{-# LANGUAGE BangPatterns #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE FunctionalDependencies #-}
{-# LANGUAGE GeneralizedNewtypeDeriving #-}
{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE RankNTypes #-}
{-# LANGUAGE Trustworthy #-}
{-# LANGUAGE TypeFamilies #-}
{-# LANGUAGE UndecidableInstances #-}

-- |
-- Module      :  System.Random.Stateful
-- Copyright   :  (c) The University of Glasgow 2001
-- License     :  BSD-style (see the file LICENSE in the 'random' repository)
-- Maintainer  :  libraries@haskell.org
-- Stability   :  stable
--
-- This library deals with the common task of pseudo-random number generation.
module System.Random.Stateful
  (
  -- * Pure Random Generator
  module System.Random
  -- * Monadic Random Generator
  -- $introduction

  -- * Usage
  -- $usagemonadic

  -- * Mutable pseudo-random number generator interfaces
  -- $interfaces
  , StatefulGen(..)
  , FrozenGen(..)
  , RandomGenM(..)
  , withMutableGen
  , withMutableGen_
  , randomM
  , randomRM
  , splitGenM

  -- * Monadic adapters for pure pseudo-random number generators #monadicadapters#
  -- $monadicadapters

  -- ** Pure adapter
  , StateGen(..)
  , StateGenM(..)
  , runStateGen
  , runStateGen_
  , runStateGenT
  , runStateGenT_
  , runStateGenST
  , runStateGenST_
  -- ** Mutable adapter with atomic operations
  , AtomicGen(..)
  , AtomicGenM(..)
  , newAtomicGenM
  , applyAtomicGen
  , globalStdGen
  -- ** Mutable adapter in 'IO'
  , IOGen(..)
  , IOGenM(..)
  , newIOGenM
  , applyIOGen
  -- ** Mutable adapter in 'ST'
  , STGen(..)
  , STGenM(..)
  , newSTGenM
  , applySTGen
  , runSTGen
  , runSTGen_
  -- ** Mutable adapter in 'STM'
  , TGen(..)
  , TGenM(..)
  , newTGenM
  , newTGenMIO
  , applyTGen

  -- * Pseudo-random values of various types
  -- $uniform
  , Uniform(..)
  , uniformListM
  , uniformViaFiniteM
  , UniformRange(..)

  -- * Generators for sequences of pseudo-random bytes
  , genShortByteStringIO
  , genShortByteStringST
  , uniformByteStringM
  , uniformDouble01M
  , uniformDoublePositive01M
  , uniformFloat01M
  , uniformFloatPositive01M
  , uniformEnumM
  , uniformEnumRM

  -- * Appendix

  -- ** How to implement 'StatefulGen'
  -- $implementmonadrandom

  -- ** Floating point number caveats #fpcaveats#
  -- $floating

  -- * References
  -- $references
  ) where

import Control.DeepSeq
import Control.Monad (replicateM)
import Control.Monad.IO.Class
import Control.Monad.ST
import GHC.Conc.Sync (STM, TVar, newTVar, newTVarIO, readTVar, writeTVar)
import Control.Monad.State.Strict (MonadState, state)
import Data.IORef
import Data.STRef
import Foreign.Storable
import System.Random
import System.Random.Internal

