Dr. Konstantinos Tatas

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Dr. Konstantinos Tatas

• FLOATING-POINT NUMBER REPRESENTATION

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The range/accuracy problem

• The range of numbers that can be represented with
  n bits is
• In 2s complement from - /2 to /2 -1
• For n8 From 128 to 127
• For n16 From 32,768 to 32,767
• Still, in many application an even larger range
  is required

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Real numbers

• Instead of representing the actual value, in the
  base system, we represent the sign, M, b and e

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FLOATING-POINT REPRESENTATION

• IEEE short real 8 bits for the exponent (in
  Ex-127), 23 bits for the mantissa
• IEEE long real 11 bits for the exponent, 52 bits
  for the mantissa

Sign (S) Biased exponent (E) Unsigned normalized mantissa (M)
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RESERVED VALUES
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Examples (IEEE short real format)
Binary value Normalized Binary value exponent Biased Exponent (Excess -127 Sign, exponent, mantissa
-1.01 -1.01 0 127 1 01111111 0100000000000000000000
1011.0101 1.0110101 3 130 0 01000010 0110101000000000000000
-0.0000011 -1.1 -6 121 1 01111001 1000000000000000000000
11010101 1.11010101 7 134 0 10000110 1101010100000000000000
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Homework

• Convert the following 2s complement values to
  IEEE short real floating-point representation
• 10011010
• 0110.0101
• 0.1111110
• 1100.0001