Arithmetic For Computers

1
Arithmetic For Computers

• Chapter 3

2
Topics for discussion

• Number system radix/base, a set of distinct
  digits, operations
• Radix conversion
• ASCII versus binary representation
• Signed arithmetic sign extension
• Bounds check validation
• Addition and subtraction algorithms
• Multiplication algorithms
• Floating point representation
• Floating point arithmetic algorithms

3
Number system

• Radix or Base 10 for decimal system, 2 for
  binary system, 8 for octal , 12 for duo-decimal
  (to count dozens), 16 for hexa-decimal
• Decimal digits 0,1,2,3,4,5,6,7,8,9
• Binary 0,1 binary digit is a bit
• Octal 0,1,2,3,4,5,6,7
• Hex 0,..9, A, B, C, D, E, F
• If we assume digits are number from right to left
  starting from 0th digit as the least significant
  digit (Little Endian), the value of the ith digit
  d in is
• d x basei
• Example Your pay is 101 per hour. Would you
  prefer it (the base) in decimal, octal,
  hexadecimal or binary?

4
Number Representation

• On the keyboard it is represented by ASCII
  American Standard Code for Information
  Interchange 7 bit code represented by a byte
  container.
• For character representation this is a nice
  system.
• How efficient is this representing numbers for
  processing?
• Consider 4 ASCII digits. What is the range of
  integers you can represent with this?
• 0 9999
• 4 X 8 32 bits
• With 32 bits and binary systems and only positive
  numbers
• 0 (232 -1). What is this value? Approx
  4,000,000,000 or 4G !

5
Signed Numbers

• When we allow negative and positive numbers, half
  the range is occupied by positive numbers and the
  other by negative numbers.
• How to represent the sign? Using a bit ? 0 for
  positive and 1 for negative?
• Then with 32 bits 0 to (231 -1) positive
  range
• - 0 to
  (231 -1) negative range
• A better representation for negative number is
  2s complement. How to compute 2s complement?
• Whats the advantage of 2s complement?
• Subtraction is equivalent to adding 2s
  complement of the second operand.
• Sign extension can be used to extend the number
  from, from 16 bits to 32 bits for example.

6
Dealing with Overflow
7
Dealing with overflow

• Signed operations that result in overflow cause
  an exception whereas unsigned operations on
  overflow do not cause exception.
• Example
• add, addi, sub will cause exception on overflow
• addu, addiu, and subu will not cause exception on
  overflow
• How to detect overflow in unsigned then?
• addu t0,t1,t2
• xor t3,t1,t2 check if signs differ
• slt t3,t3,zero signs differ?
• bne t3,zero,No_Overflow signs differ
• nop
• signs area same
• xor t3,t0,t1 find sign of sum
• slt t3,t3,zero if sum sign is diff
• bne t3,zero,Overflow

8
Branches and Jumps

• beq s1,s2,lab1 if s1 s2 jump to label
  lab1
• bne s1,s2,lab2 if s1 ? s2 jump to label
  lab2
• j lab3 unconditional
  jump to lab3
• jal proc3 jump and link to
  proc3
• jr retAddr jump back to callee
• beqz s2, lab4 is s2 0 then jump to
  lab4