import numpy as np


def IV(i):
    fib = (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597)
    inv = tuple(range(-2, 16))
    return 1 / max(3, max(b for a, b in zip(fib, inv) if i >= a))


def T2(i):
    return IV(i) * 2


def T3(i):
    return IV(i) * 3


def D2(i):
    return IV(i) / 2


def D3(i):
    return IV(i) / 3


def P3(i):
    return IV(i) ** 3


def X3(i):
    return i * 3


def C2(i):  # crude workaround for my own broken interface
    return 2.0


def H2(i):  # throwaway
    return 0.5 if i >= 65 // 5 else 1.0


def II(i):
    return 1 / (i + 1)


def S2(i):
    return 1 / (2 * np.sqrt(i + 2))


def S3(i):
    return 1 / (3 * np.sqrt(i + 3))