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1.2. EXPANSION AND SIMPLIFICATION 21
Moses [Mos71] This is a seminal paper, but describes approaches to simplification
rather than defining it. Inasmuch as it does define it, it talks about
‘utility for further operations’, which again is not effective as a definition
for top-level simplification, though it’s relevant for intermediate operations.
However, the principle is important, since a request to factor would
find the expression x 999 + · · · + 1 appropriate 9 , whereas x1000 −1
x−1
is in a
field, and factorisation is not a relevant question.
Carette [Car04] He essentially defines simplification in terms of the length of
the result, again insisting on mathematical equivalence. This would regard
examples 1 (assuming Q(x), so that the expressions were mathematically
equivalent) and 3 as simplifications, but not 2, since the expression becomes
longer, or 4, since we don’t have mathematical equivalence.
Stoutemyer [Sto11] sets out 10 goals for simplification, one of which (Goal
6) is that
Default simplification should produce candid [Definition 5] results
for rational expressions and for as many other classes as is
practical. Default simplification should try hard even for classes
where candidness cannot be guaranteed for all examples.
Candid expressions tend be the shorter than others ( x10 −1
x−1
etc. being
an obvious family of counter-examples), so this view is relatively close to
Carette’s.
Numerical Analysts A numerical analyst would be shocked at the idea that
x 2 − y 2 was ‘simpler’ than (x + y)(x − y). He would instantly quote an
example such as the following [Ham07].
For simplicity, assume decimal arithmetic, with perfect rounding
and four decimal places. Let x = 543.2 and y = 543.1. Then
x 2 evaluates to 295100 and y 2 evaluates to 295000, so x 2 − y 2
becomes 100, while (x + y)(x − y) evaluates to 108.6, a perfect
rounding of the true answer 108.63.
Furthermore, if we take x = 913.2 and y = 913.1, x 2 − y 2 is
still 100, while the true result is 182.63.
This is part of the whole area of numerical stability:
outside the scope of this text.
fascinating, but
One principle that can be extracted from the above is “if the expression is zero,
please tell me”: this would certainly meet both the teachers’ and Carette’s
views. This can be seen as a call for simplification to return a normal form
where possible [Joh71, Ric97].
Maple’s description of the ‘simplify’ command is as follows.
9 In fact, knowing the expression came from that quotient would be relevant [BD89] to
factorisation algorithms, but that’s beside the point here.