\begin{code}
{-# LANGUAGE Trustworthy #-}
{-# LANGUAGE CPP
           , NoImplicitPrelude
           , MagicHash
           , UnboxedTuples
  #-}
-- We believe we could deorphan this module, by moving lots of things
-- around, but we haven't got there yet:
{-# OPTIONS_GHC -fno-warn-orphans #-}
{-# OPTIONS_HADDOCK hide #-}

-----------------------------------------------------------------------------
-- |
-- Module      :  GHC.Float
-- Copyright   :  (c) The University of Glasgow 1994-2002
--                Portions obtained from hbc (c) Lennart Augusstson
-- License     :  see libraries/base/LICENSE
--
-- Maintainer  :  [email protected]
-- Stability   :  internal
-- Portability :  non-portable (GHC Extensions)
--
-- The types 'Float' and 'Double', and the classes 'Floating' and 'RealFloat'.
--
-----------------------------------------------------------------------------

#include "ieee-flpt.h"

module GHC.Float( module GHC.Float, Float(..), Double(..), Float#, Double#
                , double2Int, int2Double, float2Int, int2Float )
    where

import Data.Maybe

import Data.Bits
import GHC.Base
import GHC.List
import GHC.Enum
import GHC.Show
import GHC.Num
import GHC.Real
import GHC.Arr
import GHC.Float.RealFracMethods
import GHC.Float.ConversionUtils
import GHC.Integer.Logarithms ( integerLogBase# )
import GHC.Integer.Logarithms.Internals

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

    {-# INLINE (**) #-}
    {-# INLINE logBase #-}
    {-# INLINE sqrt #-}
    {-# INLINE tan #-}
    {-# INLINE tanh #-}
    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) <= 'abs' m < b^d@, where @d@ is
    -- the value of @'floatDigits' x@.
    -- In particular, @'decodeFloat' 0 = (0,0)@. If the type
    -- contains a negative zero, also @'decodeFloat' (-0.0) = (0,0)@.
    -- /The result of/ @'decodeFloat' x@ /is unspecified if either of/
    -- @'isNaN' x@ /or/ @'isInfinite' x@ /is/ 'True'.
    decodeFloat         :: a -> (Integer,Int)
    -- | 'encodeFloat' performs the inverse of 'decodeFloat' in the
    -- sense that for finite @x@ with the exception of @-0.0@,
    -- @'uncurry' 'encodeFloat' ('decodeFloat' x) = x@.
    -- @'encodeFloat' m n@ is one of the two closest representable
    -- floating-point numbers to @m*b^^n@ (or @&#177;Infinity@ if overflow
    -- occurs); usually the closer, but if @m@ contains too many bits,
    -- the result may be rounded in the wrong direction.
    encodeFloat         :: Integer -> Int -> a
    -- | 'exponent' corresponds to the second component of 'decodeFloat'.
    -- @'exponent' 0 = 0@ and for finite nonzero @x@,
    -- @'exponent' x = snd ('decodeFloat' x) + 'floatDigits' x@.
    -- If @x@ is a finite floating-point number, it is equal in value to
    -- @'significand' x * b ^^ 'exponent' x@, where @b@ is the
    -- floating-point radix.
    -- The behaviour is unspecified on infinite or @NaN@ values.
    exponent            :: a -> Int
    -- | The first component of 'decodeFloat', scaled to lie in the open
    -- interval (@-1@,@1@), either @0.0@ or of absolute value @>= 1\/b@,
    -- where @b@ is the floating-point radix.
    -- The behaviour is unspecified on infinite or @NaN@ values.
    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 0 x      =  x
    scaleFloat k x
      | isFix           =  x
      | otherwise       =  encodeFloat m (n + clamp b k)
                           where (m,n) = decodeFloat x
                                 (l,h) = floatRange x
                                 d     = floatDigits x
                                 b     = h - l + 4*d
                                 -- n+k may overflow, which would lead
                                 -- to wrong results, hence we clamp the
                                 -- scaling parameter.
                                 -- If n + k would be larger than h,
                                 -- n + clamp b k must be too, simliar
                                 -- for smaller than l - d.
                                 -- Add a little extra to keep clear
                                 -- from the boundary cases.
                                 isFix = x == 0 || isNaN x || isInfinite 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 @Float@} %* * %********************************************************* \begin{code}
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 i = F# (floatFromInteger i)

