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Efficient Computer Arithmetic: Unsigned and Signed Notations, Multiplication, Division, and Optimization

Learn about unsigned and signed notations, multiplication algorithms, division techniques, and hardware implementation strategies in computer arithmetic. Explore different methods and examples to optimize arithmetic operations efficiently.

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Efficient Computer Arithmetic: Unsigned and Signed Notations, Multiplication, Division, and Optimization

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  1. Chapter 8Computer Arithmetic

  2. Chapter Outline • Unsigned notations • Signed notations • Binary Coded Decimal • Specialized arithmetic hardware • Floating point numbers • IEEE 754 floating point standard

  3. Unsigned Notations • Unsigned non-negative • Unsigned two’s-complement

  4. Unsigned Notations • Unsigned non-negative • Unsigned two’s-complement

  5. Addition: X  X + Y

  6. Overflow

  7. Subtraction: X  X - Y

  8. Overflow

  9. Multiplication • A non-optimal method z = 0 FOR i = 1 TO y DO { z = z + x }

  10. Multiplication • A more typical method

  11. Multiplication • Calculating running totals

  12. Multiplication • Shifting partial results to align sums

  13. Shift-add Multiplication Algorithm C = 0, U = 0;

  14. Example: UV  X  Y(X = 1101, Y = 1001) C = 0, U = 0 0

  15. RTL Code C  0, U  0, i  n

  16. Example: UV  X  Y(X = 1101, Y = 1001) 0 C  0, U  0, i  4

  17. Hardware Implementation

  18. Optimizing the RTL Code • UV  X  V • Register Y not needed • One operand is lost

  19. Optimizing the RTL Code C  0, U  0

  20. Example 0 C  0, U  0, i  4

  21. Booth’s Algorithm • Multiplying unsigned 2’s-complement numbers

  22. Example • UV  X  Y, X = -3 (1101), Y = -5 (1011)

  23. Example • UV  X  Y, X = -3 (1101), Y = -5 (1011)

  24. RTL Code

  25. Example

  26. Optimized RTL Code

  27. Hardware Implementation

  28. Division • A non-optimal method

  29. Division • A more typical method

  30. Division • Shifting results to align remainders

  31. Shift-subtract Division Algorithm

  32. Example: UV  X(UV = 1001 0011, X = 1101)

  33. RTL Code

  34. Example

  35. Hardware Implementation

  36. Restoring Division Algorithm

  37. Overflow Comparison

  38. Example

  39. RTL Code

  40. Example

  41. Hardware Implementation

  42. Signed Notations • Signed-magnitude

  43. Signed Notations • Signed-magnitude • Signed-2’s complement

  44. Signed Notations • Signed-magnitude • Signed-2’s complement • Value Signed-magnitude Signed-2’s complement • +3 0 0011 0 0011 • -3 1 0011 1 1101

  45. Addition and Subtraction

  46. RTL Code

  47. RTL Code

  48. Examples

  49. Examples

  50. Examples

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