IEEE 754

Objectives

1. Develop a basic understanding of the IEEE 754 format.
2. Discuss how floating point mathematics might be more difficult than integer.

Notes

• This is from Carpinelli page 32 of the pdf 1-12 in the document.
• Institute of Electrical and Electronics Engineers
• Early History Distilled
  • There were many different formats.
  • Precision and Gap were different on all of these.
  • Moving from one machine to another caused all kinds of problems with the results.
• 754 standardized this in 1985.
• Mostly numbers are represented as in scientific notation,
  • Given a number +3.24x10^-15
    • The leading + is the sign, +/-
    • The 3.24 is the mantissa
    • the -15 is the exponent
  • In scientific notation, we normally require the mantissa to be more than 0 and less than 10.
  • This is called normalized.
• When we switch to base 2, in a fixed number of bits things change.
  • We call the mantissa the significand, a normalized mantissa
  • The significand almost always starts with a 1.
• Representation
  • In 32 bits
    • The first bit is a sign bit.
      • 0 for +, 1 for -
      • The actual computation is (-1)^bit
      • (-1)⁰ = 1
      • (-1)¹ = -1
    • The exponent is represented in 8 bits.
      • But it is represented in excess 128 format
      • Subtract 128 to find the actual exponent
      • The bit pattern 10000000 is 0, since 128-128 = 0
      • The bit pattern 10000001 is 1, since 129-128 = 0
      • The bit pattern 01111111 is -1, since 127-128 = -1
      • The bit pattern 00000000 is a special case, see below.
      • The bit pattern 11111111 is a special case, see below.
    • The next 23 bits represent the signifcand, but
      • Since the number is 1.xxx the 1 is implied.
    • Please note, his diagrams in the book are wrong
  • When the exponent is 0 (the pattern 0) the number is special
    • This is called a denormal or subnormal
    • It represents a number very close to 0
    • The representation is 0.bitpattern x 2 ^-127
    • These are apparently bad and can be turned off.
  • When the exponent is 128, the number is infinity
• See float.cpp.