# picomath

### Gamma.hs

```module Gamma (gamma, logGamma) where

-- Visit http://www.johndcook.com/stand_alone_code.html for the source of this code and more like it.

-- Note that the functions Gamma and LogGamma are mutually dependent.

gamma :: Double -> Double
gamma x
-- Split the function domain into three intervals:
-- (0, 0.001), [0.001, 12), and (12, infinity)

---------------------------------------------------------------------------
-- First interval: (0, 0.001)
--
-- For small x, 1/Gamma(x) has power series x + gamma x^2  - ...
-- So in this range, 1/Gamma(x) = x + gamma x^2 with error on the order of x^3.
-- The relative error over this interval is less than 6e-7.

| x > 0 && x < 0.001 =
let gamma = 0.577215664901532860606512090 -- Euler's gamma constant
in 1.0/(x*(1.0 + gamma*x))

---------------------------------------------------------------------------
-- Second interval: [0.001, 12)

| x >= 0.001 && x < 12.0 =
-- The algorithm directly approximates gamma over (1,2) and uses
-- reduction identities to reduce other arguments to this interval.

let arg_was_less_than_one = (x < 1.0)

-- Add or subtract integers as necessary to bring y into (1,2)
-- Will correct for this below
n = if arg_was_less_than_one then 0 else floor(x) - 1
y = if arg_was_less_than_one then x + 1 else x - (fromIntegral n)

-- numerator coefficients for approximation over the interval (1,2)
p = [
-1.71618513886549492533811E+0,
2.47656508055759199108314E+1,
-3.79804256470945635097577E+2,
6.29331155312818442661052E+2,
8.66966202790413211295064E+2,
-3.14512729688483675254357E+4,
-3.61444134186911729807069E+4,
6.64561438202405440627855E+4
]

-- denominator coefficients for approximation over the interval (1,2)
q = [
-3.08402300119738975254353E+1,
3.15350626979604161529144E+2,
-1.01515636749021914166146E+3,
-3.10777167157231109440444E+3,
2.25381184209801510330112E+4,
4.75584627752788110767815E+3,
-1.34659959864969306392456E+5,
-1.15132259675553483497211E+5
]

z = y - 1
gamma_iter z num den [] [] =
(num, den)
gamma_iter z num den ps qs =
gamma_iter z new_num new_den (tail ps) (tail qs)
where new_num = (num + (head ps)) * z
new_den = den * z + (head qs)
(num, den) = gamma_iter z 0.0 1.0 p q
result = num / den + 1.0
in
-- Apply correction if argument was not initially in (1,2)
if arg_was_less_than_one
-- Use identity gamma(z) = gamma(z+1)/z
-- The variable "result" now holds gamma of the original y + 1
-- Thus we use y-1 to get back the orginal y.
then result / (y - 1.0)
-- Use the identity gamma(z+n) = z*(z+1)* ... *(z+n-1)*gamma(z)
else let
gamma_z_n result y 0 = result
gamma_z_n result y n = gamma_z_n (result * y) (y + 1) (n - 1)
in gamma_z_n result y n

---------------------------------------------------------------------------
-- Third interval: [12, infinity)

| x >= 12.0 && x <= 171.624 =
exp(logGamma x)

| x > 171.624 =
-- Correct answer too large to display.
1.0/0 -- float infinity

logGamma :: Double -> Double
logGamma x | x > 0 =
if x < 12.0
then log(abs(gamma x))
else let
-- Abramowitz and Stegun 6.1.41
-- Asymptotic series should be good to at least 11 or 12 figures
-- For error analysis, see Whittiker and Watson
-- A Course in Modern Analysis (1927), page 252

c = reverse [
1.0/12.0,
-1.0/360.0,
1.0/1260.0,
-1.0/1680.0,
1.0/1188.0,
-691.0/360360.0,
1.0/156.0,
-3617.0/122400.0
]
z = 1.0/(x*x)
log_gamma_iter z sum [] = sum
log_gamma_iter z sum cs =
log_gamma_iter z s (tail cs)
where s = (sum * z) + (head cs)
sum = log_gamma_iter z 0 c
series = sum/x

halfLogTwoPi = 0.91893853320467274178032973640562
in (x - 0.5)*log(x) - x + halfLogTwoPi + series
```