-- $introduction
--
-- This module provides type classes and instances for the following concepts:
--
-- [Monadic pseudo-random number generators] 'StatefulGen' is an interface to
--     monadic pseudo-random number generators.
--
-- [Monadic adapters] 'StateGenM', 'AtomicGenM', 'IOGenM', 'STGenM` and 'TGenM'
--     turn a 'RandomGen' instance into a 'StatefulGen' instance.
--
-- [Drawing from a range] 'UniformRange' is used to generate a value of a
--     type uniformly within a range.
--
--     This library provides instances of 'UniformRange' for many common
--     numeric types.
--
-- [Drawing from the entire domain of a type] 'Uniform' is used to generate a
--     value of a type uniformly over all possible values of that type.
--
--     This library provides instances of 'Uniform' for many common bounded
--     numeric types.
--
-- $usagemonadic
--
-- In monadic code, use the relevant 'Uniform' and 'UniformRange' instances to
-- generate pseudo-random values via 'uniformM' and 'uniformRM', respectively.
--
-- As an example, @rollsM@ generates @n@ pseudo-random values of @Word@ in the
-- range @[1, 6]@ in a 'StatefulGen' context; given a /monadic/ pseudo-random
-- number generator, you can run this probabilistic computation as follows:
--
-- >>> :{
-- let rollsM :: StatefulGen g m => Int -> g -> m [Word]
--     rollsM n = replicateM n . uniformRM (1, 6)
-- in do
--     monadicGen <- MWC.create
--     rollsM 10 monadicGen :: IO [Word]
-- :}
-- [3,4,3,1,4,6,1,6,1,4]
--
-- Given a /pure/ pseudo-random number generator, you can run the monadic
-- pseudo-random number computation @rollsM@ in an 'IO' or 'ST' context by
-- applying a monadic adapter like 'AtomicGenM', 'IOGenM' or 'STGenM'
-- (see [monadic-adapters](#monadicadapters)) to the pure pseudo-random number
-- generator.
--
-- >>> :{
-- let rollsM :: StatefulGen g m => Int -> g -> m [Word]
--     rollsM n = replicateM n . uniformRM (1, 6)
--     pureGen = mkStdGen 42
-- in
--     newIOGenM pureGen >>= rollsM 10 :: IO [Word]
-- :}
-- [1,1,3,2,4,5,3,4,6,2]

-------------------------------------------------------------------------------
-- Pseudo-random number generator interfaces
-------------------------------------------------------------------------------

-- $interfaces
--
-- Pseudo-random number generators come in two flavours: /pure/ and /monadic/.
--
-- ['System.Random.RandomGen': pure pseudo-random number generators]
--     See "System.Random" module.
--
-- ['StatefulGen': monadic pseudo-random number generators] These generators
--     mutate their own state as they produce pseudo-random values. They
--     generally live in 'ST' or 'IO' or some transformer that implements
--     @PrimMonad@.
--

-------------------------------------------------------------------------------
-- Monadic adapters
-------------------------------------------------------------------------------

-- $monadicadapters
--
-- Pure pseudo-random number generators can be used in monadic code via the
-- adapters 'StateGenM', 'AtomicGenM', 'IOGenM', 'STGenM' and 'TGenM'
--
-- *   'StateGenM' can be used in any state monad. With strict 'StateT' there is
--     no performance overhead compared to using the 'RandomGen' instance
--     directly. 'StateGenM' is /not/ safe to use in the presence of exceptions
--     and concurrency.
--
-- *   'AtomicGenM' is safe in the presence of exceptions and concurrency since
--     it performs all actions atomically.
--
-- *   'IOGenM' is a wrapper around an 'IORef' that holds a pure generator.
--     'IOGenM' is safe in the presence of exceptions, but not concurrency.
--
-- *   'STGenM' is a wrapper around an 'STRef' that holds a pure generator.
--     'STGenM' is safe in the presence of exceptions, but not concurrency.
--
-- *   'TGenM' is a wrapper around a 'TVar' that holds a pure generator. 'TGenM'
--     can be used in a software transactional memory monad 'STM`. It is not as
--     performant as 'AtomicGenM`, but it can provide stronger guarantees in a
--     concurrent setting.

-- | Interface to operations on 'RandomGen' wrappers like 'IOGenM' and 'StateGenM'.
--
-- @since 1.2.0
class (RandomGen r, StatefulGen g m) => RandomGenM g r m | g -> r where
  applyRandomGenM :: (r -> (a, r)) -> g -> m a

-- | Splits a pseudo-random number generator into two. Overwrites the mutable
-- wrapper with one of the resulting generators and returns the other.
--
-- @since 1.2.0
splitGenM :: RandomGenM g r m => g -> m r
splitGenM = applyRandomGenM split

instance (RandomGen r, MonadIO m) => RandomGenM (IOGenM r) r m where
  applyRandomGenM = applyIOGen

instance (RandomGen r, MonadIO m) => RandomGenM (AtomicGenM r) r m where
  applyRandomGenM = applyAtomicGen

instance (RandomGen r, MonadState r m) => RandomGenM (StateGenM r) r m where
  applyRandomGenM f _ = state f

instance RandomGen r => RandomGenM (STGenM r s) r (ST s) where
  applyRandomGenM = applySTGen

instance RandomGen r => RandomGenM (TGenM r) r STM where
  applyRandomGenM = applyTGen


