Floating point arithmetic: $(x-2)^9$ This is taken from Trefethen and Bau, 13.3.
Why is there a difference in accuracy between evaluating near 2 the expression $(x-2)^9$ and this expression:
$$x^9 - 18x^8 + 144x^7 -672x^6 + 2016x^5 - 4032x^4 + 5376x^3 - 4608x^2 + 2304x - 512 $$
Where exactly is the problem?
Thanks.

I have tried to plot this 2 expresions near x=2.0. Look at the image. The behaviour ( shape) seems very different . 

Here is the Maxima CAS code :
draw2d(
  file_name = "d",
  color = blue,
  key = "second function",
  explicit(x^9 - 18*x^8 + 144*x^7 -672*x^6 + 2016*x^5 - 4032*x^4 + 5376*x^3 - 4608*x^2 + 2304*x - 512 ,x, 2-0.0000005,2.0000005),

  color = red,
  key = " y = (x-2)^9",
  explicit((x-2)^9, x, 2-0.0000005,2.0000005),

  terminal = 'svg) $   

Note that above 2 expressions are the same ! 
expand((x-2)^9)

So problem is in computer arithmethic not in math ( informally speaking)
More : stable evaluation of polynomials  . See also discussion 
