    \begin{code}
    {-# OPTIONS_GHC -fno-implicit-prelude #-}
    -----------------------------------------------------------------------------
    -- |
    -- Module      :  GHC.Float
    -- Copyright   :  (c) The University of Glasgow 1994-2002
    -- License     :  see libraries/base/LICENSE
    --
    -- Maintainer  :  cvs-ghc@haskell.org
    -- Stability   :  internal
    -- Portability :  non-portable (GHC Extensions)
    --
    -- The types 'Float' and 'Double', and the classes 'Floating' and 'RealFloat'.
    --
    -----------------------------------------------------------------------------

    #include "ieee-flpt.h"

    -- #hide
    module GHC.Float( module GHC.Float, Float#, Double# )  where

    import Data.Maybe

    import GHC.Base
    import GHC.List
    import GHC.Enum
    import GHC.Show
    import GHC.Num
    import GHC.Real
    import GHC.Arr

    infixr 8  **
    \end{code}

    %*********************************************************
    %*                                                      *
    \subsection{Standard numeric classes}
    %*                                                      *
    %*********************************************************

    \begin{code}
    -- | Trigonometric and hyperbolic functions and related functions.
    --
    -- Minimal complete definition:
    --      'pi', 'exp', 'log', 'sin', 'cos', 'sinh', 'cosh',
    --      'asin', 'acos', 'atan', 'asinh', 'acosh' and 'atanh'
    class  (Fractional a) => Floating a  where
        pi                  :: a
        exp, log, sqrt      :: a -> a
        (**), logBase       :: a -> a -> a
        sin, cos, tan       :: a -> a
        asin, acos, atan    :: a -> a
        sinh, cosh, tanh    :: a -> a
        asinh, acosh, atanh :: a -> a

        x ** y              =  exp (log x * y)
        logBase x y         =  log y / log x
        sqrt x              =  x ** 0.5
        tan  x              =  sin  x / cos  x
        tanh x              =  sinh x / cosh x

    -- | Efficient, machine-independent access to the components of a
    -- floating-point number.
    --
    -- Minimal complete definition:
    --      all except 'exponent', 'significand', 'scaleFloat' and 'atan2'
    class  (RealFrac a, Floating a) => RealFloat a  where
        -- | a constant function, returning the radix of the representation
        -- (often @2@)
        floatRadix          :: a -> Integer
        -- | a constant function, returning the number of digits of
        -- 'floatRadix' in the significand
        floatDigits         :: a -> Int
        -- | a constant function, returning the lowest and highest values
        -- the exponent may assume
        floatRange          :: a -> (Int,Int)
        -- | The function 'decodeFloat' applied to a real floating-point
        -- number returns the significand expressed as an 'Integer' and an
        -- appropriately scaled exponent (an 'Int').  If @'decodeFloat' x@
        -- yields @(m,n)@, then @x@ is equal in value to @m*b^^n@, where @b@
        -- is the floating-point radix, and furthermore, either @m@ and @n@
        -- are both zero or else @b^(d-1) <= m < b^d@, where @d@ is the value
        -- of @'floatDigits' x@.  In particular, @'decodeFloat' 0 = (0,0)@.
        decodeFloat         :: a -> (Integer,Int)
        -- | 'encodeFloat' performs the inverse of 'decodeFloat'
        encodeFloat         :: Integer -> Int -> a
        -- | the second component of 'decodeFloat'.
        exponent            :: a -> Int
        -- | the first component of 'decodeFloat', scaled to lie in the open
        -- interval (@-1@,@1@)
        significand         :: a -> a
        -- | multiplies a floating-point number by an integer power of the radix
        scaleFloat          :: Int -> a -> a
        -- | 'True' if the argument is an IEEE \"not-a-number\" (NaN) value
        isNaN               :: a -> Bool
        -- | 'True' if the argument is an IEEE infinity or negative infinity
        isInfinite          :: a -> Bool
        -- | 'True' if the argument is too small to be represented in
        -- normalized format
        isDenormalized      :: a -> Bool
        -- | 'True' if the argument is an IEEE negative zero
        isNegativeZero      :: a -> Bool
        -- | 'True' if the argument is an IEEE floating point number
        isIEEE              :: a -> Bool
        -- | a version of arctangent taking two real floating-point arguments.
        -- For real floating @x@ and @y@, @'atan2' y x@ computes the angle
        -- (from the positive x-axis) of the vector from the origin to the
        -- point @(x,y)@.  @'atan2' y x@ returns a value in the range [@-pi@,
        -- @pi@].  It follows the Common Lisp semantics for the origin when
        -- signed zeroes are supported.  @'atan2' y 1@, with @y@ in a type
        -- that is 'RealFloat', should return the same value as @'atan' y@.
        -- A default definition of 'atan2' is provided, but implementors
        -- can provide a more accurate implementation.
        atan2               :: a -> a -> a

        exponent x          =  if m == 0 then 0 else n + floatDigits x
                               where (m,n) = decodeFloat x