-- | Runs a mutable pseudo-random number generator from its 'FrozenGen' state.
--
-- ====__Examples__
--
-- >>> import Data.Int (Int8)
-- >>> withMutableGen (IOGen (mkStdGen 217)) (uniformListM 5) :: IO ([Int8], IOGen StdGen)
-- ([-74,37,-50,-2,3],IOGen {unIOGen = StdGen {unStdGen = SMGen 4273268533320920145 15251669095119325999}})
--
-- @since 1.2.0
withMutableGen :: FrozenGen f m => f -> (MutableGen f m -> m a) -> m (a, f)
withMutableGen fg action = do
  g <- thawGen fg
  res <- action g
  fg' <- freezeGen g
  pure (res, fg')

-- | Same as 'withMutableGen', but only returns the generated value.
--
-- ====__Examples__
--
-- >>> import System.Random.Stateful
-- >>> let pureGen = mkStdGen 137
-- >>> withMutableGen_ (IOGen pureGen) (uniformRM (1 :: Int, 6 :: Int))
-- 4
--
-- @since 1.2.0
withMutableGen_ :: FrozenGen f m => f -> (MutableGen f m -> m a) -> m a
withMutableGen_ fg action = fst <$> withMutableGen fg action


-- | Generates a list of pseudo-random values.
--
-- ====__Examples__
--
-- >>> import System.Random.Stateful
-- >>> let pureGen = mkStdGen 137
-- >>> g <- newIOGenM pureGen
-- >>> uniformListM 10 g :: IO [Bool]
-- [True,True,True,True,False,True,True,False,False,False]
--
-- @since 1.2.0
uniformListM :: (StatefulGen g m, Uniform a) => Int -> g -> m [a]
uniformListM n gen = replicateM n (uniformM gen)

-- | Generates a pseudo-random value using monadic interface and `Random` instance.
--
-- ====__Examples__
--
-- >>> import System.Random.Stateful
-- >>> let pureGen = mkStdGen 137
-- >>> g <- newIOGenM pureGen
-- >>> randomM g :: IO Double
-- 0.5728354935654512
--
-- @since 1.2.0
randomM :: (RandomGenM g r m, Random a) => g -> m a
randomM = applyRandomGenM random

-- | Generates a pseudo-random value using monadic interface and `Random` instance.
--
-- ====__Examples__
--
-- >>> import System.Random.Stateful
-- >>> let pureGen = mkStdGen 137
-- >>> g <- newIOGenM pureGen
-- >>> randomRM (1, 100) g :: IO Int
-- 52
--
-- @since 1.2.0
randomRM :: (RandomGenM g r m, Random a) => (a, a) -> g -> m a
randomRM r = applyRandomGenM (randomR r)

-- | Wraps an 'IORef' that holds a pure pseudo-random number generator. All
-- operations are performed atomically.
--
-- *   'AtomicGenM' is safe in the presence of exceptions and concurrency.
-- *   'AtomicGenM' is the slowest of the monadic adapters due to the overhead
--     of its atomic operations.
--
-- @since 1.2.0
newtype AtomicGenM g = AtomicGenM { unAtomicGenM :: IORef g}


-- | Frozen version of mutable `AtomicGenM` generator
--
-- @since 1.2.0
newtype AtomicGen g = AtomicGen { unAtomicGen :: g}
  deriving (Eq, Ord, Show, RandomGen, Storable, NFData)

-- | Creates a new 'AtomicGenM'.
--
-- @since 1.2.0
newAtomicGenM :: MonadIO m => g -> m (AtomicGenM g)
newAtomicGenM = fmap AtomicGenM . liftIO . newIORef


