# Let's understand sigmoid step by step: 1 / (1 + e^(-x))
x = 2.0
print(f"Input: x = {x}")
print(f"Step 1: -x = {-x}")
print(f"Step 2: e^(-x) = np.exp(-x) = {np.exp(-x):.3f}")
print(f"Step 3: 1 + e^(-x) = {1 + np.exp(-x):.3f}")
print(f"Step 4: 1 / (1 + e^(-x)) = {1 / (1 + np.exp(-x)):.3f}")
print(f"Sigmoid result: {sigmoid(x):.3f}")

# Why clipping?
print(f"\nWhy we clip large values:")
large_x = 1000
print(f"Without clipping: np.exp(-{large_x}) would cause overflow")
print(f"With clipping: np.exp(-500) = {np.exp(-500):.2e} (very small, safe)")