It is a Newton-Cotes formula of degree 2, and the error is proportional to the 4th power of the step size and an unknown value of the fourth derivative of the function being approximated.
It can be derived from first creating a Lagrange interpolating polynomial of degree 2, and then  integrating the result. It is a Newton-Cotes formula of degree 2, and the error is proportional to the 4th power of the step size and an unknown value of the fourth derivative of the function being approximated. A more accurate but complicated version is known as the 3/8ths rule, and unlike the 3/8ths rule, can only be applied over an even number of divisions. Giving an exact result for polynomials of degree three or less, this is, FTP, what technique of numerical integration, using weights of 2 and 4 and named for an English mathematician.
It can be derived from first creating a Lagrange interpolating polynomial of degree 2, and then  integrating the result.
Simpson's Rule