1. Computer Architecture & Operations I
Instructor: Ryan Florin

2. Floating Point Numbers
Morgan Kaufmann Publishers
29 May, 2018
Floating Point Numbers
6.022 x 10^23
16.2
119
What do you call a decimal point in binary?
Chapter 3 — Arithmetic for Computers

3. Scientific Notation (base 10)
Takes the form:
a x 10^b
Normalized form ensures there is only 1 non-zero number to the left of the decimal point.
300 => 3.0 x 10^2
=> x 10^2
=> 5.85 x 10^-5

4. Scientific Notation (base 2)
Takes the form
a X 2^b
Normalized form ensures there is only 1 non-zero number to the left of the “binary” point.
=> x 2^2
.05 => 1.6 x 2^-5

5. Normalizing binary R = a x 2^b
Step 1: Find the power of 2 that is the closest to the number without going over. This is the value of b.
Step 2: divide R by 2^b to get the value of a.
Example: 6.84
Step 1: b = 2 (2^2 = 4)
Step 2: 6.84 / 4 = a = 1.71
1.71 x 2^2

6. IEEE 754 Single Precision
(-1)signbit x (1 + fraction) x 2 exponent – bias Single Precision bias: 127 Range: 2^128 – 2^-127 Precision: ~7 digits

7. Single Precision Example: 3.14 3.14 is positive. Sign bit = 0 a x 2^b
3.14 is between 2^1 and 2^2 (2 and 4)
a = 3.14 / 2 = 1.57
Exponential = 1 + bias = 128 ( )
Fraction = 1.57

8. Single Precision Example: 3.14 Fraction = 1.57
Due to being normalized, there will always be a 1 to the left of the binary point. It is assumed to be a 1.
Represent the rest (.57) using negative powers of 2
.57 =
Fraction:

9. Single Precision Example: 3.14 Sign bit: 0 Exponential: 1000 0000
Fraction:

10. IEEE 754 Double Precision
(-1)signbit x (1 + fraction) x 2 exponent – bias Double Precision bias: 1023 Range: 2^1024 – 2^-1023 Precision: ~16 digits

11. Double Precision Example: .0025 .0025 is positive. Sign bit = 0
a x 2^b
b = -9:
.0025 is between 2^-9 and 2^-8 (.002 and .004)
a = / 2^-9 = 1.28
Exponential = -9 + bias = 1014( )
Fraction = 1.28

12. Double Precision Example: .0025 Fraction = 1.28
Due to being normalized, there will always be a 1 to the left of the binary point. It is assumed to be a 1.
Represent the rest (.28) using negative powers of 2
.28 =

13. Double Precision Example: .0025 Sign bit: 0 Exponential: 011 1111 0110
Fraction:

14. Reverse Process Example: 412C000016 0100 0001 0010 1100 0000 0000 0000
Sign bit: 0
Positive
Exponential:
130 – 127 = 3
Fraction:
= =
x 23 = 10.75

15. What I want you to do
Review Chapters 3.5
Review Midterm II