{-# LANGUAGE CPP, OverloadedStrings #-}
{-# LANGUAGE Trustworthy #-}

-- |
-- 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 (Int -> FPFormat
FPFormat -> Int
FPFormat -> [FPFormat]
FPFormat -> FPFormat
FPFormat -> FPFormat -> [FPFormat]
FPFormat -> FPFormat -> FPFormat -> [FPFormat]
(FPFormat -> FPFormat)
-> (FPFormat -> FPFormat)
-> (Int -> FPFormat)
-> (FPFormat -> Int)
-> (FPFormat -> [FPFormat])
-> (FPFormat -> FPFormat -> [FPFormat])
-> (FPFormat -> FPFormat -> [FPFormat])
-> (FPFormat -> FPFormat -> FPFormat -> [FPFormat])
-> Enum FPFormat
forall a.
(a -> a)
-> (a -> a)
-> (Int -> a)
-> (a -> Int)
-> (a -> [a])
-> (a -> a -> [a])
-> (a -> a -> [a])
-> (a -> a -> a -> [a])
-> Enum a
enumFromThenTo :: FPFormat -> FPFormat -> FPFormat -> [FPFormat]
$cenumFromThenTo :: FPFormat -> FPFormat -> FPFormat -> [FPFormat]
enumFromTo :: FPFormat -> FPFormat -> [FPFormat]
$cenumFromTo :: FPFormat -> FPFormat -> [FPFormat]
enumFromThen :: FPFormat -> FPFormat -> [FPFormat]
$cenumFromThen :: FPFormat -> FPFormat -> [FPFormat]
enumFrom :: FPFormat -> [FPFormat]
$cenumFrom :: FPFormat -> [FPFormat]
fromEnum :: FPFormat -> Int
$cfromEnum :: FPFormat -> Int
toEnum :: Int -> FPFormat
$ctoEnum :: Int -> FPFormat
pred :: FPFormat -> FPFormat
$cpred :: FPFormat -> FPFormat
succ :: FPFormat -> FPFormat
$csucc :: FPFormat -> FPFormat
Enum, ReadPrec [FPFormat]
ReadPrec FPFormat
Int -> ReadS FPFormat
ReadS [FPFormat]
(Int -> ReadS FPFormat)
-> ReadS [FPFormat]
-> ReadPrec FPFormat
-> ReadPrec [FPFormat]
-> Read FPFormat
forall a.
(Int -> ReadS a)
-> ReadS [a] -> ReadPrec a -> ReadPrec [a] -> Read a
readListPrec :: ReadPrec [FPFormat]
$creadListPrec :: ReadPrec [FPFormat]
readPrec :: ReadPrec FPFormat
$creadPrec :: ReadPrec FPFormat
readList :: ReadS [FPFormat]
$creadList :: ReadS [FPFormat]
readsPrec :: Int -> ReadS FPFormat
$creadsPrec :: Int -> ReadS FPFormat
Read, Int -> FPFormat -> ShowS
[FPFormat] -> ShowS
FPFormat -> [Char]
(Int -> FPFormat -> ShowS)
-> (FPFormat -> [Char]) -> ([FPFormat] -> ShowS) -> Show FPFormat
forall a.
