```{-# LANGUAGE DeriveDataTypeable #-}

-- | Decimal numbers are represented as @m*10^(-e)@ where
-- @m@ and @e@ are integers.  The exponent @e@ is an unsigned Word8.  Hence
-- the smallest value that can be represented is @10^-255@.
--
-- Unary arithmetic results have the exponent of the argument.  Binary
-- arithmetic results have an exponent equal to the maximum of the exponents
-- of the arguments.
--
-- Decimal numbers are defined as instances of @Real@.  This means that
-- conventional division is not defined.  Instead the functions @divide@ and
-- @allocate@ will split a decimal amount into lists of results.  These
-- results are guaranteed to sum to the original number.  This is a useful
-- property when doing financial arithmetic.
--
-- The arithmetic on mantissas is always done using @Integer@, regardless of
-- the type of @DecimalRaw@ being manipulated.  In practice it is recommended
-- that @Decimal@ be used, with other types being used only where necessary
-- (e.g. to conform to a network protocol).

module Data.Decimal (
-- ** Decimal Values
DecimalRaw (..),
Decimal,
realFracToDecimal,
decimalConvert,
roundTo,
(*.),
divide,
allocate,
eitherFromRational,
normalizeDecimal,
) where

import Control.DeepSeq
import Data.Char
import Data.Ratio
import Data.Word
import Data.Typeable

-- | Raw decimal arithmetic type constructor.  A decimal value consists of an
-- integer mantissa and a negative exponent which is interpreted as the number
-- of decimal places.  The value stored in a @Decimal d@ is therefore equal to:
--
-- > decimalMantissa d / (10 ^ decimalPlaces d)
--
-- The "Show" instance will add trailing zeros, so @show \$ Decimal 3 1500@
-- will return \"1.500\".  Conversely the "Read" instance will use the decimal
-- places to determine the precision.
--
-- Arithmetic and comparision operators convert their arguments to the
-- greater of the two precisions, and return a result of that precision.
-- Regardless of the type of the arguments, all mantissa arithmetic is done
-- using @Integer@ types, so application developers do not need to worry about
-- overflow in the internal algorithms.  However the result of each operator
-- will be converted to the mantissa type without checking for overflow.
data (Integral i) => DecimalRaw i = Decimal {
decimalPlaces :: ! Word8,
decimalMantissa :: ! i}
deriving (Typeable)

-- | Arbitrary precision decimal type.  As a rule programs should do decimal
-- arithmetic with this type and only convert to other instances of
-- "DecimalRaw" where required by an external interface.
--
-- Using this type is also faster because it avoids repeated conversions
-- to and from @Integer@.
type Decimal = DecimalRaw Integer

instance (Integral i, NFData i) => NFData (DecimalRaw i) where
rnf (Decimal _ i) = rnf i

-- | Convert a real fractional value into a Decimal of the appropriate
-- precision.
realFracToDecimal :: (Integral i, RealFrac r) => Word8 -> r -> DecimalRaw i
realFracToDecimal e r = Decimal e \$ round (r * (10^e))

-- Internal function to divide and return the nearest integer.
divRound :: (Integral a) => a -> a -> a
divRound n1 n2 = if abs r > abs (n2 `quot` 2) then n + signum n else n
where (n, r) = n1 `quotRem` n2

-- | Convert a @DecimalRaw@ from one base representation to another.  Does
-- not check for overflow in the new representation.
decimalConvert :: (Integral a, Integral b) => DecimalRaw a -> DecimalRaw b
decimalConvert (Decimal e n) = Decimal e \$ fromIntegral n

-- | Round a @DecimalRaw@ to a specified number of decimal places.
roundTo :: (Integral i) => Word8 -> DecimalRaw i -> DecimalRaw Integer
roundTo d (Decimal e n) = Decimal d \$ fromIntegral n1
where
n1 = case compare d e of
LT -> n `divRound` divisor
EQ -> n
GT -> n * multiplier
divisor = 10 ^ (e-d)
multiplier = 10 ^ (d-e)

-- Round the two DecimalRaw values to the largest exponent.
roundMax :: (Integral i) =>
DecimalRaw i -> DecimalRaw i -> (Word8, Integer, Integer)
roundMax d1@(Decimal e1 _) d2@(Decimal e2 _) = (e, n1, n2)
where
e = max e1 e2
(Decimal _ n1) = roundTo e d1
(Decimal _ n2) = roundTo e d2

instance (Integral i, Show i) => Show (DecimalRaw i) where
showsPrec _ (Decimal e n)
| e == 0     = (concat [signStr, strN] ++)
| otherwise  = (concat [signStr, intPart, ".", fracPart] ++)
where
strN = show \$ abs n
signStr = if n < 0 then "-" else ""
len = length strN
padded = replicate (fromIntegral e + 1 - len) '0' ++ strN
(intPart, fracPart) = splitAt (max 1 (len - fromIntegral e)) padded

(intPart, _) <- gather \$ do
optional \$ char '-'
munch1 isDigit
fractPart    <- option "" \$ do
_ <- char '.'
