A Bike Shifting Heuristic

My bicycle is a 24-speed model, with three front sprockets and eight rear sprockets. What’s a good shifting pattern to apply when riding in variable terrain?

If we number the sprockets front and back from lowest to highest gear ratio, and zero indexing, in front they count 0..3 from smallest to largest, and in the rear 0..7 from largest to smallest. In the interest of minimal chain wear, I stick to to these rear-sprocket ranges, where $f$ is the front-sprocket number:

$$ r_f = [2f,2f + 4) $$

That’s to say I use the bottom four rear sprockets for the lowest front sprocket, the middle four rear for the middle front, and the highest four rear for the highest front.

My heuristic relies on the predicted next shift. If it will go in the same direction, then prefer the front shift. If it will go in the opposite direction, then prefer the rear shift. And stay within the range constraints.

This may not give produce the ideal gear ratio at all times. The front shift is larger than the rear, so it will skew the first shift in the direction of the second. Close enough.

My wife sometimes refers to this type of material as stupid math tricks. I see her point.