The whole Enchilada

I've changed the (indexing) semantics of division to allow for all kinds of list slicings.

The new division semantics may appear to be strange at first but is carefully crafted to make it possible to do an efficient transpose of a list.

Let's say we want to transpose the following '3x4 list':

[1; 2; 3
 4; 5; 6
 7; 8; 9
10;11;12]

To do this we first right-multiply the list with the number of columns (3):

3 [ 1; 2; 3
    4; 5; 6
    7; 8; 9
   10;11;12] *

36 [ 1; 2; 3
     4; 5; 6
     7; 8; 9
    10;11;12
     1; 2; 3
     4; 5; 6
     7; 8; 9
    10;11;12
     1; 2; 3
     4; 5; 6
     7; 8; 9
    10;11;12]

After that, we left-divide the result with the number of columns:

36 [ 1; 2; 3
     4; 5; 6
     7; 8; 9
    10;11;12
     1; 2; 3
     4; 5; 6
     7; 8; 9
    10;11;12
     1; 2; 3
     4; 5; 6
     7; 8; 9
    10;11;12] 3 /

giving us the transpose we want:

36 [ 1; 4; 7;10
     2; 5; 8;11
     3; 6; 9;12] 1

To understand the new division semantics, it is best to picture it as being a digital line-drawing algorithm with x's being the pixels of a line.

The previous version of division would draw the following line:

   [ x; 2; 3
     x; 5; 6
     x; 8; 9
     x;11;12
     x; 2; 3
     x; 5; 6
     x; 8; 9
     x;11;12
     x; 2; 3
     x; 5; 6
     x; 8; 9
     x;11;12]

while the new version draws the line from top-left to bottom-right:

   [ x; 2; 3
     x; 5; 6
     x; 8; 9
     x;11;12
     1; x; 3
     4; x; 6
     7; x; 9
    10; x;12
     1; 2; x
     4; 5; x
     7; 8; x
    10;11; x]

What makes the new division interesting is that the same trick can be achieved on huge lists. For instance the following code transposes a 50000x50000 matrix:

50000 2500000000 ~ * 50000 /

producing the result:

125000000000000 [0;50000;100000;150000;200000;250000;300000;
350000;400000;450000;500000;550000;600000;650000;700000;750000
 ... {2500000000} ... 2499299999;2499349999;2499399999;2499449999;
2499499999;2499549999;2499599999;2499649999;2499699999;2499749999;
2499799999;2499849999;2499899999;2499949999;2499999999] 1

You can find the newest version (0_5_5) on the Enchilada website.