-- | Global mutable standard pseudo-random number generator. This is the same
-- generator that was historically used by `randomIO` and `randomRIO` functions.
--
-- >>> replicateM 10 (uniformRM ('a', 'z') globalStdGen)
-- "tdzxhyfvgr"
--
-- @since 1.2.1
globalStdGen :: AtomicGenM StdGen
globalStdGen = AtomicGenM theStdGen


instance (RandomGen g, MonadIO m) => StatefulGen (AtomicGenM g) m where
  uniformWord32R r = applyAtomicGen (genWord32R r)
  {-# INLINE uniformWord32R #-}
  uniformWord64R r = applyAtomicGen (genWord64R r)
  {-# INLINE uniformWord64R #-}
  uniformWord8 = applyAtomicGen genWord8
  {-# INLINE uniformWord8 #-}
  uniformWord16 = applyAtomicGen genWord16
  {-# INLINE uniformWord16 #-}
  uniformWord32 = applyAtomicGen genWord32
  {-# INLINE uniformWord32 #-}
  uniformWord64 = applyAtomicGen genWord64
  {-# INLINE uniformWord64 #-}
  uniformShortByteString n = applyAtomicGen (genShortByteString n)


instance (RandomGen g, MonadIO m) => FrozenGen (AtomicGen g) m where
  type MutableGen (AtomicGen g) m = AtomicGenM g
  freezeGen = fmap AtomicGen . liftIO . readIORef . unAtomicGenM
  thawGen (AtomicGen g) = newAtomicGenM g

-- | Atomically applies a pure operation to the wrapped pseudo-random number
-- generator.
--
-- ====__Examples__
--
-- >>> import System.Random.Stateful
-- >>> let pureGen = mkStdGen 137
-- >>> g <- newAtomicGenM pureGen
-- >>> applyAtomicGen random g :: IO Int
-- 7879794327570578227
--
-- @since 1.2.0
applyAtomicGen :: MonadIO m => (g -> (a, g)) -> AtomicGenM g -> m a
applyAtomicGen op (AtomicGenM gVar) =
  liftIO $ atomicModifyIORef' gVar $ \g ->
    case op g of
      (a, g') -> (g', a)
{-# INLINE applyAtomicGen #-}

-- | Wraps an 'IORef' that holds a pure pseudo-random number generator.
--
-- *   'IOGenM' is safe in the presence of exceptions, but not concurrency.
-- *   'IOGenM' is slower than 'StateGenM' due to the extra pointer indirection.
-- *   'IOGenM' is faster than 'AtomicGenM' since the 'IORef' operations used by
--     'IOGenM' are not atomic.
--
-- An example use case is writing pseudo-random bytes into a file:
--
-- >>> import UnliftIO.Temporary (withSystemTempFile)
-- >>> import Data.ByteString (hPutStr)
-- >>> let ioGen g = withSystemTempFile "foo.bin" $ \_ h -> uniformRM (0, 100) g >>= flip uniformByteStringM g >>= hPutStr h
--
-- and then run it:
--
-- >>> newIOGenM (mkStdGen 1729) >>= ioGen
--
-- @since 1.2.0
newtype IOGenM g = IOGenM { unIOGenM :: IORef g }

-- | Frozen version of mutable `IOGenM` generator
--
-- @since 1.2.0
newtype IOGen g = IOGen { unIOGen :: g }
  deriving (Eq, Ord, Show, RandomGen, Storable, NFData)


-- | Creates a new 'IOGenM'.
--
-- @since 1.2.0
newIOGenM :: MonadIO m => g -> m (IOGenM g)
newIOGenM = fmap IOGenM . liftIO . newIORef



instance (RandomGen g, MonadIO m) => StatefulGen (IOGenM g) m where
  uniformWord32R r = applyIOGen (genWord32R r)
  {-# INLINE uniformWord32R #-}
  uniformWord64R r = applyIOGen (genWord64R r)
  {-# INLINE uniformWord64R #-}
  uniformWord8 = applyIOGen genWord8
  {-# INLINE uniformWord8 #-}
  uniformWord16 = applyIOGen genWord16
  {-# INLINE uniformWord16 #-}
  uniformWord32 = applyIOGen genWord32
  {-# INLINE uniformWord32 #-}
  uniformWord64 = applyIOGen genWord64
  {-# INLINE uniformWord64 #-}
  uniformShortByteString n = applyIOGen (genShortByteString n)


instance (RandomGen g, MonadIO m) => FrozenGen (IOGen g) m where
  type MutableGen (IOGen g) m = IOGenM g
  freezeGen = fmap IOGen . liftIO . readIORef . unIOGenM
  thawGen (IOGen g) = newIOGenM g


