from cmath import exp, pi, sin, sqrt

p = [
    676.5203681218851,
    -1259.1392167224028,
    771.32342877765313,
    -176.61502916214059,
    12.507343278686905,
    -0.13857109526572012,
    9.9843695780195716e-6,
    1.5056327351493116e-7,
]

EPSILON = 1e-07


def drop_imag(z):
    if abs(z.imag) <= EPSILON:
        z = z.real
    return z


def gamma(z):
    z = complex(z)
    if z.real < 0.5:
        y = pi / (sin(pi * z) * gamma(1 - z))  # Reflection formula
    else:
        z -= 1
        x = 0.99999999999980993
        for (i, pval) in enumerate(p):
            x += pval / (z + i + 1)
        t = z + len(p) - 0.5
        y = sqrt(2 * pi) * t ** (z + 0.5) * exp(-t) * x
    return drop_imag(y)


"""
The above use of the reflection (thus the if-else structure) is necessary, even though 
it may look strange, as it allows to extend the approximation to values of z where 
Re(z) < 0.5, where the Lanczos method is not valid.
"""

print(gamma(1))
print(gamma(5))
print(gamma(0.5))
print(gamma(0.5* (1 + 1j)))
print(gamma(-0.5))