The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Arithmetic Circuits II Anselmo Lastra.

The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL

Arithmetic Circuits II

Anselmo Lastra

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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

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Examples

• 4-bit signed numbers• 7 + 7• 7 – 7♦ Generates carry but result OK

• -7 -7• 4 + 4♦ Generates no Cout, but overflowed

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Overflow Detection

• Condition is that either Cn-1 or Cn is high, but not both

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Multiplier

• Multiply by doing single-bit multiplies and shifts

• Look at combinational circuit to do this

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Combinational Multiplier

AND computes A0 B0

Half adder computes sum. Will need FA for larger multiplier.

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Larger Multiplier

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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

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Contraction

• Can implement other functions♦ Like increment, decrement

• By using basic arithmetic circuits♦ Adder

• And removing unused portions• This is called contraction

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Example: Incrementing

• Very common♦ Next address computation

• Specialize an adder

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Eliminate Unneeded Gates

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Multiplication by Constant

• What if constant is a power of two?

• What is the circuit?

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Multiplication by Pwr of 2

• Just wires

• Division by pwr of 2 similar

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Arbitrary Constant

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Contraction Not Always Best

• Sometimes it pays to rethink the function

• Example of decrementer in book

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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

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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