Arithmetic Micro-operations

Introduction

Arithmetic micro-operations are the basic calculations performed on data stored in registers. They're called "micro-operations" because they're the smallest unit of work the hardware can perform — each one completes in a single clock cycle. When you execute an ADD instruction, the CPU breaks it down into micro-operations like "read register A, read register B, add them, write result to register C." Let's explore each type and see how they're implemented in hardware.

Types of Arithmetic Micro-operations

The fundamental arithmetic micro-operations are:

Addition Micro-operation

RTL Description

Hardware: Binary Adder

The binary adder is built from full adder cells cascaded together:

Each full adder takes three inputs (A, B, Carry-in) and produces two outputs (Sum, Carry-out). The carry ripples from LSB to MSB — this is the ripple carry adder.

Full Adder Truth Table

Subtraction Micro-operation

The Complement Method

Instead of building a separate subtractor, we reuse the adder:

Hardware: Adder-Subtractor

By adding XOR gates to the B input, we can build a combined adder-subtractor:

This elegant design means we need only ONE hardware unit for both addition and subtraction.

Increment Micro-operation

RTL Description

Hardware Implementation

A dedicated incrementer can be simpler than a full adder. It's implemented using half adders cascaded together with the initial carry set to 1:

Half adder: Sum = A ⊕ 1 = Ā (for the first stage, complement the bit if carry-in is 1)

Decrement Micro-operation

RTL Description

Implementation

Can be done as: R1 + all_ones (adding -1 in two's complement) Or using a specialized decrementer circuit (cascade of half-subtractors with initial borrow = 1)

Negate (Two's Complement) Micro-operation

RTL Description

Hardware

1. Pass R1 through NOT gates (complement each bit)
2. Pass result through incrementer (add 1)
3. Store back in R1

This converts positive to negative and vice versa.

The Arithmetic Circuit: A Complete Unit

A practical arithmetic circuit combines these operations using a controlled adder:

This single circuit handles ALL arithmetic micro-operations by controlling the B input and carry-in.

Status Flags

Arithmetic operations produce status information stored in a condition code register:

• C (Carry): Carry out of the MSB — indicates unsigned overflow
• Z (Zero): Result is zero — all bits are 0
• N (Negative): MSB of result is 1 (negative in two's complement)
• V (Overflow): Signed overflow — carry into MSB ≠ carry out of MSB

These flags are used by conditional branch instructions to make decisions.

Key Takeaways

1. Arithmetic micro-operations are single-clock-cycle calculations on register contents
2. The binary adder (cascade of full adders) is the fundamental building block
3. Subtraction is implemented as addition with two's complement (A + B̄ + 1)
4. A single adder circuit with input logic handles ADD, SUB, INCREMENT, DECREMENT, and NEGATE
5. Status flags (C, Z, N, V) record the conditions of each arithmetic result
6. This shared hardware design is exactly how real ALUs are built — one circuit, many operations