IEEE-754
- From: Roman Töngi <roman.toengi@xxxxxxxxxx>
- Date: Mon, 17 Sep 2007 22:51:57 +0200
IEEE-754 Arithmetic:
Most real numbers can't be stored exactly on the computer, but there can
be stated the range within which a machine number lies.
For the following example, I assume 'double precision', the round mode
in effect to be 'round to nearest' and that the number lies within the
normalized range:
Definitions:
x := real number
round(x) := correctly rounded normalized number
eps := machine epsilon (2^(-52) for double precision)
abs(x) := absolute value of x
That is:
round(x) = x*(1 + delta)
with delta:
abs(delta) <= 1/2*eps (round to nearest)
i.d. abs(delta) <= 2^(-53) (double precision)
abs(delta) corresponds to the relative rounding error.
Now I can state the range including round(x):
For positive x:
-----------------------------------------
x*(1-2(-53)) <= round(x) <= x*(1+2^(-53))
-----------------------------------------
and for negative x:
-----------------------------------------
x*(1-2(-53)) >= round(x) >= x*(1+2^(-53))
-----------------------------------------
Is this the correct range according to my assumptions?
Thanks
.
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