        significand x       =  encodeFloat m (negate (floatDigits x))
                               where (m,_) = decodeFloat x

        scaleFloat k x      =  encodeFloat m (n+k)
                               where (m,n) = decodeFloat x

        atan2 y x
          | x > 0            =  atan (y/x)
          | x == 0 && y > 0  =  pi/2
          | x <  0 && y > 0  =  pi + atan (y/x)
          |(x <= 0 && y < 0)            ||
           (x <  0 && isNegativeZero y) ||
           (isNegativeZero x && isNegativeZero y)
                             = -atan2 (-y) x
          | y == 0 && (x < 0 || isNegativeZero x)
                              =  pi    -- must be after the previous test on zero y
          | x==0 && y==0      =  y     -- must be after the other double zero tests
          | otherwise         =  x + y -- x or y is a NaN, return a NaN (via +)
    \end{code}

    %*********************************************************
    %*                                                      *
    \subsection{Type @Integer@, @Float@, @Double@}
    %*                                                      *
    %*********************************************************

    \begin{code}
    -- | Single-precision floating point numbers.
    -- It is desirable that this type be at least equal in range and precision
    -- to the IEEE single-precision type.
    data Float      = F# Float#

    -- | Double-precision floating point numbers.
    -- It is desirable that this type be at least equal in range and precision
    -- to the IEEE double-precision type.
    data Double     = D# Double#
    \end{code}

    %*********************************************************
    %*                                                      *
    \subsection{Type @Float@}
    %*                                                      *
    %*********************************************************

    \begin{code}
    instance Eq Float where
        (F# x) == (F# y) = x `eqFloat#` y

    instance Ord Float where
        (F# x) `compare` (F# y) | x `ltFloat#` y = LT
                                | x `eqFloat#` y = EQ
                                | otherwise      = GT

        (F# x) <  (F# y) = x `ltFloat#`  y
        (F# x) <= (F# y) = x `leFloat#`  y
        (F# x) >= (F# y) = x `geFloat#`  y
        (F# x) >  (F# y) = x `gtFloat#`  y

    instance  Num Float  where
        (+)         x y     =  plusFloat x y
        (-)         x y     =  minusFloat x y
        negate      x       =  negateFloat x
        (*)         x y     =  timesFloat x y
        abs x | x >= 0.0    =  x
              | otherwise   =  negateFloat x
        signum x | x == 0.0  = 0
                 | x > 0.0   = 1
                 | otherwise = negate 1

        {-# INLINE fromInteger #-}
        fromInteger (S# i#)    = case (int2Float# i#) of { d# -> F# d# }
        fromInteger (J# s# d#) = encodeFloat# s# d# 0
            -- previous code: fromInteger n = encodeFloat n 0
            -- doesn't work too well, because encodeFloat is defined in
            -- terms of ccalls which can never be simplified away.  We
            -- want simple literals like (fromInteger 3 :: Float) to turn
            -- into (F# 3.0), hence the special case for S# here.

    instance  Real Float  where
        toRational x        =  (m%1)*(b%1)^^n
                               where (m,n) = decodeFloat x
                                     b     = floatRadix  x

    instance  Fractional Float  where
        (/) x y             =  divideFloat x y
        fromRational x      =  fromRat x
        recip x             =  1.0 / x

    {-# RULES "truncate/Float->Int" truncate = float2Int #-}
    instance  RealFrac Float  where

        {-# SPECIALIZE properFraction :: Float -> (Int, Float) #-}
        {-# SPECIALIZE round    :: Float -> Int #-}

        {-# SPECIALIZE properFraction :: Float  -> (Integer, Float) #-}
        {-# SPECIALIZE round    :: Float -> Integer #-}

            -- ceiling, floor, and truncate are all small
        {-# INLINE ceiling #-}
        {-# INLINE floor #-}
        {-# INLINE truncate #-}

        properFraction x
          = case (decodeFloat x)      of { (m,n) ->
            let  b = floatRadix x     in
            if n >= 0 then
                (fromInteger m * fromInteger b ^ n, 0.0)
            else
                case (quotRem m (b^(negate n))) of { (w,r) ->
                (fromInteger w, encodeFloat r n)
                }
            }

        truncate x  = case properFraction x of
                         (n,_) -> n

        round x     = case properFraction x of
                         (n,r) -> let
                                    m         = if r < 0.0 then n - 1 else n + 1
                                    half_down = abs r - 0.5
                                  in
                                  case (compare half_down 0.0) of
                                    LT -> n
                                    EQ -> if even n then n else m
                                    GT -> m

        ceiling x   = case properFraction x of
                        (n,r) -> if r > 0.0 then n + 1 else n

        floor x     = case properFraction x of
                        (n,r) -> if r < 0.0 then n - 1 else n

