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# Converting bit patterns to IEEE 754 Single Precision equivalents

Updated on May 8, 2011

## 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.

## Test Case

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.

Sign
Exponent (biased 127)
Fraction (without the leading '1')
0
100 0010 0
010 1001 1110 0100 0000 0000

The IEEE 754 Single Precision Standard mapping of the bits

## Calculate the exponent

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.

## Calculate the Fraction

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.

## Apply the 'hidden' digit to the right of the radix point

One solution is 1.010 1001 1110 0100 0000 00002 X 25

This solution is unwieldy since it is in base 2.

## Apply the exponent to the fraction by sliding the radix point

Another solution is 1010 10.01 1110 0100 0000 0000

This solution is unwieldy since it is in base 2.

## Convert the resulting bit pattern to powers of 2

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

## Apply the exponent to the powers of 2

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`

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## Comments

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• Author

nicomp really 8 years ago from Ohio, USA

@ICnow: Thanks.

• ICnow 8 years ago

Awesome! Best explanation on the web! Perfect!

working