import numpy as np

# Objective function (the function we want to minimize)
def objective(x):
    return x**2  # simple parabola, minimum is at x = 0

# Lévy flight function (random step generator)
def levy_flight(Lambda):
    u = np.random.normal(0, 1)
    v = np.random.normal(0, 1)
    step = u / abs(v)**(1/Lambda)
    return step

# Basic Cuckoo Search
def cuckoo_search(n=10, max_iter=50, pa=0.25):
    nests = np.random.uniform(-10, 10, n)  # random initial nests
    fitness = objective(nests)
    
    for t in range(max_iter):
        for i in range(n):
            new_nest = nests[i] + levy_flight(1.5)  # new solution via Lévy flight
            new_fit = objective(new_nest)
            if new_fit < fitness[i]:  # if better, replace
                nests[i] = new_nest
                fitness[i] = new_fit
        
        # Discovery and random replacement (like host birds rejecting eggs)
        rand_idx = np.random.rand(n) > pa
        nests[rand_idx] = np.random.uniform(-10, 10, np.sum(rand_idx))
        fitness[rand_idx] = objective(nests[rand_idx])
    
    best_nest = nests[np.argmin(fitness)]
    best_value = np.min(fitness)
    return best_nest, best_value

# Run it
best, value = cuckoo_search()
print(f"Best solution: x = {best:.4f}, f(x) = {value:.4f}")