    instance  Floating Float  where
        pi                  =  3.141592653589793238
        exp x               =  expFloat x
        log x               =  logFloat x
        sqrt x              =  sqrtFloat x
        sin x               =  sinFloat x
        cos x               =  cosFloat x
        tan x               =  tanFloat x
        asin x              =  asinFloat x
        acos x              =  acosFloat x
        atan x              =  atanFloat x
        sinh x              =  sinhFloat x
        cosh x              =  coshFloat x
        tanh x              =  tanhFloat x
        (**) x y            =  powerFloat x y
        logBase x y         =  log y / log x

        asinh x = log (x + sqrt (1.0+x*x))
        acosh x = log (x + (x+1.0) * sqrt ((x-1.0)/(x+1.0)))
        atanh x = log ((x+1.0) / sqrt (1.0-x*x))

    instance  RealFloat Float  where
        floatRadix _        =  FLT_RADIX        -- from float.h
        floatDigits _       =  FLT_MANT_DIG     -- ditto
        floatRange _        =  (FLT_MIN_EXP, FLT_MAX_EXP) -- ditto

        decodeFloat (F# f#)
          = case decodeFloat# f#    of
              (# exp#, s#, d# #) -> (J# s# d#, I# exp#)

        encodeFloat (S# i) j     = int_encodeFloat# i j
        encodeFloat (J# s# d#) e = encodeFloat# s# d# e

        exponent x          = case decodeFloat x of
                                (m,n) -> if m == 0 then 0 else n + floatDigits x

        significand x       = case decodeFloat x of
                                (m,_) -> encodeFloat m (negate (floatDigits x))

        scaleFloat k x      = case decodeFloat x of
                                (m,n) -> encodeFloat m (n+k)
        isNaN x          = 0 /= isFloatNaN x
        isInfinite x     = 0 /= isFloatInfinite x
        isDenormalized x = 0 /= isFloatDenormalized x
        isNegativeZero x = 0 /= isFloatNegativeZero x
        isIEEE _         = True

    instance  Show Float  where
        showsPrec   x = showSignedFloat showFloat x
        showList = showList__ (showsPrec 0)
    \end{code}

    %*********************************************************
    %*                                                      *
    \subsection{Type @Double@}
    %*                                                      *
    %*********************************************************

    \begin{code}
    instance Eq Double where
        (D# x) == (D# y) = x ==## y

    instance Ord Double where
        (D# x) `compare` (D# y) | x <## y   = LT
                                | x ==## y  = EQ
                                | otherwise = GT

        (D# x) <  (D# y) = x <##  y
        (D# x) <= (D# y) = x <=## y
        (D# x) >= (D# y) = x >=## y
        (D# x) >  (D# y) = x >##  y

    instance  Num Double  where
        (+)         x y     =  plusDouble x y
        (-)         x y     =  minusDouble x y
        negate      x       =  negateDouble x
        (*)         x y     =  timesDouble x y
        abs x | x >= 0.0    =  x
              | otherwise   =  negateDouble x
        signum x | x == 0.0  = 0
                 | x > 0.0   = 1
                 | otherwise = negate 1

        {-# INLINE fromInteger #-}
            -- See comments with Num Float
        fromInteger (S# i#)    = case (int2Double# i#) of { d# -> D# d# }
        fromInteger (J# s# d#) = encodeDouble# s# d# 0

    instance  Real Double  where
        toRational x        =  (m%1)*(b%1)^^n
                               where (m,n) = decodeFloat x
                                     b     = floatRadix  x

    instance  Fractional Double  where
        (/) x y             =  divideDouble x y
        fromRational x      =  fromRat x
        recip x             =  1.0 / x

    instance  Floating Double  where
        pi                  =  3.141592653589793238
        exp x               =  expDouble x
        log x               =  logDouble x
        sqrt x              =  sqrtDouble x
        sin  x              =  sinDouble x
        cos  x              =  cosDouble x
        tan  x              =  tanDouble x
        asin x              =  asinDouble x
        acos x              =  acosDouble x
        atan x              =  atanDouble x
        sinh x              =  sinhDouble x
        cosh x              =  coshDouble x
        tanh x              =  tanhDouble x
        (**) x y            =  powerDouble x y
        logBase x y         =  log y / log x

        asinh x = log (x + sqrt (1.0+x*x))
        acosh x = log (x + (x+1.0) * sqrt ((x-1.0)/(x+1.0)))
        atanh x = log ((x+1.0) / sqrt (1.0-x*x))

    {-# RULES "truncate/Double->Int" truncate = double2Int #-}
    instance  RealFrac Double  where

        {-# SPECIALIZE properFraction :: Double -> (Int, Double) #-}
        {-# SPECIALIZE round    :: Double -> Int #-}

        {-# SPECIALIZE properFraction :: Double -> (Integer, Double) #-}
        {-# SPECIALIZE round    :: Double -> Integer #-}

