40593 Computer Arithmetic Fall 2003 Damu Radhakrishnan 204 Resnick Engineering Hall [email protected] damu Lecture:Tuesday/Thursday.

40593 Computer Arithmetic 

Fall 2003

• Damu Radhakrishnan• 204 Resnick Engineering Hall• [email protected]• http://www.engr.newpaltz.edu/~damu• Lecture: Tuesday/Thursday 12.30 – 1.45PM,

REH 111• Office Hours: Wednesday 3.30 - 5.00PM• Prerequisites: Graduate Standing –

background in logic design of digital systems and basic computer organization.

What is Computer Arithmetic?• Major field in Computer architecture

• Implementation of arithmetic functions– arithmetic algorithms for firmware/software implementation

– Hardware realizations

– High-speed circuits for computation

• Applications– Design of top-of-the-line CPUs

– High-performance arithmetic circuits

– Designs for embedded application-specific circuits

– Arithmetic algorithms for software

– Understand what went wrong in the Pentium…

• Major Focus– development of high-speed arithmetic algorithms – design of application specific circuits to enhance the speed of numerical

calculations – understanding implementation in VLSI technology

• Course consists of– a detailed study of the theory, specification and design of basic arithmetic

algorithms and architectures used in digital systems

• First a review: digit representation, number systems and digit encoding. • Basic arithmetic operations (algorithms and architectures) such as addition,

multiplication and division will be presented and discussed. • Particular emphasis is given to the actual design and implementation of the

logic circuitry associated with the underlying algorithms. • Advanced topics like high throughput arithmetic, low power arithmetic, and

fault tolerant arithmetic will also be covered. Students will design several arithmetic circuits.

Topics Covered• Numbers and arithmetic

• Representing Signed numbers

• Redundant number systems, Residue number systems

• Basic addition and counting

• Carry-lookahead adders

• Variations in fast adders

• Multi-operand addition

• Basic multiplication schemes, High radix multipliers

• Tree and array multipliers, Variations in multipliers

• Basic division schemes, High-radix dividers

• Floating point representations, Errors and error control

Topics Covered (Contd.)

• Square rooting methods

• The cordic algorithms

• Arithmetic by Table lookup

• High-throughput arithmetic, Low power arithmetic

• Fault-tolerant arithmetic, Future challenges

• Project presentations

• FINAL EXAM

References

• Computer Arithmetic: Algorithms and Hardware Designs, Behrooz Parhami, Oxford University Press, 2000

• Computer Arithmetic Algorithms, Israel Koren, Prentice-Hall , 2002.

• Journal articles

• Conference papers

Grading Policy

• Homework: 20%

• Quiz: 10%

• Midterm exams: 30%

• Project: 15%

• Final exam: 25% 

Total Points Final Grade

90-100 A

85-89 A-

80-84 B+

75-79 B

70-74 B-

65-69 C+

60-64 C

55-59 C-

50-54 D

Below 50 F

Course rules and general comments

        Examinations are closed book, closed notes and no calculators allowed.

        Homework assignments are generally from the text book and are given on a weekly basis; the due date is one week from the distribution date (unless otherwise specified). No late homework solutions will be accepted except under extreme non-academic conditions with the prior approval of the instructor.

        There is a course project. Each student is required to give a formal presentation of the project in class.    

      Any disputed grade must be resolved within 7 days of the return of the graded item.

        You are responsible for all the course materials and lecture contents unless specified otherwise. If you miss a class, it is your responsibility to obtain assignments and other information given on that day.

      All your course work (homework, project, and exams) is expected to be your own. Evidence indicating copying of work or other cooperation will be dealt with based on the University academic conduct rules. General instructions such as assisting in problem interpretation, and giving occasional hints on problem attack (i.e., the kind of help you would get from the instructor or the teaching assistant in the course!), however, are permitted. On the other hand, you are encouraged to form informal study groups to solve homework problems.       

    If you have questions on course materials, the instructor will be available for consultation. Please try to get answers before serious difficulties in your understanding of course material arise. In particular, it is much better to get your questions answered before an exam than after!

      Regular class attendance is very important and attendance will be taken at the beginning of each lecture. A bonus of 50 points will be credited for those who attend classes regularly, and do not miss any lecture. Those who do not miss more than 2 lecture sessions during the whole semester will receive 40 bonus points. No bonus credit will be given otherwise. Those who come to the lecture more than 10 minutes late will be marked as absent. If you come late within the permitted 10 minute time period, it is your responsibility to make sure that your attendance is recorded properly.

       Please pay attention to the following requirements regarding your homework assignment

        Always use standard size (81/2 11) paper. Do not use torn-off paper from spiral bound notebooks (max penalty 10%)

        Write the course #, homework #, and your name on top of the first page, as shown below (max penalty 10%)

 Course # Homework # Your Name         Write clearly, neatly and in an orderly fashion (max

penalty 10%)         Draw schematics and circuit diagrams (max penalty

20%)         Show steps involved in getting to the final answer,

no credit may be given for the work not shown.         Staple all homework pages together before you turn

them in (max penalty 10%)

Disasters caused by Numerical Errors

• Patriot Missile Failure• Explosion of the Ariane 5• EURO page: Conversion Arithmetics• The Vancouver Stock Exchange• Rounding error changes Parliament makeup• The sinking of the Sleipner A offshore platform• Tacoma bridge failure (wrong design)• Collection of software bugs

Patriot Missile Failure – software problem led to system failure

• Feb. 1991, during Gulf War, an American Patriot Missile battery in Saudi Arabia failed to intercept an incoming Iraqi Scud missile. The Scud struck an American army Barracks and killed 28 soldiers. The cause was an inaccurate calculation of the time since boot due to computer arithmetic errors.

Explosion of the Ariane 5 – software error in the inertial reference system

• On June 4, 1996 an unmanned Ariane 5 rocket launched by the European Space Agency exploded just forty seconds after lift-off. The rocket was on its first voyage, after a decade of development costing $7 billion. The cause was a software error. A 64 bit floating point number relating to the horizontal velocity of the rocket wrt the platform was converted to a 16 bit signed integer. The number was larger than 32678, the largest integer storeable in a 16 bit signed integer, and thus the conversion failed.

EURO page: Conversion Arithmetics

• Introduction of EURO and the rounding of currency amounts. The French National Council has calculated that due to the rounding process, all giant supermarkets in France together will gain about 0.5 million EUR extra.

The Vancouver Stock Exchange

• In 1982 the Vancouver stock exchange instituted a new index initialized to a value of 1000.000. The index was updated after each transaction. Twenty two months later it had fallen to 520. The cause was that the updated value was truncated rather than rounded. The rounded calculation gave a value of 1098.892.

Rounding error changes Parliament makeup

• On 5th April 1992 a German election was shattered due to a computer error. German elections are quite complicated to calculate. No party with less than 5% of the vote may be seated in parliament.

• When the votes were being counted, it looked like the Green party was hanging on by their teeth to a vote percentage of exactly 5%. After midnight someone discovered that the Greens actually only had 4.97% of the vote. The program that prints out the percentages only uses one place after the decimal, and had rounded the count up to 5%! This software had been used for years.