instance  Real Float  where
    toRational (F# x#)  =
        case decodeFloat_Int# x# of
          (# m#, e# #)
            | isTrue# (e# >=# 0#)                               ->
                    (smallInteger m# `shiftLInteger` e#) :% 1
            | isTrue# ((int2Word# m# `and#` 1##) `eqWord#` 0##) ->
                    case elimZerosInt# m# (negateInt# e#) of
                      (# n, d# #) -> n :% shiftLInteger 1 d#
            | otherwise                                         ->
                    smallInteger m# :% shiftLInteger 1 (negateInt# e#)

instance  Fractional Float  where
    (/) x y             =  divideFloat x y
    {-# INLINE fromRational #-}
    fromRational (n:%d) = rationalToFloat n d
    recip x             =  1.0 / x

rationalToFloat :: Integer -> Integer -> Float
{-# NOINLINE [1] rationalToFloat #-}
rationalToFloat n 0
    | n == 0        = 0/0
    | n < 0         = (-1)/0
    | otherwise     = 1/0
rationalToFloat n d
    | n == 0        = encodeFloat 0 0
    | n < 0         = -(fromRat'' minEx mantDigs (-n) d)
    | otherwise     = fromRat'' minEx mantDigs n d
      where
        minEx       = FLT_MIN_EXP
        mantDigs    = FLT_MANT_DIG

-- RULES for Integer and Int
{-# RULES
"properFraction/Float->Integer"     properFraction = properFractionFloatInteger
"truncate/Float->Integer"           truncate = truncateFloatInteger
"floor/Float->Integer"              floor = floorFloatInteger
"ceiling/Float->Integer"            ceiling = ceilingFloatInteger
"round/Float->Integer"              round = roundFloatInteger
"properFraction/Float->Int"         properFraction = properFractionFloatInt
"truncate/Float->Int"               truncate = float2Int
"floor/Float->Int"                  floor = floorFloatInt
"ceiling/Float->Int"                ceiling = ceilingFloatInt
"round/Float->Int"                  round = roundFloatInt
  #-}
instance  RealFrac Float  where

        -- ceiling, floor, and truncate are all small
    {-# INLINE [1] ceiling #-}
    {-# INLINE [1] floor #-}
    {-# INLINE [1] truncate #-}

-- We assume that FLT_RADIX is 2 so that we can use more efficient code
#if FLT_RADIX != 2
#error FLT_RADIX must be 2
#endif
    properFraction (F# x#)
      = case decodeFloat_Int# x# of
        (# m#, n# #) ->
            let m = I# m#
                n = I# n#
            in
            if n >= 0
            then (fromIntegral m * (2 ^ n), 0.0)
            else let i = if m >= 0 then                m `shiftR` negate n
                                   else negate (negate m `shiftR` negate n)
                     f = m - (i `shiftL` negate n)
                 in (fromIntegral i, encodeFloat (fromIntegral f) 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 = 0.5 * log ((1.0+x) / (1.0-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_Int# f# of
                          (# i, e #) -> (smallInteger i, I# e)

    encodeFloat i (I# e) = F# (encodeFloatInteger i 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 0 x      = x
    scaleFloat k x
      | isFix           = x
      | otherwise       = case decodeFloat x of
                            (m,n) -> encodeFloat m (n + clamp bf k)
                        where bf = FLT_MAX_EXP - (FLT_MIN_EXP) + 4*FLT_MANT_DIG
                              isFix = x == 0 || isFloatFinite x == 0

    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  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 #-}
    fromInteger i = D# (doubleFromInteger i)


instance  Real Double  where
    toRational (D# x#)  =
        case decodeDoubleInteger x# of
          (# m, e# #)
            | isTrue# (e# >=# 0#)                                  ->
                shiftLInteger m e# :% 1
            | isTrue# ((integerToWord m `and#` 1##) `eqWord#` 0##) ->
                case elimZerosInteger m (negateInt# e#) of
                    (# n, d# #) ->  n :% shiftLInteger 1 d#
            | otherwise                                            ->
                m :% shiftLInteger 1 (negateInt# e#)

instance  Fractional Double  where
    (/) x y             =  divideDouble x y
    {-# INLINE fromRational #-}
    fromRational (n:%d) = rationalToDouble n d
    recip x             =  1.0 / x

rationalToDouble :: Integer -> Integer -> Double
{-# NOINLINE [1] rationalToDouble #-}
rationalToDouble n 0
    | n == 0        = 0/0
    | n < 0         = (-1)/0
    | otherwise     = 1/0
rationalToDouble n d
    | n == 0        = encodeFloat 0 0
    | n < 0         = -(fromRat'' minEx mantDigs (-n) d)
    | otherwise     = fromRat'' minEx mantDigs n d
      where
        minEx       = DBL_MIN_EXP
        mantDigs    = DBL_MANT_DIG

