Inside Floating Point

• The floating point representation is like scientific notation in that it breaks the real number up into an exponent and a mantissa. In scientific notation, you write the number as a * 10^b ( with 1 <= a < 10 ). In floating point, powers of two are used instead: a * 2^b ( with 1 <= a < 2 ).

• The number of bits used to store the mantissa and the exponent depends on both the computer and compiler used, but here are some common values:

  Format          Exponent Bits     Mantissa Bits     Total Bits
  ----------------------------------------------------------------
  IEEE 754            11                 52                64
  IEEE 854            15                 64                80
  Intel 387            8                 23                32
  Motorola 68881       8                 23                32
  

• Total Bits takes into account the sign bit used to keep track of whether the number is negative or not. The exponent bits count includes the bit used to keep track of whether the exponent is negative or not. You always have one more bit of mantissa accuracy than the number of bits used because since it is always of the form 1.xxxxxxxxx, you don't have to store the leading 1.

• What are the largest real numbers allowed in the above formats? What are the smallest positive real numbers allowed? How many significant figures does each give?