Arithmetic

1
Arithmetic

• CPSC 321 Computer Architecture
• Andreas Klappenecker

2
Overview

• Number representations
• Overflows
• Floating point numbers
• Arithmetic logic units

3
Unsigned Numbers

• 32 bits are available
• Range 0..232 -1
• 11012 232220 1310
• Upper bound 232 1 4 294 967 295

4
Number representations

• What signed integer number
• representations do you know?

5
Signed Numbers

• Sign-magnitude representation
• MSB represents sign, 31bits for magnitude
• Ones complement
• Use 0..231-1 for non-negative range
• Invert all bits for negative numbers
• Twos complement
• Same as ones complement except
• negative numbers are obtained by inverting all
  bits and adding 1

6
Advantages and Disadvantages

• sign-magnitude representation
• ones complement representation
• twos complement representation

7
Signed Numbers (3bits)
8
Twos complement

• The unsigned sum of an n-bit number its negative
  yields?
• Example with 3 bits
• 0112
• 1012
• 10002 2n gt negate(x) 2n-x
• Explain ones complement

9
MIPS 32bit signed numbers

• 0000 0000 0000 0000 0000 0000 0000 0000two 0ten
• 0000 0000 0000 0000 0000 0000 0000 0001two
  1ten
• 0000 0000 0000 0000 0000 0000 0000 0010two
  2ten
• ...
• 0111 1111 1111 1111 1111 1111 1111 1110two
  2,147,483,646ten
• 0111 1111 1111 1111 1111 1111 1111 1111two
  2,147,483,647ten
• 1000 0000 0000 0000 0000 0000 0000 0000two
  2,147,483,648ten
• 1000 0000 0000 0000 0000 0000 0000 0001two
  2,147,483,647ten
• 1000 0000 0000 0000 0000 0000 0000 0010two
  2,147,483,646ten
• ...
• 1111 1111 1111 1111 1111 1111 1111 1101two
  3ten
• 1111 1111 1111 1111 1111 1111 1111 1110two
  2ten
• 1111 1111 1111 1111 1111 1111 1111 1111two
  1ten

10
Conversions

• How do you convert an n-bit number
• into a 2n-bit number?
• (Assume twos complement representation)

11
Conversions

• Suppose that you have 3bit twos complement
  number
• 1012 -3
• Convert into a 6bit twos complement number
• 1111012 -3
• Replicate most significant bit!

12
Comparisons

• What can go wrong if you accidentally
• compare unsigned with signed numbers?

13
Comparisons for unsigned

• Register s0
• 1111 1111 1111 1111 1111 1111 1111 1111
• Register s1
• 0000 0000 0000 0000 0000 0000 0000 0001
• Compare registers (set less than)
• slt t0, s0, s1 yes, since 1 lt 1
• sltu t1, s0, s1 no, since 232-1gt1

14
Addition Subtraction

• Just like in grade school (carry/borrow 1s)
  0111 0111 0110  0110 - 0110 - 0101
• Two's complement operations easy
• subtraction using addition of negative numbers
  0111  1010

15
Overflow

• Overflow means that the result is too large for
• a finite computer word
• for example, adding two n-bit numbers does not
  yield an n-bit number 0111  0001
• 1000
• the term overflow is somewhat misleading

16
Detecting Overflow

• No overflow when adding a positive and a negative
  number
• No overflow when signs are the same for
  subtraction
• Overflow occurs when the value affects the sign
• overflow when adding two positives yields a
  negative
• or, adding two negatives gives a positive
• or, subtract a negative from a positive and get a
  negative
• or, subtract a positive from a negative and get a
  positive

17
Detecting Overflow
18
Effects of Overflow

• An exception (interrupt) occurs
• Control jumps to predefined address for exception
• Interrupted address is saved for possible
  resumption
• Don't always want to detect overflow
• MIPS instructions addu, addiu, subu note
  addiu still sign-extends!

19
What next?

• More MIPS assembly operations
• How does an ALU work?
• Simple digital logic design
• How can we speed-up addition?
• What about multiplication?