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16-bit Division
Written by Jorg Rockstroh

16-bit division is the division of one 16-bit value by another 16-bit value, returning a 16-bit quotient and a 16-bit remainder. I used r1/r0 for dividend/remainder and r3/r2 for divisor/quotient.

Programming Tip: The number of bits in the quotient and the remainder can never be larger than the number of bits in the original divident. For example, if you are dividing a 16-bit value by a 2-bit value, both the quotient and the remainder must be able to handle a 16-bit result. If you are dividing a 24-bit value by a 16-bit value, the quotient and remainder must both be able to handle a 24-bit result.

So, again, let's remember how we did division in elementary school. For example, 179 divided by 8: 

  1 7 9 / 8 = 22 (quotient)
  1 6 
  ---
    1 9 
    1 6
    ---
      3 (remainder)

It's necessary necessary to follow this same process step by step. There is a 3-digit-dividend, so we expect 3 digits maximum for quotient. We "shift left" the divisor 2 digits (3-1) such that the number of digits in the divisor is the same as the number of digits in the dividend. So we get:

We divide the two numbers, multiply the result by the divisor and substract this result from the dividend. In this first step 179 can't be divided by 800, so the the result is 0. We subtract 0 from 179 and still have 179:


 1 7 9 / 8 0 0 = 0 ? ?
     0
 -----
 1 7 9

We then "shift right" the divisor 1 digit and repeat the process. 179 divided by 80 results in an answer of 2. After we subtract 160 (2x80) we are left with a remainder of 19:


 1 7 9 / 8 0 = 0 2 ?
 1 6 0
 -----
   1 9

We repeat the process again until the divisor has shifted into its original position:


 1 7 9 : 8 = 0 2 2
 1 6 0
 -----
   1 9
   1 6
 -----
     3

This may have been an unnecessary review of elementary school math, but it is important to remember exactly how the process is performed because we do exactly the same with the 8052 in binary system.

In this routine we will place the original dividend into R1 (high-byte) and R0 (low-byte) and the divisor in R3 (high-byte) and R2 (low-byte).

In the case of our example (179 divided by 8), the initial registers would be:

Step 1. Shift left the divisor. 

  MOV B,#00h ;Clear B since B will count the number of left-shifted bits
div1:
  INC B     ;Increment counter for each left shift
  MOV A,R2  ;Move the current divisor low byte into the accumulator
  RLC A     ;Shift low-byte left, rotate through carry to apply highest bit to high-byte
  MOV R2,A  ;Save the updated divisor low-byte
  MOV A,R3  ;Move the current divisor high byte into the accumulator
  RLC A     ;Shift high-byte left high, rotating in carry from low-byte
  MOV R3,A  ;Save the updated divisor high-byte
  JNC div1  ;Repeat until carry flag is set from high-byte

In the case of our example, once the above code is executed the registers will be as follows (including the carry bit 'C'):

At this point we can do the division itself. As we are in binary mode there is no need for a real division--it's just a comparison. At this point it's important to know the steps from above.

Step 2. Shift left the divisor. 

div2:        ;Shift right the divisor
  MOV A,R3   ;Move high-byte of divisor into accumulator
  RRC A      ;Rotate high-byte of divisor right and into carry
  MOV R3,A   ;Save updated value of high-byte of divisor
  MOV A,R2   ;Move low-byte of divisor into accumulator
  RRC A      ;Rotate low-byte of divisor right, with carry from high-byte
  MOV R2,A   ;Save updated value of low-byte of divisor
  CLR C      ;Clear carry, we don't need it anymore
  MOV 07h,R1 ;Make a safe copy of the dividend high-byte
  MOV 06h,R0 ;Make a safe copy of the dividend low-byte
  MOV A,R0   ;Move low-byte of dividend into accumulator
  SUBB A,R2  ;Dividend - shifted divisor = result bit (no factor, only 0 or 1)
  MOV R0,A   ;Save updated dividend 
  MOV A,R1   ;Move high-byte of dividend into accumulator
  SUBB A,R3  ;Subtract high-byte of divisor (all together 16-bit substraction)
  MOV R1,A   ;Save updated high-byte back in high-byte of divisor
  JNC div3   ;If carry flag is NOT set, result is 1
  MOV R1,07h ;Otherwise result is 0, save copy of divisor to undo subtraction
  MOV R0,06h
div3:
  CPL C      ;Invert carry, so it can be directly copied into result
  MOV A,R4 
  RLC A      ;Shift carry flag into temporary result
  MOV R4,A   
  MOV A,R5
  RLC A
  MOV R5,A		
  DJNZ B,div2 ;Now count backwards and repeat until "B" is zero

To see how the loop works here are the registers after each step:

1 r1/0 00000000 10110011 ;dividend
  r3/2 10000000 00000000 ;divisor
  r5/4 00000000 00000000 ;result

2 r1/0 00000000 10110011 ;dividend
  r3/2 01000000 00000000 ;divisor
  r5/4 00000000 00000000 ;result

...