            -- ceiling, floor, and truncate are all small
        {-# INLINE ceiling #-}
        {-# INLINE floor #-}
        {-# INLINE truncate #-}

        properFraction x
          = case (decodeFloat x)      of { (m,n) ->
            let  b = floatRadix x     in
            if n >= 0 then
                (fromInteger m * fromInteger b ^ n, 0.0)
            else
                case (quotRem m (b^(negate n))) of { (w,r) ->
                (fromInteger w, encodeFloat r n)
                }
            }

        truncate x  = case properFraction x of
                         (n,_) -> n

        round x     = case properFraction x of
                         (n,r) -> let
                                    m         = if r < 0.0 then n - 1 else n + 1
                                    half_down = abs r - 0.5
                                  in
                                  case (compare half_down 0.0) of
                                    LT -> n
                                    EQ -> if even n then n else m
                                    GT -> m

        ceiling x   = case properFraction x of
                        (n,r) -> if r > 0.0 then n + 1 else n

        floor x     = case properFraction x of
                        (n,r) -> if r < 0.0 then n - 1 else n

    instance  RealFloat Double  where
        floatRadix _        =  FLT_RADIX        -- from float.h
        floatDigits _       =  DBL_MANT_DIG     -- ditto
        floatRange _        =  (DBL_MIN_EXP, DBL_MAX_EXP) -- ditto

        decodeFloat (D# x#)
          = case decodeDouble# x#   of
              (# exp#, s#, d# #) -> (J# s# d#, I# exp#)

        encodeFloat (S# i) j     = int_encodeDouble# i j
        encodeFloat (J# s# d#) e = encodeDouble# s# d# e

        exponent x          = case decodeFloat x of
                                (m,n) -> if m == 0 then 0 else n + floatDigits x

        significand x       = case decodeFloat x of
                                (m,_) -> encodeFloat m (negate (floatDigits x))

        scaleFloat k x      = case decodeFloat x of
                                (m,n) -> encodeFloat m (n+k)

        isNaN x             = 0 /= isDoubleNaN x
        isInfinite x        = 0 /= isDoubleInfinite x
        isDenormalized x    = 0 /= isDoubleDenormalized x
        isNegativeZero x    = 0 /= isDoubleNegativeZero x
        isIEEE _            = True

    instance  Show Double  where
        showsPrec   x = showSignedFloat showFloat x
        showList = showList__ (showsPrec 0)
    \end{code}

    %*********************************************************
    %*                                                      *
    \subsection{@Enum@ instances}
    %*                                                      *
    %*********************************************************

    The @Enum@ instances for Floats and Doubles are slightly unusual.
    The @toEnum@ function truncates numbers to Int.  The definitions
    of @enumFrom@ and @enumFromThen@ allow floats to be used in arithmetic
    series: [0,0.1 .. 1.0].  However, roundoff errors make these somewhat
    dubious.  This example may have either 10 or 11 elements, depending on
    how 0.1 is represented.

    NOTE: The instances for Float and Double do not make use of the default
    methods for @enumFromTo@ and @enumFromThenTo@, as these rely on there being
    a `non-lossy' conversion to and from Ints. Instead we make use of the
    1.2 default methods (back in the days when Enum had Ord as a superclass)
    for these (@numericEnumFromTo@ and @numericEnumFromThenTo@ below.)

    \begin{code}
    instance  Enum Float  where
        succ x         = x + 1
        pred x         = x - 1
        toEnum         = int2Float
        fromEnum       = fromInteger . truncate   -- may overflow
        enumFrom       = numericEnumFrom
        enumFromTo     = numericEnumFromTo
        enumFromThen   = numericEnumFromThen
        enumFromThenTo = numericEnumFromThenTo

    instance  Enum Double  where
        succ x         = x + 1
        pred x         = x - 1
        toEnum         =  int2Double
        fromEnum       =  fromInteger . truncate   -- may overflow
        enumFrom       =  numericEnumFrom
        enumFromTo     =  numericEnumFromTo
        enumFromThen   =  numericEnumFromThen
        enumFromThenTo =  numericEnumFromThenTo
    \end{code}

    %*********************************************************
    %*                                                      *
    \subsection{Printing floating point}
    %*                                                      *
    %*********************************************************

    \begin{code}
    -- | Show a signed 'RealFloat' value to full precision
    -- using standard decimal notation for arguments whose absolute value lies
    -- between @0.1@ and @9,999,999@, and scientific notation otherwise.
    showFloat :: (RealFloat a) => a -> ShowS
    showFloat x  =  showString (formatRealFloat FFGeneric Nothing x)

    -- These are the format types.  This type is not exported.

    data FFFormat = FFExponent | FFFixed | FFGeneric

    formatRealFloat :: (RealFloat a) => FFFormat -> Maybe Int -> a -> String
    formatRealFloat fmt decs x
       | isNaN x                   = "NaN"
       | isInfinite x              = if x < 0 then "-Infinity" else "Infinity"
       | x < 0 || isNegativeZero x = '-':doFmt fmt (floatToDigits (toInteger base) (-x))
       | otherwise                 = doFmt fmt (floatToDigits (toInteger base) x)
     where
      base = 10

