-- TODO: Add QuickCheck-ness, though beware of the fuzz. -- TODO: Make sure rewrite rules really fire -- TODO: profile to make sure we don't waste too much time constructing -- dictionaries -- TODO: investigate adding strictness annotations for register -- unboxing -- TODO: write the signed variant -- -- To turn on optimizations and look at the optimization records, cf: -- http://www.haskell.org/ghc/docs/latest/html/users_guide/rewrite-rules.html -- http://www.randomhacks.net/articles/2007/02/10/map-fusion-and-haskell-performance -- {-# OPTIONS_GHC -ddump-simpl-stats #-} {-# OPTIONS_GHC -Wall -Werror #-} {-# OPTIONS_GHC -O2 -fvia-C -optc-O3 #-} -- Version History -- (v0.8.5) Gave up and converted from lhs to hs so Hackage docs work -- (v0.8.4) Broke out Transfinite -- (v0.8.3) Documentation updates -- (v0.8.2) Announced release -- (v0.8) Did a bunch of tweaking. Things should be decent now -- (v0.7) Haddockified -- (v0.6) Fixed monomorphism. -- (v0.5) Added optimization rules. -- (v0.4) Translated to Haskell at revision 2007.12.20. -- (v0.3) Converted extensive comments to POD format. -- (v0.2) Did a bunch of profiling, optimizing, and debugging. -- (v0.1) Initial version created for hw5 for NLP with Jason Eisner. -- ---------------------------------------------------------------- -- ~ 2008.08.16 -- | -- Module : Data.Number.LogFloat -- Copyright : Copyright (c) 2007--2008 wren ng thornton -- License : BSD3 -- Maintainer : wren@community.haskell.org -- Stability : stable -- Portability : portable -- -- This module presents a type for storing numbers in the log-domain. -- The main reason for doing this is to prevent underflow when -- multiplying many small probabilities as is done in Hidden Markov -- Models and other statistical models often used for natural -- language processing. The log-domain also helps prevent overflow -- when multiplying many large numbers. In rare cases it can speed -- up numerical computation (since addition is faster than -- multiplication, though logarithms are exceptionally slow), but -- the primary goal is to improve accuracy of results. A secondary -- goal has been to maximize efficiency since these computations -- are frequently done within a /O(n^3)/ loop. -- -- The 'LogFloat' of this module is restricted to non-negative -- numbers for efficiency's sake, see the forthcoming -- "Data.Number.LogFloat.Signed" for doing signed log-domain -- calculations. ---------------------------------------------------------------- module Data.Number.LogFloat ( -- * Basic functions log, toFractional -- * @LogFloat@ data type and conversion functions , LogFloat , logFloat, logToLogFloat , fromLogFloat, logFromLogFloat -- * Exceptional numeric values , module Data.Number.Transfinite ) where import Prelude hiding (log, isNaN) import qualified Prelude (log, isNaN) import Data.Number.Transfinite ---------------------------------------------------------------- -- -- Try to add in some optimizations. Why the first few need to be -- down here and localized to the module, I don't know. We don't -- do anything foolish like this, but our clients might, or they -- might be generated by other code transformations. Note that due -- to the fuzz, these equations are not actually true, even though -- they are mathematically correct. {-# 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 #-} -- These are general rule versions of our operators for 'LogFloat'. -- I had some issues inducing 'Ord' on @x@ and @y@, even though -- they're 'Num' so I can't do "(+)\/log" and "(-)\/log" so easily. {-# RULES "(*)/log" forall x y. log x * log y = log (x + y) "(/)/log" forall x y. log x / log y = log (x - y) #-} ---------------------------------------------------------------- -- -- | 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 when dealing with @LogFloat@s. {-# SPECIALIZE log :: Double -> Double #-} log :: (Floating a, Transfinite a) => a -> a log x = case compare x 0 of GT -> Prelude.log x EQ -> negativeInfinity LT -> errorOutOfRange "log" -- | The most generic numeric converter I can come up with. All the -- built-in numeric types are 'Real', though 'Int' and 'Integer' -- aren't 'Fractional'. Beware that converting transfinite values -- into @Ratio@ types is error-prone and non-portable, as discussed -- in "Data.Number.Transfinite". {-# SPECIALIZE toFractional :: (Real a) => a -> Double #-} {-# SPECIALIZE toFractional :: (Fractional b) => Double -> b #-} toFractional :: (Real a, Fractional b) => a -> b toFractional = fromRational . toRational -- This should only fire when it's type-safe {-# RULES "toFractional/id" toFractional = id #-} -- This should happen already, but who knows -- TODO: see if it ever fires {-# RULES "toFractional/toFractional" forall x. toFractional (toFractional x) = toFractional x "toFractional.toFractional" toFractional . toFractional = toFractional #-} ---------------------------------------------------------------- -- -- | Reduce the number of constant string literals we need to store. errorOutOfRange :: String -> a errorOutOfRange fun = error $ "Data.Number.LogFloat."