Fast Cube Root ( < 2^32 )

/*                                                                                                            **
 The time to take a cube root from a 32 bits uint was: < 26 ns (mean: < 22 ns)
                                             new time: < 14 ns (mean: < 13 ns)
 
 The trick used for small square roots is used again: "unroll the loop".
 First step: A binary search, the "if" statements, large values ( >= 1u << 24 )
 are found relatively faster, small values ( < 64 ) are handled during the search.
 Second step: A jump into the unrolled loop, the "labels".
 
      |--------------------------------| 
      |         cro32(x)               |                         
      |-----|-----||------------|------|
      |   x |  ns ||          x |   ns |
      |-----|-----||------------|------|
      |   0 | 2,4 ||        511 |  4,6 |
      |   1 | 2,3 ||       1023 |  5,3 |
      |   3 | 2,3 ||       2047 |  5,3 |
      |   7 | 2,3 ||       4095 |  5,0 |
      |  15 | 3,6 ||      32768 |  7,2 |
      |  31 | 3,0 ||    1048576 |  8,5 |
      |  63 | 3,0 ||  268435456 | 12,0 |
      |  64 | 5,0 ||  536870912 | 12,3 |     heaviest case,
      |  65 | 5,0 || 1070599167 | 13,7 | <== 1070599167 is the
      | 100 | 5,0 || 1070599168 | 13,7 |     3th power of 1023, 
      | 127 | 5,0 || 1073741824 | 12,3 |     binary: 1111111111
      | 128 | 5,0 || 1073774592 | 12,0 |
      | 255 | 4,6 || 2147483648 | 12,3 |
      | 256 | 4,6 || 4294967295 | 12,0 |
      |-----|-----||------------|------|

       4294967295 roots in 53884 ms, mean time 12,55 ns, faster than square root!
         UPDATE 2013/11/04 50686 ms, mean time 11,80 ns, see: part 2
 
 ** Original version: Figure 2: Architecture of the iteration (alternative A)
    A digit-by-digit algorithm for radix-2 cube root and its implementation (2004)
    J.-A.Pineiro, J.D.Bruguera, L.Ciminiera, P.Montuschi
  
*/
using System;
using System.Diagnostics;

class cro_32
{
    private static uint cro32(uint x)
    {
        uint y = 4u, z = 16u, b = 0u;
        if (x < 1u << 24)
            if (x < 1u << 12)
                if (x < 1u << 06)
                    if (x < 1u << 03)
                        return x == 0u ? 0u : 1u;
                    else
                        return x < 27u ? 2u : 3u;
                else
                    if (x < 1u << 09) goto L8; else goto L7;
            else
                if (x < 1u << 18)
                    if (x < 1u << 15) goto L6; else goto L5;
                else
                    if (x < 1u << 21) goto L4; else goto L3;
        else
            if (x < 1u << 30)
                if (x < 1u << 27) goto L2; else goto L1;

        if (x >= 27u << 27) { x -= 27u << 27; z = 36u; y = 6u; } else { x -= 1u << 30; } goto M0;
    L1: if (x >= 27u << 24) { x -= 27u << 24; z = 36u; y = 6u; } else { x -= 1u << 27; } goto M1;
    L2: if (x >= 27u << 21) { x -= 27u << 21; z = 36u; y = 6u; } else { x -= 1u << 24; } goto M2;
    L3: if (x >= 27u << 18) { x -= 27u << 18; z = 36u; y = 6u; } else { x -= 1u << 21; } goto M3;
    L4: if (x >= 27u << 15) { x -= 27u << 15; z = 36u; y = 6u; } else { x -= 1u << 18; } goto M4;
    L5: if (x >= 27u << 12) { x -= 27u << 12; z = 36u; y = 6u; } else { x -= 1u << 15; } goto M5;
    L6: if (x >= 27u << 09) { x -= 27u << 09; z = 36u; y = 6u; } else { x -= 1u << 12; } goto M6;
    L7: if (x >= 27u << 06) { x -= 27u << 06; z = 36u; y = 6u; } else { x -= 1u << 09; } goto M7;
    L8: if (x >= 27u << 03) { x -= 27u << 03; z = 36u; y = 6u; } else { x -= 1u << 06; } goto M8;

