local Reciprocal_gamma_coeff = {1,0.577215664901532860607,-0.655878071520253881077,-0.0420026350340952355290,0.166538611382291489502,-0.0421977345555443367482,-0.00962197152787697356211,0.00721894324666309954240,-0.00116516759185906511211,-0.000215241674114950972816,0.000128050282388116186153,-0.0000201348547807882386557,-1.25049348214267065735e-6,1.13302723198169588237e-6,-2.05633841697760710345e-7,6.11609510448141581786e-9,5.00200764446922293006e-9,-1.18127457048702014459e-9,1.04342671169110051049e-10,7.78226343990507125405e-12,-3.69680561864220570819e-12,5.10037028745447597902e-13,-2.05832605356650678322e-14,-5.34812253942301798237e-15,1.22677862823826079016e-15,-1.18125930169745876951e-16,1.18669225475160033258e-18,1.41238065531803178156e-18,-2.29874568443537020659e-19,1.71440632192733743338e-20}
--https://oeis.org/A001163 、 https://oeis.org/A001164
local stirling_series_coeff = {1,0.0833333333333333333333333,0.00347222222222222222222222,-0.00268132716049382716049383,-0.000229472093621399176954733,0.000784039221720066627474035,0.0000697281375836585777429399,-0.000592166437353693882864836,-0.0000517179090826059219337058,0.000839498720672087279993358,0.0000720489541602001055908572,-0.00191443849856547752650090,-0.000162516262783915816898635,0.00640336283380806979482364,0.000540164767892604515180468,-0.0295278809456991205054407,-0.00248174360026499773091566,0.179540117061234856107699,0.0150561130400264244123842,-1.39180109326533748139915,-0.116546276599463200850734}
function p._gamma_high_imag(cal_z)
	local z = to_number(cal_z)
	if z ~= nil and math_lib.abs(math_lib.nonRealPart(z)) > 1 then
		local inv_z = math_lib.inverse(z)
		return math_lib.sqrt((math_lib.pi * 2) * inv_z) * math_lib.pow(z * math_lib.exp(-1) *
			math_lib.sqrt( (z * math_lib.sinh(inv_z) ) + math_lib.inverse(to_number(810) * z * z * z * z * z * z) ),z)
	end
	return nil
end
function p._gamma_morethen_lua_int(cal_z)
	local z = to_number(cal_z) - to_number(1)
	local lua_int_term = 18.1169 --FindRoot[ Factorial[ x ] == 2 ^ 53, {x, 20} ]
	if math_lib.abs(z) > (lua_int_term - 1) or (math_lib.re(z) < 0 and math_lib.abs(math_lib.nonRealPart(z)) > 1 ) then
		local sum = 1
		for i = 1, #stirling_series_coeff - 1 do
			local a, n = to_number(z), tonumber(i) local y, k, f = to_number(1), n, to_number(a)
			while k ~= 0 do 
				if k % 2 == 1 then y = y * f end 
				k = math.floor(k / 2); f = f * f
			end
			sum = sum + stirling_series_coeff[i + 1] * math_lib.inverse(y)
		end
		return math_lib.sqrt( (2 * math.pi) * z ) * math_lib.pow( z * math.exp(-1), z ) * sum
	end
	return nil
end
function p._gamma_abs_less1(cal_z)
	local z = to_number(cal_z)
	if math_lib.abs(z) <=1.001 then
		if math_lib.abs(math_lib.nonRealPart(z)) < 1e-14 and ( (math.abs(math_lib.re(z) - 1) < 1e-14) or (math.abs(math_lib.re(z) - 2) < 1e-14) ) then return to_number(1)end
		return math_lib.inverse(p._recigamma_abs_less1(z))
	end
	return nil
end
function p._recigamma_abs_less1(z)
	local result = to_number(0)
	for i=1,#Reciprocal_gamma_coeff do
		result = result + Reciprocal_gamma_coeff[i] * math_lib.pow(z,i)
	end
	return result
end
function p._gamma(cal_z)
	local z = to_number(cal_z)
	if math_lib.abs(math_lib.nonRealPart(z)) < 1e-14 and ((math_lib.re(z) < 0 or math.abs(math_lib.re(z)) < 1e-14)
		and math.abs(math.floor(math_lib.re(z)) - math_lib.re(z)) < 1e-14 ) then return tonumber("nan") end
	local pre_result = p._gamma_morethen_lua_int(z) or p._gamma_high_imag(z) or p._gamma_abs_less1(z)
	if pre_result then return pre_result end
	local real_check = math_lib.re(z)
	local loop_count = math.floor(real_check)
	local start_number, zero_flag = z - loop_count, false
	if math_lib.abs(start_number) <= 1e-14 then start_number = to_number(1);zero_flag = true end
	local result = math_lib.inverse(p._recigamma_abs_less1(start_number))
	if math_lib.abs(math_lib.nonRealPart(z)) < 1e-14 and ((math_lib.re(z) > 1e-14 )and math.abs(math.floor(math_lib.re(z)) - math_lib.re(z)) < 1e-14 ) then result = to_number(1)  end
	local j = to_number(start_number)
	for i=1,math.abs(loop_count) do
		if loop_count > 0 then result = result * j else result = result * math_lib.inverse(j-1) end
		if zero_flag==true and loop_count > 0 then zero_flag=false else if loop_count > 0 then j = j + 1 else j = j - 1 end end
	end
	if math_lib.abs(math_lib.nonRealPart(z)) < 1e-14 and ((math_lib.re(z) > 1e-14 )and math.abs(math.floor(math_lib.re(z)) - math_lib.re(z)) < 1e-14 ) then return math_lib.floor(result) end
	return result
end