```{-# LANGUAGE Rank2Types #-}
--------------------------------------------------------------------------------
-- |
-- Module      :  Numeric.Lens
-- Copyright   :  (C) 2012-14 Edward Kmett
-- Maintainer  :  Edward Kmett <ekmett@gmail.com>
-- Stability   :  provisional
-- Portability :  portable
-------------------------------------------------------------------------------
module Numeric.Lens
( base
, integral
-- * Predefined bases
, binary
, octal
, decimal
, hex
-- * Arithmetic lenses
, subtracting
, multiplying
, dividing
, exponentiating
) where

import Control.Lens
import Data.Char (chr, ord, isAsciiLower, isAsciiUpper, isDigit)
import Data.Maybe (fromMaybe)

-- \$setup
-- >>> import Data.Monoid (Sum(..))

-- | This 'Prism' extracts can be used to model the fact that every 'Integral'
-- type is a subset of 'Integer'.
--
-- Embedding through the 'Prism' only succeeds if the 'Integer' would pass
-- through unmodified when re-extracted.
integral :: (Integral a, Integral b) => Prism Integer Integer a b
integral = prism toInteger \$ \ i -> let a = fromInteger i in
if toInteger a == i
then Right a
else Left i

-- | A prism that shows and reads integers in base-2 through base-36
--
-- >>> "100" ^? base 16
-- Just 256
--
-- >>> 1767707668033969 ^. re (base 36)
-- "helloworld"
base :: Integral a => Int -> Prism' String a
base b
| b < 2 || b > 36 = error ("base: Invalid base " ++ show b)
| otherwise       = prism intShow intRead
where
intShow n = showSigned' (showIntAtBase (toInteger b) intToDigit') (toInteger n) ""

[(n,"")] -> Right n
_ -> Left s
{-# INLINE base #-}

-- | Like 'Data.Char.intToDigit', but handles up to base-36
intToDigit' :: Int -> Char
intToDigit' i
| i >= 0  && i < 10 = chr (ord '0' + i)
| i >= 10 && i < 36 = chr (ord 'a' + i - 10)
| otherwise = error ("intToDigit': Invalid int " ++ show i)

-- | Like 'Data.Char.digitToInt', but handles up to base-36
digitToInt' :: Char -> Int
digitToInt' c = fromMaybe (error ("digitToInt': Invalid digit " ++ show c))
(digitToIntMay c)

-- | A safe variant of 'digitToInt''
digitToIntMay :: Char -> Maybe Int
digitToIntMay c
| isDigit c      = Just (ord c - ord '0')
| isAsciiLower c = Just (ord c - ord 'a' + 10)
| isAsciiUpper c = Just (ord c - ord 'A' + 10)
| otherwise = Nothing

-- | Select digits that fall into the given base
isDigit' :: Int -> Char -> Bool
isDigit' b c = case digitToIntMay c of
Just i -> i < b
_ -> False

-- | A simpler variant of 'Numeric.showSigned' that only prepends a dash and
showSigned' :: Real a => (a -> ShowS) -> a -> ShowS
showSigned' f n
| n < 0     = showChar '-' . f (negate n)
| otherwise = f n

-- | A simpler variant of 'Numeric.readSigned' that supports any base, only
-- recognizes an initial dash and doesn't know about parentheses
readSigned' f ('-':xs) = f xs & mapped . _1 %~ negate
readSigned' f xs       = f xs

-- | @'binary' = 'base' 2@
binary :: Integral a => Prism' String a
binary = base 2

-- | @'octal' = 'base' 8@
octal :: Integral a => Prism' String a
octal = base 8

-- | @'decimal' = 'base' 10@
decimal :: Integral a => Prism' String a
decimal = base 10

-- | @'hex' = 'base' 16@
hex :: Integral a => Prism' String a
hex = base 16

-- | @'adding' n = 'iso' (+n) (subtract n)@
--
-- [1001,1002,1003]
adding :: Num a => a -> Iso' a a
adding n = iso (+n) (subtract n)

-- | @
-- 'subtracting' n = 'iso' (subtract n) ((+n)
-- 'subtracting' n = 'from' ('adding' n)
-- @
subtracting :: Num a => a -> Iso' a a
subtracting n = iso (subtract n) (+n)

-- | @'multiplying' n = iso (*n) (/n)@
--
-- Note: This errors for n = 0
--
-- >>> 5 & multiplying 1000 +~ 3
-- 5.003
--
-- >>> let fahrenheit = multiplying (9/5).adding 32 in 230^.from fahrenheit
-- 110.0
multiplying :: (Fractional a, Eq a) => a -> Iso' a a
multiplying 0 = error "Numeric.Lens.multiplying: factor 0"
multiplying n = iso (*n) (/n)

-- | @
-- 'dividing' n = 'iso' (/n) (*n)
-- 'dividing' n = 'from' ('multiplying' n)@
--
-- Note: This errors for n = 0
dividing :: (Fractional a, Eq a) => a -> Iso' a a
dividing 0 = error "Numeric.Lens.dividing: divisor 0"
dividing n = iso (/n) (*n)

-- | @'exponentiating' n = 'iso' (**n) (**recip n)@
--
-- Note: This errors for n = 0
--
-- >>> au (exponentiating 2._Unwrapping Sum) (foldMapOf each) (3,4)
-- 5.0
exponentiating :: (Floating a, Eq a) => a -> Iso' a a
exponentiating 0 = error "Numeric.Lens.exponentiating: exponent 0"
exponentiating n = iso (**n) (**recip n)
```