import numpy as np

def cross_entropy_loss(outputs, targets, clip=True):
    """
    outputs: [
        [ 0.32, 0.12, 0.04 ],
        [ 0.62, 0.02, 0.14 ]
    ]
    targets: [ 2, 1 ]
    :param outputs: np.array: Vector of all the predicted probabilities vectors
    :param targets: np.array: Vector of one-hot vectors representing the actual values
    :param clip: boolean, whether to clip the output probabilities
    :return:
    """
    if clip:
        # Clipping the predictions for numerical stability
        outputs = np.clip(outputs, 1e-12, 1 - 1e-12)
    # Calculate cross-entropy loss and average over batch size
    m = targets.shape[0]
    log_likelihood = -np.log(outputs[range(m), targets])
    return np.sum(log_likelihood) / m  # Average loss

def cross_entropy_derivative_loss(outputs, targets):
    # One-hot encode the labels
    y_true = np.eye(outputs.shape[1])[targets]
    # Derivative of cross-entropy with respect to softmax inputs
    return outputs - y_true