(Int -> a -> ShowS) -> (a -> [Char]) -> ([a] -> ShowS) -> Show a
showList :: [FPFormat] -> ShowS
$cshowList :: [FPFormat] -> ShowS
show :: FPFormat -> [Char]
$cshow :: FPFormat -> [Char]
showsPrec :: Int -> FPFormat -> ShowS
$cshowsPrec :: Int -> FPFormat -> ShowS
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 :: forall a. RealFloat a => a -> Builder
realFloat a
x = FPFormat -> Maybe Int -> a -> Builder
forall a. RealFloat a => FPFormat -> Maybe Int -> a -> Builder
formatRealFloat FPFormat
Generic Maybe Int
forall a. Maybe a
Nothing a
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 :: forall a. RealFloat a => FPFormat -> Maybe Int -> a -> Builder
formatRealFloat FPFormat
fmt Maybe Int
decs a
x
   | a -> Bool
forall a. RealFloat a => a -> Bool
isNaN a
x                   = Builder
"NaN"
   | a -> Bool
forall a. RealFloat a => a -> Bool
isInfinite a
x              = if a
x a -> a -> Bool
forall a. Ord a => a -> a -> Bool
< a
0 then Builder
"-Infinity" else Builder
"Infinity"
   | a
x a -> a -> Bool
forall a. Ord a => a -> a -> Bool
< a
0 Bool -> Bool -> Bool
|| a -> Bool
forall a. RealFloat a => a -> Bool
isNegativeZero a
x = Char -> Builder
singleton Char
'-' Builder -> Builder -> Builder
<> FPFormat -> ([Int], Int) -> Builder
doFmt FPFormat
fmt (a -> ([Int], Int)
forall a. RealFloat a => a -> ([Int], Int)
floatToDigits (-a
x))
   | Bool
otherwise                 = FPFormat -> ([Int], Int) -> Builder
doFmt FPFormat
fmt (a -> ([Int], Int)
forall a. RealFloat a => a -> ([Int], Int)
floatToDigits a
x)
 where
  doFmt :: FPFormat -> ([Int], Int) -> Builder
doFmt FPFormat
format ([Int]
is, Int
e) =
    let ds :: [Char]
ds = (Int -> Char) -> [Int] -> [Char]
forall a b. (a -> b) -> [a] -> [b]
map Int -> Char
i2d [Int]
is in
    case FPFormat
format of
     FPFormat
Generic ->
      FPFormat -> ([Int], Int) -> Builder
doFmt (if Int
e Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
< Int
0 Bool -> Bool -> Bool
|| Int
e Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
> Int
7 then FPFormat
Exponent else FPFormat
Fixed)
            ([Int]
is,Int
e)
     FPFormat
Exponent ->
      case Maybe Int
decs of
       Maybe Int
Nothing ->
        let show_e' :: Builder
show_e' = Int -> Builder
forall a. Integral a => a -> Builder
decimal (Int
eInt -> Int -> Int
forall a. Num a => a -> a -> a
-Int
1) in
        case [Char]
ds of
          [Char]
"0"     -> Builder
"0.0e0"
          [Char
d]     -> Char -> Builder
singleton Char
d Builder -> Builder -> Builder
<> Builder
".0e" Builder -> Builder -> Builder
<> Builder
show_e'
          (Char
d:[Char]
ds') -> Char -> Builder
singleton Char
d Builder -> Builder -> Builder
<> Char -> Builder
singleton Char
'.' Builder -> Builder -> Builder
<> [Char] -> Builder
fromString [Char]
ds' Builder -> Builder -> Builder
<> Char -> Builder
singleton Char
'e' Builder -> Builder -> Builder
<> Builder
show_e'
          []      -> [Char] -> Builder
forall a. HasCallStack => [Char] -> a
error [Char]
"formatRealFloat/doFmt/Exponent/Nothing: []"
       Just Int
dec ->
        let dec' :: Int
dec' = Int -> Int -> Int
forall a. Ord a => a -> a -> a
max Int
dec Int
1 in
        case [Int]
is of
         [Int
0] -> Builder
"0." Builder -> Builder -> Builder
<> Text -> Builder
fromText (Int -> Text -> Text
T.replicate Int
dec' Text
"0") Builder -> Builder -> Builder
<> Builder
"e0"
         [Int]
_ ->
          let (Int
ei,[Int]
is') = Int -> [Int] -> (Int, [Int])
roundTo (Int
dec'Int -> Int -> Int
forall a. Num a => a -> a -> a
+Int
1) [Int]
is
              is'' :: [Char]
