Computer Arithmetic: Multiplication and Division in CS447 Lecture

1. CS 447 – Computer Architecture Lecture 4Computer Arithmetic (2) August 27, 2008 www.qatar.cmu.edu/~msakr/15447-f08/

2. Last Time • Representation of finite-width unsigned and signed integers • Addition and subtraction of integers • Multiplication of unsigned integers

3. Unsigned Binary Multiplication

4. Execution of Example

5. Flowchart for Unsigned Binary Multiplication

6. Multiplying Negative Numbers • This does not work! • Solution 1 • Convert to positive if required • Multiply as above • If signs were different, negate answer • Solution 2 • Booth’s algorithm

7. Observation • Which of these two multiplications is more difficult? 98,765 x 10,001 98,765 x 9,999 • Note: 98,765 x 10,001 = 98,765 x (10,000 + 1) 98,765 x 9,999 = 98,765 x (10,000 – 1)

8. In Binary Let Q = d d d 0 1 1 . . . 1 1 0 d d QL = d d d 1 0 0 . . . 0 0 0 d d QR = 0 0 0 0 0 0 . . . 0 1 0 0 0

9. In Binary Let Q = d d d 01 1 . . . 1 10 d d QL = d d d 10 0 . . . 0 0 0 d d QR = 0 0 0 0 0 0 . . . 01 0 0 0 Then Q = QL – QR And M x Q = M x QL – M x QR

10. A bit more explanation P = M x Qwhere M and Q are n-bit two’s complement integers e.g., Q = -23 q3 + 22 q2 + 21 q1 + 20 q0 which can be re-written as follows: Q = 20(q-1–q0)+21(q0-q1)+22(q1-q2)+23(q2-q3) leading to P = M x 20 x (q-1 - q0) + M x 21 x (q0 - q1) + M x 22 x (q1 - q2) + M x 23 x (q2 - q3)

11. Finally! P = M x 20 x (q-1 - q0) + M x 21 x (q0 - q1) + M x 22 x (q1 - q2) + M x 23 x (q2 - q3)

12. Example of Booth’s Algorithm

13. Booth’s Algorithm

14. Division • More complex than multiplication • Negative numbers are really bad! • Based on long division

15. Quotient 00001101 Divisor 1011 10010011 Dividend 1011 001110 Partial Remainders 1011 001111 1011 Remainder 100 Division of Unsigned Binary Integers

16. Flowchart for Unsigned Binary Division

17. Real Numbers • Numbers with fractions • Could be done in pure binary • 1001.1010 = 24 + 20 +2-1 + 2-3 =9.625 • Where is the binary point? • Fixed? • Very limited • Moving? • How do you show where it is?

18. Biased Exponent Significand or Mantissa Sign bit Floating Point • +/- .significand x 2exponent • Misnomer • Point is actually fixed between sign bit and body of mantissa • Exponent indicates place value (point position)

19. Floating Point Examples

20. Signs for Floating Point • Sign Magnitude for Mantissa • Exponent is in excess or biased notation • e.g. Excess (bias) 127 • 8 bit exponent field • Pure value range 0-255 • Subtract 127 to get correct value • Range -127 to +128

21. Normalization • FP numbers are usually normalized • i.e. exponent is adjusted so that leading bit (MSB) of mantissa is 1 • Since it is always 1 there is no need to store it • (c.f. Scientific notation where numbers are normalized to give a single digit before the decimal point • e.g. 3.123 x 103)

22. FP Ranges • For a 32 bit number • 8 bit exponent • +/- 2128  3.4 x 1038 • Accuracy • The effect of changing lsb of mantissa • 23 bit mantissa 2-23  1.2 x 10-7 • About 6 decimal places

23. Examples of 32-bit Numbers (wikipedia)

24. Range and Precision (wikipedia)

25. Expressible Numbers

26. Density of Floating Point Numbers

27. IEEE 754 Formats

28. FP Arithmetic +/- • Check for zeros • Align significands (adjusting exponents) • Add or subtract significands • Normalize result

29. FP Addition & Subtraction Flowchart

30. Floating point adder

33. FP Arithmetic x/ • Check for zero • Add/subtract exponents • Multiply/divide significands (watch sign) • Normalize • Round • All intermediate results should be in double length storage

34. Floating Point Multiplication

35. Floating Point Division