ECE 8053 Introduction to Computer Arithmetic (Website: 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

Course Learning Objectives Computer Arithmetic students will be able to 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

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

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

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 Note: added from right to left. Why?

Ripple-carry Structure STEP 2 – Design a circuit c 32 z 31 z1z1 z0z0 c 31 c2c2 c1c1 c 0 =0 x 31 y 31 x1x1 y1y1 x0x0 y0y0 – Each box is a full-adder – Implement it

Full-Adder Implementation xy c in xy c in c out = c in x + c in y + x y

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!!!!

Number Systems Roman Numeral System Symbolic DigitsRULES: If symbol is repeated or lies to the right of another higher-valued symbol, value is additive XX=10+10=20 CXX= =120 If symbol is repeated or lies to the left of a higher-valued symbol, value is subtractive XXC = - (10+10) = 80 XLVIII = -(10) = 48

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

Addition Paradigms right to left serial right to left, parallel random

Binary Number System n-ordered sequence: each x i  {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

Number System A Number System is defined if the following exist Example: The binary number system Addition operator defined by addition table 4.Multiplication operator defined by multiplication table 1. A digit set 2.A radix or base value 3.An addition operation 4.A multiplication operation

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

Example Assume 4-bit registers, unsigned binary numbers Answer in register is =3 10 Overflow

Machine Representations Most familiar number systems are: 1.nonredundant – every value is uniquely represented by a radix polynomial 2.weighted – sequence of weights determines the value of the n-tuple formed from the digit set 3.positional – w i depends only on position i 4.conventional number systems where ß is a constant. These are fixed-radix systems.

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 w 2 =8 2

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

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

Unit in the Last Position (ulp) Given w 0 =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 Example ulp = 1  = ulp is the smallest amount a fixed point number may increase or decrease ulp = 1

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

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, X I, in the  d system Can Repeatedly Divide to Obtain Converted Value Consider the integral part of the number, X I, in the  d system If X I is divided by  d, we obtain x 0 as a remainder and quotient

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

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

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