is'' = (Int -> Char) -> [Int] -> [Char]
forall a b. (a -> b) -> [a] -> [b]
map Int -> Char
i2d (if Int
ei Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
> Int
0 then [Int] -> [Int]
forall a. [a] -> [a]
init [Int]
is' else [Int]
is')
          in case [Char]
is'' of
               [] -> [Char] -> Builder
forall a. HasCallStack => [Char] -> a
error [Char]
"formatRealFloat/doFmt/Exponent/Just: []"
               (Char
d:[Char]
ds') -> Char -> Builder
singleton Char
d Builder -> Builder -> Builder
<> Char -> Builder
singleton Char
'.' Builder -> Builder -> Builder
<> [Char] -> Builder
fromString [Char]
ds' Builder -> Builder -> Builder
<> Char -> Builder
singleton Char
'e' Builder -> Builder -> Builder
<> Int -> Builder
forall a. Integral a => a -> Builder
decimal (Int
eInt -> Int -> Int
forall a. Num a => a -> a -> a
-Int
1Int -> Int -> Int
forall a. Num a => a -> a -> a
+Int
ei)
     FPFormat
Fixed ->
      let
       mk0 :: [Char] -> Builder
mk0 [Char]
ls = case [Char]
ls of { [Char]
"" -> Builder
"0" ; [Char]
_ -> [Char] -> Builder
fromString [Char]
ls}
      in
      case Maybe Int
decs of
       Maybe Int
Nothing
          | Int
e Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
<= Int
0    -> Builder
"0." Builder -> Builder -> Builder
<> Text -> Builder
fromText (Int -> Text -> Text
T.replicate (-Int
e) Text
"0") Builder -> Builder -> Builder
<> [Char] -> Builder
fromString [Char]
ds
          | Bool
otherwise ->
             let
                f :: t -> [Char] -> [Char] -> Builder
f t
0 [Char]
s    [Char]
rs  = [Char] -> Builder
mk0 (ShowS
forall a. [a] -> [a]
reverse [Char]
s) Builder -> Builder -> Builder
<> Char -> Builder
singleton Char
'.' Builder -> Builder -> Builder
<> [Char] -> Builder
mk0 [Char]
rs
                f t
n [Char]
s    [Char]
""  = t -> [Char] -> [Char] -> Builder
f (t
nt -> t -> t
forall a. Num a => a -> a -> a
-t
1) (Char
'0'Char -> ShowS
forall a. a -> [a] -> [a]
:[Char]
s) [Char]
""
                f t
n [Char]
s (Char
r:[Char]
rs) = t -> [Char] -> [Char] -> Builder
f (t
nt -> t -> t
forall a. Num a => a -> a -> a
-t
1) (Char
rChar -> ShowS
forall a. a -> [a] -> [a]
:[Char]
s) [Char]
rs
             in
                Int -> [Char] -> [Char] -> Builder
forall {t}. (Eq t, Num t) => t -> [Char] -> [Char] -> Builder
f Int
e [Char]
"" [Char]
ds
       Just Int
dec ->
        let dec' :: Int
dec' = Int -> Int -> Int
forall a. Ord a => a -> a -> a
max Int
dec Int
0 in
        if Int
e Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
>= Int
0 then
         let
          (Int
ei,[Int]
is') = Int -> [Int] -> (Int, [Int])
roundTo (Int
dec' Int -> Int -> Int
forall a. Num a => a -> a -> a
+ Int
e) [Int]
is
          ([Char]
ls,[Char]
rs)  = Int -> [Char] -> ([Char], [Char])
forall a. Int -> [a] -> ([a], [a])
splitAt (Int
eInt -> Int -> Int
forall a. Num a => a -> a -> a
+Int
ei) ((Int -> Char) -> [Int] -> [Char]
forall a b. (a -> b) -> [a] -> [b]
map Int -> Char
i2d [Int]
is')
         in
         [Char] -> Builder
mk0 [Char]
ls Builder -> Builder -> Builder
<> (if [Char] -> Bool
forall (t :: * -> *) a. Foldable t => t a -> Bool
null [Char]
rs then Builder
"" else Char -> Builder
singleton Char
'.' Builder -> Builder -> Builder
<> [Char] -> Builder
fromString [Char]
rs)
        else
         let (Int
ei,[Int]
is') = Int -> [Int] -> (Int, [Int])
roundTo Int
dec' (Int -> Int -> [Int]
forall a. Int -> a -> [a]
replicate (-Int
e) Int
0 [Int] -> [Int] -> [Int]
forall a. [a] -> [a] -> [a]
++ [Int]
is)
             is'' :: [Char]
