IEEE 754

IEEE 754 is a standardised way of storing floating point numbers with three components

• A sign bit
• A biased exponent
• A normalised mantissa

The examples below all refer to 32 bit numbers, but the principles apply to 64 bit.

• The exponent is an 8 bit unsigned number in biased form
  • To get the true exponent, subtract 127 from the binary value
• The mantissa is a binary fraction, with the first bit representing $1/2$, second bit $1/4$, etc.
  • The mantissa has an implicit $1.$, so 1 must always be added to the mantissa

Formula

$$-1^{sign}\times 2^{E-127}\times (1+\sum _{i=1}^{23}{b}_{23-i}{2}^{-i})$$

Decimal to Float

The number is converted to a binary fractional format, then adjusted to fit into the form we need. Take 12.375 for example:

• Integer part $(12{)}_{10}=(1100{)}_2$
• Fraction part $(0.375{)}_{10}=(0.011{)}_2$

Combining the two parts yields $1100.011$. However, the standard requires that the mantissa have an implicit 1, so it must be shifted to the right until the number is normalised (ie has only 1 as an integer part). This yields $(1.100011{)}_2$. As this has been shifted, it is actually $(1.100011{)}_2\times 2^3$. The three $(10)$ is therefore the exponent, but this has to be normalised (+127) to yield 130 $(10000010)$. The number is positive (sign bit zero) so this yields:

$$01000001010001100000000000000000$$

Float to Decimal

Starting with the value 0x41C80000 = 01000001110010000000000000000000:

• The exponent is 131, biasing (-127) gives 4
• The mantissa is 0.5625, adding 1 (normalising) gives 1.5625
• $2^4\times 1.5625$ gives 25

Special Values

• Zero
  • When both exponent and mantissa are zero, the number is zero
  • Can have both positive and negative zero
• Infinity
  • Exponent is all 1s, mantissa is zero
  • Can be either positive or negative
• Denormalised
  • If the exponent is all zeros but the mantissa is non-zero, then the value is a denormalised number
  • The mantissa does not have an assumed leading one
• NaN (Not a Number)
  • Exponent is all 1s, mantissa is non-zero
  • Represents error values