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import numpy as np


def sigmoid(x: float) -> float:
    return 1.0 / (1.0 + np.exp(-x))

def sigmoid_prime(x: float) -> float:
    return sigmoid(x) / (1.0 - sigmoid())

def relu(x):
    return np.maximum(x, 0)

def relu_prime(x):
    return np.where(x > 0, 1, 0)