Conversions decimal to binary and binary to and from hexadecimal

Some examples of machine codes for particular real numbers (1, -1, 2) and largest machine number.

(C codes for hexadecimal machine code to and from scientific notation: https://blue.math.buffalo.edu:public_html/537_f25/hexcodeout.c https://blue.math.buffalo.edu:public_html/537_f25/hexcodein.c)

All the non-negative machine numbers

if we had a 3-bit mantissa and a 2-bit exponent biased by 1:

Undesirable feature: there's a big "hole at zero". This can be mitigated using "denormalization" of the numbers with the smallest exponent:

Denormalization

Floating point types supported by Nvidia's current chips (used for AI):

Machine epsilon

a very important characteristic - the maximum relative spacing of machine numbers: