What is the smallest decimal that can be represented using IEEE754 single precision?

Question

Is seems like it would be IE-38, which would make the exponent 127-127. But when I add 1E10 and 1E-32, I get 1E10 which makes me think that 1E-32 is so small it's not being properly represented.

Answer 1

You are mixing two different terms:

• minimal value representable in given format ( ~10^-38 ) and
• machine epsilon - such minimal value that 1.0 + eps != 1.0 , for single precision it is about 2^-23

For bigger value 1e10 minimal summand is ~1e10 * macheps

Simplified example: for 4 decimal digits you can represent values 100.0 and 0.001 , but 100.0 + 0.001 = 100.0 because true sum 100.001 cannot fit into 4 decimal places.