```-- | 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,
-- ** QuickCheck Properties
prop_fromIntegerZero,
prop_increaseDecimals,
prop_decreaseDecimals,
prop_divisionParts,
prop_divisionUnits,
prop_allocateParts,
prop_allocateUnits,
prop_abs,
prop_signum
) where

import Data.Char
import Data.Ratio
import Data.Word
import Test.QuickCheck

-- | 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}

-- | 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

-- | 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 (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 = Decimal e \$ fromIntegral \$
(n1 * n2) `divRound` (10 ^ e)
where (e, n1, n2) = roundMax d1 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, Arbitrary i) => Arbitrary (DecimalRaw i) where
arbitrary = do
e <- sized (\n -> resize (n `div` 10) arbitrary) :: Gen Int
m <- sized (\n -> resize (n * 10) arbitrary)
return \$ Decimal (fromIntegral \$ abs e) m
coarbitrary (Decimal e m) gen =
variant (fromIntegral e + fromIntegral m) gen

-- | 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 -> [Int] -> [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

-- | "read" is the inverse of "show".
--
-- > read (show n) == n

-- | Read and show preserve decimal places.
--
-- > decimalPlaces (read (show n)) == decimalPlaces n
== decimalPlaces d

-- | "fromInteger" definition.
--
-- > decimalPlaces (fromInteger n) == 0 &&
-- > decimalMantissa (fromInteger n) == n
prop_fromIntegerZero :: Integer -> Bool
prop_fromIntegerZero n =  decimalPlaces (fromInteger n :: Decimal) == 0 &&
decimalMantissa (fromInteger n :: Decimal) == n

-- | Increased precision does not affect equality.
--
-- > decimalPlaces d < maxBound ==> roundTo (decimalPlaces d + 1) d == d
prop_increaseDecimals :: Decimal -> Property
prop_increaseDecimals d =
decimalPlaces d < maxBound ==> roundTo (decimalPlaces d + 1) d == d

-- | Decreased precision can make two decimals equal, but it can never change
-- their order.
--
-- > forAll d1, d2 :: Decimal -> legal beforeRound afterRound
-- >      where
-- >         beforeRound = compare d1 d2
-- >         afterRound = compare (roundTo 0 d1) (roundTo 0 d2)
-- >         legal GT x = x `elem` [GT, EQ]
-- >         legal EQ x = x `elem` [EQ]
-- >         legal LT x = x `elem` [LT, EQ]
prop_decreaseDecimals :: Decimal -> Decimal -> Bool
prop_decreaseDecimals d1 d2 =  legal beforeRound afterRound
where
beforeRound = compare d1 d2
afterRound = compare (roundTo 0 d1) (roundTo 0 d2)
legal GT x = x `elem` [GT, EQ]
legal EQ x = x `elem` [EQ]
legal LT x = x `elem` [LT, EQ]

-- | > (x + y) - y == x
prop_inverseAdd :: Decimal -> Decimal -> Bool
prop_inverseAdd x y =  (x + y) - y == x

-- | Multiplication is repeated addition.
--
-- > (sum \$ replicate i d) == d * fromIntegral (max i 0)
prop_repeatedAdd :: Decimal -> Int -> Bool
prop_repeatedAdd d i = (sum \$ replicate i d) == d * fromIntegral (max i 0)

-- | Division produces the right number of parts.
--
-- > i > 0 ==> (sum \$ map fst \$ divide d i) == i
prop_divisionParts :: Decimal -> Int -> Property
prop_divisionParts d i =  i > 0 ==> (sum \$ map fst \$ divide d i) == i

-- | Division doesn't drop any units.
--
-- > i > 0 ==> (sum \$ map (\(n,d1) -> fromIntegral n * d1) \$ divide d i) == d
prop_divisionUnits :: Decimal -> Int -> Property
prop_divisionUnits d i =
i > 0 ==> (sum \$ map (\(n,d1) -> fromIntegral n * d1) \$ divide d i) == d

-- | Allocate produces the right number of parts.
--
-- > (not . null) ps ==> length ps == length (allocate d ps)
prop_allocateParts :: Decimal -> [Int] -> Property
prop_allocateParts d ps =
sum ps /= 0 ==> length ps == length (allocate d ps)

-- | Allocate doesn't drop any units.
--
-- >     (not . null) ps ==> sum (allocate d ps) == d
prop_allocateUnits :: Decimal -> [Int] -> Property
prop_allocateUnits d ps =
sum ps /= 0 ==> sum (allocate d ps) == d

-- | Absolute value definition
--
-- > decimalPlaces a == decimalPlaces d &&
-- > decimalMantissa a == abs (decimalMantissa d)
-- >    where a = abs d
prop_abs :: Decimal -> Bool
prop_abs d =  decimalPlaces a == decimalPlaces d &&
decimalMantissa a == abs (decimalMantissa d)
where a = abs d

-- | Sign number defintion
--
-- > signum d == (fromInteger \$ signum \$ decimalMantissa d)
prop_signum :: Decimal -> Bool
prop_signum d =  signum d == (fromInteger \$ signum \$ decimalMantissa d)
```