“fast xie integral”: faster method of calculating the trapezoidal estimate of an integral

Instead of calculating the left and right Riemann sums separately and then averaging them together, solve for the average of 1/2 of the leftmost value, then all the middle values, and lastly, 1/2 of the rightmost value. This will save up to 50% of the work you need to do, and always gives the exact same answer as the trapezoidal sum.

This can be helpful in optimizing the math in logic circuits (i.e., computers) and can also help you work up to twice as fast when taking a timed test in a high school or college math class