Re: ALU, negationx, y, add16 off by one?

Posted by cadet1620 on Dec 11, 2011; 5:16pm
URL: http://nand2tetris-questions-and-answers-forum.52.s1.nabble.com/ALU-negationx-y-add16-off-by-one-tp3564869p3577682.html

While rogerdodger91 was writing:

Whats wrong with my ALU basically theoretically speaking.

I emailed him:

The Hack ALU is a very clever design.  It's not obvious why you don't need to use 2's complement negation to do subtraction.  Here's why it works to use binary NOT.

First, the definition of 2's complement gives:
    -x = NOT(x) + 1
    -x - 1 = NOT(x)

The x-y op code for the ALU is
    zx=0 nx=1 zy=0 ny=0 f=1 no=1
so the output is:
    out = NOT( NOT(x) + y )
    out = NOT( (-x - 1) + y )
    out = NOT( y - x - 1 )
    out = -(y - x - 1) - 1
    out = -y + x + 1 - 1
    out = x - y

--Mark