Floating Point

• Review scientific notation
  • [+|-]d.d*[e[+|-]dd*]
  • Normalized
  • 3.45x10⁴ + 4.56x10¹
• A simple representation
  • 1 sign bit (s)
  • 5 bits for exponent, excess 16 (e) (00000^e-16 = -16, 10000^e-16 = 0, 11111^e-16 = 15)
  • 8 bits for mantissa (m)
  • seeeeemmmmmmmm
  • (-1)^sm.mmmmmmm x 2^eeeee
  • 4.125
  • What is the smallest number in this system?
  • What is the largest?
• But when we normalize
  • It is always in the form 1.b*x2^e
  • So there is no need to store the 1
  • New representation (letters as above)
  • (-1)^s1.mmmmmmmm x 2^eeeee
  • 4.125
  • Look at biggest and smallest numbers again
  • Look at the gap
  • What did we gain? What did we loose?
• IEEE 754 floating point format
  • Single Precision
    • 32 bits
    • 1 sign bit
    • 8 bit excess 127 exponent
    • 23 bit mantissa
    • exponent = 11111111 => NAN (m not all 1s) or infinity (m all 1)
    • exponent = 00000000 => denormals, (-1)^s 0.m x 2 ^-126
    • See some code
    • The denormals give us a large set of numbers very close to zero
    • NANs are the result of an illeagal operation sqrt(-5)
  • Double Precision
    • 64 bits
    • 1 sign bit
    • 11 bit excess 1023 exponent
    • 52 bit mantissa
  • Quad Precisison
    • 128 bits
    • 1 sign bit
    • 15 bit excess 16384 exponent
    • 112 bit mantissa
  • And some more code
• Adding floating point numbers
  • Find the larger exponent
  • Shift the smaller in terms of the larger
  • Add the mantissas
  • Normalize if required.
  • 3.26 x 10 ² + 6.04 x 10 ^-1
  • Same for our format
  • Add 6.5 to 12.3125 (in 14 bit fp format)
  • Add 6 and 1/32 to 12.3125, notice here we have an error.