-- | Applies a pure operation to the wrapped pseudo-random number generator.
--
-- ====__Examples__
--
-- >>> import System.Random.Stateful
-- >>> let pureGen = mkStdGen 137
-- >>> g <- newIOGenM pureGen
-- >>> applyIOGen random g :: IO Int
-- 7879794327570578227
--
-- @since 1.2.0
applyIOGen :: MonadIO m => (g -> (a, g)) -> IOGenM g -> m a
applyIOGen f (IOGenM ref) = liftIO $ do
  g <- readIORef ref
  case f g of
    (!a, !g') -> a <$ writeIORef ref g'
{-# INLINE applyIOGen #-}

-- | Wraps an 'STRef' that holds a pure pseudo-random number generator.
--
-- *   'STGenM' is safe in the presence of exceptions, but not concurrency.
-- *   'STGenM' is slower than 'StateGenM' due to the extra pointer indirection.
--
-- @since 1.2.0
newtype STGenM g s = STGenM { unSTGenM :: STRef s g }

-- | Frozen version of mutable `STGenM` generator
--
-- @since 1.2.0
newtype STGen g = STGen { unSTGen :: g }
  deriving (Eq, Ord, Show, RandomGen, Storable, NFData)

-- | Creates a new 'STGenM'.
--
-- @since 1.2.0
newSTGenM :: g -> ST s (STGenM g s)
newSTGenM = fmap STGenM . newSTRef


instance RandomGen g => StatefulGen (STGenM g s) (ST s) where
  uniformWord32R r = applySTGen (genWord32R r)
  {-# INLINE uniformWord32R #-}
  uniformWord64R r = applySTGen (genWord64R r)
  {-# INLINE uniformWord64R #-}
  uniformWord8 = applySTGen genWord8
  {-# INLINE uniformWord8 #-}
  uniformWord16 = applySTGen genWord16
  {-# INLINE uniformWord16 #-}
  uniformWord32 = applySTGen genWord32
  {-# INLINE uniformWord32 #-}
  uniformWord64 = applySTGen genWord64
  {-# INLINE uniformWord64 #-}
  uniformShortByteString n = applySTGen (genShortByteString n)

instance RandomGen g => FrozenGen (STGen g) (ST s) where
  type MutableGen (STGen g) (ST s) = STGenM g s
  freezeGen = fmap STGen . readSTRef . unSTGenM
  thawGen (STGen g) = newSTGenM g


-- | Applies a pure operation to the wrapped pseudo-random number generator.
--
-- ====__Examples__
--
-- >>> import System.Random.Stateful
-- >>> let pureGen = mkStdGen 137
-- >>> (runSTGen pureGen (\g -> applySTGen random g)) :: (Int, StdGen)
-- (7879794327570578227,StdGen {unStdGen = SMGen 11285859549637045894 7641485672361121627})
--
-- @since 1.2.0
applySTGen :: (g -> (a, g)) -> STGenM g s -> ST s a
applySTGen f (STGenM ref) = do
  g <- readSTRef ref
  case f g of
    (!a, !g') -> a <$ writeSTRef ref g'
{-# INLINE applySTGen #-}

-- | Runs a monadic generating action in the `ST` monad using a pure
-- pseudo-random number generator.
--
-- ====__Examples__
--
-- >>> import System.Random.Stateful
-- >>> let pureGen = mkStdGen 137
-- >>> (runSTGen pureGen (\g -> applySTGen random g)) :: (Int, StdGen)
-- (7879794327570578227,StdGen {unStdGen = SMGen 11285859549637045894 7641485672361121627})
--
-- @since 1.2.0
runSTGen :: RandomGen g => g -> (forall s . STGenM g s -> ST s a) -> (a, g)
runSTGen g action = unSTGen <$> runST (withMutableGen (STGen g) action)

