176
Appendix: Accuracy
of
Numerical Calculations
This value
is so
small
that
the
calculated value
F(x)
—
S(R(x))
underflows
to 0. So the
HP-15C
isn't
broken;
it is
doing
the
best
that
can be
done with
10
significant
digits
of
precision
and 2
exponent
digits.
We
have explained example
1
using
no
more information about
the
HP-15C
than
that
it
performs each arithmetic operation
Rx~|
and [71
fully
as
accurately
as is
possible within
the
limitations
of 10
significant digits
and 2
exponent digits.
The
rest
of the
information
we
needed
was
mathematical knowledge about
the
functions
f, r,
and
s. For
instance,
the
value
r(10100)
above
was
evaluated
as
r(10lon)
=
(io100)(l/;!50)
= exp
(In
(10100)/250)
=
exp(100(lnlO)/250)
= exp
(2.045
X
lO'13)
= 1 +
(2.045
X
10-13)
+
>/2(2.045
X
lO"13)2
+ ...
by
using
the
series
exp (z) = 1 + z +
Vzz2
+
Vez3
+
....
Similarly,
the
binomial theorem
was
used
for
\70.9999999999
= (1 -
lO'10)'72
=
1 -
^(IQ-10)
-
V8(10-10)2
-
....
These mathematical facts
lie
well beyond
the
kind
of
knowledge
that
might have been considered adequate
to
cope with
a
calculation containing only
a
handful
of
multiplications
and
square roots.
In
this
respect, example
1
illustrates
an
unhappy
truism: Errors make computation very much harder
to
analyze.
That
is why a
well-designed calculator, like
the
HP-15C, will
introduce
errors
of its own as
sparingly
as is
possible
at a
tolerable
cost. Much more error
than
that
would turn
an
already
difficult
task into something hopeless.
Example
1
should
lay two
common misconceptions
to
rest:
•
Rounding errors
can
overwhelm
a
computation only
if
vast
numbers
of
them accumulate.
«
A few
rounding errors
can
overwhelm
a
computation only
if
accompanied
by
massive cancellation.
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