8  r1/0	00000000 10110011 ;dividend
   r3/2	00000001 00000000 ;divisor
   r5/4	00000000 00000000 ;result

9  r1/0	00000000 00110011 ;dividend
   r3/2	00000000 10000000 ;divisor
   r5/4	00000000 00000001 ;result

10 r1/0 00000000 00110011 ;dividend
   r3/2 00000000 01000000 ;divisor
   r5/4 00000000 00000010 ;result

11 r1/0 00000000 00010011 ;dividend
   r3/2 00000000 00100000 ;divisor
   r5/4 00000000 00000101 ;result

12 r1/0 00000000 00000011 ;dividend
   r3/2 00000000 00010000 ;divisor
   r5/4 00000000 00001011 ;result

13 r1/0 00000000 00000011 ;dividend
   r3/2 00000000 00001000 ;divisor
   r5/4 00000000 00010110 ;result
STOP!

Register "B" is zero at this point. The remainder is already in R1/R0, and it is 3 decimal, same as above. The result is still in R5/R4, but we can see it's correct, too (10110b=22d). To finish the routine, we just "clean up" by moving R5/R4 to R3/R2.

Step 3. Final Clean-up. 

  MOV R3,05h  ;Move result to R3/R2
  MOV R2,04h  ;Move result to R3/R2

We used small numbers here for easier explanation. Of course it works also with 16-bit numbers, that's what it was designed to do.

Taken as a whole, the above division algorithm can be converted into an easy-to-use function that can be called from your program. To call this function, you should pre-load R1/R0 with the high/low value to be divided, and R3/R2 with the high/low value that the number is to be divided by.

div16_16:
  CLR C       ;Clear carry initially
  MOV R4,#00h ;Clear R4 working variable initially
  MOV R5,#00h ;CLear R5 working variable initially
  MOV B,#00h  ;Clear B since B will count the number of left-shifted bits
div1:
  INC B      ;Increment counter for each left shift
  MOV A,R2   ;Move the current divisor low byte into the accumulator
  RLC A      ;Shift low-byte left, rotate through carry to apply highest bit to high-byte
  MOV R2,A   ;Save the updated divisor low-byte
  MOV A,R3   ;Move the current divisor high byte into the accumulator
  RLC A      ;Shift high-byte left high, rotating in carry from low-byte
  MOV R3,A   ;Save the updated divisor high-byte
  JNC div1   ;Repeat until carry flag is set from high-byte
div2:        ;Shift right the divisor
  MOV A,R3   ;Move high-byte of divisor into accumulator
  RRC A      ;Rotate high-byte of divisor right and into carry
  MOV R3,A   ;Save updated value of high-byte of divisor
  MOV A,R2   ;Move low-byte of divisor into accumulator
  RRC A      ;Rotate low-byte of divisor right, with carry from high-byte
  MOV R2,A   ;Save updated value of low-byte of divisor
  CLR C      ;Clear carry, we don't need it anymore
  MOV 07h,R1 ;Make a safe copy of the dividend high-byte
  MOV 06h,R0 ;Make a safe copy of the dividend low-byte
  MOV A,R0   ;Move low-byte of dividend into accumulator
  SUBB A,R2  ;Dividend - shifted divisor = result bit (no factor, only 0 or 1)
  MOV R0,A   ;Save updated dividend 
  MOV A,R1   ;Move high-byte of dividend into accumulator
  SUBB A,R3  ;Subtract high-byte of divisor (all together 16-bit substraction)
  MOV R1,A   ;Save updated high-byte back in high-byte of divisor
  JNC div3   ;If carry flag is NOT set, result is 1
  MOV R1,07h ;Otherwise result is 0, save copy of divisor to undo subtraction
  MOV R0,06h
div3:
  CPL C      ;Invert carry, so it can be directly copied into result
  MOV A,R4 
  RLC A      ;Shift carry flag into temporary result
  MOV R4,A   
  MOV A,R5
  RLC A
  MOV R5,A		
  DJNZ B,div2 ;Now count backwards and repeat until "B" is zero
  MOV R3,05h  ;Move result to R3/R2
  MOV R2,04h  ;Move result to R3/R2
  RET

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