# picomath

## Erlang

### gamma.erl

```-module(gamma).
-export([gamma/1, log_gamma/1]).

% 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(X) when X > 0 ->

% Split the function domain into three intervals:
% (0, 0.001), [0.001, 12), and (12, infinity)

if

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 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 < 0.001 ->
Gamma = 0.577215664901532860606512090, % Euler's gamma constant
1.0/(X*(1.0 + Gamma*X));

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

X < 12.0 ->
% The algorithm directly approximates gamma over (1,2) and uses
% reduction identities to reduce other arguments to this interval.

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
[Y, N] = if
Arg_was_less_than_one ->
[X + 1.0, 0];
true ->
N1 = trunc(X) - 1,
[X - N1, N1]
end,

% 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,
[Num, Den] = gamma_iter(Z, 0.0, 1.0, P, Q),
Result = Num/Den + 1.0,

% 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.
Result / (Y - 1.0);
true ->
% Use the identity gamma(z+n) = z*(z+1)* ... *(z+n-1)*gamma(z)
gamma_z_n(Result, Y, N)
end;

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

X < 171.624 ->
math:exp(log_gamma(X));

true ->
999.0e299

end.

gamma_iter(_Z, Num, Den, [], []) ->
[Num, Den];
gamma_iter(Z, Num, Den, [P|Ps], [Q|Qs]) ->
NewNum = (Num + P) * Z,
NewDen = Den * Z + Q,
gamma_iter(Z, NewNum, NewDen, Ps, Qs).

gamma_z_n(Result, _Y, 0) -> Result;
gamma_z_n(Result, Y, N) ->
gamma_z_n(Result * Y, Y + 1, N - 1).

log_gamma(X) when X > 0 ->

if
X < 12.0 ->
math:log(abs(gamma(X)));

true ->

% 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 = lists: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),
Sum = log_gamma_iter(Z, 0, C),
Series = Sum/X,

HalfLogTwoPi = 0.91893853320467274178032973640562,
LogGamma = (X - 0.5)*math:log(X) - X + HalfLogTwoPi + Series,
LogGamma
end.

log_gamma_iter(_Z, Sum, []) ->
Sum;
log_gamma_iter(Z, Sum, [C|Cs]) ->
S = (Sum * Z) + C,
log_gamma_iter(Z, S, Cs).
```