Boolean algebra

Boolean values are true and false values stored as binary bits 1 and 0 respectively. In the same way that integers have operations such as addition and subtraction, boolean values have the operators of and, or, and exclusive or.

Boolean Operators

AND (&)

A & B is true if and only if A is true and B is true.

AND	1	0
1	1	0
0	0	0

  1001 1011  A
& 0010 1001  B
-----------
  0000 1001  A&B

BOR (|)

A | B is true if A is true, B is true, or both are true.

BOR	1	0
1	1	1
0	1	0

  1001 1011  A
| 0010 1001  B
-----------
  1011 1011  A|B

XOR (^)

A ^ B is true if and only if A is true and B is false OR A is false and B is true. In other words, A ^ B is true when A and B are different.

XOR	1	0
1	0	1
0	1	0

  1001 1011  A
^ 0010 1001  B
-----------
  1011 0010  A^B

Shifting

The Shift Left (SHL, <<) and Shift Right (SHR, >>) push all of the bits to the left and right, respectively. The bits that go beyond the boundary of a 16-bit word will be placed in the Excess (EX) register.

   SET A, 0x670F
   ; A = 0110 0111 0000 1111
   SHL A, 3
   ; A = 0011 1000 0111 1000
   ; EX = 0000 0000 0000 0110
   SHR A, 6
   ; A = 0000 0000 1110 0001
   ; EX = 1110 0000 0000 0000

A shifting operation has the effect of multiplying (left), or dividing (right) by 2.

Using Boolean Algebra in Code

The IFB Conditional Opcode

The IFB conditional will execute the next instruction if A&B != 0. This is to say that if any of the bits between A and B are are both 1, then the next operation will be set. This is useful when checking for bit flags. For example, let's say you have a variable where the highest bit means to run a certain subroutine.

   IFB A, 0x8000    ; If the highest bit is true
   JSR something    ; Perform a subroutine
   ...              ; other code here

The IFC Conditional Opcode

The IFC conditional is very similar to IFB in that it tests A&B, but it will only execute the next instruction if A&B == 0. That is, if none of the bit positions are 1 in both A and B, the next instruction is executed. It is effectively the opposite of IFB.

Bitmasks

In some cases, you may wish to try to pack as much information into a single word as possible. Let's say instead of wanting to store a single 16-bit value in a register, you wish to use it to store two 8-bit values. For this, you can create a mask that has 1's in the bits that represent each component. So the mask for one value would be 0xff00, while the other would be 0x00ff. AND-ing a value with a mask will erase all irrelevant bits, leaving the ones that are part of the mask.

      A = 1101 0011 0110 0001   Two 8-bit values
& mask1 = 1111 1111 0000 0000
          -------------------
          1101 0011             The first value. Shift right 8 bits to save it.

      A = 1101 0011 0110 0001
& mask2 = 0000 0000 1111 1111
          -------------------
                    0110 0001   The second value.

    ; Lets say A holds two 8-bit values. We want to put the first in B and the second in C.
    SET B, A
    AND B, 0xff00 ; AND the mask.
    SHR B, 8      ; Shift the bits to the right.
 
    SET C, A
    AND C, 0x00ff ; AND the mask.

Inverting Bits

XOR is a special command in that any value XOR 1 is equal to the opposite value. Therefore, it's possible to flip every bit in a register by XOR-ing it with 0xffff.