-- | Runs a monadic generating action in the `ST` monad using a pure
-- pseudo-random number generator. Returns only the resulting pseudo-random
-- value.
--
-- ====__Examples__
--
-- >>> import System.Random.Stateful
-- >>> let pureGen = mkStdGen 137
-- >>> (runSTGen_ pureGen (\g -> applySTGen random g)) :: Int
-- 7879794327570578227
--
-- @since 1.2.0
runSTGen_ :: RandomGen g => g -> (forall s . STGenM g s -> ST s a) -> a
runSTGen_ g action = fst $ runSTGen g action


-- | Wraps a 'TVar' that holds a pure pseudo-random number generator.
--
-- @since 1.2.1
newtype TGenM g = TGenM { unTGenM :: TVar g }

-- | Frozen version of mutable `TGenM` generator
--
-- @since 1.2.1
newtype TGen g = TGen { unTGen :: g }
  deriving (Eq, Ord, Show, RandomGen, Storable, NFData)

-- | Creates a new 'TGenM' in `STM`.
--
-- @since 1.2.1
newTGenM :: g -> STM (TGenM g)
newTGenM = fmap TGenM . newTVar


-- | Creates a new 'TGenM' in `IO`.
--
-- @since 1.2.1
newTGenMIO :: MonadIO m => g -> m (TGenM g)
newTGenMIO g = liftIO (TGenM <$> newTVarIO g)


-- | @since 1.2.1
instance RandomGen g => StatefulGen (TGenM g) STM where
  uniformWord32R r = applyTGen (genWord32R r)
  {-# INLINE uniformWord32R #-}
  uniformWord64R r = applyTGen (genWord64R r)
  {-# INLINE uniformWord64R #-}
  uniformWord8 = applyTGen genWord8
  {-# INLINE uniformWord8 #-}
  uniformWord16 = applyTGen genWord16
  {-# INLINE uniformWord16 #-}
  uniformWord32 = applyTGen genWord32
  {-# INLINE uniformWord32 #-}
  uniformWord64 = applyTGen genWord64
  {-# INLINE uniformWord64 #-}
  uniformShortByteString n = applyTGen (genShortByteString n)

-- | @since 1.2.1
instance RandomGen g => FrozenGen (TGen g) STM where
  type MutableGen (TGen g) STM = TGenM g
  freezeGen = fmap TGen . readTVar . unTGenM
  thawGen (TGen g) = newTGenM g


-- | Applies a pure operation to the wrapped pseudo-random number generator.
--
-- ====__Examples__
--
-- >>> import Control.Concurrent.STM
-- >>> import System.Random.Stateful
-- >>> import Data.Int (Int32)
-- >>> let pureGen = mkStdGen 137
-- >>> stmGen <- newTGenMIO pureGen
-- >>> atomically $ applyTGen uniform stmGen :: IO Int32
-- 637238067
--
-- @since 1.2.1
applyTGen :: (g -> (a, g)) -> TGenM g -> STM a
applyTGen f (TGenM tvar) = do
  g <- readTVar tvar
  case f g of
    (a, !g') -> a <$ writeTVar tvar g'
{-# INLINE applyTGen #-}

-- $uniform
--
-- This library provides two type classes to generate pseudo-random values:
--
-- *   'UniformRange' is used to generate a value of a type uniformly within a
--     range.
-- *   'Uniform' is used to generate a value of a type uniformly over all
--     possible values of that type.
--
-- Types may have instances for both or just one of 'UniformRange' and
-- 'Uniform'. A few examples illustrate this:
--
-- *   'Int', 'Data.Word.Word16' and 'Bool' are instances of both 'UniformRange' and
--     'Uniform'.
-- *   'Integer', 'Float' and 'Double' each have an instance for 'UniformRange'
--     but no 'Uniform' instance.
-- *   A hypothetical type @Radian@ representing angles by taking values in the
--     range @[0, 2π)@ has a trivial 'Uniform' instance, but no 'UniformRange'
--     instance: the problem is that two given @Radian@ values always span /two/
--     ranges, one clockwise and one anti-clockwise.
-- *   It is trivial to construct a @Uniform (a, b)@ instance given
--     @Uniform a@ and @Uniform b@ (and this library provides this tuple
--     instance).
-- *   On the other hand, there is no correct way to construct a
--     @UniformRange (a, b)@ instance based on just @UniformRange a@ and
--     @UniformRange b@.

