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Integer Arithmetic Floating Point Representation Floating Point Arithmetic

Topics. Integer Arithmetic Floating Point Representation Floating Point Arithmetic. 2’s Complement Integers. 2’s Complement Addition/Subtraction. A + B  A. What is Overflow? How is it identified?. Unsigned Integer Multiplication. Unsigned Integer Multiplication. Q x M  AQ.

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Integer Arithmetic Floating Point Representation Floating Point Arithmetic

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  1. Topics Integer Arithmetic Floating Point Representation Floating Point Arithmetic

  2. 2’s Complement Integers

  3. 2’s Complement Addition/Subtraction A + B  A What is Overflow? How is it identified?

  4. Unsigned Integer Multiplication

  5. Unsigned Integer Multiplication Q x M  AQ

  6. Unsigned Integer Multiplication Flow Diagram

  7. 2’s Comp MultiplicationBooth’s Algorithm Add one extra bit to the Q register: Q-1

  8. 2’s Comp MultiplicationBooth’s Algorithm Q-1

  9. 2’s Comp MultiplicationBooth’s Algorithm

  10. Booth : (7) x (3) A Q M 3 7 0000 0011 0 0111 -------------------- 1001 0011 0 0111 A <- (A - M) 1st 1100 1001 1 0111 Shift -------------------- 2nd 1110 0100 1 0111 Shift -------------------- 0101 0100 1 0111 A <- (A + M) 3rd 0010 1010 0 0111 Shift -------------------- 4th 0001 0101 0 0111 Shift --------------------

  11. Booth : (7) x (-3) A Q M -3 7 0000 1101 0 0111 -------------------- 1001 1101 0 0111 A <- (A - M) 1st 1100 1110 1 0111 Shift -------------------- 0011 1110 1 0111 A <- (A + M) 2nd 0001 1111 0 0111 Shift -------------------- 1010 1111 0 0111 A <- (A - M) 3rd 1101 0111 1 0111 Shift -------------------- 4th 1110 1011 1 0111 Shift --------------------

  12. Booth : (-7) x (3) A Q M 3 -7 0000 0011 0 1001 -------------------- 0111 0011 0 1001 A <- (A - M) 1st 0011 1001 1 1001 Shift -------------------- 2nd 0001 1100 1 1001 Shift -------------------- 1010 1100 1 1001 A <- (A + M) 3rd 1101 0110 0 1001 Shift -------------------- 4th 1110 1011 0 1001 Shift --------------------

  13. Booth : (-7) x (-3) A Q M -3 -7 0000 1101 0 1001 -------------------- 0111 1101 0 1001 A <- (A - M) 1st 0011 1110 1 1001 Shift -------------------- 1100 1110 1 1001 A <- (A + M) 2nd 1110 0111 0 1001 Shift -------------------- 0101 0111 0 1001 A <- (A - M) 3rd 0010 1011 1 1001 Shift -------------------- 4th 0001 0101 1 1001 Shift --------------------

  14. Unsigned Integer Division

  15. Unsigned Integer Division

  16. Unsigned Integer Division

  17. Unsigned Integer Division Divisor  M Dividend  Q Quotient in Q Remainder in A

  18. What about 2’s Comp Division ? • Can we do something like Booth’s Algorithm? • Why might we use Booth’s Algorithm for multiplication but not for Division?

  19. Single Precision Floating Point NumbersIEEE Standard • 32 bit Single Precision Floating Point Numbers are stored as: • S EEEEEEEE FFFFFFFFFFFFFFFFFFFFFFF • S: Sign – 1 bit • E: Exponent – 8 bits • F: Fraction – 23 bits • The value V: • If E=255 and F is nonzero, then V= NaN ("Not a Number") • If E=255 and F is zero and S is 1, then V= - Infinity • If E=255 and F is zero and S is 0, then V= Infinity • If 0<E<255 then V= (-1)**S * 2 ** (E-127) * (1.F) (exponent range = -127 to +128) • If E=0 and F is nonzero, then V= (-1)**S * 2 ** (-126) * (0.F) ("unnormalized" values”) • If E=0 and F is zero and S is 1, then V= - 0 • If E=0 and F is zero and S is 0, then V = 0 Significand Note: 255 decimal = 11111111 in binary (8 bits)

  20. FP Examples

  21. Double Precision Floating Point NumbersIEEE Standard • 64 bit Double Precision Floating Point Numbers are stored as: • S EEEEEEEEEEE FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF • S: Sign – 1 bit • E: Exponent – 11 bits • F: Fraction – 52 bits • The value V: • If E=2047 and F is nonzero, then V= NaN ("Not a Number") • If E=2047 and F is zero and S is 1, then V= - Infinity • If E=2047 and F is zero and S is 0, then V= Infinity • If 0<E<2047 then V= (-1)**S * 2 ** (E-1023) * (1.F) (exponent range = -1023 to +1024) • If E=0 and F is nonzero, then V= (-1)**S * 2 ** (-1022) * (0.F) ("unnormalized" values) • If E=0 and F is zero and S is 1, then V= - 0 • If E=0 and F is zero and S is 0, then V= 0 Significand Note: 2047 decimal = 11111111111 in binary (11 bits)

  22. 32 bit 2’s Complement Integer Numbers All the Integers from -2,147,483,648 to + 2,147,483,647, i.e. - 2 Gig to + 2 Gig-1

  23. 32 bit FP Numbers

  24. “Density” of 32 bit FP Numbers Note: ONLY 232 FP numbers are representable There are only 232 distinct combinations of bits in 32 bits !

  25. The Added Denormalized FP Numbers

  26. Floating Point Addition / Subtraction Steps: • Check for zero • Align the significands (fractions) • Add or Subtract the significands • Normalize the Result Bad Results: • Exponent Overflow • Exponent Underflow • Significand Overflow • Significand Underflow

  27. Floating Point Addition/Subraction

  28. Floating Point Multiplication • Check for zero • Multiply significands • Add exponents • Normalize Overflow/underflow?

  29. Floating Point Multiplication

  30. Floating Point Division • Check for zero • Divide significands • Subract exponents • Normalize Overflow/underflow?

  31. Floating Point Division

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