munch1 isDigit
return \$ Decimal (fromIntegral \$ length fractPart) \$ read \$
intPart ++ fractPart

instance (Integral i) => Eq (DecimalRaw i) where
d1 == d2   =   n1 == n2 where (_, n1, n2) = roundMax d1 d2

instance (Integral i) => Ord (DecimalRaw i) where
compare d1 d2 = compare n1 n2 where (_, n1, n2) = roundMax d1 d2

instance (Integral i) => Num (DecimalRaw i) where
d1 + d2 = Decimal e \$ fromIntegral (n1 + n2)
where (e, n1, n2) = roundMax d1 d2
d1 - d2 = Decimal e \$ fromIntegral (n1 - n2)
where (e, n1, n2) = roundMax d1 d2
d1 * d2 = normalizeDecimal \$ realFracToDecimal maxBound \$ (toRational d1) * (toRational d2)

abs (Decimal e n) = Decimal e \$ abs n
signum (Decimal _ n) = fromIntegral \$ signum n
fromInteger n = Decimal 0 \$ fromIntegral n

instance (Integral i) => Real (DecimalRaw i) where
toRational (Decimal e n) = fromIntegral n % (10 ^ e)

instance (Integral i) => Fractional (DecimalRaw i) where
fromRational r = normalizeDecimal \$ realFracToDecimal maxBound r
a / b = fromRational \$ (toRational a) / (toRational b)

instance (Integral i) => RealFrac (DecimalRaw i) where
properFraction a = (rnd, fromRational rep)
where
(rnd, rep) = properFraction \$ toRational a

-- | Divide a @DecimalRaw@ value into one or more portions.  The portions
-- will be approximately equal, and the sum of the portions is guaranteed to
-- be the original value.
--
-- The portions are represented as a list of pairs.  The first part of each
-- pair is the number of portions, and the second part is the portion value.
-- Hence 10 dollars divided 3 ways will produce @[(2, 3.33), (1, 3.34)]@.
divide :: (Integral i) => DecimalRaw i -> Int -> [(Int, DecimalRaw i)]
divide (Decimal e n) d
| d > 0 =
case n `divMod` fromIntegral d of
(result, 0) -> [(fromIntegral d, Decimal e result)]
(result, r) -> [(fromIntegral d - fromIntegral r,
Decimal e result),
(fromIntegral r, Decimal e (result+1))]
| otherwise = error "Data.Decimal.divide: Divisor must be > 0."

-- | Allocate a @DecimalRaw@ value proportionately with the values in a list.
-- The allocated portions are guaranteed to add up to the original value.
--
-- Some of the allocations may be zero or negative, but the sum of the list
-- must not be zero.  The allocation is intended to be as close as possible
-- to the following:
--
-- > let result = allocate d parts
-- > in all (== d / sum parts) \$ zipWith (/) result parts
allocate :: (Integral i) => DecimalRaw i -> [Integer] -> [DecimalRaw i]
allocate (Decimal e n) ps
| total == 0  =
error "Data.Decimal.allocate: allocation list must not sum to zero."
| otherwise   = map (Decimal e) \$ zipWith (-) ts (tail ts)
where
ts = map fst \$ scanl nxt (n, total) ps
nxt (n1, t1) p1 = (n1 - (n1 * fromIntegral p1) `zdiv` t1,
t1 - fromIntegral p1)
zdiv 0 0 = 0
zdiv x y = x `divRound` y
total = fromIntegral \$ sum ps

-- | Multiply a @DecimalRaw@ by a @RealFrac@ value.
(*.) :: (Integral i, RealFrac r) => DecimalRaw i -> r -> DecimalRaw i
(Decimal e m) *. d = Decimal e \$ round \$ fromIntegral m * d

-- | Count the divisors, i.e. the count of 2 divisors in 18 is 1 because 18 = 2 * 3 * 3
factorN :: (Integral a)
=> a                  -- ^ Denominator base
-> a                  -- ^ dividing value
-> (a, a)             -- ^ The count of divisors and the result of division
factorN d val = factorN' val 0
where
factorN' 1 acc = (acc, 1)
factorN' v acc = if md == 0
then factorN' vd (acc + 1)
else (acc, v)
where
(vd, md) = v `divMod` d

-- | Try to convert Rational to Decimal with absolute precision
-- return string with fail description if not converted
eitherFromRational :: (Integral i) => Rational -> Either String (DecimalRaw i)
eitherFromRational r = if done == 1
then do
wres <- we
return \$ Decimal wres (fromIntegral m)
else Left \$ show r ++ " has no decimal denominator"
where
den = denominator r
num = numerator r
(f2, rest) = factorN 2 den
(f5, done) = factorN 5 rest
e = max f2 f5
m = num * ((10^e) `div` den)
we = if e > (fromIntegral (maxBound :: Word8)) --  FIXME: will fail if DecimalRaw changed
then Left \$ show e ++ " is too big ten power to represent as Decimal"
else Right \$ fromIntegral e

-- | Reduce the exponent of the decimal numer to the minimal posible value
normalizeDecimal :: (Integral i) => (DecimalRaw i) -> (DecimalRaw i)
normalizeDecimal r = case eitherFromRational \$ toRational r of
Right x -> x
Left e -> error \$ "Imposible happened: " ++ e
```