is'' = (Int -> Char) -> [Int] -> [Char]
forall a b. (a -> b) -> [a] -> [b]
map Int -> Char
i2d (if Int
ei Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
> Int
0 then [Int]
is' else Int
0Int -> [Int] -> [Int]
forall a. a -> [a] -> [a]
:[Int]
is')
         in case [Char]
is'' of
              [] -> [Char] -> Builder
forall a. HasCallStack => [Char] -> a
error [Char]
"formatRealFloat/doFmt/Fixed: []"
              (Char
d:[Char]
ds') -> Char -> Builder
singleton Char
d Builder -> Builder -> Builder
<> (if [Char] -> Bool
forall (t :: * -> *) a. Foldable t => t a -> Bool
null [Char]
ds' then Builder
"" else Char -> Builder
singleton Char
'.' Builder -> Builder -> Builder
<> [Char] -> Builder
fromString [Char]
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 :: forall a. RealFloat a => a -> ([Int], Int)
floatToDigits a
0 = ([Int
0], Int
0)
floatToDigits a
x =
 let
  (Integer
f0, Int
e0) = a -> (Integer, Int)
forall a. RealFloat a => a -> (Integer, Int)
decodeFloat a
x
  (Int
minExp0, Int
_) = a -> (Int, Int)
forall a. RealFloat a => a -> (Int, Int)
floatRange a
x
  p :: Int
p = a -> Int
forall a. RealFloat a => a -> Int
floatDigits a
x
  b :: Integer
b = a -> Integer
forall a. RealFloat a => a -> Integer
floatRadix a
x
  minExp :: Int
minExp = Int
minExp0 Int -> Int -> Int
forall a. Num a => a -> a -> a
- Int
p -- the real minimum exponent
  -- Haskell requires that f be adjusted so denormalized numbers
  -- will have an impossibly low exponent.  Adjust for this.
  (Integer
f, Int
e) =
   let n :: Int
n = Int
minExp Int -> Int -> Int
forall a. Num a => a -> a -> a
- Int
e0 in
   if Int
n Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
> Int
0 then (Integer
f0 Integer -> Integer -> Integer
forall a. Integral a => a -> a -> a
`quot` (Integer -> Int -> Integer
expt Integer
b Int
n), Int
e0Int -> Int -> Int
forall a. Num a => a -> a -> a
+Int
n) else (Integer
f0, Int
e0)
  (Integer
r, Integer
s, Integer
mUp, Integer
mDn) =
   if Int
e Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
>= Int
0 then
    let be :: Integer
be = Integer -> Int -> Integer
expt Integer
b Int
e in
    if Integer
f Integer -> Integer -> Bool
forall a. Eq a => a -> a -> Bool
== Integer -> Int -> Integer
expt Integer
b (Int
pInt -> Int -> Int
forall a. Num a => a -> a -> a
-Int
1) then
      (Integer
fInteger -> Integer -> Integer
forall a. Num a => a -> a -> a
*Integer
beInteger -> Integer -> Integer
forall a. Num a => a -> a -> a
*Integer
bInteger -> Integer -> Integer
forall a. Num a => a -> a -> a
*Integer
2, Integer
2Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
*Integer
b, Integer
beInteger -> Integer -> Integer
forall a. Num a => a -> a -> a
*Integer
b, Integer
be)     -- according to Burger and Dybvig
    else
      (Integer
fInteger -> Integer -> Integer
forall a. Num a => a -> a -> a
*Integer
beInteger -> Integer -> Integer
forall a. Num a => a -> a -> a
*Integer
2, Integer
2, Integer
be, Integer
be)
   else
    if Int
e Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
> Int
minExp Bool -> Bool -> Bool
&& Integer
f Integer -> Integer -> Bool
forall a. Eq a => a -> a -> Bool
== Integer -> Int -> Integer
expt Integer
b (Int
pInt -> Int -> Int
forall a. Num a => a -> a -> a
-Int
1) then
      (Integer
fInteger -> Integer -> Integer
forall a. Num a => a -> a -> a
*Integer
bInteger -> Integer -> Integer
forall a. Num a => a -> a -> a
*Integer
2, Integer -> Int -> Integer
expt Integer
b (-Int
eInt -> Int -> Int
forall a. Num a => a -> a -> a
+Int
1)Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