      doFmt format (is, e) =
        let ds = map intToDigit is in
        case format of
         FFGeneric ->
          doFmt (if e < 0 || e > 7 then FFExponent else FFFixed)
                (is,e)
         FFExponent ->
          case decs of
           Nothing ->
            let show_e' = show (e-1) in
            case ds of
              "0"     -> "0.0e0"
              [d]     -> d : ".0e" ++ show_e'
              (d:ds') -> d : '.' : ds' ++ "e" ++ show_e'
           Just dec ->
            let dec' = max dec 1 in
            case is of
             [0] -> '0' :'.' : take dec' (repeat '0') ++ "e0"
             _ ->
              let
               (ei,is') = roundTo base (dec'+1) is
               (d:ds') = map intToDigit (if ei > 0 then init is' else is')
              in
              d:'.':ds' ++ 'e':show (e-1+ei)
         FFFixed ->
          let
           mk0 ls = case ls of { "" -> "0" ; _ -> ls}
          in
          case decs of
           Nothing
              | e <= 0    -> "0." ++ replicate (-e) '0' ++ ds
              | otherwise ->
                 let
                    f 0 s    rs  = mk0 (reverse s) ++ '.':mk0 rs
                    f n s    ""  = f (n-1) ('0':s) ""
                    f n s (r:rs) = f (n-1) (r:s) rs
                 in
                    f e "" ds
           Just dec ->
            let dec' = max dec 0 in
            if e >= 0 then
             let
              (ei,is') = roundTo base (dec' + e) is
              (ls,rs)  = splitAt (e+ei) (map intToDigit is')
             in
             mk0 ls ++ (if null rs then "" else '.':rs)
            else
             let
              (ei,is') = roundTo base dec' (replicate (-e) 0 ++ is)
              d:ds' = map intToDigit (if ei > 0 then is' else 0:is')
             in
             d : (if null ds' then "" else '.':ds')

    roundTo :: Int -> Int -> [Int] -> (Int,[Int])
    roundTo base d is =
      case f d is of
        x@(0,_) -> x
        (1,xs)  -> (1, 1:xs)
     where
      b2 = base `div` 2

      f n []     = (0, replicate n 0)
      f 0 (x:_)  = (if x >= b2 then 1 else 0, [])
      f n (i:xs)
         | i' == base = (1,0:ds)
         | otherwise  = (0,i':ds)
          where
           (c,ds) = f (n-1) xs
           i'     = c + i

    -- Based on "Printing Floating-Point Numbers Quickly and Accurately"
    -- by R.G. Burger and R.K. Dybvig in PLDI 96.
    -- This version uses a much slower logarithm estimator. It should be improved.

    -- | 'floatToDigits' takes a base and a non-negative 'RealFloat' number,
    -- and returns a list of digits and an exponent.
    -- In particular, if @x>=0@, and
    --
    -- > floatToDigits base x = ([d1,d2,...,dn], e)
    --
    -- then
    --
    --      (1) @n >= 1@
    --
    --      (2) @x = 0.d1d2...dn * (base**e)@
    --
    --      (3) @0 <= di <= base-1@

    floatToDigits :: (RealFloat a) => Integer -> a -> ([Int], Int)
    floatToDigits _ 0 = ([0], 0)
    floatToDigits base x =
     let
      (f0, e0) = decodeFloat x
      (minExp0, _) = floatRange x
      p = floatDigits x
      b = floatRadix x
      minExp = minExp0 - p -- the real minimum exponent
      -- Haskell requires that f be adjusted so denormalized numbers
      -- will have an impossibly low exponent.  Adjust for this.
      (f, e) =
       let n = minExp - e0 in
       if n > 0 then (f0 `div` (b^n), e0+n) else (f0, e0)
      (r, s, mUp, mDn) =
       if e >= 0 then
        let be = b^ e in
        if f == b^(p-1) then
          (f*be*b*2, 2*b, be*b, b)
        else
          (f*be*2, 2, be, be)
       else
        if e > minExp && f == b^(p-1) then
          (f*b*2, b^(-e+1)*2, b, 1)
        else
          (f*2, b^(-e)*2, 1, 1)
      k :: Int
      k =
       let
        k0 :: Int
        k0 =
         if b == 2 && base == 10 then
            -- logBase 10 2 is slightly bigger than 3/10 so
            -- the following will err on the low side.  Ignoring
            -- the fraction will make it err even more.
            -- Haskell promises that p-1 <= logBase b f < p.
            (p - 1 + e0) * 3 `div` 10
         else
            ceiling ((log (fromInteger (f+1)) +
                     fromInteger (int2Integer e) * log (fromInteger b)) /
                       log (fromInteger base))
    --WAS:            fromInt e * log (fromInteger b))

        fixup n =
          if n >= 0 then
            if r + mUp <= expt base n * s then n else fixup (n+1)
          else
            if expt base (-n) * (r + mUp) <= s then n else fixup (n+1)
       in
       fixup k0

      gen ds rn sN mUpN mDnN =
       let
        (dn, rn') = (rn * base) `divMod` sN
        mUpN' = mUpN * base
        mDnN' = mDnN * base
       in
       case (rn' < mDnN', rn' + mUpN' > sN) of
        (True,  False) -> dn : ds
        (False, True)  -> dn+1 : ds
        (True,  True)  -> if rn' * 2 < sN then dn : ds else dn+1 : ds
        (False, False) -> gen (dn:ds) rn' sN mUpN' mDnN'

      rds =
       if k >= 0 then
          gen [] r (s * expt base k) mUp mDn
       else
         let bk = expt base (-k) in
         gen [] (r * bk) s (mUp * bk) (mDn * bk)
     in
     (map fromIntegral (reverse rds), k)

