Addition & Subtraction

Addition: just do the binary addition with given numbers
Subtraction: do the binary addition with the negation of the second operand

Half adder

1-bit adder without carry-input

• Input: two one bit-data A, B
• Output: sum(S), carry(C)

Full adder

1-bit adder with carry-input

• Input: two one bit-data A, B, carry(C$_{in}$)
• Output: sum(S), carry(C$_{out}$)

Circuit design for addition and subtraction

N-bit parallel binary adder

• Initial carry input is 0
• The i-th adder waits for the carry until it is generated by the (i-1)-th adder

Adder with subtraction (in two's complement)

• Subtract = 0 or 1 (if subtract == 1, B$_i$ is inverted)
• Subtract is also added as the initial carry (if subtract ==1, 1 is added)

Overflow

Overflow occurs when computation results are too large (out of range)

When does it occur?

• Add two positives or negatives + the sign of result is different with sources
• Subtract a negative from a positive + the sign of result is 1
• Subtract a positive from a negative + the sign of result is 0

How to detect Overflow?

• Use add, addi, sub instructions
• They cause exceptions on overflow
  • A program jumps to predefined exception handler address
• e.g., Fortran does not allow overflows. So MIPS Fortran compilers always use add, addi, sub

How to ignore Overflow?

• Use addu, addui, subu instructions (u means unsigned)
• They do not cause exceptions on overflow
• e.g., C ignores overflows. So MIPS C compilers always use addu, addui, subu