3. Arithmetic with IEEE-754 format

Arithmetic with IEEE-754 format

Addition

To add two floating-point numbers:

• Adjust position of binary points so significands have the same exponent.
• If numbers have the same sign, add significands. Otherwise subtract smaller from larger.
• If required, normalise significand at the end.
• Select correct sign depending on values.

Subtraction

Use the relationship $a-b=a+(-b)$.

Multiplication

To multiply two floating-point numbers:

• Add the exponents.
• Multiply the significands.
• If required, normalise significand at the end.
• Select correct sign depending on values.

Division

Same as mult, but in step 1 subtract exponents and in set 2 divide significands.

Common Errors

• Overflow: magnitude is too large to store the format
• Underflow: magnitude is too small to store in the format Can often be confused with overflow in negative direction.
• Catastrophic cancellation: lack of significant figures to represent the result of colliding large and small numbers
• Lack of exact representations for common Base-10 numbers: $0.1+0.2=0.300000000000004$