2. Short Cuts for Multiply and Divide For Positive Numbers 1. Multiply by 2 k is the same as shift k to the left, 0 fill 2. Divide by 2 k is the same as shift k to the right, 0 fill

3. Short Cuts for Multiply and Divide For Positive Numbers 1. Multiply by 2 k is the same as shift k to the left, 0 fill 2. Divide by 2 k is the same as shift k to the right, 0 fill For 2’s Complement Numbers It does not always work!

4. Short Cuts for Multiply and Divide For Positive Numbers 1. Multiply by 2 k is the same as shift k to the left, 0 fill 2. Divide by 2 k is the same as shift k to the right, 0 fill For 2’s Complement Numbers It does not always work! Ex: -5 = -0101, 2’s Complement = 1011 Divide by 4 by shift right 2, fill with ?

5. Short Cuts for Multiply and Divide For Positive Numbers 1. Multiply by 2 k is the same as shift k to the left, 0 fill 2. Divide by 2 k is the same as shift k to the right, 0 fill For 2’s Complement Numbers It does not always work! Ex: -5 = -0101, 2’s Complement = 1011 Divide by 4 by shift right 2, fill with ? 1110 is 0001+1=0010 = 2 WRONG!

6. Floating Point Numbers Normalized Mantissa or Significand and Exponent Add 0.713 x 10 -2 and 0.964 x 10 -1

7. Floating Point Numbers Normalized Mantissa or Significand and Exponent Add 0.713 x 10 -2 and 0.964 x 10 -1 1.Align the exponents by shifting the smaller 0.0713 x 10 -1 0.9640 x 10 -1

8. Floating Point Numbers Normalized Mantissa or Significand and Exponent Add 0.713 x 10 -2 and 0.964 x 10 -1 1.Align the exponents by shifting the smaller 0.0713 x 10 -1 0.9640 x 10 -1 2. Add 1.0353 x 10 -1

9. Floating Point Numbers Add 0.713 x 10 -2 and 0.964 x 10 -1 1.Align the exponents by shifting the smaller 0.0713 x 10 -1 0.9640 x 10 -1 2. Add 1.0353 x 10 -1 3. Normalize0.10353 x 10 0

10. Floating Point Numbers Add 0.713 x 10 -2 and 0.964 x 10 -1 1.Align the exponents by shifting the smaller 0.0713 x 10 -1 0.9640 x 10 -1 2. Add 1.0353 x 10 -1 3. Normalize0.10353 x 10 0 4. Round off 0.104 x 10 0

11. Normalized REAL Binary Number: ± 1.yyyyyyyyy x 2 eeee

12. Normalized REAL Binary Number: ± 1.yyyyyyyyy x 2 eeee Leading 1 Binary Point Significand = 1.yyyyyyyyy Exponent(signed) Arithmetic

13. Normalized REAL Binary Number: ± 1.yyyyyyyyy x 2 eeee Leading 1 Binary Point Significand = 1.yyyyyyyyy Exponent(signed) Arithmetic

14. Floating Point Numbers Add 1.0101 x 2 3 and 1.1011 x 2 1

15. Floating Point Numbers Add 1.0101 x 2 3 and 1.1011 x 2 1 1.Align the exponents by shifting the smaller 1.0101 x 2 3 0.011011 x 2 3

16. Floating Point Numbers Add 1.0101 x 2 3 and 1.1011 x 2 1 1.Align the exponents by shifting the smaller 1.0101 x 2 3 0.011011 x 2 3 2. Add 1.101111 x 2 3

17. Floating Point Numbers Add 1.0101 x 2 3 and 1.1011 x 2 1 1.Align the exponents by shifting the smaller 1.0101 x 2 3 0.011011 x 2 3 2. Add 1.101111 x 2 3 3. Normalize – Check for Overflow and Underflow 1.101111 x 2 3

