1. Digital System Design II 数字系统设计2
Weidong Wang (王维东)
Dept. of Information Science & Electronic Engineering
ISEE
Zhejiang University

2. Course Information Instructor: Weidong WANG TA:
Tel(O): ;
Office Hours: TBD, Yuquan Campus, Xindian (High-Tech) Building 306, / whenever
TA:
mobile，
Hanqi Shen沈翰祺, ;
Jingjing
Office Hours: Tuesday & Saturday 14:00-16:30 PM.
Xindian (High-Tech) Building 308.

3. Lecture 6 Floating Point Arithmetic
For Computer

4. Arithmetic for Computers
Computer numbers
Binary numbers: word, half word, byte, bit
What about fractions and real numbers?
How bigger number?
Arithmetic
Addition, Subtraction, Multiplication, Division
How does hardware really do?
Real numbers
Binary, decimal, real form
Float number
Floating-point arithmetic

5. Addition Binary addition Negation Added bit by bit From right to left
overflow
Negation
Signed, unsigned
Signed: two= -1 ten
Unsigned: two= 4,294,967,295 ten
Sign extension

6. Subtraction
x-y=x+(-y)
7-6=7+(-6)=1
overflow

7. Multiplication 1000x1001=? Hardware Paper and pencil method
32-bit by 32-bit
Product is 64-bit
shift

8. Multiplication Algorithm

9. Multiplication version 2
Refined version
32 bits wide registers
Product is shifted right
Multiplier is in the right half of product register

10. Multiplication version 3
Faster version
Unrolls the loop
31 adders
Or Multiply in ARM ?

11. Division
÷1000=?
Hardware

12. Division Algorithm

13. Division version 2 Improved version 32 bits wide registers
Product is shifted right
Quotient is in the right half of remainder register
Hardware architecture same as multiplication

14. Faster division version?
Use many adders?
Not
Produce more bits of the quotient per step
Guess
Lookup table
Correct wrong guess
Not all computer ISA had division instruction

15. Floating Point Scientific notation Fractions or reals in mathematics
Bigger than 32-bit integer
Smaller than 1
Floating point normalized form: binary point

16. Representation
Compromise
F: fraction
E: exponent
S: sign

17. Double Precision Two words IEEE754 floating-point standard
Exponent: 11-bit /8
Fraction: 52-bit /23
IEEE754 floating-point standard

18. Example:
-0.75

19. Example: binary
Single precision
Double precision representation

20. Floating-Point Addition
Step 1
Align point of the number with smaller exponent
Step 2
Addition of the significands
Step 3
Adjust the sum to normalized scientific notation
Step 4
Round the number

21. Addition Algorithm

22. Binary Floating-Point Addition
Step 1
Shift right
Step 2
Add
Step 3
normalize the sum
Step 4
Round the sum

23. Block diagram of addition

24. Floating-Point Multiplication
Step 1
Adding the exponents
Correct biased sum
Step 2
Multiplication of the significands
Step 3
normalize scientific notation
Step 4
Round the number
Step 5
Sign of product

25. Multiplication Algorithm

26. Binary Floating-Point Multiplication
Step 1
Adding the exponents with biase
Step 2
Multiplication of the significands
Step 3
normalize scientific notation
Step 4
Round the number
Step 5
Sign of product

27. Floating-Point Instructions in MIPS
Addition
Single add.s
Double add.d
Subtraction
sub.s, sub.d
Multiplication
mul.s, mul.d
Division
div.s, div.d
Comparison
c.x.s, c.x.d
Where x maybe eq, neq, lt(less than), le(less than or equal), gt(greater than), ge(greater than or equal)
Branch
bclt(true), bclf(false)
On separate floating-point registers
$f0, $f1, $f2, ……, $f31
Load
lwcl
Store
swcl

28. MIPS floating-point ISA

29. C to MIPS C
MIPS
Assume fahr in $f12, result in $f0, global pointer is $gp

30. Floating-Point Architecture in X86
xmm
double precision
8 SSE2 registers as floating-point registers
For multimedia
8-bit each color of pixel
Eight 8-bit or Four 16-bit Arithmetic

31. Rounding Reading IEEE754 What is the Max number in 32-bit MIPS?
What is the Max negative number in 32-bit MIPS?

32. summary
IEEE754 floating-point standard
Number in computer system

33. HomeWork Readings: HW4 Read IEEE754; Chapter 4.1 to 4.4
p.286: Exercise 3.4
P.289: Exercise 3.7
p.293: Exercise 3.12
p.296: Exercise 3.14