*Integer
2, Integer
b, Integer
1)
    else
      (Integer
fInteger -> Integer -> Integer
forall a. Num a => a -> a -> a
*Integer
2, Integer -> Int -> Integer
expt Integer
b (-Int
e)Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
*Integer
2, Integer
1, Integer
1)
  k :: Int
  k :: Int
k =
   let
    k0 :: Int
    k0 :: Int
k0 =
     if Integer
b Integer -> Integer -> Bool
forall a. Eq a => a -> a -> Bool
== Integer
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 :: Int
lx = Int
p Int -> Int -> Int
forall a. Num a => a -> a -> a
- Int
1 Int -> Int -> Int
forall a. Num a => a -> a -> a
+ Int
e0
            k1 :: Int
k1 = (Int
lx Int -> Int -> Int
forall a. Num a => a -> a -> a
* Int
8651) Int -> Int -> Int
forall a. Integral a => a -> a -> a
`quot` Int
28738
        in if Int
lx Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
>= Int
0 then Int
k1 Int -> Int -> Int
forall a. Num a => a -> a -> a
+ Int
1 else Int
k1
     else
        -- f :: Integer, log :: Float -> Float,
        --               ceiling :: Float -> Int
        Float -> Int
forall a b. (RealFrac a, Integral b) => a -> b
ceiling ((Float -> Float
forall a. Floating a => a -> a
log (Integer -> Float
forall a. Num a => Integer -> a
fromInteger (Integer
fInteger -> Integer -> Integer
forall a. Num a => a -> a -> a
+Integer
1) :: Float) Float -> Float -> Float
forall a. Num a => a -> a -> a
+
                 Int -> Float
intToFloat Int
e Float -> Float -> Float
forall a. Num a => a -> a -> a
* Float -> Float
forall a. Floating a => a -> a
log (Integer -> Float
forall a. Num a => Integer -> a
fromInteger Integer
b)) Float -> Float -> Float
forall a. Fractional a => a -> a -> a
/
                   Float -> Float
forall a. Floating a => a -> a
log Float
10)
--WAS:            fromInt e * log (fromInteger b))

    fixup :: Int -> Int
fixup Int
n =
      if Int
n Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
>= Int
0 then
        if Integer
r Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
+ Integer
mUp Integer -> Integer -> Bool
forall a. Ord a => a -> a -> Bool
<= Integer -> Int -> Integer
expt Integer
10 Int
n Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
* Integer
s then Int
n else Int -> Int
fixup (Int
nInt -> Int -> Int
forall a. Num a => a -> a -> a
+Int
1)
      else
        if Integer -> Int -> Integer
expt Integer
10 (-Int
n) Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
* (Integer
r Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
+ Integer
mUp) Integer -> Integer -> Bool
forall a. Ord a => a -> a -> Bool
<= Integer
s then Int
n else Int -> Int
fixup (Int
nInt -> Int -> Int
forall a. Num a => a -> a -> a
+Int
1)
   in
   Int -> Int
fixup Int
k0

  gen :: [t] -> t -> t -> t -> t -> [t]
gen [t]
ds t
rn t
sN t
mUpN t
mDnN =
   let
    (t
dn, t
rn') = (t
rn t -> t -> t
forall a. Num a => a -> a -> a
* t
10) t -> t -> (t, t)
forall a. Integral a => a -> a -> (a, a)
`quotRem` t
sN
    mUpN' :: t
mUpN' = t
mUpN t -> t -> t
forall a. Num a => a -> a -> a
* t
10
    mDnN' :: t
mDnN' = t
mDnN t -> t -> t
forall a. Num a => a -> a -> a
* t
10
   in
   case (t
rn' t -> t -> Bool
forall a. Ord a => a -> a -> Bool
< t
mDnN', t
rn' t -> t -> t
forall a. Num a => a -> a -> a
+ t
mUpN' t -> t -> Bool
forall a. Ord a => a -> a -> Bool
> t
sN) of
    (Bool
True,  Bool
False) -> t
dn t -> [t] -> [t]
forall a. a -> [a] -> [a]
: [t]
ds
    (Bool
False, Bool
True)  -> t
dnt -> t -> t
forall a. Num a => a -> a -> a
+t
1 t -> [t] -> [t]
forall a. a -> [a] -> [a]
: [t]
ds
    (Bool
True,  Bool