    \end{code}

    %*********************************************************
    %*                                                      *
    \subsection{Converting from a Rational to a RealFloat
    %*                                                      *
    %*********************************************************

    [In response to a request for documentation of how fromRational works,
    Joe Fasel writes:] A quite reasonable request!  This code was added to
    the Prelude just before the 1.2 release, when Lennart, working with an
    early version of hbi, noticed that (read . show) was not the identity
    for floating-point numbers.  (There was a one-bit error about half the
    time.)  The original version of the conversion function was in fact
    simply a floating-point divide, as you suggest above. The new version
    is, I grant you, somewhat denser.

    Unfortunately, Joe's code doesn't work!  Here's an example:

    main = putStr (shows (1.82173691287639817263897126389712638972163e-300::Double) "\n")

    This program prints
            0.0000000000000000
    instead of
            1.8217369128763981e-300

    Here's Joe's code:

    \begin{pseudocode}
    fromRat :: (RealFloat a) => Rational -> a
    fromRat x = x'
            where x' = f e

    --              If the exponent of the nearest floating-point number to x
    --              is e, then the significand is the integer nearest xb^(-e),
    --              where b is the floating-point radix.  We start with a good
    --              guess for e, and if it is correct, the exponent of the
    --              floating-point number we construct will again be e.  If
    --              not, one more iteration is needed.

                  f e   = if e' == e then y else f e'
                          where y      = encodeFloat (round (x * (1 % b)^^e)) e
                                (_,e') = decodeFloat y
                  b     = floatRadix x'

    --              We obtain a trial exponent by doing a floating-point
    --              division of x's numerator by its denominator.  The
    --              result of this division may not itself be the ultimate
    --              result, because of an accumulation of three rounding
    --              errors.

                  (s,e) = decodeFloat (fromInteger (numerator x) `asTypeOf` x'
                                            / fromInteger (denominator x))
    \end{pseudocode}

    Now, here's Lennart's code (which works)

    \begin{code}
    -- | Converts a 'Rational' value into any type in class 'RealFloat'.
    {-# SPECIALISE fromRat :: Rational -> Double,
                              Rational -> Float #-}
    fromRat :: (RealFloat a) => Rational -> a

    -- Deal with special cases first, delegating the real work to fromRat'
    fromRat (n :% 0) | n > 0  =  1/0        -- +Infinity
                     | n == 0 =  0/0        -- NaN
                     | n < 0  = -1/0        -- -Infinity

    fromRat (n :% d) | n > 0  = fromRat' (n :% d)
                     | n == 0 = encodeFloat 0 0             -- Zero
                     | n < 0  = - fromRat' ((-n) :% d)

    -- Conversion process:
    -- Scale the rational number by the RealFloat base until
    -- it lies in the range of the mantissa (as used by decodeFloat/encodeFloat).
    -- Then round the rational to an Integer and encode it with the exponent
    -- that we got from the scaling.
    -- To speed up the scaling process we compute the log2 of the number to get
    -- a first guess of the exponent.

    fromRat' :: (RealFloat a) => Rational -> a
    -- Invariant: argument is strictly positive
    fromRat' x = r
      where b = floatRadix r
            p = floatDigits r
            (minExp0, _) = floatRange r
            minExp = minExp0 - p            -- the real minimum exponent
            xMin   = toRational (expt b (p-1))
            xMax   = toRational (expt b p)
            p0 = (integerLogBase b (numerator x) - integerLogBase b (denominator x) - p) `max` minExp
            f = if p0 < 0 then 1 % expt b (-p0) else expt b p0 % 1
            (x', p') = scaleRat (toRational b) minExp xMin xMax p0 (x / f)
            r = encodeFloat (round x') p'

    -- Scale x until xMin <= x < xMax, or p (the exponent) <= minExp.
    scaleRat :: Rational -> Int -> Rational -> Rational -> Int -> Rational -> (Rational, Int)
    scaleRat b minExp xMin xMax p x
     | p <= minExp = (x, p)
     | x >= xMax   = scaleRat b minExp xMin xMax (p+1) (x/b)
     | x < xMin    = scaleRat b minExp xMin xMax (p-1) (x*b)
     | otherwise   = (x, p)

    -- Exponentiation with a cache for the most common numbers.
    minExpt, maxExpt :: Int
    minExpt = 0
    maxExpt = 1100

    expt :: Integer -> Int -> Integer
    expt base n =
        if base == 2 && n >= minExpt && n <= maxExpt then
            expts!n
        else
            base^n

    expts :: Array Int Integer
    expts = array (minExpt,maxExpt) [(n,2^n) | n <- [minExpt .. maxExpt]]

