# import necessary packages
import numpy as np
import matplotlib.pyplot as plt

def feed_forward(x):
    # weighted sum of inputs to the output layer
    z_1 = x*output_weights + output_bias
    # Output from output node (one node only)
    # Here the output is equal to the input
    a_1 = z_1
    return a_1

def backpropagation(x, y):
    a_1 = feed_forward(x)
    # derivative of cost function
    derivative_cost = a_1 - y
    # the variable delta in the equations, note that output a_1 = z_1, its derivatives wrt z_o is thus 1
    delta_1 = derivative_cost
    # gradients for the output layer
    output_weights_gradient = delta_1*x
    output_bias_gradient = delta_1
    # The cost function is 0.5*(a_1-y)^2. This gives a measure of the error for each iteration
    return output_weights_gradient, output_bias_gradient

# ensure the same random numbers appear every time
np.random.seed(0)
# Input variable
x = 4.0
# Target values
y = 2*x+1.0

# Defining the neural network
n_inputs = 1
n_outputs = 1
# Initialize the network
# weights and bias in the output layer
output_weights = np.random.randn()
output_bias = np.random.randn()

# implementing a simple gradient descent approach with fixed learning rate
eta = 0.01
for i in range(40):
    # calculate gradients from back propagation
    derivative_w1, derivative_b1 = backpropagation(x, y)
    # update weights and biases
    output_weights -= eta * derivative_w1
    output_bias -= eta * derivative_b1
# our final prediction after training
ytilde = output_weights*x+output_bias
print(0.5*((ytilde-y)**2))