for equality, there may be -- more than one machine representation of this `value'. notANumber :: a -- | Return true for both @infinity@ and @negativeInfinity@, -- false for all other values. isInfinite :: a -> Bool -- | Return true only for @notANumber@. isNaN :: a -> Bool instance Transfinite Double where infinity = 1/0 negativeInfinity = negate (1/0) notANumber = 0/0 isInfinite = Prelude.isInfinite isNaN = Prelude.isNaN instance Transfinite Float where infinity = 1/0 negativeInfinity = negate (1/0) notANumber = 0/0 isInfinite = Prelude.isInfinite isNaN = Prelude.isNaN ---------------------------------------------------------------- -- | Since the normal 'Prelude.log' throws an error on zero, we -- have to redefine it in order for things to work right. Arguing -- from limits we can see that @log 0 == negativeInfinity@. Newer -- versions of GHC have this behavior already, but older versions -- and Hugs do not. -- -- This function will raise an error when taking the log of negative -- numbers, rather than returning 'notANumber' as the newer GHC -- implementation does. The reason being that typically this is a -- logical error, and @notANumber@ allows the error to propegate -- silently. -- -- In order to improve portability, the 'Transfinite' class is -- required to indicate that the 'Floating' type does in fact have -- a representation for negative infinity. Both native floating -- types ('Double' and 'Float') are supported. If you define your -- own instance of @Transfinite@, verify the above equation holds -- for your @0@ and @negativeInfinity@. If it doesn't, then you -- should avoid importing our @log@ and will probably want converters -- to handle the discrepancy. -- -- For GHC, this version of @log@ has rules for fusion with @exp@. -- These can give different behavior by preventing overflow to -- @infinity@ and preventing errors for taking the logarithm of -- negative values. For 'Double' and 'Float' they can also give -- different answers due to eliminating floating point fuzz. The -- rules strictly improve mathematical accuracy, however they should -- be noted in case your code depends on the implementation details. log :: (Floating a, Transfinite a) => a -> a {-# SPECIALIZE log :: Double -> Double #-} {-# SPECIALIZE log :: Float -> Float #-} log x = case x `cmp` 0 of Just GT -> Prelude.log x Just EQ -> negativeInfinity Just LT -> err "argument out of range" Nothing -> err "argument not comparable to 0" where err e = error $! "Data.Number.Transfinite.log: "++e -- Note, Floating ultimately requires Num, but not Ord. If PartialOrd -- proves to be an onerous requirement on Transfinite, we could -- hack our way around without using PartialOrd by using isNaN, (== -- 0), ((>0).signum) but that would be less efficient. ---------------------------------------------------------------- -- These rules moved here from "LogFloat" in v0.11.2 {-# RULES "log/exp" forall x. log (exp x) = x "log.exp" log . exp = id "exp/log" forall x. exp (log x) = x "exp.log" exp . log = id #-} -- We'd like to be able to take advantage of general rule versions -- of our operators for 'LogFloat', with rules like @log x + log y -- = log (x * y)@ and @log x - log y = log (x / y)@. However the -- problem is that those equations could be driven in either direction -- depending on whether we think time performance or non-underflow -- performance is more important, and the answers may be different -- at every call site. -- -- Since we implore users to do normal-domain computations whenever -- it would not degenerate accuracy, we should not rewrite their -- decisions in any way. The log\/exp fusion strictly improves both -- time and accuracy, so those are safe. But the buck stops with -- them. ---------------------------------------------------------------- ----------------------------------------------------------- fin.