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floating-pointieee-754subnormal-numbers
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Reputation: 29096

Confusing IEEE-754 Subnormal/Normal: why FLT_MIN is bigger than 1.40e-45?

I noticed in C/C++ that FLT_MIN is about 1.1754943508222875e-38 while FLT_MAX is 3.40282346639e+38, I was expecting more symmetric values around zero until I realized that FLT_MIN represents the minimum value close to zero, not the absolute minimum value which is in this case -FLT_MAX. So I wrote this:

>>> exp = 0
>>> mantissa = 0
>>> 2**(exp - 126) * (1 + mantissa / 2**23)
1.1754943508222875e-38

I was confident, but when I played around with this excellent FloatConverter I found that my interpretation is wrong:

When exp==0 and mantissa==0 I don't get 1.1754943508222875e-38, but 0:

enter image description here

Moreover, the above FLT_MIN value isn't quite achieved in the same way:

enter image description here

It seems if the exponent is equal to 0, we are in a special case named denormalized in which we get rid of the +1:

2**((exp if exp else 1) - 127) * ((1 if exp else 0) + mantissa / 2**23)

In this case, I could have: mantissa = 1, and therefore my number equals: 1.401298464324817e-45 which is smaller than FLT_MIN. Oh damned!

So I don't understand why I can get smaller values than the smallest value.

Upvotes: 1

Views: 34

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