Floating Point Addition & Subtraction (OCR A Level Computer Science)

Floating Point Arithmetic

How do you represent Floating Point Numbers?

• Floating point numbers are represented with a sign, mantissa, and exponent

• Arithmetic operations must take into account these three components

Sections of a floating point number

Steps for adding or subtracting floating point numbers

1. Ensure the numbers have the same exponent before performing arithmetic

   • This might involve shifting the decimal point of one number and adjusting its exponent until both numbers have matching exponents.

   • Example:

     • Number A: 1.101×

     • Number B: 1.010×

     • Number A has an exponent of and B has an exponent of , we need to adjust B to have the same exponent as A

     • This is achieved by moving the point one space to the left in Number B and increasing the exponent by 1

     • Resulting in: 0.101×

2. Perform the binary addition or subtraction on the mantissa

   • 

3. Ensure the result is in a normalised form

   • The sum 10.010 exceeds the normal range for mantissa (1.0 to 1.111... in binary)

   • To normalise it, we shift the mantissa one position to the right and increment the exponent by 1

   • New Mantissa: 1.0010

   • New Exponent: Increment the exponent from  to 

   • The final result would be 1.0010×.

4. Determine Sign

   • For addition: If both numbers are positive or negative, the result takes the common sign

   • If they have different signs, the result's sign depends on the larger absolute value

   • For subtraction: The sign is determined by the sign of the number you're subtracting from and the result of the subtraction

Example addition

1.  +

2. Align exponents:  +

3. Add mantissa: ​

4. Normalise (if required) and determine the sign.

Example subtraction

1.  -

2. Align exponents:  -

3. Subtract mantissas: ​

4. Normalise (if required) and determine the sign.