src/Data/Text/Lazy/Builder/RealFloat.hs

    {-# LANGUAGE CPP, OverloadedStrings #-}
    #if __GLASGOW_HASKELL__ >= 702
    {-# LANGUAGE Trustworthy #-}
    #endif

    -- |
    -- Module:    Data.Text.Lazy.Builder.RealFloat
    -- Copyright: (c) The University of Glasgow 1994-2002
    -- License:   see libraries/base/LICENSE
    --
    -- Write a floating point value to a 'Builder'.

    module Data.Text.Lazy.Builder.RealFloat
        (
          FPFormat(..)
        , realFloat
        , formatRealFloat
        ) where

    import Data.Array.Base (unsafeAt)
    import Data.Array.IArray
    import Data.Text.Internal.Builder.Functions ((<>), i2d)
    import Data.Text.Lazy.Builder.Int (decimal)
    import Data.Text.Internal.Builder.RealFloat.Functions (roundTo)
    import Data.Text.Lazy.Builder
    import qualified Data.Text as T
    #if MIN_VERSION_base(4,11,0)
    import Prelude hiding ((<>))
    #endif

    -- | Control the rendering of floating point numbers.
    data FPFormat = Exponent
                  -- ^ Scientific notation (e.g. @2.3e123@).
                  | Fixed
                  -- ^ Standard decimal notation.
                  | Generic
                  -- ^ Use decimal notation for values between @0.1@ and
                  -- @9,999,999@, and scientific notation otherwise.
                    deriving (Enum, Read, Show)

    -- | 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.
    realFloat :: (RealFloat a) => a -> Builder
    {-# SPECIALIZE realFloat :: Float -> Builder #-}
    {-# SPECIALIZE realFloat :: Double -> Builder #-}
    realFloat x = formatRealFloat Generic Nothing x

    -- | Encode a signed 'RealFloat' according to 'FPFormat' and optionally requested precision.
    --
    -- This corresponds to the @show{E,F,G}Float@ operations provided by @base@'s "Numeric" module.
    --
    -- __NOTE__: The functions in @base-4.12@ changed the serialisation in
    -- case of a @Just 0@ precision; this version of @text@ still provides
    -- the serialisation as implemented in @base-4.11@. The next major
    -- version of @text@ will switch to the more correct @base-4.12@ serialisation.
    formatRealFloat :: (RealFloat a) =>
                       FPFormat
                    -> Maybe Int  -- ^ Number of decimal places to render.
                    -> a
                    -> Builder
    {-# SPECIALIZE formatRealFloat :: FPFormat -> Maybe Int -> Float -> Builder #-}
    {-# SPECIALIZE formatRealFloat :: FPFormat -> Maybe Int -> Double -> Builder #-}
    formatRealFloat fmt decs x
       | isNaN x                   = "NaN"
       | isInfinite x              = if x < 0 then "-Infinity" else "Infinity"
       | x < 0 || isNegativeZero x = singleton '-' <> doFmt fmt (floatToDigits (-x))
       | otherwise                 = doFmt fmt (floatToDigits x)
     where
      doFmt format (is, e) =
        let ds = map i2d is in
        case format of
         Generic ->
          doFmt (if e < 0 || e > 7 then Exponent else Fixed)
                (is,e)
         Exponent ->
          case decs of
           Nothing ->
            let show_e' = decimal (e-1) in
            case ds of
              "0"     -> "0.0e0"
              [d]     -> singleton d <> ".0e" <> show_e'
              (d:ds') -> singleton d <> singleton '.' <> fromString ds' <> singleton 'e' <> show_e'
              []      -> error "formatRealFloat/doFmt/Exponent/Nothing: []"
           Just dec ->
            let dec' = max dec 1 in
            case is of
             [0] -> "0." <> fromText (T.replicate dec' "0") <> "e0"
             _ ->
              let (ei,is') = roundTo (dec'+1) is
                  is'' = map i2d (if ei > 0 then init is' else is')
              in case is'' of
                   [] -> error "formatRealFloat/doFmt/Exponent/Just: []"
                   (d:ds') -> singleton d <> singleton '.' <> fromString ds' <> singleton 'e' <> decimal (e-1+ei)
         Fixed ->
          let
           mk0 ls = case ls of { "" -> "0" ; _ -> fromString ls}
          in
          case decs of
           Nothing
              | e <= 0    -> "0." <> fromText (T.replicate (-e) "0") <> fromString ds
              | otherwise ->
                 let
                    f 0 s    rs  = mk0 (reverse s) <> singleton '.' <> 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 (dec' + e) is
              (ls,rs)  = splitAt (e+ei) (map i2d is')
             in
             mk0 ls <> (if null rs then "" else singleton '.' <> fromString rs)
            else
             let (ei,is') = roundTo dec' (replicate (-e) 0 ++ is)
                 is'' = map i2d (if ei > 0 then is' else 0:is')
             in case is'' of
                  [] -> error "formatRealFloat/doFmt/Fixed: []"
                  (d:ds') -> singleton d <> (if null ds' then "" else singleton '.' <> fromString ds')

    -- 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) => a -> ([Int], Int)
    {-# SPECIALIZE floatToDigits :: Float -> ([Int], Int) #-}
    {-# SPECIALIZE floatToDigits :: Double -> ([Int], Int) #-}
    floatToDigits 0 = ([0], 0)
    floatToDigits 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 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 10)
    --WAS:            fromInt e * log (fromInteger b))

        fixup n =
          if n >= 0 then
            if r + mUp <= expt 10 n * s then n else fixup (n+1)
          else
            if expt 10 (-n) * (r + mUp) <= s then n else fixup (n+1)
       in
       fixup k0

      gen ds rn sN mUpN mDnN =
       let
        (dn, rn') = (rn * 10) `quotRem` sN
        mUpN' = mUpN * 10
        mDnN' = mDnN * 10
       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 10 k) mUp mDn
       else
         let bk = expt 10 (-k) in
         gen [] (r * bk) s (mUp * bk) (mDn * bk)
     in
     (map fromIntegral (reverse rds), k)

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

    expt :: Integer -> Int -> Integer
    expt base n
        | base == 2 && n >= minExpt && n <= maxExpt = expts `unsafeAt` n
        | base == 10 && n <= maxExpt10              = expts10 `unsafeAt` n
        | otherwise                                 = 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]]