Converting bit patterns to IEEE 754 Single Precision equivalents

The IEEE 754 standard defines how floating point numbers are stored in binary form. In this tutorial we convert a bit pattern into its' base 10 equivalent.

Given a 32-bit base 2 number:

0100 0010 0010 1001 1110 0100 0000 00002

This bit pattern has no inherent encoding, but if we assume that it represents an IEEE 754 single precision number, we can convert it to base 10.

Exponent = 128 + 4 = 132

Subtracting the bias of 127, we get 5. Therefore we will need to shift the fraction 5 places to the right.

Fraction = 1.010 1001 1110 0100 0000 0000 _{(base 2)}

The leading '1' is not stored in the original bit pattern. According to the IEEE 754 standard, a leading '1' is always assumed.

One solution is 1.010 1001 1110 0100 0000 0000₂ X 2⁵

This solution is unwieldy since it is in base 2.

Another solution is 1010 10.01 1110 0100 0000 0000

This solution is unwieldy since it is in base 2.

Another solution is (2⁰+2^-2+2^-4+2^-7+2^-8+2^-9+2^-10+2^-13) * 2⁵

Another solution is (2⁵+2³+2¹+2^-2+2^-3+2^-4+2^-5+2^-8)

Converting each position to base 10 and adding gives us:

32.

 8.

 2.

  .25

  .125

  .0625

  .03125

  .00390625
============

42.47265625  

Therefore,

0100 0010 0010 1001 1110 0100 0000 00002 converts to = 42.4726562510