# Partitioned Add

this principle also applies to subtract and negate (-)

the basic principle is: the partition bits, when inverted, can actually
be inserted into an (expanded) add, and, if the bit is set, it has
the side-effect of "rolling through" the carry bit of the MSB from
the previous partition.

this is a really neat trick, basically, that allows the use of a
straight "add" (DSP in an FPGA, add in a simulator) where
otherwise it would be extraordinarily complex, CPU-intensive
and take up large resources.

    partition:     P    P    P     (3 bits)
    a        : .... .... .... .... (32 bits)
    b        : .... .... .... .... (32 bits)
    exp-a    : ....P....P....P.... (32+3 bits, P=0 if no partition)
    exp-b    : ....0....0....0.... (32 bits plus 3 zeros)
    exp-o    : ....xN...xN...xN... (32+3 bits - x to be discarded)
    o        : .... N... N... N... (32 bits - x ignored, N is carry-over)

new version:

    partition:      p    p    p       (3 bits)
    carry-in :      c    c    c    c (4 bits)
    C = c & P:      C    C    C    c (4 bits)
    I = P=>c :      I    I    I    c (4 bits)
    a        :  AAAA AAAA AAAA AAAA  (32 bits)
    b        :  BBBB BBBB BBBB BBBB  (32 bits)
    exp-a    : 0AAAACAAAACAAAACAAAAc (32+3+2 bits, P=0 if no partition)
    exp-b    : 0BBBBIBBBBIBBBBIBBBBc (32+2 bits plus 3 zeros)
    exp-o    : o....oN...oN...oN...x (32+3+2 bits - x to be discarded)
    o        :  .... N... N... N...  (32 bits - x ignored, N is carry-over)
    carry-out: o    o    o    o      (4 bits)

the new version

* brings in the carry-in (C) bits which, in combination with
  the Partition bits, are ANDed to create "C & p".
* C is positioned twice (in both A and B‌) intermediates, which
  has the effect of preserving carry-out, yet only performing a
  carry-over if the carry-in bit (c) is set and this is part of
  a partition
* o (carry-out) must be "cascaded" down to the relevant partition
  start-point.  this can be done with a Mux-cascade.

carry-out-cascade example:

    partition:      1    0    0    1      (4 bits)
    actual   : <--->|<------------>|<---> actual numbers
    carryotmp: o4   o3   o2   o1   o0     (5 bits)
    cascade  : |    |    x    x    |      o2 and o1 ignored
    carry-out: o4   \->  -->  o3   o0     (5 bits)

because the partitions subdivide the 5-wide input into 8-24-8, o4 is already
in "both" the MSB-and-LSB position for the top 8-bit result; o3 is the
carry-out for the 24-bit result and must be cascaded down to the *beginning*
of the 24-bit partitioned result (the LSB), and o0, like o4, is already in
position because the partition is only 1 wide.