-------------------------------------------------------------------------------
-- Notes
-------------------------------------------------------------------------------

-- $floating
--
-- The 'UniformRange' instances for 'Float' and 'Double' use the following
-- procedure to generate a random value in a range for @uniformRM (a, b) g@:
--
-- If \(a = b\), return \(a\). Otherwise:
--
-- 1.  Generate \(x\) uniformly such that \(0 \leq x \leq 1\).
--
--     The method by which \(x\) is sampled does not cover all representable
--     floating point numbers in the unit interval. The method never generates
--     denormal floating point numbers, for example.
--
-- 2.  Return \(x \cdot a + (1 - x) \cdot b\).
--
--     Due to rounding errors, floating point operations are neither
--     associative nor distributive the way the corresponding operations on
--     real numbers are. Additionally, floating point numbers admit special
--     values @NaN@ as well as negative and positive infinity.
--
-- For pathological values, step 2 can yield surprising results.
--
-- *   The result may be greater than @max a b@.
--
--     >>> :{
--     let (a, b, x) = (-2.13238e-29, -2.1323799e-29, 0.27736077)
--         result = x * a + (1 - x) * b :: Float
--     in (result, result > max a b)
--     :}
--     (-2.1323797e-29,True)
--
-- *   The result may be smaller than @min a b@.
--
--     >>> :{
--     let (a, b, x) = (-1.9087862, -1.908786, 0.4228573)
--         result = x * a + (1 - x) * b :: Float
--     in (result, result < min a b)
--     :}
--     (-1.9087863,True)
--
-- What happens when @NaN@ or @Infinity@ are given to 'uniformRM'? We first
-- define them as constants:
--
-- >>> nan = read "NaN" :: Float
-- >>> inf = read "Infinity" :: Float
--
-- *   If at least one of \(a\) or \(b\) is @NaN@, the result is @NaN@.
--
--     >>> let (a, b, x) = (nan, 1, 0.5) in x * a + (1 - x) * b
--     NaN
--     >>> let (a, b, x) = (-1, nan, 0.5) in x * a + (1 - x) * b
--     NaN
--
-- *   If \(a\) is @-Infinity@ and \(b\) is @Infinity@, the result is @NaN@.
--
--     >>> let (a, b, x) = (-inf, inf, 0.5) in x * a + (1 - x) * b
--     NaN
--
-- *   Otherwise, if \(a\) is @Infinity@ or @-Infinity@, the result is \(a\).
--
--     >>> let (a, b, x) = (inf, 1, 0.5) in x * a + (1 - x) * b
--     Infinity
--     >>> let (a, b, x) = (-inf, 1, 0.5) in x * a + (1 - x) * b
--     -Infinity
--
-- *   Otherwise, if \(b\) is @Infinity@ or @-Infinity@, the result is \(b\).
--
--     >>> let (a, b, x) = (1, inf, 0.5) in x * a + (1 - x) * b
--     Infinity
--     >>> let (a, b, x) = (1, -inf, 0.5) in x * a + (1 - x) * b
--     -Infinity
--
-- Note that the [GCC 10.1.0 C++ standard library](https://gcc.gnu.org/git/?p=gcc.git;a=blob;f=libstdc%2B%2B-v3/include/bits/random.h;h=19307fbc3ca401976ef6823e8fda893e4a263751;hb=63fa67847628e5f358e7e2e7edb8314f0ee31f30#l1859),
-- the [Java 10 standard library](https://docs.oracle.com/javase/10/docs/api/java/util/Random.html#doubles%28double,double%29)
-- and [CPython 3.8](https://github.com/python/cpython/blob/3.8/Lib/random.py#L417)
-- use the same procedure to generate floating point values in a range.
--
-- $implementmonadrandom
--
-- Typically, a monadic pseudo-random number generator has facilities to save
-- and restore its internal state in addition to generating pseudo-random numbers.
--
-- Here is an example instance for the monadic pseudo-random number generator
