# Building Math Hardware

## Parameterized Adder and Subtractor in Verilog

The inquiry “How do computers do Math?” introduced the carry-propagate adder and the borrow-propagate subtractor. Here we will recapitulate and implement these using Verilog HDL.

The full adder (fa) forms the basic building block. This full adder adds two 1-bit values a and b to the incoming carry (ci), and outputs a 1-bit sum (s) and a 1-bit outgoing carry (co). The circuit and Boolean equations, shown below, give the relations between these inputs and outputs.
 \begin{aligned} s&=a \oplus b \oplus c_i\\ c_o&=a \cdot b + c_i \cdot (a \oplus b) \end{aligned}
To build a n-bit propagate-adder, we combine n fa blocks. The circuits adds the least significant bits and passes the carry on to the next bit, and so on. Combining the output bits forms the sum s. The circuit shown below gives an example of a 4-bit carry-propagate adder.
We describe the circuit using array instantiation. In this a[] and b[] are the summands, ci[] is the in-coming carry, co[] is the out-going carry and s[] is the sum. >math_adder_fa_block fa [N-1:0] ( .a ( a ), .b ( b ), .ci ( {c[N-2:0], 1’b0} ), .s ( s[N-1:0] ), .co ( {s[N], c[N-2:0]}) ); The compiler will optimize the fa blocks with forced inputs, but most fa blocks will compile to a RTL netlist as shown below.
The 4-bit adder compiles into the daisy chained fa blocks as shown.
The complete Verilog HDL code along with the test bench and constraint files are available through GitHub for the Xilinx (Spartan-6 LX9) and Altera (Cyclone IV DE0-Nano) boards.

#### Results

The propagation delay $$t_{pd}$$ depends on size N and the value of operands. For a given size N, adding the value 1 to an operand that contains all zeroes causes the longest propagation delay. The worst-case propagation delays for the Altera Cyclone IV on the DE0-Nano are found using the post-map Timing Analysis tool. The exact values depend on the model and speed grade of the FPGA, the silicon itself, voltage and the die temperature.
$$N$$ Timing Analysis Measured
slow 85°C slow 0°C fast 0°C actual
4-bits 6.9 ns 6.2 ns 4.3 ns
8-bits 8.6 ns 7.6 ns 5.4 ns
16-bits 15.7 ns 13.9 ns 9.7 ns
27-bits 20.2 ns 18.0 ns 12.3 ns
32-bits 26.0 ns 23.2 ns 15.7 ns
42-bits 30.2 ns 26.9 ns 18.1 ns

### Borrow-propagate subtractor

Similar to addition, the simplest subtraction method is borrow-propagate, as introduced in Chapter 7 of the inquiry “How do computers do Math?”. Again, we will start by building a 1-bit subtractor (fs). The inputs a and b represent the 1-bit binary numbers being added. Output d, the difference. li and lo are the incoming and outgoing borrow/loan signals. The outputs d and lo can be expressed as a function of the inputs. The difference d is an Exclusive-OR function, just as the sum was for addition.
 \begin{aligned} d&=a \oplus b \oplus l_i\\ l_o&=\overline{a} \cdot b + l_i \cdot (\overline{a \oplus b}) \end{aligned}
To build a 4-bit subtractor we combine four of these building blocks.
The implementation of the borrow-propagate subtractor is very similar to the adder shown earlier and is left as an exercise for the reader.

Another method to subtract is based around the fact that $$a – b = a + (-b)$$. This allows us to build a circuit that can add or subtract. When the operation input op equals 1, it subtracts b from a, otherwise it adds the values. Under two’s complement, subtracting b is the same as adding the bit-wise complement of b and adding 1. The inputs b is negated by inverting its bits (using an XOR with signal op), and 1 is added by setting the least significant carry input to 1. We can build this a 4-bit adder/subtractor using fa blocks as shown below, where r is the result.