# Re: IEEE-754

Roman Töngi wrote On 08/23/07 06:45,:
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 and 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):

-----------------------------------------
x*(1-2(-53)) <= round(x) <= x*(1+2^(-53))
-----------------------------------------

Is this the correct range according to my assumptions?

It looks right to me for x >= 0 (for x < 0 the
inequalities are backwards), and given suitable hand-
waving for abs(x) very small or very large. It might
be possible (I'm not sure) to sharpen the analysis a
tiny bit and change a `<=' to a `<', but whether that's
worth trying depends on your purpose in obtaining the
bound in the first place.

Note that the C language does not require IEEE
floating-point, nor does it require round-to-nearest,
nor does it specify the value of eps.

--
Eric.Sosman@xxxxxxx

.

• References:

## Relevant Pages

• Re: IEEE-754
... I assume double precision and the round mode in effect to be 'round to nearest' and that the number lies within the ... round= x*(1 + delta) ... How did you arrive at a machine epsilon of 2^? ...
(comp.lang.c)
• Re: IEEE-754
... I assume double precision and the round mode in effect to be 'round to nearest' and that the number lies within the ... round= x*(1 + delta) ... How did you arrive at a machine epsilon of 2^? ...
(comp.lang.c)
• IEEE-754
... be stated the range within which a machine number lies. ... For the following example, I assume 'double precision', the round mode ... round= x*(1 + delta) ...
(sci.math.num-analysis)
• IEEE-754
... I assume double precision and the round mode in effect to be 'round to nearest' and that the number lies within the ... round= x*(1 + delta) ...
(comp.lang.c)
• IEEE-754
... I assume double precision and the round mode in effect to be 'round to nearest' and that the number lies within the ... round= x*(1 + delta) ...
(comp.programming)