Maximum number of temporary variables needed for math expressions

Hi all,


Is there any upper bounds on the number of temporary variables that are
needed to translate any given mathematical expression? By intuition, I
guess the answer is 2, but I'm looking for a (possibly formal) proof to
make sure. To make clear what I mean, here is a simple three address
code translation of a math expression using 2 temporary variables:


x+((a+b)*(c/d)-(e*f))


# opcode, operand1, operand2, result
+, a, b, t1
/, c, d, t2
*, t1, t2, t1
*, e, f, t2
-, t1, t2, t1
+, x, t1, t1
# the result is in t1, and t2 is free


Thanks in advance,
Ehsan
[It is my vague recollection that you can use up arbitrarily large
numbers of temporaries by composing operators that aren't commutative
-John]

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