• The idea of floating point numbers is that we have to compress a certain number range to a smaller one. We have to move from an infinite to a finite system so to say.

• But no matter how many bits we have, there will always be gaps!
• IEEE 754 - is a compression algorithm that used to compress or describe possibly infinite floating point number in the finite system. ****

• Sign - takes up 1 bit, indicates +- sign (who would doubt that?)
• Exponent - takes up 5 bits, signifies the integer power to which the radix (number system base) is raised. In our case 2, e.g. $2^5$ if exponent is 5 in binary.
• Mantissa - it’s like our magnifying glass which we zoom in with to determine the exact location of our number in the -Infinity to +Infinity interval.

1. Since exponent represents the power we have to raise our radix to, we can represent each power range with a corresponding numerical integer range (take a look at the picture below)

So far, we’re modeling a 16-bit system, where 5 bits are dedicated for exponent, which means we can hold 32 numbers in that exponent position, or using a sign from $[-16, 15]$

($2^N, where\;N\;-\;number\;of\;bits)$

2. As we’ve selected 12.52571, the whole-part of the number falls into the range of $[8, 16]$.

   Exponent range - $[3, 4]$

   Numerical range - $[8, 16]$

   Exponent - 3

   Then we basically have to somehow zoom in our interval to the corresponding number.

   

   We have to figure out where in this range the aforementioned number is placed using percentage. That’s where Mantissa comes in!

3. In order to calculate percentage of our number in the numerical range we’re gonna normalise it!

   Normalisation formula:

   $$ N = \frac{value - min}{max - min} $$

   In our case:

   $$ N = \frac{12.52571 - 8}{16 - 8} = 0.56571375 ; or ; 56.571375 % ; into;the;range $$

   Basically, we can use our exponent and the percentage to calculate the exact position of our number in the range.

   E.g:

   $$ N = 8 * (1 + 0.56571375) = 12.52571 $$

4. The question left intact is how do we actually encode that percentage into a binary format?

   For Mantissa we have only 10 bits available:

   10 bits are 1024 possible numbers, so we can point out a closest whole number corresponding to 56.571375%:

   $$ N = 0.56571375 * 1024 = 579.29088 $$

   $$ round(579.29088) = 579 $$

   Therefore, 579 can be represented as 1001000011 in the binary format, BUT!

   This is exactly the step when we performed our compression and lost some precision!

   If try to conduct this operation in the opposite way: $579 / 1024 = 0.5654296875$ and then try to replicate the number:

   $$ 8 * (1 + 0.5654296875) = 12.5234375 $$

   As you can see 12.5234375 ≠ 12.52571

   

   Actual formula to decode floating point number using exponent, sign and mantissa:

   

Special thanks to the author of the original video.