(Knuth's power tree is used for computing   xn   efficiently.)




Task

Compute and show the list of Knuth's power tree integers necessary for the computation of:

   xn   for any real   x   and any non-negative integer   n.


Then, using those integers, calculate and show the exact values of (at least) the integer powers below:

   2n     where   n   ranges from   0 ──► 17   (inclusive) 

   3191
   1.181


A  zero  power is often handled separately as a special case.

Optionally, support negative integer powers.



Example

An example of a small power tree for some low integers:

                                                                    1
                                                                     \
                                                                      2
                          ___________________________________________/ \
                         /                                              \
                        3                                                4
                       / \____________________________________            \
                      /                                       \            \
                     5                                         6            8
                    / \____________                           / \            \
                   /               \                         /   \            \
                  7                 10                      9     12           16
                 /                 //\\                     │      │           /\
                /            _____//  \\________            │      │          /  \
              14            /     /    \        \           │      │         /    \
             /│ \         11    13      15       20        18     24        17    32
            / │  \         │    /\      /\        │        /\      │        /\     │
           /  │   \        │   /  \    /  \       │       /  \     │       /  \    │
         19  21    28     22 23   26  25   30    40     27   36    48     33 34   64
         │   /\    /│\     │  │   /\   │   /\    /│\     │   /\    /│\     │  │   /\
         │  /  \  / │ \    │  │  /  \  │  /  \  / │ \    │  /  \  / │ \    │  │  /  \
        38 35 42 29 31 56 44 46 39 52 50 45 60 41 43 80 54 37 72 49 51 96 66 68 65 128

Where, for the power   43,   following the tree "downwards" from   1:

   (for   2)   compute square of   X,   store X2
   (for   3)   compute   X * X2,   store X3
   (for   5)   compute   X3 * X2,   store X5
   (for 10)   compute square of   X5,   store X10
   (for 20)   compute square of   X10,   store X20
   (for 40)   compute square of   X20,   store X40
   (for 43)   compute   X40 * X3   (result).
Note that for every even integer (in the power tree),   one just squares the previous value.

For an odd integer, multiply the previous value with an appropriate odd power of   X   (which was previously calculated).
  For the last multiplication in the above example, it would be   (43-40),   or   3. 


According to Dr. Knuth (see below),   computer tests have shown that this power tree gives optimum results for all of the   n
  listed above in the graph.

For   n   ≤ 100,000,   the power tree method:

   bests the factor method   88,803   times,
   ties   11,191   times,
   loses   6   times.


References

   Donald E. Knuth's book:   The Art of Computer Programming, Vol. 2, Second Edition, Seminumerical Algorithms, section 4.6.3: Evaluation of Powers.
   link   codegolf.stackexchange.com/questions/3177/knuths-power-tree     It shows a   Haskell,   Python,   and a   Ruby   computer program example   (but they are mostly   code golf).
   link   comeoncodeon.wordpress.com/tag/knuth/     (See the section on Knuth's Power Tree.)     It shows a   C++   computer program example.
   link to Rosetta Code   addition-chain exponentiation.


V p = [1 = 0]
V lvl = [[1]]
 
F path(n)
   I !n
      R [Int]()
   L n !C :p
      [Int] q
      L(x) :lvl[0]
         L(y) path(x)
            I !(x + y C :p)
               :p[x + y] = x
               q.append(x + y)
      :lvl[0] = q
 
   R path(:p[n]) [+] [n]
 
F tree_pow(x, n)
   T Ty = T(x)
   V (r, p) = ([0 = Ty(1), 1 = x], 0)
   L(i) path(n)
      r[i] = r[i - p] * r[p]
      p = i
   R r[n]
 
F show_pow_i(x, n)
   print("#.: #.\n#.^#. = #.\n".format(n, path(n), x, n, tree_pow(BigInt(x), n)))
 
F show_pow_f(x, n)
   print("#.: #.\n#.^#. = #.6\n".format(n, path(n), x, n, tree_pow(x, n)))
 
L(x) 18
   show_pow_i(2, x)
show_pow_i(3, 191)
show_pow_f(1.1, 81)