# picomath

### gamma.a

```generic
type Float_Type is digits <>;
package Gamma is

function Gamma(x: Float_Type'Base) return Float_Type'Base;
function Log_Gamma(x: Float_Type'Base) return Float_Type'Base;

end Gamma;

package body Gamma is

use Elementary_Functions;

function Positive_Infinity return Float_Type'Base is

result: Float_Type'Base := Float_Type'Base'Last;

begin
return Float_Type'Base'Succ(result);
exception
when Constraint_Error =>
return result;
end Positive_Infinity;

function Gamma(x: Float_Type'Base) return Float_Type'Base is

gamma: constant := 0.577215664901532860606512090; -- Euler's gamma constant

-- numerator coefficients for approximation over the interval (1,2)
p: constant array(1..8) of Float_Type'Base := (
-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: constant array(1..8) of Float_Type'Base := (
-3.08402300119738975254353E+1,
3.15350626979604161529144E+2,
-1.01515636749021914166146E+3,
-3.10777167157231109440444E+3,
2.25381184209801510330112E+4,
4.75584627752788110767815E+3,
-1.34659959864969306392456E+5,
-1.15132259675553483497211E+5
);

y, num, den, z, result: Float_Type'Base;
n: Natural;
arg_was_less_than_one: Boolean;

begin
if x <= 0.0 then
end if;

-- 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.

if x < 0.001 then
return 1.0/(x*(1.0 + gamma*x));

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

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

y := x;
n := 0;
arg_was_less_than_one := (y < 1.0);

-- Add or subtract integers as necessary to bring y into (1,2)
-- Will correct for this below
if arg_was_less_than_one then
y := y + 1.0;
else
n := Natural(Float_Type'Base'Floor(y)) - 1;  -- will use n later
y := y - Float_Type'Base(n);
end if;

num := 0.0;
den := 1.0;

z := y - 1.0;
for i in 1..8 loop
num := (num + p(i))*z;
den := den*z + q(i);
end loop;
result := num/den + 1.0;

-- Apply correction if argument was not initially in (1,2)
if arg_was_less_than_one then
-- 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 := result / (y-1.0);
else
-- Use the identity gamma(z+n) = z*(z+1)* ... *(z+n-1)*gamma(z)
for i in 1..n loop
result := result * y;
y := y + 1.0;
end loop;
end if;

return result;

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

elsif x <= 171.624 then
return Exp(Log_Gamma(x));

else
-- Correct answer too large to display.
return Positive_Infinity; -- float infinity

end if;
end Gamma;

function Log_Gamma(x: Float_Type'Base) return Float_Type'Base is

c: constant array(1..8) of Float_Type'Base := (
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
);
halfLogTwoPi: constant := 0.91893853320467274178032973640562;

z, sum, series: Float_Type'Base;

begin
if x <= 0.0 then
end if;

if x < 12.0 then
return Log(Abs(Gamma(x)));
end if;

-- 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

z := 1.0/(x*x);
sum := c(8);
for i in reverse 1..7 loop
sum := sum * z;
sum := sum + c(i);
end loop;
series := sum/x;

return (x - 0.5)*Log(x) - x + halfLogTwoPi + series;
end Log_Gamma;

end Gamma;
```