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 = 0.5 * log ((1.0+x) / (1.0-x))

-- RULES for Integer and Int
{-# RULES
"properFraction/Double->Integer"    properFraction = properFractionDoubleInteger
"truncate/Double->Integer"          truncate = truncateDoubleInteger
"floor/Double->Integer"             floor = floorDoubleInteger
"ceiling/Double->Integer"           ceiling = ceilingDoubleInteger
"round/Double->Integer"             round = roundDoubleInteger
"properFraction/Double->Int"        properFraction = properFractionDoubleInt
"truncate/Double->Int"              truncate = double2Int
"floor/Double->Int"                 floor = floorDoubleInt
"ceiling/Double->Int"               ceiling = ceilingDoubleInt
"round/Double->Int"                 round = roundDoubleInt
  #-}
instance  RealFrac Double  where

        -- ceiling, floor, and truncate are all small
    {-# INLINE [1] ceiling #-}
    {-# INLINE [1] floor #-}
    {-# INLINE [1] truncate #-}

    properFraction x
      = case (decodeFloat x)      of { (m,n) ->
        if n >= 0 then
            (fromInteger m * 2 ^ n, 0.0)
        else
            case (quotRem m (2^(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 decodeDoubleInteger x#   of
          (# i, j #) -> (i, I# j)

    encodeFloat i (I# j) = D# (encodeDoubleInteger i j)

    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 0 x      = x
    scaleFloat k x
      | isFix           = x
      | otherwise       = case decodeFloat x of
                            (m,n) -> encodeFloat m (n + clamp bd k)
                        where bd = DBL_MAX_EXP - (DBL_MIN_EXP) + 4*DBL_MANT_DIG
                              isFix = x == 0 || isDoubleFinite x == 0

    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

-- This is just a compatibility stub, as the "alt" argument formerly
-- didn't exist.
formatRealFloat :: (RealFloat a) => FFFormat -> Maybe Int -> a -> String
formatRealFloat fmt decs x = formatRealFloatAlt fmt decs False x

formatRealFloatAlt :: (RealFloat a) => FFFormat -> Maybe Int -> Bool -> a
                 -> String
formatRealFloatAlt fmt decs alt 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'
          []      -> error "formatRealFloat/doFmt/FFExponent: []"
       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 && not alt 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' && not alt then "" else '.':ds')


roundTo :: Int -> Int -> [Int] -> (Int,[Int])
roundTo base d is =
  case f d True is of
    x@(0,_) -> x
    (1,xs)  -> (1, 1:xs)
    _       -> error "roundTo: bad Value"
 where
  b2 = base `quot` 2

  f n _ []     = (0, replicate n 0)
  f 0 e (x:xs) | x == b2 && e && all (== 0) xs = (0, [])   -- Round to even when at exactly half the base
               | otherwise = (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) (even i) 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 `quot` (expt b n), e0+n) else (f0, e0)
  (r, s, mUp, mDn) =
   if e >= 0 then
    let be = expt b e in
    if f == expt b (p-1) then
      (f*be*b*2, 2*b, be*b, be)     -- according to Burger and Dybvig
    else
      (f*be*2, 2, be, be)
   else
    if e > minExp && f == expt b (p-1) then
      (f*b*2, expt b (-e+1)*2, b, 1)
    else
      (f*2, expt b (-e)*2, 1, 1)
  k :: Int
  k =
   let
    k0 :: Int
    k0 =
     if b == 2 && base == 10 then
        -- logBase 10 2 is very slightly larger than 8651/28738
        -- (about 5.3558e-10), so if log x >= 0, the approximation
        -- k1 is too small, hence we add one and need one fixup step less.
        -- If log x < 0, the approximation errs rather on the high side.
        -- That is usually more than compensated for by ignoring the
        -- fractional part of logBase 2 x, but when x is a power of 1/2
        -- or slightly larger and the exponent is a multiple of the
        -- denominator of the rational approximation to logBase 10 2,
        -- k1 is larger than logBase 10 x. If k1 > 1 + logBase 10 x,
        -- we get a leading zero-digit we don't want.
        -- With the approximation 3/10, this happened for
        -- 0.5^1030, 0.5^1040, ..., 0.5^1070 and values close above.
        -- The approximation 8651/28738 guarantees k1 < 1 + logBase 10 x
        -- for IEEE-ish floating point types with exponent fields
        -- <= 17 bits and mantissae of several thousand bits, earlier
        -- convergents to logBase 10 2 would fail for long double.
        -- Using quot instead of div is a little faster and requires
        -- fewer fixup steps for negative lx.
        let lx = p - 1 + e0
            k1 = (lx * 8651) `quot` 28738
        in if lx >= 0 then k1 + 1 else k1
     else
	-- f :: Integer, log :: Float -> Float,
        --               ceiling :: Float -> Int
        ceiling ((log (fromInteger (f+1) :: Float) +
                 fromIntegral 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) `quotRem` 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'.
{-# RULES
"fromRat/Float"     fromRat = (fromRational :: Rational -> Float)
"fromRat/Double"    fromRat = (fromRational :: Rational -> Double)
  #-}

