1. 15-213 “The course that gives CMU its Zip!”
Floating Point Sept 2, 2008
Topics
IEEE Floating Point Standard
Rounding
Floating Point Operations
Mathematical properties
class03.ppt
15-213, F’08

2. Creating Floating Point Number
Steps
Normalize to have leading 1
Round to fit within fraction
Postnormalize to deal with effects of rounding
Case Study
Convert 8-bit unsigned numbers to tiny floating point format
Example Numbers s
exp
frac 2 3 6 7

3. Normalize Requirement Set binary point so that numbers of form 1.xxxxx
exp
frac 2 3 6 7
Requirement
Set binary point so that numbers of form 1.xxxxx
Adjust all to have leading one
Decrement exponent as shift left
Value Binary Fraction Exponent

4. Rounding 1.BBGRXXX Round up conditions
Guard bit: LSB of result
Sticky bit: OR of remaining bits
Round bit: 1st bit removed
Round up conditions
Round = 1, Sticky = 1  > 0.5
Guard = 1, Round = 1, Sticky = 0  Round to even
Value Fraction GRS Incr? Rounded N N N Y Y Y

5. Postnormalize Issue Rounding may have caused overflow
Handle by shifting right once & incrementing exponent
Value Rounded Exp Adjusted Result
/6 64