I'm reading the following table (from https://www.tutorialspoint.com/cprogramming/c_data_types.htm )
Why the precision of the float is 6 decimal places, but I see that the float are in the interval [1.2E-38 , 3.4E38] ? So, I think that I can have 38 decimals of precision.
Which is my error?
Prof. Bangerth's comment is completely correct. To add more detail, we can refer to IEEE754 standard which defines the floats as
| sign bit | exponent bits | mantissa |
| 1 bit | 8 bits | 23 bits |
Sign bit represents the sign of the number $+$ or $-$
Exponent bits are signed (using two's complement) and ranges from $-128$ to $127$
Lastly, mantissa has an implicit 1 assumption (without getting into intricacies like subnormal -or denormal- number), so a number in this representation has the form: $\pm 2^{E} \times 1.\text{mantissa}$. For example;
$3.1415=$0 10000000 10010010000111001010110
or equivalently, $+ 2^{(10000000)_2-128} \times (\color{red}1.10010010000111001010110)_2$ where red coloured $1$ is implicitly assumed to be there.
Now, if you transform the floating point representation back to decimal, you will notice that it is actually equal to $3.14149996185302734375$ which is not equal to $3.1415$. This is because mantissa has only so much space (23 bits in case of floats) and we have to round. This rounding may introduce an error of at most $2^{-23}\approx 10^{-7}$.
Depending on how you define precision, this means that you have either 6 decimal places or 6-7 decimal places of precision.
I wrote this in a rush, I may have made some mistakes. Please be critical of what I am saying here and refer to other sources. And if I said anything wrong, please let me know so I can fix it.
