(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.


class PowerTree {
    private static Map<Integer, Integer> p = new HashMap<>()
    private static List<List<Integer>> lvl = new ArrayList<>()
 
    static {
        p[1] = 0
 
        List<Integer> temp = new ArrayList<Integer>()
        temp.add 1
        lvl.add temp
    }
 
    private static List<Integer> path(int n) {
        if (n == 0) return new ArrayList<Integer>()
        while (!p.containsKey(n)) {
            List<Integer> q = new ArrayList<>()
            for (Integer x in lvl.get(0)) {
                for (Integer y in path(x)) {
                    if (p.containsKey(x + y)) break
                    p[x + y] = x
                    q.add x + y
                }
            }
            lvl[0].clear()
            lvl[0].addAll q
        }
        List<Integer> temp = path p[n]
        temp.add n
        temp
    }
 
    private static BigDecimal treePow(double x, int n) {
        Map<Integer, BigDecimal> r = new HashMap<>()
        r[0] = BigDecimal.ONE
        r[1] = BigDecimal.valueOf(x)
 
        int p = 0
        for (Integer i in path(n)) {
            r[i] = r[i - p] * r[p]
            p = i
        }
        r[n]
    }
 
    private static void showPos(double x, int n, boolean isIntegral) {
        printf("%d: %s\n", n, path(n))
        String f = isIntegral ? "%.0f" : "%f"
        printf(f, x)
        printf(" ^ %d = ", n)
        printf(f, treePow(x, n))
        println()
        println()
    }
 
    static void main(String[] args) {
        for (int n = 0; n <= 17; ++n) {
            showPos 2.0, n, true
        }
        showPos 1.1, 81, false
        showPos 3.0, 191, true
    }
}