    -- Compute the (floor of the) log of i in base b.
    -- Simplest way would be just divide i by b until it's smaller then b, but that would
    -- be very slow!  We are just slightly more clever.
    integerLogBase :: Integer -> Integer -> Int
    integerLogBase b i
       | i < b     = 0
       | otherwise = doDiv (i `div` (b^l)) l
           where
            -- Try squaring the base first to cut down the number of divisions.
             l = 2 * integerLogBase (b*b) i

             doDiv :: Integer -> Int -> Int
             doDiv x y
                | x < b     = y
                | otherwise = doDiv (x `div` b) (y+1)

    \end{code}

    %*********************************************************
    %*                                                      *
    \subsection{Floating point numeric primops}
    %*                                                      *
    %*********************************************************

    Definitions of the boxed PrimOps; these will be
    used in the case of partial applications, etc.

    \begin{code}
    plusFloat, minusFloat, timesFloat, divideFloat :: Float -> Float -> Float
    plusFloat   (F# x) (F# y) = F# (plusFloat# x y)
    minusFloat  (F# x) (F# y) = F# (minusFloat# x y)
    timesFloat  (F# x) (F# y) = F# (timesFloat# x y)
    divideFloat (F# x) (F# y) = F# (divideFloat# x y)

    negateFloat :: Float -> Float
    negateFloat (F# x)        = F# (negateFloat# x)

    gtFloat, geFloat, eqFloat, neFloat, ltFloat, leFloat :: Float -> Float -> Bool
    gtFloat     (F# x) (F# y) = gtFloat# x y
    geFloat     (F# x) (F# y) = geFloat# x y
    eqFloat     (F# x) (F# y) = eqFloat# x y
    neFloat     (F# x) (F# y) = neFloat# x y
    ltFloat     (F# x) (F# y) = ltFloat# x y
    leFloat     (F# x) (F# y) = leFloat# x y

    float2Int :: Float -> Int
    float2Int   (F# x) = I# (float2Int# x)

    int2Float :: Int -> Float
    int2Float   (I# x) = F# (int2Float# x)

    expFloat, logFloat, sqrtFloat :: Float -> Float
    sinFloat, cosFloat, tanFloat  :: Float -> Float
    asinFloat, acosFloat, atanFloat  :: Float -> Float
    sinhFloat, coshFloat, tanhFloat  :: Float -> Float
    expFloat    (F# x) = F# (expFloat# x)
    logFloat    (F# x) = F# (logFloat# x)
    sqrtFloat   (F# x) = F# (sqrtFloat# x)
    sinFloat    (F# x) = F# (sinFloat# x)
    cosFloat    (F# x) = F# (cosFloat# x)
    tanFloat    (F# x) = F# (tanFloat# x)
    asinFloat   (F# x) = F# (asinFloat# x)
    acosFloat   (F# x) = F# (acosFloat# x)
    atanFloat   (F# x) = F# (atanFloat# x)
    sinhFloat   (F# x) = F# (sinhFloat# x)
    coshFloat   (F# x) = F# (coshFloat# x)
    tanhFloat   (F# x) = F# (tanhFloat# x)

    powerFloat :: Float -> Float -> Float
    powerFloat  (F# x) (F# y) = F# (powerFloat# x y)

    -- definitions of the boxed PrimOps; these will be
    -- used in the case of partial applications, etc.

    plusDouble, minusDouble, timesDouble, divideDouble :: Double -> Double -> Double
    plusDouble   (D# x) (D# y) = D# (x +## y)
    minusDouble  (D# x) (D# y) = D# (x -## y)
    timesDouble  (D# x) (D# y) = D# (x *## y)
    divideDouble (D# x) (D# y) = D# (x /## y)

    negateDouble :: Double -> Double
    negateDouble (D# x)        = D# (negateDouble# x)

    gtDouble, geDouble, eqDouble, neDouble, leDouble, ltDouble :: Double -> Double -> Bool
    gtDouble    (D# x) (D# y) = x >## y
    geDouble    (D# x) (D# y) = x >=## y
    eqDouble    (D# x) (D# y) = x ==## y
    neDouble    (D# x) (D# y) = x /=## y
    ltDouble    (D# x) (D# y) = x <## y
    leDouble    (D# x) (D# y) = x <=## y

    double2Int :: Double -> Int
    double2Int   (D# x) = I# (double2Int#   x)

    int2Double :: Int -> Double
    int2Double   (I# x) = D# (int2Double#   x)

    double2Float :: Double -> Float
    double2Float (D# x) = F# (double2Float# x)

    float2Double :: Float -> Double
    float2Double (F# x) = D# (float2Double# x)

