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ECE 8053 Introduction to Computer Arithmetic

ECE 8053 Introduction to Computer Arithmetic. (Website: http://www.ece.msstate.edu/classes/ece8053/fall_2003/). Course & Text Content: Part 1: Number Representation Part 2: Addition/Subtraction Part 3: Multiplication Part 4: Division Part 5: Real Arithmetic (Floating-Point)

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ECE 8053 Introduction to Computer Arithmetic

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  1. ECE 8053 Introduction to Computer Arithmetic (Website: http://www.ece.msstate.edu/classes/ece8053/fall_2003/) Course & Text Content: Part 1: Number Representation Part 2: Addition/Subtraction Part 3: Multiplication Part 4: Division Part 5: Real Arithmetic (Floating-Point) Part 6: Function Evaluation Part 7: Implementation Topics

  2. Course Learning Objectives Computer Arithmetic students will be able to ... 1. explain the relative merits of number systems used by arithmetic circuits including both fixed- and floating-point number systems 2. demonstrate the use of key acceleration algorithms and hardware for addition/subtraction, multiplication, and division, plus certain functions 3. distinguish between the relative theoretical merits of the different acceleration schemes

  3. Course Learning Objectives(Continued) 4. identify the implementation limitations constraining the speed of acceleration schemes 5. evaluate, design, and optimize arithmetic circuits for low-power 6. evaluate, design, and optimize arithmetic circuits for precision

  4. Course Learning Objectives(Continued) 7. design, simulate, and evaluate an arithmetic circuit using appropriate references including current journal and conference literature 8. write a paper compatible with journal format standards on an arithmetic design 9. make a professional presentation with strong technical content and audience interaction

  5. Importance of Computer Arithmetic 3.2 GHz Pentium has a clock cycle of 0.31 ns Can one integer addition be done < 0.31 ns in execution stage of Pipeline? What if you had to build a 32-bit adder – ripple carry and a gate delay was approximately 0.1 ns? STEP 1 1101 1110 11011 -Note: added from right to left. Why?

  6. Ripple-carry Structure STEP 2 – Design a circuit x31 y31 x1 y1 x0 y0 c31 c2 c1 c0=0 c32 z31 z1 z0 – Each box is a full-adder – Implement it

  7. Full-Adder Implementation xy 00 01 11 10 cin 0 0 1 0 1 1 1 0 1 0 xy 00 01 11 10 cin 0 0 0 1 0 1 0 1 1 1 cout = cinx + ciny + x y

  8. Adder Circuit Analysis STEP 3 – Analysis Critical Path is carry chain, 2 gate delays/bit 2  32 = 64 (64)(0.1ns) = 6.4ns 6.4ns >> 0.31ns Must Use Faster Adder and/or Pipeline More!!!!

  9. Number SystemsRoman Numeral System Symbolic Digits • RULES: • If symbol is repeated or lies to the right of another higher-valued symbol, value is additive • XX=10+10=20 • CXX=100+10+10=120 • If symbol is repeated or lies to the left of a higher-valued symbol, value is subtractive • XXC = - (10+10) + 100 = 80 • XLVIII = -(10) + 50 + 5 + 3 = 48

  10. Weighted Positional Number System Example: Arabic Number System (first used by Chinese)

  11. Addition Paradigms • right to left serial 1 147865+30921 178786 • right to left, parallel 147865+30921 177786001000 178786 • random 461325 147865+30921 177786001000 178786

  12. Binary Number System • n-ordered sequence: • each xi{0,1} is a BInary digiT (BIT) • magnitude of n is important • sequence is a short-hand notation • more precise definition is: • This is a radix-polynomial form

  13. Number System • A Number System is defined if the following exist • A digit set • A radix or base value • An addition operation • A multiplication operation • Example: The binary number system • Addition operator defined by addition table • Multiplication operator defined by multiplication table

  14. Number System Observations • Cardinality of digit set (2) is equal to radix value • Addition operator XOR, Multiplication is AND • How many integers exist? Mathematically, there are an infinite number,  In computer, finite range due to register length smallest representable number largest representable number range of representable numbers[-inclusive; (-exclusive interval bounds • When ALU produces a result >Xmax or <Xmin, incorrect result occurs • ALU should produce an error signal

  15. Example • Assume 4-bit registers, unsigned binary numbers Answer in register is 00112=310 Overflow

  16. Machine Representations • Most familiar number systems are: • nonredundant – every value is uniquely represented by a radix polynomial • weighted – sequence of weights determines the value of the n-tuple formed from the digit set • positional – wi depends only on position i • conventional number systemswhere ß is a constant. These are fixed-radix systems.

  17. Example Since octal is fixed-radix and positional,we can rewrite this value using shorthand notation Note the importance of the use of 0 to serve as a coefficient of the weight value w2=82

  18. Fixed-Radix Systems Register of length n can represent a number with a fractional part and an integral part k – number of integral digits m – number of fractional digits n = k + m radix point A programmer can use an implied radix point in any position

  19. Scaling Factors Fixed point arithmetic can utilize scaling factors to adjust radix point position a – scaling factor no correction must divide by a must multiply by a

  20. Unit in the Last Position (ulp) • Given w0=r-m and n, the position of the radix point is determined • Simpler to disregard position of the radix point in fixed point by using unit in least (significant) position ulp • For fractions • For integers ulp = 1 Example 98.67510 1 ulp = 110-3=0.001 1 ulp is the smallest amount a fixed point number may increase or decrease

  21. Radix Conversion Given a number in old radix r, conversion to the new radix R representation - can be accomplished doing the arithmetic in the old or new radix - old and new representations may not be exactly equal

  22. Radix Conversion Given a value X represented in source system with radix s, represent the same number in a destination system with radix d Consider the integral part of the number, XI, in the d system Consider the integral part of the number, XI, in the d system If XI is divided by d , we obtain x0 as a remainder and quotient Can Repeatedly Divide to Obtain Converted Value

  23. Radix Conversion Example XI = 34610s=10 d =3 Fixed-point Decimal to Ternary Integer Conversion, (arithmetic in old radix) XI = 1102113 Check by evaluating the radix polynomial

  24. Radix Conversion (fractional) Consider the fractional part of the value in d Fixed point system Thus, PI is the desired digit We can repeatedly multiply by the d value

  25. Radix Conversion Example XI = 0.29110s=10 d =5 Fixed-point Decimal to Pentary Fractional Conversion (arithmetic in old radix) 0.29110 is Finite Fraction for s=10, but infinite fraction for d =5

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