18. Round - Off 4.563 significant digits +.0357 4.5957

19. Round - Off 4.563 significant digits +.0357 4.5957 4.60 Rounded Off

20. Round - Off 4.563 significant digits +.0357 4.5957 4.60 Rounded Off What if 4.5950 ?

21. Round - Off 4.563 significant digits +.0357 4.5957 4.60 Rounded Off What if 4.5950 ? 4.60 Round to even

22. Round - Off BinaryDecimal.00.01.25.10.50.11.75

23. Round - Off BinaryDecimal.00.01.25.10.50 Are there any trailing 1’s ? If not, round to even.11.75

24. Floating Point Numbers Add 1.0101 x 2 3 and 1.1011 x 2 1 1.Align the exponents by shifting the smaller 1.0101 x 2 3 0.011011 x 2 3 2. Add 1.101111 x 2 3 3. Normalize – Check for Overflow and Underflow 1.101111 x 2 3 4. Round to 4 bits Guard bit = 1 and Round bit = 1

25. Floating Point Numbers Add 1.0101 x 2 3 and 1.1011 x 2 1 1.Align the exponents by shifting the smaller 1.0101 x 2 3 0.011011 x 2 3 2. Add 1.101111 x 2 3 3. Normalize – Check for Overflow and Underflow 1.101111 x 2 3 4. Round to 4 bits Guard bit = 1 and Round bit = 1 1.1100 x 2 3

26. IEEE 754 Floating Point Standard Normalized REAL Binary Number: ± 1.yyyyyyyyy x 2 eeee IEEE 754 (-1) S x ( 1 + F) x 2 E-127 s exponent+127 significand - 1 1 bit E (8 bits) F (23 bits)

27. IEEE 754 Floating Point Standard Normalized REAL Binary Number: ± 1.yyyyyyyyy x 2 eeee IEEE 754 (-1) S x ( 1 + F) x 2 E-127 s exponent+127 significand - 1 1 bit E (8 bits) F (23 bits) Bias Exponent E>0

28. IEEE 754 Floating Point Standard Normalized REAL Binary Number: ± 1.yyyyyyyyy x 2 eeee IEEE 754 (-1) S x ( 1 + F) x 2 E-127 s exponent+127 significand - 1 1 bit E (8 bits) F (23 bits) Only Zero is F = 0 and E = 0 Simplifies data exchange Compare using integer processes Accuracy and round-off & Overflow and Underflow

29. IEEE 754 Floating Point Standard 1 10000010 01010000000000000000000 =.25+.0625 =.3125

30. IEEE 754 Floating Point Standard 1 10000010 01010000000000000000000 =.25+.0625 =.3125 E = 2 7 + 2 = 128 + 2 = 130

31. IEEE 754 Floating Point Standard 1 10000010 01010000000000000000000 =.25+.0625 =.3125 E = 2 7 + 2 = 128 + 2 = 130 Number = (-1) S x ( 1 + F) x 2 E-127

32. IEEE 754 Floating Point Standard 1 10000010 01010000000000000000000 =.25+.0625 =.3125 E = 2 7 + 2 = 128 + 2 = 130 Number = (-1) S x ( 1 + F) x 2 E-127 = – ( 1 +.3125) x 2 130 – 127

33. IEEE 754 Floating Point Standard 1 10000010 01010000000000000000000 =.25+.0625 =.3125 E = 2 7 + 2 = 128 + 2 = 130 Number = (-1) S x ( 1 + F) x 2 E-127 = – ( 1 +.3125) x 2 130 – 127 = – 1.3125 x 2 3 = – 1.3125 x 8 = – 10.5

34. IEEE 754 Floating Point Standard Consider representing –5 in IEEE 754 Floating Point Format

35. IEEE 754 Floating Point Standard Consider representing –5 in IEEE 754 Floating Point Format 5 = 101.0 x 2 0 = 1.01 x 2 2

36. IEEE 754 Floating Point Standard Consider representing –5 in IEEE 754 Floating Point Format 5 = 101.0 x 2 0 = 1.01 x 2 2 IEEE 754 (-1) S x ( 1 + F) x 2 E-127 s exponent+127 significand - 1 1bit E (8 bits) F (23 bits)

