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# Binary Integer Adder/Subtractor, with Saturation

A signed saturating integer adder/subtractor, with `carry_in`, `carry_out`, internal `carries` into each bit. The operation is selected with `add_sub`: setting it to 0 for an add (A+B), and to 1 for a subtract (A-B). This assignment conveniently matches the convention of sign bits.

## Saturation

If the result of the addition/subtraction falls outside of the inclusive minimum or maximum limits, the result is clipped (saturated) to the nearest exceeded limit. The maximum limit must be greater or equal than the minimum limit. If the limits are reversed, such that limit_max < limit_min, the result will be meaningless.

Internally, we perform the addition/subtraction on WORD_WIDTH + 1 bits. Since the limits must be within the range of WORD_WIDTH-wide numbers, there can never be an overflow or underflow. Instead, we signal if we have reached or would have exceeded the limits at the last incrementation. The saturation logic is a pair of simple signed comparisons in the larger range. This is also likely optimal, as the delay from one extra bit of carry is less than that of any extra logic to handle overflows.

Also, we internally perform the addition/subtraction as unsigned so we can easily handle the carry_in bit. The signed comparisons are done in a separate module which implements signed/unsigned comparisons as raw logic, to avoid having to make sure all compared values are declared signed, else the comparison silently defaults to unsigned!

## Maintaining high operating frequency

You will very likely need to pipeline the inputs (for better retiming) of this module inside the enclosing module since we are chaining adder/subtractors together (there is a subtraction inside the Arithmetic Predicates modules), so the total carry-chain is twice as long as expected, plus 2 more bits to avoid overflow. Most of the time, this will take longer than your clock cycle since the carry-chain of arithmetic logic is often a limiting factor in timing closure.

````default_nettype none

#(
parameter       WORD_WIDTH          = 0
)
(
input   wire    [WORD_WIDTH-1:0]    limit_max,
input   wire    [WORD_WIDTH-1:0]    limit_min,
input   wire                        add_sub,    // 0/1 -> A+B/A-B
input   wire                        carry_in,
input   wire    [WORD_WIDTH-1:0]    A,
input   wire    [WORD_WIDTH-1:0]    B,
output  reg     [WORD_WIDTH-1:0]    sum,
output  reg                         carry_out,
output  reg     [WORD_WIDTH-1:0]    carries,
output  wire                        at_limit_max,
output  wire                        over_limit_max,
output  wire                        at_limit_min,
output  wire                        under_limit_min
);

localparam WORD_ZERO            = {WORD_WIDTH{1'b0}};
localparam WORD_WIDTH_EXTENDED  = WORD_WIDTH + 1;
localparam WORD_ZERO_EXTENDED   = {WORD_WIDTH_EXTENDED{1'b0}};

initial begin
sum         = WORD_ZERO;
carry_out   = 1'b0;
carries     = WORD_ZERO;
end
```

Extend the inputs to prevent overflow over their original range. We extend them as signed integers, despite declaring them as unsigned.

```    wire [WORD_WIDTH_EXTENDED-1:0] A_extended;

#(
.WORD_WIDTH_IN  (WORD_WIDTH),
.SIGNED         (1),
.WORD_WIDTH_OUT (WORD_WIDTH_EXTENDED)
)
extend_A
(
.original_input     (A),
);

wire [WORD_WIDTH_EXTENDED-1:0] B_extended;

#(
.WORD_WIDTH_IN  (WORD_WIDTH),
.SIGNED         (1),
.WORD_WIDTH_OUT (WORD_WIDTH_EXTENDED)
)
extend_B
(
.original_input     (B),
);
```

Extend the limits in the same way, as if signed integers.

```    wire [WORD_WIDTH_EXTENDED-1:0] limit_max_extended;

#(
.WORD_WIDTH_IN  (WORD_WIDTH),
.SIGNED         (1),
.WORD_WIDTH_OUT (WORD_WIDTH_EXTENDED)
)
extend_limit_max
(
.original_input     (limit_max),
);

wire [WORD_WIDTH_EXTENDED-1:0] limit_min_extended;

#(
.WORD_WIDTH_IN  (WORD_WIDTH),
.SIGNED         (1),
.WORD_WIDTH_OUT (WORD_WIDTH_EXTENDED)
)
extend_limit_min
(
.original_input     (limit_min),
);
```

Then select and perform the addition or subtraction in the usual way. NOTE: we don't capture the extended `carry_out`, as it will never be set properly since the inputs are too small for the `WORD_WIDTH_EXTENDED`. We compute the real `carry_out` separately.

```    wire [WORD_WIDTH_EXTENDED-1:0] sum_extended;
wire [WORD_WIDTH_EXTENDED-1:0] carries_extended;

#(
.WORD_WIDTH (WORD_WIDTH_EXTENDED)
)
(
.carry_in   (carry_in),
.A          (A_extended),
.B          (B_extended),
.sum        (sum_extended),
// verilator lint_off PINCONNECTEMPTY
.carry_out  (),
.carries    (carries_extended),
.overflow   ()
// verilator lint_on  PINCONNECTEMPTY
);
```

Since we extended the width by one bit, the original `carry_out` is now the carry into that extra bit. Let's also get the original `carries` into each bit.

```    always @(*) begin
carry_out = carries_extended [WORD_WIDTH_EXTENDED-1];
carries   = carries_extended [WORD_WIDTH-1:0];
end
```

Check if `sum_extended` is past the min/max limits. Using these arithmetic predicate modules removes the need to get all the signed declarations correct, else we accidentally and silently fall back to unsigned comparisons!

```    Arithmetic_Predicates_Binary
#(
.WORD_WIDTH (WORD_WIDTH_EXTENDED)
)
limit_max_check
(
.A                  (sum_extended),
.B                  (limit_max_extended),

// verilator lint_off PINCONNECTEMPTY
.A_eq_B             (at_limit_max),

.A_lt_B_unsigned    (),
.A_lte_B_unsigned   (),
.A_gt_B_unsigned    (),
.A_gte_B_unsigned   (),

.A_lt_B_signed      (),
.A_lte_B_signed     (),
.A_gt_B_signed      (over_limit_max),
.A_gte_B_signed     ()
// verilator lint_on  PINCONNECTEMPTY
);

Arithmetic_Predicates_Binary
#(
.WORD_WIDTH (WORD_WIDTH_EXTENDED)
)
limit_min_check
(
.A                  (sum_extended),
.B                  (limit_min_extended),

// verilator lint_off PINCONNECTEMPTY
.A_eq_B             (at_limit_min),

.A_lt_B_unsigned    (),
.A_lte_B_unsigned   (),
.A_gt_B_unsigned    (),
.A_gte_B_unsigned   (),

.A_lt_B_signed      (under_limit_min),
.A_lte_B_signed     (),
.A_gt_B_signed      (),
.A_gte_B_signed     ()
// verilator lint_on  PINCONNECTEMPTY
);
```

After, clip the sum to the limits. This must be done as a signed comparison so we can place the limits anywhere in the positive or negative integers, so long as `limit_max >= limit_min`, as signed integers. And finally, truncate the output back to the input `WORD_WIDTH`.

```    reg [WORD_WIDTH_EXTENDED-1:0] sum_extended_clipped = WORD_ZERO_EXTENDED;

always @(*) begin
sum_extended_clipped = (over_limit_max  == 1'b1) ? limit_max_extended : sum_extended;
sum_extended_clipped = (under_limit_min == 1'b1) ? limit_min_extended : sum_extended_clipped;
sum                  = sum_extended_clipped [WORD_WIDTH-1:0];
end

endmodule
```

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