Re: How Computers Represent Floats

["William B. Clodius" <wclodius(at)lanl.gov>]

"William B. Clodius" wrote:
> <snip> IEEE 754 requires that all intermediate calculations
> be performed a higher precision so
Ignore the above incomplete sentence. What I originally attempted to
write was covered later.
> 
> <snip>

Some other surprises. 

The definition of the IEEE 754 mantisa, an integer with values from
2^n_mant to 2*2^n_mant-1, where n_mant is the number of bits available
for the mantisa, is termed a normalized number. This is error prone for
very small numbers. IEEE 754 mandates that there be available for such
small numbers what are termed denorms where the mantissa is  interpreted
as an integer from 0 to 2^n_mant, so that accuracy degrades gradually
for such values. However, this complicates the implementation of the
floating point, so some processors, e.g., the DEC Alpha make this
available only in software at a greatly reduced performance.

The special values for IEEE 754 were chosen with specific applications
in mind and are not always the best for other applications. In
particular some applications benefit from unsigned zeros and infinities
others from signed NaNs none of which are provided by IEEE 754. Only
sophisticated users tend to complain about this.

The default rounding behavior for IEEE 754 from an "extended" precision
intermediate that is exactly halfway between values in the result
representation, always to the same final bit (effectively alternately up
and down), often surpises very observant users, although it is designed
to reduce the propagation of systematic errors in most applications.