<SYSTEM_TASK:>
Solve the following problem using Python, implementing the functions described below, one line at a time
<END_TASK>
<USER_TASK:>
Description:
 def approximate_density( dist, xloc, parameters=None, cache=None, eps=1.e-7 ):
 """ Approximate the probability density function. Args: dist : Dist Distribution in question. May not be an advanced variable. xloc : numpy.ndarray Location coordinates. Requires that xloc.shape=(len(dist), K). eps : float Acceptable error level for the approximations retall : bool If True return Graph with the next calculation state with the approximation. Returns: numpy.ndarray: Local probability density function with ``out.shape == xloc.shape``. To calculate actual density function, evaluate ``numpy.prod(out, 0)``. Example: [[0.0242 0.0399 0.0242]] [[0.0242 0.0399 0.0242]] """

    if parameters is None:
        parameters = dist.prm.copy()
    if cache is None:
        cache = {}

    xloc = numpy.asfarray(xloc)
    lo, up = numpy.min(xloc), numpy.max(xloc)
    mu = .5*(lo+up)
    eps = numpy.where(xloc < mu, eps, -eps)*xloc

    floc = evaluation.evaluate_forward(
        dist, xloc, parameters=parameters.copy(), cache=cache.copy())
    for d in range(len(dist)):
        xloc[d] += eps[d]
        tmp = evaluation.evaluate_forward(
            dist, xloc, parameters=parameters.copy(), cache=cache.copy())
        floc[d] -= tmp[d]
        xloc[d] -= eps[d]

    floc = numpy.abs(floc / eps)
    return floc