A piece of beautiful and trick bitwise operation code.

A piece of beautiful and trick bitwise operation code.

A detailed reading process of a piece of beautiful and trick bitwise operation code.
 
The following code is from MMIXWare, it is used to implement the Wyde difference between two octabyte.
 
     in file: "mmix-arith.w"
     423 tetra wyde_diff(y,z)
     424   tetra y,z;
     425 {
     426   register tetra a=((y>>16)-(z>>16))&0x10000;
     427   register tetra b=((y&0xffff)-(z&0xffff))&0x10000;
     428   return y-(z^((y^z)&(b-a-(b>>16))));
     429 }
 
It is hard to understand it without any thinking or verification, here is the process I used
to check the correctness of this algorithm.

let y = 0xuuuuvvvv;
     z = 0xccccdddd; (please note the [c]s may be different hex number.)
    
then y>>16 = 0x0000uuuu;
     z>>16 = 0x0000cccc;
     
then ((y>>16)-(z>>16)) = 0x1111gggg if #uuuu < #cccc or
     ((y>>16)-(z>>16)) = 0x0000gggg if #uuuu >= #cccc   

so variable a = 0x00010000 if #uuuu < #cccc or
   variable a = 0x00000000 if #uuuu >= #cccc
  
similarly, we can get
   variable b = 0x00010000 if #vvvv < #dddd or
   variable b = 0x00000000 if #vvvv >= #dddd

for (b-a-(b>>16)))), there are four different result depending on the relation between a and b.
when #uuuu >= #cccc and #vvvv >= #dddd, (b-a-(b>>16)))) = 0x00000000;
when #uuuu >= #cccc and #vvvv < #dddd, (b-a-(b>>16)))) = 0x00001111;
when #uuuu < #cccc and #vvvv >= #dddd, (b-a-(b>>16)))) = 0x11110000;
when #uuuu < #cccc and #vvvv < #dddd, (b-a-(b>>16)))) = 0x11111111;
You can see that >= map to #0000 and < map to #1111

for y-(z^((y^z)&(b-a-(b>>16)))), when (b-a-(b>>16)))) is 0x00000000, z^((y^z)&(b-a-(b>>16))) is
z^((y^z)& 0) = z^0=z, so y-(z^((y^z)&(b-a-(b>>16))))=y-z.
similarily, when (b-a-(b>>16)))) is 0x11111111, z^((y^z)&(b-a-(b>>16))) is
z^((y^z)& 1) = z^(y^z)=y, so y-(z^((y^z)&(b-a-(b>>16))))=0.

when (b-a-(b>>16)))) is 0x11110000 or 0x11110000, we can treat the y and z as two separate wydes.
each wyde in the result is correct.

You may think it is a little stupid to verify such kind of details. but for my point of view,
without such detailed analysis, I can not understand the algorithm in the code. with the hard
work like this, I successfully understand it. The pleasure deserve the effort.
 
I am wondering how can the author discover such a genius algorithm.