{-# NOINLINE [1] fromRat #-}
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     = -1/0        -- -Infinity
                 | otherwise =  0/0        -- NaN

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

-- 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
        xMax   = toRational (expt b p)
        p0 = (integerLogBase b (numerator x) - integerLogBase b (denominator x) - p) `max` minExp
        -- if x = n/d and ln = integerLogBase b n, ld = integerLogBase b d,
        -- then b^(ln-ld-1) < x < b^(ln-ld+1)
        f = if p0 < 0 then 1 :% expt b (-p0) else expt b p0 :% 1
        x0 = x / f
        -- if ln - ld >= minExp0, then b^(p-1) < x0 < b^(p+1), so there's at most
        -- one scaling step needed, otherwise, x0 < b^p and no scaling is needed
        (x', p') = if x0 >= xMax then (x0 / toRational b, p0+1) else (x0, p0)
        r = encodeFloat (round 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
        if base == 10 && n <= maxExpt10 then
            expts10!n
        else
            base^n

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

maxExpt10 :: Int
maxExpt10 = 324

expts10 :: Array Int Integer
expts10 = array (minExpt,maxExpt10) [(n,10^n) | n <- [minExpt .. maxExpt10]]

-- 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, except for base 2, where
-- we take advantage of the representation of Integers.
-- The general case could be improved by a lookup table for
-- approximating the result by integerLog2 i / integerLog2 b.
integerLogBase :: Integer -> Integer -> Int
integerLogBase b i
   | i < b     = 0
   | b == 2    = I# (integerLog2# i)
   | otherwise = I# (integerLogBase# b i)