    expDouble, logDouble, sqrtDouble :: Double -> Double
    sinDouble, cosDouble, tanDouble  :: Double -> Double
    asinDouble, acosDouble, atanDouble  :: Double -> Double
    sinhDouble, coshDouble, tanhDouble  :: Double -> Double
    expDouble    (D# x) = D# (expDouble# x)
    logDouble    (D# x) = D# (logDouble# x)
    sqrtDouble   (D# x) = D# (sqrtDouble# x)
    sinDouble    (D# x) = D# (sinDouble# x)
    cosDouble    (D# x) = D# (cosDouble# x)
    tanDouble    (D# x) = D# (tanDouble# x)
    asinDouble   (D# x) = D# (asinDouble# x)
    acosDouble   (D# x) = D# (acosDouble# x)
    atanDouble   (D# x) = D# (atanDouble# x)
    sinhDouble   (D# x) = D# (sinhDouble# x)
    coshDouble   (D# x) = D# (coshDouble# x)
    tanhDouble   (D# x) = D# (tanhDouble# x)

    powerDouble :: Double -> Double -> Double
    powerDouble  (D# x) (D# y) = D# (x **## y)
    \end{code}

    \begin{code}
    foreign import ccall unsafe "__encodeFloat"
            encodeFloat# :: Int# -> ByteArray# -> Int -> Float
    foreign import ccall unsafe "__int_encodeFloat"
            int_encodeFloat# :: Int# -> Int -> Float

    foreign import ccall unsafe "isFloatNaN" isFloatNaN :: Float -> Int
    foreign import ccall unsafe "isFloatInfinite" isFloatInfinite :: Float -> Int
    foreign import ccall unsafe "isFloatDenormalized" isFloatDenormalized :: Float -> Int
    foreign import ccall unsafe "isFloatNegativeZero" isFloatNegativeZero :: Float -> Int

    foreign import ccall unsafe "__encodeDouble"
            encodeDouble# :: Int# -> ByteArray# -> Int -> Double
    foreign import ccall unsafe "__int_encodeDouble"
            int_encodeDouble# :: Int# -> Int -> Double

    foreign import ccall unsafe "isDoubleNaN" isDoubleNaN :: Double -> Int
    foreign import ccall unsafe "isDoubleInfinite" isDoubleInfinite :: Double -> Int
    foreign import ccall unsafe "isDoubleDenormalized" isDoubleDenormalized :: Double -> Int
    foreign import ccall unsafe "isDoubleNegativeZero" isDoubleNegativeZero :: Double -> Int
    \end{code}

    %*********************************************************
    %*                                                      *
    \subsection{Coercion rules}
    %*                                                      *
    %*********************************************************

    \begin{code}
    {-# RULES
    "fromIntegral/Int->Float"   fromIntegral = int2Float
    "fromIntegral/Int->Double"  fromIntegral = int2Double
    "realToFrac/Float->Float"   realToFrac   = id :: Float -> Float
    "realToFrac/Float->Double"  realToFrac   = float2Double
    "realToFrac/Double->Float"  realToFrac   = double2Float
    "realToFrac/Double->Double" realToFrac   = id :: Double -> Double
    "realToFrac/Int->Double"    realToFrac   = int2Double	-- See Note [realToFrac int-to-float]
    "realToFrac/Int->Float"     realToFrac   = int2Float	-- 	..ditto
        #-}
    \end{code}

    Note [realToFrac int-to-float]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    Don found that the RULES for realToFrac/Int->Double and simliarly
    Float made a huge difference to some stream-fusion programs.  Here's
    an example

          import Data.Array.Vector

          n = 40000000

          main = do
                let c = replicateU n (2::Double)
                    a = mapU realToFrac (enumFromToU 0 (n-1) ) :: UArr Double
                print (sumU (zipWithU (*) c a))

    Without the RULE we get this loop body:

          case $wtoRational sc_sY4 of ww_aM7 { (# ww1_aM9, ww2_aMa #) ->
          case $wfromRat ww1_aM9 ww2_aMa of tpl_X1P { D# ipv_sW3 ->
          Main.$s$wfold
            (+# sc_sY4 1)
            (+# wild_X1i 1)
            (+## sc2_sY6 (*## 2.0 ipv_sW3))

    And with the rule:

         Main.$s$wfold
            (+# sc_sXT 1)
            (+# wild_X1h 1)
            (+## sc2_sXV (*## 2.0 (int2Double# sc_sXT)))

    The running time of the program goes from 120 seconds to 0.198 seconds
    with the native backend, and 0.143 seconds with the C backend.

    A few more details in Trac #2251, and the patch message
    "Add RULES for realToFrac from Int".

    %*********************************************************
    %*                                                      *
    \subsection{Utils}
    %*                                                      *
    %*********************************************************

    \begin{code}
    showSignedFloat :: (RealFloat a)
      => (a -> ShowS)       -- ^ a function that can show unsigned values
      -> Int                -- ^ the precedence of the enclosing context
      -> a                  -- ^ the value to show
      -> ShowS
    showSignedFloat showPos p x
       | x < 0 || isNegativeZero x
           = showParen (p > 6) (showChar '-' . showPos (-x))
       | otherwise = showPos x
    \end{code}