How to determine that 2.0 == (float) 2 in a front-end.

[Robert Dewar]

<<Why is that?  Naively, I would think that x*x*x*x (evaluated left to right)
has 3 multiplications in a row, while (x**2)**2 has only two.  So, from an
error propagation point of view, the latter should be more accurate.
>>

It is always hard to get an intuitive feel for these results (the above
result is fairly easy to prove). Here is an attempt. In the case of
x*x*x*x, you have 3 multiplications but in the 2nd and 3rd, one of the
operands at least is exact. In the case of (x**2)**2 the second multiplication
has two potentially inexact operands so the error is multiplied.

<<Wrt your first statement: I assume that a "well written pow routine" would
calculate that as or similar to exp(6.0 * log(x)).  This surely involves many
more floating point operations.  Is it really more accurate?  [Checking
glibc...  Moving on to to http://oss.software.ibm.com/mathlib/...]  Okay, it
seems to be.  I'm impressed.
>>

no no you definitely cannot calculate x**y using exp and log, even with
perfectly accurate (last bit rounded) exp and log, this is a terrible
way of doing things. For a detailed analysis of this case see chapter
5 of my book on Microprocessors (McGraw Hill 1990).