1. Example 1 (P46) What’s the binary form of x=2/3
Example 1 (P46) What’s the binary form of x=2/3? What are two nearby machine numbers x- and x+ in the Marc-32? Which one is taken to be fl(x)? What are the absolute roundoff error and relative roundoff error in representing x by fl(x)?

3. Stored as machine numbers fl(a),fl(b),… Rounded off (舍入)
Input numbers a,b,c,…
Normalized (标准化)
Stored as machine numbers fl(a),fl(b),…
Rounded off (舍入)
Do one arithmetic operation/calculation
Obtain a number (result) e.g. fl(a)fl(b)

5. The computer with 5 decimal digits stores those results in rounded form as
The relative errors are respectively

6. The computer with 5 decimal digits stores those results in rounded form as
The relative errors are respectively

7. Denote  one of the four basic arithmetic operations:   , 
Assume x,y are machine numbers, then
there is some constant  s.t.
fl(xy) = [xy] (1+)
where ||; here,  can be taken to be the unit roundoff error for the machine. In Marc-32, =2-24.
Q: How to compute xy if x,y are not machine numbers?

8. If x,y are not machine numbers, then
there is still some constant  s.t.
fl(x) = x (1+1)
fl(y) = y (1+2)
fl(xy) = fl(fl(x)fl(y))
= (fl(x)fl(y)) (1+3)
= [(x(1+1))  (y(1+2))](1+3)
= (xy)(1+1+2+12) (1+3)  xy
where |1|,|2|,|3|; still,  can be taken to be the unit roundoff error for the machine.

9. Q: How about compand arithmetic operations?
Assume x,y,z  A={machine numbers of Marc-32}.
fl(x(y+z)) =
= [x fl(y+z)] (1+1) |1| 2-24
= [x (y+z) (1+2)] (1+1) |2| 2-24
= x (y+z) (1+2+1 +12)
 x (y+z) (1+2+1)
= x (y+z) (1+3) |3| ?
Here 3=2+1

10. Find fl(x(y+z)) for
x, y, z  A={machine numbers of Marc-32}.

11. Theorem on Relative Roundoff Error in Adding

12. Theorem on Relative Roundoff Error in Adding