Re: How Computers Represent Floats

[Karl Schultz <karl_schultz(at)my-deja.com>]

Here's another useful table if anyone is interested in pondering the
limits of floating point representations:


                   32-bit(single)     64-bit(double)
Mantissa               24 bits           53 bits
Exponent                8 bits           11 bits
EPS                 1.19209e-007       2.22045e-016
Min                 1.17549e-038       2.22507e-308
Max                 3.40282e+038       1.79769e+308
Decimal Places           6                  15

                       80-bit          128-bit(quad)
Mantissa               65 bits           113 bits
Exponent               15 bits            15 bits
EPS                 1.08420e-019       1.92593e-034
Min                 3.36210e-4932      3.36210e-4932
Max                 1.18973e+4932      1.18973e+4932
Decimal Places           18                 33

The mantissa bit counts include the sign bit.

I find it a little interesting that the number of bits in the exponent
remains the same between 80-bit and 128-bit.

I also think that the EPS (machine epsilon or machine precision) is
probably one of the most important values.  It is the smallest value
that you can add to 1.0 and still have the result be something other
than 1.0.  This can give you an idea of how closely you can resolve
(differentiate) floating point values at a given magnitude.

For example, see what this does in IDL 5.3:
PLOT,FINDGEN(100),FINDGEN(100)+2d8,YSTYLE=3

IDL 5.4 gives different results because PLOT works in double
precision.  You can "simulate" the old 5.3 behavior with:

PLOT,FINDGEN(100),FLOAT(FINDGEN(100)+2d8), YSTYLE=3

Karl


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