```\begin{code}
{-# OPTIONS_GHC -fno-implicit-prelude #-}
-----------------------------------------------------------------------------
-- |
-- Module      :  GHC.Float
-- Copyright   :  (c) The University of Glasgow 1994-2002
--
-- 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

-- 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@)
-- | a constant function, returning the number of digits of
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

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
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

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
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
minExp = minExp0 - p -- the real minimum exponent
-- 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
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),
--              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

--              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
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}
```