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CS 325: CS Hardware and Software Organization and Architecture

CS 325: CS Hardware and Software Organization and Architecture. Integers and Arithmetic Part 3. Outline. 2’s Complement Binary Addition 2’s Complement Binary Subtraction Binary Multiplication Unsigned 2’s Complement Multiplication Booth’s Algorithm Binary Division Unsigned

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CS 325: CS Hardware and Software Organization and Architecture

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  1. CS 325: CS Hardware and SoftwareOrganization and Architecture Integers and Arithmetic Part 3

  2. Outline • 2’s Complement Binary Addition • 2’s Complement Binary Subtraction • Binary Multiplication • Unsigned • 2’s Complement Multiplication • Booth’s Algorithm • Binary Division • Unsigned • 2’s Complement Division

  3. 2’s Complement Binary Addition • -39 + 92 = 53 1 1 1 1 1 1 0 1 1 0 1 1 +0 1 0 1 1 1 0 0 0 0 1 1 0 1 0 1 Carryout without overflow. Sum is correct.

  4. 2’s Complement Binary Addition • 104 + 45 = 149 1 1 1 0 1 1 0 1 0 0 0 +0 0 1 0 1 1 0 1 1 0 0 1 0 1 0 1 No carryout, Overflow. Sum is not correct.

  5. 2’s Complement Binary Addition • -75 + 59 = -16 1 1 1 1 1 1 1 0 1 1 0 1 0 1 +0 0 1 1 1 0 1 1 1 1 1 1 0 0 0 0 No carryout, no overflow. Sum is correct.

  6. 2’s Complement Binary Addition • 127 + 1 = 128 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 +0 0 0 00 0 0 1 1 0 0 0 0 0 0 0 No carryout, overflow. Sum is not correct.

  7. 2’s Complement Binary Subtraction • Subtraction Rule: • To subtract one number, S, from another number, M, take the 2’s complement of the number S and add it to the number M. • Example: • 11 – 4, we will negate 5 using 8-bit 2’s comp and add 11: 0000 1011 4: 0000 0100  1111 1100 (negate 4) and add 1 1 1 1 0 0 0 0 1 0 1 1 +1 1 1 1 1 1 0 0 0 0 0 0 0 11 1  +710

  8. 2’s Complement Binary Subtraction • Example: • Subtract 7 from 2 (2 – 7): • Convert to 4-bit 2s comp: • 7  0111 • 2  0010 • Perform 2s comp on 7: • 0111  1001 • Now add: 0 0 1 0 +1 00 1 1 0 1 1 • The value 1011 represented in 2’s comp converts to -510

  9. 2’s Complement Binary Subtraction • Another Example: • -23 – 6, we will negate 6 using 8-bit 2’s comp and add -23: 1110 1001 6: 0000 0110  1111 1010 (negate 6) and add 1 1 1 1 1 1 1 0 1 0 0 1 +1 1 1 1 1 0 1 0 1 1 1 0 0 0 1 1  -2910

  10. 2’s Complement Binary Subtraction • Try: • 17 – (-6), we will negate 6 using 8-bit 2’s comp and add 17: 0001 0001 -6: 1111 1010  0000 0110 (negate 6) and add 0 0 0 1 0 0 0 1 +0 0 0 0 0 1 1 0 00 0 1 0 1 1 1  +2310

  11. 2’s Complement Binary Subtraction • Another Example: • 12 – 123, we will negate 123 using 8-bit 2’s comp and add 12: 0000 1100 123: 0111 1011  1000 0101 (negate 123) and add 0 0 0 0 1 1 0 0 +1 0 0 0 0 1 0 1 1 0 0 1 0 0 0 1  +11110 No Carry, with overflow. Difference is not correct.

  12. Unsigned Binary Multiplication • Quite Easy. • Rules to remember: • 0 * 1 = 0 • 1 * 1 = 1 • Same as logical “and” operation • Multiplying an m-bit number by and n-bit number results in an n+m-bit number. This n+m-bit width ensures overflow cannot occur. • Simple Example: m = n = 2 • 2x3 = 6 • 102 x 112 = 1102 • Largest 2-bit value: 11 or 310 • 112 x 112 = 10012 or 910

  13. Unsigned Binary Multiplication • Example: 1 0 1 0 x0 1 1 0 0 0 0 0 1 0 1 0 1 0 1 0 +0 0 0 0______ 0 1 1 1 1 0 0

  14. Unsigned Binary Multiplication • Try: 1 1 0 1 1 x1 1 0 0

  15. 2’s Complement Binary Multiplication • Still Easy: • Only difference between 2’s comp multiply and unsigned multiply: • Sign extend both values to twice as many bits. Ex: 1 1 0 0  1 1 1 1 1 1 0 0 x0 1 0 1 x0 0 0 0 0 1 0 0 • The result will be stored in m+n least significant bits.

  16. 2’s Complement Binary Multiplication • Example: 1 1 1 0  1 1 1 1 1 1 1 0 0 0 1 1  x0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 + 1 1 1 1 1 1 1 0__ 0 1 1 1 1 1 0 1 0 • Result located in m+n (4+4) least significant bits. 1 1 1 1 1 0 1 0  -610

  17. 2’s Complement Binary Multiplication • Try: 0 0 1 0 x0 1 0 1

  18. 2’s Complement Binary Multiplication • Efficiency of this method? • To perform a multiply instruction, product bit-width needs to be 2N when using N-bit values. • Multiply instruction? • Requires more complex hardware with signed values. • Better way?

  19. 2’s Complement Binary Multiplication – Booth’s Algorithm • Multiplication by bit shifting and addition. • Removes the need for multiply circuit • Requires: • A way to compute 2’s Complement • Available as fast hardware instructions • X86 assembly instruction: NEG • A way to compare two values for equality • How to do this quickly? Exclusive Not OR (NXOR) Gate Compare all sequential bits of bit string A and bit string B. Values are equal if the comparison process produces all 1s. • A way to shift bit strings. • Arithmetic bit shift, which preserves the sign bit when shifting to the right. 10110110arithmetic shift right 11011011 • x86 assembly instruction: SAR

  20. 2’s Complement Binary Multiplication – Booth’s Algorithm • Example: 5 x -3 • First, convert to Bin: • 5 = 0101 • -3 = 1101 • If we add 0 to the right of both values, there are 4 0-1 or 1-0 switches in 0101, and 3 in 1101. Pick 1101 as X value, and 0101 as Y value • Next, 2s Comp of Y: 1011 • Next, set 2 registers, U and V, to 0. Make a table using U, V, and 2 additional registers X, and X-1.

  21. 2’s Complement Binary Multiplication – Booth’s Algorithm • Register X is set to the predetermined value of x, and X-1 is set to 0 • Rules: Look at the LSB of X and the number in the X-1 register. • If the LSB of X is 1, and X-1 is 0, we subtract Y from U. • If LSB of X is 0, and X-1 is 1, then we add Y to U. • If both LSB of X and X-1 are equal, do nothing and skip to shifting stage.

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