Knuth's power tree

var lvl = [[1]]
var p = Hash(1 => 0)
 
func path(n) is cached {
    n || return []
    while (n !~ p) {
        var q = []
        for x in lvl[0] {
            for y in path(x) {
                break if (x+y ~~ p)
                y = x+y
                p{y} = x
                q << y
            }
        }
        lvl[0] = q
    }
    path(p{n}) + [n]
}
 
func tree_pow(x, n) {
    var r = Hash(0 => 1, 1 => x)
    var p = 0
    for i in path(n) {
        r{i} = (r{i-p} * r{p})
        p = i
    }
    r{n}
}
 
func show_pow(x, n) {
    var fmt = ("%d: %s\n" + ["%g^%s = %f", "%s^%s = %s"][x.is_int] + "\n")
    print(fmt % (n, path(n), x, n, tree_pow(x, n)))
}
 
for x in ^18 { show_pow(2, x) }
show_pow(1.1, 81)
show_pow(3, 191)

Output:

0: []
2^0 = 1
1: [1]
2^1 = 2
2: [1, 2]
2^2 = 4
3: [1, 2, 3]
2^3 = 8
4: [1, 2, 4]
2^4 = 16
5: [1, 2, 4, 5]
2^5 = 32
6: [1, 2, 4, 6]
2^6 = 64
7: [1, 2, 4, 6, 7]
2^7 = 128
8: [1, 2, 4, 8]
2^8 = 256
9: [1, 2, 4, 8, 9]
2^9 = 512
10: [1, 2, 4, 8, 10]
2^10 = 1024
11: [1, 2, 4, 8, 10, 11]
2^11 = 2048
12: [1, 2, 4, 8, 12]
2^12 = 4096
13: [1, 2, 4, 8, 12, 13]
2^13 = 8192
14: [1, 2, 4, 8, 12, 14]
2^14 = 16384
15: [1, 2, 4, 8, 12, 14, 15]
2^15 = 32768
16: [1, 2, 4, 8, 16]
2^16 = 65536
17: [1, 2, 4, 8, 16, 17]
2^17 = 131072
81: [1, 2, 4, 8, 16, 32, 64, 80, 81]
1.1^81 = 2253.240236
191: [1, 2, 4, 8, 16, 32, 64, 128, 160, 176, 184, 188, 190, 191]
3^191 = 13494588674281093803728157396523884917402502294030101914066705367021922008906273586058258347