CMP.3:5.2 - Share an expression while preserving its interpretation

For f(x,y)=(x+y)*(x+y)+(x+y), build one node t=x+y with three uses, then obtain t*t+t. With x=2,y=3, the result is 30. This replaces three additions of x+y by one and shares its stored value until the final addition.

If y changes to 4, t and its consumers must change; the result becomes 42. If each occurrence instead meant “read the next measurement and add x,” the shared expression would change the computation’s meaning. The subproblem must be a fixed pure addition of supplied values for this identification to hold.

In differentiation of a longer expression graph, intermediate values may be needed again in reverse order. Keeping all of them can exceed memory. Retain selected restart values and recompute intervening pure operations when needed, comparing the extra work with reduced storage. This applies the same method to a different receiving algorithm.