factorial.py 文件源码

def ramanujan(n, prec=100):
    r"""
    Approximates the factorial :math:`n!` given an integer :math:`n` using Ramanujan's formula.
    Ramanujan's formula is just as or more accurate than several other factorial approximation
    formulas.

    Parameters
    ----------
    n
        Integer to approximate factorial
    prec
        Defines level of precision for factorials over 100. Default 100. Optional

    Returns
    -------
    int or Decimal
        Factorial of :math:`n` as approximated by Ramanujan's formula.

    Notes
    -----
    Ramanujan's formula is another factorial approximation method known for its accuracy
    in comparison to other factorial approximation approaches including Stirling's and
    Gosper's approximations. Ramanujan's formula is defined as:

    .. math::

        n! \approx \sqrt{\pi} \left(\frac{n}{e}\right)^n \sqrt[6]{8n^3 + 4n^2 + n + \frac{1}{30}}

    Examples
    --------
    >>> ramanujan(10)
    3628800.3116126074
    >>> ramanujan(5)
    120.00014706585664

    References
    ----------
    Mortici, Cristinel. On Gosper's Formula for the Gamma Function. Valahia University of Targoviste,
        Department of Mathematics. Retrieved from http://files.ele-math.com/articles/jmi-05-53.pdf

    """
    if n != np.floor(n):
        n = np.floor(n)

    if n >= 100:
        with localcontext() as ctx:
            ctx.prec = prec
            f = Decimal(
                np.sqrt(np.pi) * n ** n * np.exp(-n) * (8. * n ** 3. + 4. * n ** 2. + n + 1. / 30.) ** (1. / 6.))
    else:
        f = np.sqrt(np.pi) * n ** n * np.exp(-n) * (8. * n ** 3. + 4. * n ** 2. + n + (1. / 30.)) ** (1. / 6.)

    return f