Floating Point Numbers

• Floating point numbers are MUCH more complex than integers.
  • Between 1 and 2 are no other integers.
    • Between any two real numbers are infinitely many real numbers.
  • It is easy to see a maximum integer (as we did in the last section)
    • Floating points are much different.
• We represent floating point numbers much like scientific notation.
  • $[+/-]d.d \times 10^{e}$ : Scientific notation
  • $[+/-]1.n \times 2^{e}$ : floating point numbers.
• The standard is actually IEEE 754
• With integers we decided to two's compliment, we use sign/magnitude for floating point numbers
• sign 1.significant x 2^ exponent
  • The sign is a single bit.
  • The significant is between 11 and 237 bits.
  • The exponent is between 5 and 19 bits.
• There are special cases for numbers that are very close to 0
• A simple example.
  • This is a simplified version of ieee 754, don't worry too much about details.
    • THIS IS NOT AN ACCURATE REPRESENTATION.
    • THERE ARE MISSING DETAILS.
  • Assume 4 exponent bits.
  • Assume 3 decimal bits.
  • The first bit is a sign bit.
  • The next 4 bits are exponent.
  • The next 3 bits are decimal
  • 0 0000 000
    • +, 0, 0
    • + 1.0 x 2^0 or 1.0
  • 1 0000 100
    • -, 0, 1/2
    • - 1.1 x 2^0
    • -1.5
  • 0 0001 100
    • +, 1, 1/2
    • + 1.100 x 2^1
    • or 11.0 x 2 = 3.
  • Our biggest number would be 0 0111 111
    • + , 7, 1/2 + 1/4 + 1/8
    • 1.111 x 2^7
    • $11110000_2 = 240$
  • Our next biggest number would be 0 0111 110
    • + , 7, 1/2 + 1/4
    • 1.110 x 2^7
    • $11100000_2 = 224$
• Note there is a space between the biggest number and the next biggest.
  • This is called the gap.
  • It is a round off error problem.
  • There is no way to represent numbers that fall in the gap.
• Floating point is slow.
  • How do you add numbers in scientific notation?
  • $1.23\times10^4 + 2.01\times10^2$
• Note that floats can be used to represent
  • Numbers very close to 0: $1.0\times2^{-100}$
  • Very large numbers : $1.0\times2^{300}$
• This has been a very quick look at floating point.
• Floating point has the same limitations as integer
  • Overflow
• But a much more complex representation
  • So it can be much slower.
• And an additional round off error problem.
• Always chose an int if you can.