Arithmetic Circuits II

1. Arithmetic Circuits II Anselmo Lastra

2. Overflow • Two cases of overflow for addition of signed numbers • Two large positive numbers overflow into sign bit • Not enough room for result • Two large negative numbers added • Same – not enough bits • Carry out can be OK

3. Examples • 4-bit signed numbers • 7 + 7 • 7 – 7 • Generates carry but result OK • -7 -7 • 4 + 4 • Generates no Cout, but overflowed

4. Overflow Detection • Condition is that either Cn-1 or Cn is high, but not both

5. Multiplier • Multiply by doing single-bit multiplies and shifts • Look at combinational circuit to do this

6. Combinational Multiplier AND computes A0 B0 Half adder computes sum. Will need FA for larger multiplier.

7. Larger Multiplier

8. Sequential Multiply • Imagine doing over time rather than in parallel • Bitwise multiply • Shift • Add • If we have time later in semester we’ll look at fancier multipliers

9. Contraction • Can implement other functions • Like increment, decrement • By using basic arithmetic circuits • Adder • And removing unused portions • This is called contraction

10. Example: Incrementing • Very common • Next address computation • Specialize an adder

11. Eliminate Unneeded Gates

12. Multiplication by Constant • What if constant is a power of two? • What is the circuit?

13. Multiplication by Pwr of 2 • Just wires • Division by pwr of 2 similar

14. Arbitrary Constant

15. Contraction Not Always Best • Sometimes it pays to rethink the function • Example of decrementer in book

16. Sign Extension • Changing size of number common in instruction sets • 16-bit immediate to register, for example • Can’t just add zeros (zero fill) • Would turn 4-bit -5 (1011) to 8-bit +11 (00001011) • Sign extend (fill with left digit) • 1011 to 11111011

17. We’ve Covered • Adders • Ripple carry • Carry lookahead • Subtracting unsigned numbers • New design for adder-subtractor • Signed numbers • Signed addition/subtraction • Multiplication – just basic • Modified Circuits