```{-# LANGUAGE
MultiParamTypeClasses,
FlexibleInstances, FlexibleContexts,
UndecidableInstances, BangPatterns
#-}

module Data.Random.Distribution.Gamma
( Gamma(..)
, gamma, gammaT

, Erlang(..)
, erlang, erlangT

, mtGamma
) where

import Data.Random.RVar
import Data.Random.Distribution
import Data.Random.Distribution.Uniform
import Data.Random.Distribution.Normal

import Data.Ratio

import Math.Gamma (p)

-- |derived from  Marsaglia & Tang, "A Simple Method for generating gamma
-- variables", ACM Transactions on Mathematical Software, Vol 26, No 3 (2000), p363-372.
{-# SPECIALIZE mtGamma :: Double -> Double -> RVarT m Double #-}
{-# SPECIALIZE mtGamma :: Float  -> Float  -> RVarT m Float  #-}
mtGamma
:: (Floating a, Ord a,
Distribution StdUniform a,
Distribution Normal a)
=> a -> a -> RVarT m a
mtGamma a b
| a < 1     = do
u <- stdUniformT
mtGamma (1+a) \$! (b * u ** recip a)
| otherwise = go
where
!d = a - fromRational (1%3)
!c = recip (sqrt (9*d))

go = do
x <- stdNormalT
let !v   = 1 + c*x

if v <= 0
then go
else do
u  <- stdUniformT
let !x_2 = x*x; !x_4 = x_2*x_2
v3 = v*v*v
dv = d * v3
if      u < 1 - 0.0331*x_4
|| log u < 0.5 * x_2 + d - dv + d*log v3
then return (b*dv)
else go

{-# SPECIALIZE gamma :: Float  -> Float  -> RVar Float  #-}
{-# SPECIALIZE gamma :: Double -> Double -> RVar Double #-}
gamma :: (Distribution Gamma a) => a -> a -> RVar a
gamma a b = rvar (Gamma a b)

gammaT :: (Distribution Gamma a) => a -> a -> RVarT m a
gammaT a b = rvarT (Gamma a b)

erlang :: (Distribution (Erlang a) b) => a -> RVar b
erlang a = rvar (Erlang a)

erlangT :: (Distribution (Erlang a) b) => a -> RVarT m b
erlangT a = rvarT (Erlang a)

data    Gamma a    = Gamma a a
newtype Erlang a b = Erlang a

instance (Floating a, Ord a, Distribution Normal a, Distribution StdUniform a) => Distribution Gamma a where
{-# SPECIALIZE instance Distribution Gamma Double #-}
{-# SPECIALIZE instance Distribution Gamma Float #-}
rvarT (Gamma a b) = mtGamma a b

instance (Real a, Distribution Gamma a) => CDF Gamma a where
cdf (Gamma a b) x = p (realToFrac a) (realToFrac x / realToFrac b)

instance (Integral a, Floating b, Ord b, Distribution Normal b, Distribution StdUniform b) => Distribution (Erlang a) b where
rvarT (Erlang a) = mtGamma (fromIntegral a) 1

instance (Integral a, Real b, Distribution (Erlang a) b) => CDF (Erlang a) b where
cdf (Erlang a) x = p (fromIntegral a) (realToFrac x)

```