37. IEEE 754 Floating Point Standard Consider representing –5 in IEEE 754 Floating Point Format 5 = 101.0 x 2 0 = 1.01 x 2 2 IEEE 754 (-1) S x ( 1 + F) x 2 E-127 s exponent+127 significand - 1 1bit E (8 bits) F (23 bits) E –127 = 2 E = 2 + 127 = 129 = 1000 0001

38. IEEE 754 Floating Point Standard Consider representing –5 in IEEE 754 Floating Point Format 5 = 101.0 x 2 0 = 1.01 x 2 2 IEEE 754 (-1) S x ( 1 + F) x 2 E-127 s exponent+127 significand - 1 1bit E (8 bits) F (23 bits) E –127 = 2 E = 2 + 127 = 129 = 1000 0001 1 +F = 1.01000...

39. IEEE 754 Floating Point Standard Consider representing –5 in IEEE 754 Floating Point Format 5 = 101.0 x 2 0 = 1.01 x 2 2 IEEE 754 (-1) S x ( 1 + F) x 2 E-127 s exponent+127 significand - 1 1bit E (8 bits) F (23 bits) 1 1000 0001 0100 0000.... 0000 E = 2 + 127 = 129 = 1000 0001 1 +F = 1.01000...

40. IEEE 754 Floating Point Standard Normalized REAL Binary Number: ± 1.yyyyyyyyy x 2 eeee Double Precision IEEE 754 (-1) S x ( 1 + F) x 2 E-1023 s exponent+1023 significand - 1 1 bit E (11 bits) F (20 bits) significand – 1 (continued) F (32 bits)

41. Sequential Network Structures - Review X1 X2 Xn Y1 Y2 Ym Q1 Q2 Qm Z1 Z2 Zk Combinational Logic Register Q Clock Stability Condition clk i Y Q Input Output

42. Flip - Flop with NOR Gates Q = R+Q Q = S+Q R S Present State RS Q 00 01 10 11 0 1 Next State Q

43. Flip - Flop with NOR Gates Q = R+Q Q = S+Q R S Present State RS Q 00 01 10 11 0 0 1 1 Next State Q

44. Flip - Flop with NOR Gates Q = R+Q Q = S+Q R S Present State RS Q 00 01 10 11 0 0 1 1 1 1 Next State Q

45. Flip - Flop with NOR Gates Q = R+Q Q = S+Q R S Present State RS Q 00 01 10 11 0 0 1 0 1 1 1 0 Next State Q

46. Flip - Flop with NOR Gates Q = R+Q Q = S+Q R S Present State RS Q 00 01 10 11 0 0 1 0 ? 1 1 1 0 ? Next State Q

47. D-latch C D Q D C Q Q S R

48. C D Q D C Q Q S R

49. D flip-flop Output changes only on the trailing clock edge QQ _ Q D latch D C D latch D C D C Q D C Q

50. D flip-flop Output changes only on the trailing clock edge QQ _ Q D latch D C D latch D C D C Q D C Q

51. Sequential Network Structures - Review X1 X2 Xn Y1 Y2 Ym Q1 Q2 Qm Z1 Z2 Zk Combinational Logic Register Q Clock Stability Condition clk i Y Q Input Output

52. Five Components of Computers Input Output Memory Control Datapath Processor

53. Start by Building the Datapath 1.Access the Instruction from Memory 2.Access the Data from Registers 3.Perform the Instruction 4.Write the Result

54. PC Instruction Memory Next PC Logic Instruction Address Simplified Overview Access the Instruction from Memory

55. PC Instruction Memory Next PC Logic Instruction Address Register File Simplified Overview Access the Data from Registers

56. PC Instruction Memory Next PC Logic Instruction Address Register File ALU Simplified Overview Perform the Instruction

57. PC Instruction Memory Next PC Logic Instruction Address Register File ALU Data Memory Addr Data In Data Out Simplified Overview Write the Result

58. PC Instruction Memory Next PC Logic Instruction Address Register File ALU Data Memory Addr Data In Data Out Simplified Overview Timing Assumption