Other Arithmetic Functions Section 4-5

Other Arithmetic FunctionsSection 4-5

Combinational Multiplier Example: • 4-Bit Multiplicand B = {B3 B2 B1 B0} • 3-Bit Multiplier A = {A2 A1 A0} • Product is a 7-Bit number, C = {C6 C5 C4 C3 C2 C1 C0 }

Computing the Product • Let us work out the multiplication on the board:

4-Bit by 3-Bit Binary Multiplier Multiplier

Contraction Can implement other functions ( Like increment, decrement ) by using basic arithmetic circuits (e.g. Adder) And removing unused portions • This is called contraction

Hiding Box Hiding Box Hiding Box Hiding Box Hiding Box Incrementing • Very common • Computation of memory address for the next instruction to execute by CPU • Use a customized adder To increment A = { A2 A1 A0 }, do A2 A1 A0 + 001; Do not use C3 (why?)

Reference C2 Customized adder To increment A = { A2 A1 A0 }, do A2 A1 A0 + 001; Ignore C3

C2 C2 C1 Cell 0 Eliminate Unneeded Gates No need for C3 0 C2 Cell 2 Cell 1

Multiplication by Constant Case 1: Constant is a power of two Case 2: Constant is NOT a power of two

C = 4 * B C = B / 4 Multiplication by Power of 2 • Just wires • Division by power of 2 similar

AND gates removed because A is a constant First adder removed by contraction Multiplication by A ConstantWhich Is Not A Power Of 2

Contraction Not Always Best • Sometimes it pays to rethink the function • Example: study decrementer in book (Example 4-6 page 167)

Sign Extension • Changing the number of bits used to store a number is common • Example: • -5 in stored in 4-bits (1011) • to store -5 in 8 bits, extend the “Sign Bit” in the 4 left most bits • 1011 is changed to 1111 1011 • 1011 = -5 • 1111 1011 is also -5

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

Other Arithmetic Functions Section 4-5