True)  -> if t
rn' t -> t -> t
forall a. Num a => a -> a -> a
* t
2 t -> t -> Bool
forall a. Ord a => a -> a -> Bool
< t
sN then t
dn t -> [t] -> [t]
forall a. a -> [a] -> [a]
: [t]
ds else t
dnt -> t -> t
forall a. Num a => a -> a -> a
+t
1 t -> [t] -> [t]
forall a. a -> [a] -> [a]
: [t]
ds
    (Bool
False, Bool
False) -> [t] -> t -> t -> t -> t -> [t]
gen (t
dnt -> [t] -> [t]
forall a. a -> [a] -> [a]
:[t]
ds) t
rn' t
sN t
mUpN' t
mDnN'

  rds :: [Integer]
rds =
   if Int
k Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
>= Int
0 then
      [Integer] -> Integer -> Integer -> Integer -> Integer -> [Integer]
forall {t}. Integral t => [t] -> t -> t -> t -> t -> [t]
gen [] Integer
r (Integer
s Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
* Integer -> Int -> Integer
expt Integer
10 Int
k) Integer
mUp Integer
mDn
   else
     let bk :: Integer
bk = Integer -> Int -> Integer
expt Integer
10 (-Int
k) in
     [Integer] -> Integer -> Integer -> Integer -> Integer -> [Integer]
forall {t}. Integral t => [t] -> t -> t -> t -> t -> [t]
gen [] (Integer
r Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
* Integer
bk) Integer
s (Integer
mUp Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
* Integer
bk) (Integer
mDn Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
* Integer
bk)
 in
 ((Integer -> Int) -> [Integer] -> [Int]
forall a b. (a -> b) -> [a] -> [b]
map Integer -> Int
forall a. Num a => Integer -> a
fromInteger ([Integer] -> [Integer]
forall a. [a] -> [a]
reverse [Integer]
rds), Int
k)

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

expt :: Integer -> Int -> Integer
expt :: Integer -> Int -> Integer
expt Integer
base Int
n
    | Integer
base Integer -> Integer -> Bool
forall a. Eq a => a -> a -> Bool
== Integer
2 Bool -> Bool -> Bool
&& Int
n Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
>= Int
minExpt Bool -> Bool -> Bool
&& Int
n Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
<= Int
maxExpt = Array Int Integer
expts Array Int Integer -> Int -> Integer
forall (a :: * -> * -> *) e i.
(IArray a e, Ix i) =>
a i e -> Int -> e
`unsafeAt` Int
n
    | Integer
base Integer -> Integer -> Bool
forall a. Eq a => a -> a -> Bool
== Integer
10 Bool -> Bool -> Bool
&& Int
n Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
<= Int
maxExpt10              = Array Int Integer
expts10 Array Int Integer -> Int -> Integer
forall (a :: * -> * -> *) e i.
(IArray a e, Ix i) =>
a i e -> Int -> e
`unsafeAt` Int
n
    | Bool
otherwise                                 = Integer
baseInteger -> Int -> Integer
forall a b. (Num a, Integral b) => a -> b -> a
^Int
n

expts :: Array Int Integer
expts :: Array Int Integer
expts = (Int, Int) -> [(Int, Integer)] -> Array Int Integer
forall (a :: * -> * -> *) e i.
(IArray a e, Ix i) =>
(i, i) -> [(i, e)] -> a i e
array (Int
minExpt,Int
maxExpt) [(Int
n,Integer
2Integer -> Int -> Integer
forall a b. (Num a, Integral b) => a -> b -> a
^Int
n) | Int
n <- [Int
minExpt .. Int
maxExpt]]

maxExpt10 :: Int
maxExpt10 :: Int
maxExpt10 = Int
324

expts10 :: Array Int Integer
expts10 :: Array Int Integer
expts10 = (Int, Int) -> [(Int, Integer)] -> Array Int Integer
forall (a :: * -> * -> *) e i.
(IArray a e, Ix i) =>
(i, i) -> [(i, e)] -> a i e
array (Int
minExpt,Int
maxExpt10) [(Int
n,Integer
10Integer -> Int -> Integer
forall a b. (Num a, Integral b) => a -> b -> a
^Int
n) | Int
n <- [Int
minExpt .. Int
maxExpt10]]

intToFloat :: Int -> Float
intToFloat :: Int -> Float
intToFloat = Int -> Float
forall a b. (Integral a, Num b) => a -> b
fromIntegral