++fun ++ ": argument out of range" -- | We need these guards in order to ensure some invariants. guardNonNegative :: String -> Double -> Double guardNonNegative fun x | x >= 0 = x | otherwise = errorOutOfRange fun -- | It's unfortunate that notANumber is not equal to itself, but -- we can hack around that. GHC gives NaN for the log of negatives -- and so we could ideally take advantage of @log . guardNonNegative -- fun = guardIsANumber fun . log@ to simplify things, but Hugs -- raises an error so that's non-portable. guardIsANumber :: String -> Double -> Double guardIsANumber fun x | Prelude.isNaN x = errorOutOfRange fun | otherwise = x ---------------------------------------------------------------- -- -- | A @LogFloat@ is just a 'Double' with a special interpretation. -- The 'logFloat' function is presented instead of the constructor, -- in order to ensure semantic conversion. At present the 'Show' -- instance will convert back to the normal-domain, and so will -- underflow at that point. This behavior may change in the future. -- -- Performing operations in the log-domain is cheap, prevents -- underflow, and is otherwise very nice for dealing with miniscule -- probabilities. However, crossing into and out of the log-domain -- is expensive and should be avoided as much as possible. In -- particular, if you're doing a series of multiplications as in -- @lp * logFloat q * logFloat r@ it's faster to do @lp * logFloat -- (q * r)@ if you're reasonably sure the normal-domain multiplication -- won't underflow, because that way you enter the log-domain only -- once, instead of twice. -- -- Even more particularly, you should /avoid addition/ whenever -- possible. Addition is provided because it's necessary at times -- and the proper implementation is not immediately transparent. -- However, between two @LogFloat@s addition requires crossing the -- exp\/log boundary twice; with a @LogFloat@ and a regular number -- it's three times since the regular number needs to enter the -- log-domain first. This makes addition incredibly slow. Again, -- if you can parenthesize to do plain operations first, do it! newtype LogFloat = LogFloat Double deriving (Eq, Ord) -- | A constructor which does semantic conversion from normal-domain -- to log-domain. {-# SPECIALIZE logFloat :: Double -> LogFloat #-} logFloat :: (Real a) => a -> LogFloat logFloat = LogFloat . log . guardNonNegative "logFloat" . toFractional -- This is simply a polymorphic version of the 'LogFloat' data -- constructor. We present it mainly because we hide the constructor -- in order to make the type a bit more opaque. If the polymorphism -- turns out to be a performance liability because the rewrite rules -- can't remove it, then we need to rethink all four -- constructors\/destructors. -- -- | Constructor which assumes the argument is already in the -- log-domain. {-# SPECIALIZE logToLogFloat :: Double -> LogFloat #-} logToLogFloat :: (Real a) => a -> LogFloat logToLogFloat = LogFloat . guardIsANumber "logToLogFloat" . toFractional -- | Return our log-domain value back into normal-domain. Beware -- of overflow\/underflow. {-# SPECIALIZE fromLogFloat :: LogFloat -> Double #-} fromLogFloat :: (Fractional a, Transfinite a) => LogFloat -> a fromLogFloat (LogFloat x) = toFractional (exp x) -- | Return the log-domain value itself without costly conversion {-# SPECIALIZE logFromLogFloat :: LogFloat -> Double #-} logFromLogFloat :: (Fractional a, Transfinite a) => LogFloat -> a logFromLogFloat (LogFloat x) = toFractional x -- These are our module-specific versions of "log\/exp" and "exp\/log"; -- They do the same things but also have a @LogFloat@ in between -- the logarithm and exponentiation. {-# RULES -- Out of log-domain and back in "log/fromLogFloat" forall x. log (fromLogFloat x) = logFromLogFloat x "log.fromLogFloat" log . fromLogFloat = logFromLogFloat "logFloat/fromLogFloat" forall x. logFloat (fromLogFloat x) = x "logFloat.fromLogFloat" logFloat . fromLogFloat = id -- Into log-domain and back out "fromLogFloat/logFloat" forall x. fromLogFloat (logFloat x) = x "fromLogFloat.logFloat" fromLogFloat . logFloat = id #-} ---------------------------------------------------------------- -- To show it, we want to show the normal-domain value rather than -- the log-domain value. Also, if someone managed to break our -- invariants (e.g. by passing in a negative and noone's pulled on -- the thunk yet) then we want to crash before printing the -- constructor, rather than after. N.B. This means the show will -- underflow\/overflow in the same places as normal doubles since -- we underflow at the @exp@. Perhaps this means we should show the -- log-domain value instead. instance Show LogFloat where show (LogFloat x) = let y = exp x in y `seq` "LogFloat "++show y ---------------------------------------------------------------- -- These all work without causing underflow. However, do note that -- they tend to induce more of the floating-point fuzz than using -- regular floating numbers because @exp . log@ doesn't really equal -- @id@. In any case, our main aim is for preventing underflow when -- multiplying many small numbers (and preventing overflow for -- multiplying many large numbers) so we're not too worried about -- +\/- 4e-16. instance Num LogFloat where (*) (LogFloat x) (LogFloat y) = LogFloat (x+y) (+) (LogFloat x) (LogFloat y) | x >= y = LogFloat (x + log (1 + exp (y - x))) | otherwise = LogFloat (y + log (1 + exp (x - y))) -- Without the guard this would return NaN instead of error (-) (LogFloat x) (LogFloat y) | x >= y = LogFloat (x + log (1 - exp (y - x))) | otherwise = errorOutOfRange "(-)" signum (LogFloat x) | x == negativeInfinity = 0 | x > negativeInfinity = 1 | otherwise = errorOutOfRange "signum" -- The extra guard protects against NaN, in case someone -- broke the invariant. That shouldn't be possible and -- so noone else bothers to check, but we check here just -- in case. negate _ = errorOutOfRange "negate" abs = id fromInteger = LogFloat . log . guardNonNegative "fromInteger" . fromInteger instance Fractional LogFloat where -- n/0 is handled seamlessly for us; we must catch 0/0 though (/) (LogFloat x) (LogFloat y) | x == negativeInfinity && y == negativeInfinity = errorOutOfRange "(/)" -- protect vs NaN | otherwise = LogFloat (x-y) fromRational = LogFloat . log . guardNonNegative "fromRational" . fromRational -- Just for fun. The more coersion functions the better. Though -- it can underflow... instance Real LogFloat where toRational (LogFloat x) = toRational (exp x) ---------------------------------------------------------------- ----------------------------------------------------------- fin.