-- from the @mwc-random@ package:
--
-- > instance (s ~ PrimState m, PrimMonad m) => StatefulGen (MWC.Gen s) m where
-- >   uniformWord8 = MWC.uniform
-- >   uniformWord16 = MWC.uniform
-- >   uniformWord32 = MWC.uniform
-- >   uniformWord64 = MWC.uniform
-- >   uniformShortByteString n g = unsafeSTToPrim (genShortByteStringST n (MWC.uniform g))
--
-- > instance PrimMonad m => FrozenGen MWC.Seed m where
-- >   type MutableGen MWC.Seed m = MWC.Gen (PrimState m)
-- >   thawGen = MWC.restore
-- >   freezeGen = MWC.save
--
-- === @FrozenGen@
--
-- `FrozenGen` gives us ability to use any stateful pseudo-random number generator in its
-- immutable form, if one exists that is. This concept is commonly known as a seed, which
-- allows us to save and restore the actual mutable state of a pseudo-random number
-- generator. The biggest benefit that can be drawn from a polymorphic access to a
-- stateful pseudo-random number generator in a frozen form is the ability to serialize,
-- deserialize and possibly even use the stateful generator in a pure setting without
-- knowing the actual type of a generator ahead of time. For example we can write a
-- function that accepts a frozen state of some pseudo-random number generator and
-- produces a short list with random even integers.
--
-- >>> import Data.Int (Int8)
-- >>> :{
-- myCustomRandomList :: FrozenGen f m => f -> m [Int8]
-- myCustomRandomList f =
--   withMutableGen_ f $ \gen -> do
--     len <- uniformRM (5, 10) gen
--     replicateM len $ do
--       x <- uniformM gen
--       pure $ if even x then x else x + 1
-- :}
--
-- and later we can apply it to a frozen version of a stateful generator, such as `STGen`:
--
-- >>> print $ runST $ myCustomRandomList (STGen (mkStdGen 217))
-- [-50,-2,4,-8,-58,-40,24,-32,-110,24]
--
-- or a @Seed@ from @mwc-random@:
--
-- >>> import Data.Vector.Primitive as P
-- >>> print $ runST $ myCustomRandomList (MWC.toSeed (P.fromList [1,2,3]))
-- [24,40,10,40,-8,48,-78,70,-12]
--
-- Alternatively, instead of discarding the final state of the generator, as it happens
-- above, we could have used `withMutableGen`, which together with the result would give
-- us back its frozen form. This would allow us to store the end state of our generator
-- somewhere for the later reuse.
--
--
-- $references
--
-- 1. Guy L. Steele, Jr., Doug Lea, and Christine H. Flood. 2014. Fast
-- splittable pseudorandom number generators. In Proceedings of the 2014 ACM
-- International Conference on Object Oriented Programming Systems Languages &
-- Applications (OOPSLA '14). ACM, New York, NY, USA, 453-472. DOI:
-- <https://doi.org/10.1145/2660193.2660195>

-- $setup
-- >>> import Control.Monad.Primitive
-- >>> import qualified System.Random.MWC as MWC
-- >>> writeIORef theStdGen $ mkStdGen 2021
--
-- >>> :set -XFlexibleContexts
-- >>> :set -XFlexibleInstances
-- >>> :set -XMultiParamTypeClasses
-- >>> :set -XTypeFamilies
-- >>> :set -XUndecidableInstances
--
-- >>> :{
-- instance (s ~ PrimState m, PrimMonad m) => StatefulGen (MWC.Gen s) m where
--   uniformWord8 = MWC.uniform
--   uniformWord16 = MWC.uniform
--   uniformWord32 = MWC.uniform
--   uniformWord64 = MWC.uniform
--   uniformShortByteString n g = unsafeSTToPrim (genShortByteStringST n (MWC.uniform g))
-- instance PrimMonad m => FrozenGen MWC.Seed m where
--   type MutableGen MWC.Seed m = MWC.Gen (PrimState m)
--   thawGen = MWC.restore
--   freezeGen = MWC.save
-- :}
--