    M0: if (x >= 61u << 24) { x -= 61u << 24; z += y * 2 + 1u; y += 1u; } y *= 2; z *= 4;
    M1: b = (y + z) * 3 + 1u << 21; if (x >= b) { x -= b; z += y * 2 + 1u; y += 1u; } y *= 2; z *= 4;
    M2: b = (y + z) * 3 + 1u << 18; if (x >= b) { x -= b; z += y * 2 + 1u; y += 1u; } y *= 2; z *= 4;
    M3: b = (y + z) * 3 + 1u << 15; if (x >= b) { x -= b; z += y * 2 + 1u; y += 1u; } y *= 2; z *= 4;
    M4: b = (y + z) * 3 + 1u << 12; if (x >= b) { x -= b; z += y * 2 + 1u; y += 1u; } y *= 2; z *= 4;
    M5: b = (y + z) * 3 + 1u << 09; if (x >= b) { x -= b; z += y * 2 + 1u; y += 1u; } y *= 2; z *= 4;
    M6: b = (y + z) * 3 + 1u << 06; if (x >= b) { x -= b; z += y * 2 + 1u; y += 1u; } y *= 2; z *= 4;
    M7: b = (y + z) * 3 + 1u << 03; if (x >= b) { x -= b; z += y * 2 + 1u; y += 1u; } y *= 2; z *= 4;
    M8: return x <= (y + z) * 3 ? y : y + 1u;
    }

    private static Stopwatch sw = new Stopwatch();
    static void Main()
    {
        cro32(225);
        cro32(1070599167);
        check_perfect_cubes32();
        time_cro32();
        cro32_all();
        Console.ReadLine();
    }

    private static void check_perfect_cubes32()
    {
        uint n = 0, a = 0, b = 1, c = 6;
        while (n <= 1625)
        {
            if (n != cro32(a)) Console.WriteLine("WRONG");
            a += b; b += c; c += 6; n++;
        }
        Console.WriteLine("CHECKED PERFECT CUBES");
    }

    private static void time_cro32()
    {
        int i, j; uint x; double t;
        uint[] a = { 0, 1, 3, 7, 15, 31, 63, 64, 65,
                     100, 127, 128, 255, 256, 511, 1023, 2047, 4095,
                     1u << 15, 1u << 20, 1u << 28 ,1u << 29 , 1070599167,
                     1070599168,1u << 30,(1u << 30) + (1u << 15), 1u << 31, ~0u };
        Console.WriteLine("|------------|------|");
        Console.WriteLine("|          x |  ns  |");
        Console.WriteLine("|------------|------|");
        for (j = 0; j < a.Length; j++)
        {
            x = a[j];
            cro32(x);
            sw.Restart();
            for (i = 0; i < 10000000; i++) cro32(x);
            sw.Stop();
            t = sw.ElapsedMilliseconds;
            sw.Restart();
            for (i = 0; i < 10000000; i++) ; // nada
            sw.Stop();
            t -= sw.ElapsedMilliseconds;
            Console.WriteLine("| {0,10} | {1:00.0} |", x, t / 10);
        }
        Console.WriteLine("|------------|------|");
        Console.WriteLine();
    }

    private static void cro32_all()
    {
        uint x; double t;
        cro32(~0u);
        sw.Restart();
        for (x = 0; x < ~0u; x++) cro32(x);
        sw.Stop();
        t = sw.ElapsedMilliseconds;
        sw.Restart();
        for (x = 0; x < ~0u; x++) ; // nada
        sw.Stop();
        t -= sw.ElapsedMilliseconds;
        Console.Write(x + " roots in " + t + " ms, ");
        Console.WriteLine("mean time {0:.00} ns", t * 1000000 / x);
        // 4294967295 roots in 53884 ms, mean time 12,55 ns
    }
}