1. CSCI206 - Computer Organization & Programming
Floating Point Arithmetic
zyBook: 10.9, 10.10

2. Review IEEE754 S Exponent Mantissa
Special values, else normalized numbers
Exponent
Mantissa (fraction)
Value
+/- zero
nonzero
denormalized number
all 1’s
+/- infinity
NaN (not a number)

3. Floating Point Addition
Match exponents
Add the significants
Normalize the sum
Check overflow/underflow
Round
Done

4. FP Addition Example
Show the FP addition of: using 4 bits of precision

5. FP Addition Example
Show the FP addition of:

6. 1. Match Exponents larger exponent is , rewrite
shift number right, increment exponent
shift number left,
decrement exponent
1. Match Exponents
larger exponent is , rewrite

7. 2. Add

8. 3. Renormalize
(nothing to do in this example)

9. 4. Check overflow/underflow
(nothing to do in this example)

10. 5. Round 6. Done (check) (nothing to do in this example)=
6. Done (check)

11. Accurate Arithmetic
IEEE754 guarantees accuracy within 0.5 ulp (units in last place)
error is no more than ½ of least significant digit
To achieve this, 3 extra bits are added to the fractional part in arithmetic, they are called: guard, round, and sticky bits (from most significant to least).

12. Guard, Round, Sticky
Guard and round bits act like any other bit to to the right of the decimal.
The sticky bit is set if any nonzero value to the right of this place has ever existed.
once set, it stays set (it’s sticky)

13. Addition Example with GRS
Use 4-bits of precision plus GRS
5. Round GRS: 111 rounds up:
1. Normalize: G
GRS
Nonzero value to right of round bit, so SET sticky bit
2. Add:
GRS

15. Multiplication Similar to addition Add exponents Multiply significands
Normalize
Check/round
Set sign