def K_to_discharge_coefficient(D, Do, K):
    r'''Converts a standard loss coefficient to a discharge coefficient.
    
    .. math::
        C = \sqrt{\frac{1}{2 \sqrt{K} \beta^{4} + K \beta^{4}}
        - \frac{\beta^{4}}{2 \sqrt{K} \beta^{4} + K \beta^{4}} }
        
    Parameters
    ----------
    D : float
        Upstream internal pipe diameter, [m]
    Do : float
        Diameter of orifice at flow conditions, [m]
    K : float
        Loss coefficient with respect to the velocity and density of the fluid
        just upstream of the orifice, [-]

    Returns
    -------
    C : float
        Coefficient of discharge of the orifice, [-]

    Notes
    -----
    If expansibility is used in the orifice calculation, the result will not
    match with the specified pressure drop formula in [1]_; it can almost
    be matched by dividing the calculated mass flow by the expansibility factor
    and using that mass flow with the loss coefficient. 
    
    This expression was derived with SymPy, and checked numerically. There were
    three other, incorrect roots.
    
    Examples
    --------
    >>> K_to_discharge_coefficient(D=0.07366, Do=0.05, K=5.2314291729754)
    0.6151200000000001
    
    References
    ----------
    .. [1] American Society of Mechanical Engineers. Mfc-3M-2004 Measurement 
       Of Fluid Flow In Pipes Using Orifice, Nozzle, And Venturi. ASME, 2001.
    .. [2] ISO 5167-2:2003 - Measurement of Fluid Flow by Means of Pressure 
       Differential Devices Inserted in Circular Cross-Section Conduits Running
       Full -- Part 2: Orifice Plates.
    '''
    beta = Do/D
    beta2 = beta*beta
    beta4 = beta2*beta2
    root_K = K**0.5
    common_term = 2.0*root_K*beta4 + K*beta4
    return (-beta4/(common_term) + 1.0/(common_term))**0.5
r'''Converts a standard loss coefficient to a discharge coefficient.
    
    .. math::
        C = \sqrt{\frac{1}{2 \sqrt{K} \beta^{4} + K \beta^{4}}
        - \frac{\beta^{4}}{2 \sqrt{K} \beta^{4} + K \beta^{4}} }
        
    Parameters
    ----------
    D : float
        Upstream internal pipe diameter, [m]
    Do : float
        Diameter of orifice at flow conditions, [m]
    K : float
        Loss coefficient with respect to the velocity and density of the fluid
        just upstream of the orifice, [-]

    Returns
    -------
    C : float
        Coefficient of discharge of the orifice, [-]

    Notes
    -----
    If expansibility is used in the orifice calculation, the result will not
    match with the specified pressure drop formula in [1]_; it can almost
    be matched by dividing the calculated mass flow by the expansibility factor
    and using that mass flow with the loss coefficient. 
    
    This expression was derived with SymPy, and checked numerically. There were
    three other, incorrect roots.
    
    Examples
    --------
    >>> K_to_discharge_coefficient(D=0.07366, Do=0.05, K=5.2314291729754)
    0.6151200000000001
    
    References
    ----------
    .. [1] American Society of Mechanical Engineers. Mfc-3M-2004 Measurement 
       Of Fluid Flow In Pipes Using Orifice, Nozzle, And Venturi. ASME, 2001.
    .. [2] ISO 5167-2:2003 - Measurement of Fluid Flow by Means of Pressure 
       Differential Devices Inserted in Circular Cross-Section Conduits Running
       Full -- Part 2: Orifice Plates.