Implement Basic Arithmetic Without Inbuilt Operators

Problem Statement

Implement the four basic arithmetic operations (+, -, *, /) for integers without using the built-in arithmetic operators.

You may only use:

• Bitwise operators (&, |, ^, ~, <<, >>)
• Comparison operators
• Loops / recursion

Constraints

• Input integers fit in 32-bit signed range
• Division is integer division (truncate toward zero)
• Division by zero is undefined
• Handle negative numbers correctly

Expected Implementation

add(5, 3)       → 8
subtract(10, 4) → 6
multiply(3, 4)  → 12
divide(10, 3)   → 3
divide(-7, 2)   → -3

What the Interviewer Expects

1. Addition with bit manipulation:
   • XOR gives sum without carry: a ^ b
   • AND + left shift gives carry: (a & b) << 1
   • Repeat until carry is 0
2. Subtraction: Negate second operand using two's complement (~b + 1), then add.
3. Multiplication: Repeated addition with bit shifting optimization (Russian peasant multiplication).
4. Division: Repeated subtraction with bit shifting to speed up (similar to long division in binary).
5. Edge cases: Negative numbers, overflow, divide by zero.

Follow-ups

1. How do you handle integer overflow in your add function?
2. Can you implement modulo (%) using only your divide and multiply?
3. What's the time complexity of your multiply and divide implementations?
4. How would you extend this to support floating-point arithmetic?