\end{code} Unfortunately, the old conversion code was awfully slow due to a) a slow integer logarithm b) repeated calculation of gcd's For the case of Rational's coming from a Float or Double via toRational, we can exploit the fact that the denominator is a power of two, which for these brings a huge speedup since we need only shift and add instead of division. The below is an adaption of fromRat' for the conversion to Float or Double exploiting the known floatRadix and avoiding divisions as much as possible. \begin{code}
{-# SPECIALISE fromRat'' :: Int -> Int -> Integer -> Integer -> Float,
                            Int -> Int -> Integer -> Integer -> Double #-}
fromRat'' :: RealFloat a => Int -> Int -> Integer -> Integer -> a
-- Invariant: n and d strictly positive
fromRat'' minEx@(I# me#) mantDigs@(I# md#) n d =
    case integerLog2IsPowerOf2# d of
      (# ld#, pw# #)
        | isTrue# (pw# ==# 0#) ->
          case integerLog2# n of
            ln# | isTrue# (ln# >=# (ld# +# me# -# 1#)) ->
                  -- this means n/d >= 2^(minEx-1), i.e. we are guaranteed to get
                  -- a normalised number, round to mantDigs bits
                  if isTrue# (ln# <# md#)
                    then encodeFloat n (I# (negateInt# ld#))
                    else let n'  = n `shiftR` (I# (ln# +# 1# -# md#))
                             n'' = case roundingMode# n (ln# -# md#) of
                                    0# -> n'
                                    2# -> n' + 1
                                    _  -> case fromInteger n' .&. (1 :: Int) of
                                            0 -> n'
                                            _ -> n' + 1
                         in encodeFloat n'' (I# (ln# -# ld# +# 1# -# md#))
                | otherwise ->
                  -- n/d < 2^(minEx-1), a denorm or rounded to 2^(minEx-1)
                  -- the exponent for encoding is always minEx-mantDigs
                  -- so we must shift right by (minEx-mantDigs) - (-ld)
                  case ld# +# (me# -# md#) of
                    ld'# | isTrue# (ld'# <=# 0#) -> -- we would shift left, so we don't shift
                           encodeFloat n (I# ((me# -# md#) -# ld'#))
                         | isTrue# (ld'# <=# ln#) ->
                           let n' = n `shiftR` (I# ld'#)
                           in case roundingMode# n (ld'# -# 1#) of
                                0# -> encodeFloat n' (minEx - mantDigs)
                                1# -> if fromInteger n' .&. (1 :: Int) == 0
                                        then encodeFloat n' (minEx-mantDigs)
                                        else encodeFloat (n' + 1) (minEx-mantDigs)
                                _  -> encodeFloat (n' + 1) (minEx-mantDigs)
                         | isTrue# (ld'# ># (ln# +# 1#)) -> encodeFloat 0 0 -- result of shift < 0.5
                         | otherwise ->  -- first bit of n shifted to 0.5 place
                           case integerLog2IsPowerOf2# n of
                            (# _, 0# #) -> encodeFloat 0 0  -- round to even
                            (# _, _ #)  -> encodeFloat 1 (minEx - mantDigs)
        | otherwise ->
          let ln = I# (integerLog2# n)
              ld = I# ld#
              -- 2^(ln-ld-1) < n/d < 2^(ln-ld+1)
              p0 = max minEx (ln - ld)
              (n', d')
                | p0 < mantDigs = (n `shiftL` (mantDigs - p0), d)
                | p0 == mantDigs = (n, d)
                | otherwise     = (n, d `shiftL` (p0 - mantDigs))
              -- if ln-ld < minEx, then n'/d' < 2^mantDigs, else
              -- 2^(mantDigs-1) < n'/d' < 2^(mantDigs+1) and we
              -- may need one scaling step
              scale p a b
                | (b `shiftL` mantDigs) <= a = (p+1, a, b `shiftL` 1)
                | otherwise = (p, a, b)
              (p', n'', d'') = scale (p0-mantDigs) n' d'
              -- n''/d'' < 2^mantDigs and p' == minEx-mantDigs or n''/d'' >= 2^(mantDigs-1)
              rdq = case n'' `quotRem` d'' of
                     (q,r) -> case compare (r `shiftL` 1) d'' of
                                LT -> q
                                EQ -> if fromInteger q .&. (1 :: Int) == 0
                                        then q else q+1
                                GT -> q+1
          in  encodeFloat rdq p'
\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) = isTrue# (gtFloat# x y)
geFloat     (F# x) (F# y) = isTrue# (geFloat# x y)
eqFloat     (F# x) (F# y) = isTrue# (eqFloat# x y)
neFloat     (F# x) (F# y) = isTrue# (neFloat# x y)
ltFloat     (F# x) (F# y) = isTrue# (ltFloat# x y)
leFloat     (F# x) (F# y) = isTrue# (leFloat# x y)

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) = isTrue# (x >##  y)
geDouble    (D# x) (D# y) = isTrue# (x >=## y)
eqDouble    (D# x) (D# y) = isTrue# (x ==## y)
neDouble    (D# x) (D# y) = isTrue# (x /=## y)
ltDouble    (D# x) (D# y) = isTrue# (x <##  y)
leDouble    (D# x) (D# y) = isTrue# (x <=## y)

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 "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 "isFloatFinite" isFloatFinite :: Float -> Int

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
foreign import ccall unsafe "isDoubleFinite" isDoubleFinite :: Double -> Int
\end{code} %********************************************************* %* * \subsection{Coercion rules} %* * %********************************************************* \begin{code}

word2Double :: Word -> Double
word2Double (W# w) = D# (word2Double# w)

word2Float :: Word -> Float
word2Float (W# w) = F# (word2Float# w)

{-# RULES
"fromIntegral/Int->Float"   fromIntegral = int2Float
"fromIntegral/Int->Double"  fromIntegral = int2Double
"fromIntegral/Word->Float"  fromIntegral = word2Float
"fromIntegral/Word->Double" fromIntegral = word2Double
"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} We need to prevent over/underflow of the exponent in encodeFloat when called from scaleFloat, hence we clamp the scaling parameter. We must have a large enough range to cover the maximum difference of exponents returned by decodeFloat. \begin{code}
clamp :: Int -> Int -> Int
clamp bd k = max (-bd) (min bd k)
\end{code}