Algorithm-

Crypto synthetic numbers dirty bits import numpy as np

Define custom activation functions

def custom_softmax(x): return np.exp(x) / np.sum(np.exp(x), axis=0)

def custom_sin(x): return np.sin(x)

def custom_cos(x): return np.cos(x)

Define loss function

def cross_entropy(y_true, y_pred): epsilon = 1e-15 y_pred = np.clip(y_pred, epsilon, 1 - epsilon) return -np.mean(y_true * np.log(y_pred) + (1 - y_true) * np.log(1 - y_pred))

Define model architecture

class DeepLearningModel: def init(self, input_dim, output_dim): self.weights = np.random.randn(output_dim, input_dim) self.bias = np.zeros((output_dim, 1))

def forward(self, X):
    Z = np.dot(self.weights, X) + self.bias
    A = custom_softmax(Z)
    return A

def backward(self, X, y_true, learning_rate):
    m = X.shape[1]
    dZ = A - y_true
    dW = 1/m * np.dot(dZ, X.T)
    db = 1/m * np.sum(dZ, axis=1, keepdims=True)

    self.weights -= learning_rate * dW
    self.bias -= learning_rate * db

Initialize random number generators

rng1 = RandomNumberGenerator(seed=42) rng2 = RandomNumberGenerator(seed=123)

Generate random numbers

random_numbers_1 = rng1.generate_random_numbers(100) random_numbers_2 = rng2.generate_random_numbers(100)

Initialize deep learning model

input_dim = 1 # Adjust based on your dataset output_dim = 1 # Adjust based on your dataset model = DeepLearningModel(input_dim, output_dim)

Gradient descent parameters

learning_rate = 0.999999999 epochs = 1000

Train model to bring synthetic numbers closer together using gradient descent

for epoch in range(epochs): for x1, x2 in zip(random_numbers_1, random_numbers_2): dirty_bit = np.random.rand() * 2 - 1 # Generate dirty bit between -1 and 1 y_true = np.sqrt(x1) + np.sqrt(x2) + dirty_bit # Adjusted spacing of numbers with square root and dirty bit model.backward(np.array([[x1 + x2]]).T, np.array([[y_true]]), learning_rate)

Linear descent to reduce error rate

for epoch in range(epochs): for x1, x2 in zip(random_numbers_1, random_numbers_2): dirty_bit = np.random.rand() * 2 - 1 # Generate dirty bit between -1 and 1 y_true = np.sqrt(x1) + np.sqrt(x2) + dirty_bit # Adjusted spacing of numbers with square root and dirty bit y_pred = model.forward(np.array([[x1 + x2]]).T) error = y_pred - y_true model.weights -= learning_rate * error * (x1 + x2) model.bias -= learning_rate * error

Print final weights and bias

print("Final weights:", model.weights) print("Final bias:", model.bias)