Explain, in general, why equivalent algebraic expressions may lead to different answers. Can you devise an interval-arithmetic package that does not have this shortcoming, or is this task impossible? (Warning: This problem is very difficult.)
In exercise 2.14, the different arithmetic operations have different effects upon the center values and tolerances of the produced intervals. We also saw that our system as implemented has no concept of identity. I.e. A/A ≠ 1 ± 0%. These issues mean that we cannot apply standard algebraic transformations to a formula and expect it to produce an interval with the same bounds afterwards. This was demonstrated quite clearly in exercise 2.15, where two algebraically equivalent formulae were shown to produce intervals with different bounds.
In general we can state that the issue is that the more interval-arithmetic operations we perform during a calculation, the wider the resulting interval will be.
As to devising an interval-arithmetic package that does not have this shortcoming... As the authors state, "This problem is very difficult". So much so that the IEEE started working on a standard for interval arithmetic back in 2008... and they're still going as I write this. So, yes, it's very difficult - and certainly beyond the scope of our current exposure to Scheme at this point in the book!
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