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September 5, 2007 Karem Sakallah ksakalla@qatar.cmu qatar.cmu/~msakr/15447-f07/

CS 447 – Computer Architecture Lecture 3 Computer Arithmetic (2). September 5, 2007 Karem Sakallah ksakalla@qatar.cmu.edu www.qatar.cmu.edu/~msakr/15447-f07/. Last Time. Representation of finite-width unsigned and signed integers Addition and subtraction of integers

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September 5, 2007 Karem Sakallah ksakalla@qatar.cmu qatar.cmu/~msakr/15447-f07/

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  1. CS 447 – Computer Architecture Lecture 3Computer Arithmetic (2) September 5, 2007 Karem Sakallah ksakalla@qatar.cmu.edu www.qatar.cmu.edu/~msakr/15447-f07/

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

  3. Today • Review of multiplication of unsigned integers • Multiplication of signed integers • Division of unsigned integers • Representation of real numbers • Addition, subtraction, multiplication, and division of real numbers

  4. Multiplication • Complex • Work out partial product for each digit • Take care with place value (column) • Add partial products

  5. Multiplication Example • 1011 Multiplicand (11 dec) • x 1101 Multiplier (13 dec) • 1011 Partial products • 0000 Note: if multiplier bit is 1 copy • 1011 multiplicand (place value) • 1011 otherwise zero • 10001111 Product (143 dec) • Note: need double length result

  6. Unsigned Binary Multiplication

  7. Execution of Example

  8. Flowchart for Unsigned Binary Multiplication

  9. 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

  10. 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)

  11. 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

  12. 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

  13. 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)

  14. 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)

  15. Example of Booth’s Algorithm

  16. Booth’s Algorithm

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

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

  19. Flowchart for Unsigned Binary Division

  20. 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?

  21. 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)

  22. Floating Point Examples

  23. Signs for Floating Point • Mantissa is stored in two’s complement • Exponent is in excess or biased notation • e.g. Excess (bias) 128 means • 8 bit exponent field • Pure value range 0-255 • Subtract 128 to get correct value • Range -128 to +127

  24. 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)

  25. FP Ranges • For a 32 bit number • 8 bit exponent • +/- 2256  1.5 x 1077 • Accuracy • The effect of changing lsb of mantissa • 23 bit mantissa 2-23  1.2 x 10-7 • About 6 decimal places

  26. Expressible Numbers

  27. Density of Floating Point Numbers

  28. IEEE 754 Formats

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

  30. FP Addition & Subtraction Flowchart

  31. 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

  32. Floating Point Multiplication

  33. Floating Point Division

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