Version with some breathing room...

CodeGolf_NotLogic.nim

proc s(x, y, b: var int)=
  x = x div 2
  y = y div 2
  b *= 2
 
proc n(x: var int): int =
  return -(x+1)
 
proc a(x, y: var int): int =
  var b = 1
  while x > 0 and y > 0:
    if (x mod 2  + y mod 2) == 2:
      result += b
 
    s(x,y,b)

 
proc o(x, y: var int): int =
  var b = 1
  while x + y > 0:
    if (x mod 2 + y mod 2) >= 1:
      result += b
 
    s(x,y,b)
 
proc r(x, y: var int): int =
  var b = 1
  while x + y > 0:
    if (x mod 2 + y mod 2) == 1:
      result += b
 
    s(x,y,b)
 


# Test harness

proc test(i,j, exp: int; p: proc; op: string){.procvar.}=
  var
    act: int
  var x = i
  var y = j
  act = p(x,y)
  if(exp != act):
    echo("Error! ", i, op, j," ", exp," ", act)  


import threadpool
var
  x, exp, act: int
for i in countup(0, 0xffffffff):
  x = i
  exp = not x
  act = n(x)
  if(exp != act):
    echo("Error! ~", i)
  parallel:
    for j in countup(0,0xffffffff):
      spawn test(i,j,i and j, a," & ")
      spawn test(i,j,i or j, o," | ")
      spawn test(i,j,i xor j, r," ^ ")

Added test harness