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Adder with Accumulator and Status Flags					[TOC]
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The bit width of the adder/subtractor will extended to 16 bits. We also add a
16-bit register (accumulator) for storing the result of the computation when we
get a clock signal. Furthermore, status flags are provided (see also the
description below).

---- VIDEO ------------------------------
https://www.youtube.com/embed/sOM0y2LTsoU
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Status Flags
============
After a clock signal the computed result gets stored in the accumulator and the
following status flags are updated:

- The _zero flag (ZF)_ is set if and only if the computed bit pattern 

- The _carry flag (CF)_ is set if and only if an unsigned integer overflow
  occurred.

  With $a := u(a_15, \dots, a_0)$ and $b := u(b_15, \dots, b_0)$ the condition
  for setting the flag can be described more formal:
  
  - In case of an addition an overflow is equivalent to $a \oplus b = (a + b)
    \bmod 2^{16} \neq a + b$.
  - And in case subtraction equivalent to $a \ominus b = (a - b) \bmod 2^{16}
    \neq a + b$

- The _overflow flag (OF)_ is set if and only if a signed integer overflow
  occurred.

  With $a := s(a_15, \dots, a_0), b := s(b_15, \dots, b_0)$ and $\tilde{s} :=
  s(s_15, \dots, s_0)$ the condition for a signed overflow
  
  - in an addition is equivalent to $a < 0 \land b < 0 \land \tilde{s} \geq 0
    \lor a \geq 0 \land b \geq 0 \land \tilde{s} < 0$.
  - And in a subtraction equivalent to $a < 0
    \land b \geq 0 \land \tilde{s} \geq 0 \lor a \geq 0 \land b < 0 \land
    \tilde{s} < 0$.

- The _sign flag (SF)_ is set if and only if the signed value of the computed
  bit pattern is negative, i.e. its most significant bit equals one.


Relational Conditions
=====================
When we have two integer values $a$ and $b$ we can use the status flags to check
relational conditions, e.g. $a = b$, $a < b$. This always happens in two steps:

- First the difference $a - b$ is computed with a subtraction,
- Second the status flags are checked.

Note that the bit pattern of the difference is not used. Only the status flags
set by the subtractor.

Equality and Inequality
-----------------------
In this case it is not relevant whether the bit patterns are supposed to
represent an unsigned or signed value.

+-------------------------------------------+---------------------------+
| Relational Condition			    | Unsigned / Signed		|
+-------------------------------------------+---------------------------+
| $ a = b \Leftrightarrow a - b = 0 $       |	$\text{ZF} = 1$		|
|					    |				|
|					    | _equal_/_zero_	        |
+-------------------------------------------+---------------------------+
| $a \neq b \Leftrightarrow a - b \neq 0 $  |	$\text{ZF} = 0$		|
|					    |				|
|					    | _not equal_/_not zero_	|
+-------------------------------------------+---------------------------+

Ordered Relations
-----------------
In this case one has to distinguish whether both operands are either
representing an unsigned or signed value.

+-------------------------------------------+---------------------------------------+-----------------------------------------------+
| Relational Condition			    |	Unsigned			    |	Signed					    |
+-------------------------------------------+---------------------------------------+-----------------------------------------------+
| $a > b \Leftrightarrow a - b > 0$         |	$\text{ZF=0} \land \text{CF} = 0$   |	$\text{ZF}=0 \land \text{OF} = \text{SF}$   |
|					    |					    |						    |
|					    |   _above_ / _not below equal_	    |	_greater_ / _not less equal_		    |
+-------------------------------------------+---------------------------------------+-----------------------------------------------+
| $a \geq b \Leftrightarrow a - b \geq 0$   |	$\text{CF} = 0$			    |	$\text{OF} = \text{SF}$			    |
|					    |					    |						    |
|					    |   _above equal_ / _not below_	    |	_greater equal_ / _not less_		    |
+-------------------------------------------+---------------------------------------+-----------------------------------------------+
| $a < b \Leftrightarrow a - b < 0$	    |	$\text{CF} = 1$			    |	$\text{OF} \neq \text{SF}$		    |
|					    |					    |						    |
|					    |   _not above equal_ / _below_	    |	_not greater equal_ / _less_		    |
+-------------------------------------------+---------------------------------------+-----------------------------------------------+
| $a \leq b  \Leftrightarrow a - b \leq 0$  |	$\text{ZF=1} \lor \text{CF} = 1$    |	$\text{ZF}=1 \lor \text{OF} \neq \text{SF}$ |
|					    |					    |						    |
|					    |   _not above_ / _below equal_	    |	_not greater_ /_less equal_		    |
+-------------------------------------------+---------------------------------------+-----------------------------------------------+