floating point - How to represent FLOAT number in memory in C

Question

While reading a tutorial I came across how to represent Float number in memory. The tutorial had an example with a floating point number.

   float a=5.2  with below Diagram

Can anyone please tell how this 5.2 is converted in to binary and how it is represented in memory in above the above diagram?

Answer 1

As was said, 5.2 is represented as a sign bit, an exponent and a mantissa. How do you encode 5.2?

5 is easy:

101. 

The rest, 0.2 is 1/5, so divide 1.00000... (hex) by 5 and you get 0.3333333... (hex).

(This can be followed more easily if you consider one bit less: 0.FFFF... → F / 5 = 3, so it is easy to see that 0.FFFF... / 5 = 0.33333.... That one missing bit doesn't matter when dividing by 5, so 1.0000... / 5 = 0.3333... too).

That should give you

0.0011001100110011001100110011... 

Add 5, and you get

101.00110011001100110011...         exp 0    (== 5.2 * 2^0)

Now shift it right (normalize it, i.e. make sure the top bit is just before the decimal point) and adjust the exponent accordingly:

1.010011001100110011001100110011... exp +2   (== 1.3 * 2^2 == 5.2)

Now you only have to add the bias of 127 (i.e. 129 = 0b10000001) to the exponent and store it:

0 10000001 1010 0110 0110 0110 0110 0110 

Forget the top 1 of the mantissa (which is always supposed to be 1, except for some special values, so it is not stored), and you get:

01000000 10100110 01100110 01100110

Now you only have to decide little or big endian.

This is not exactly how it works, but that is more or less what happens when a number like 5.2 is converted to binary.