Appendix:
Accuracy
of
Numerical
Calculations
185
|TRIG|(:c)
=
trig(xTr/p)
to
within ±0.6 units
in its
10th significant digit.
This
formula
has
important practical implications:
•
Since
rr/p
= 1 -
2.0676...
X
lO'13//)
=
0.9999999999999342...,
the
value produced
by
[TRIG
\(x)
differs
from
trig(x)
by no
more
than
can be
attributed
to two
perturbations:
one in the
10th
significant digit
of the
output
trig(o;),
and one in the
13th
significant digit
of the
input
x.
If
x has
been calculated
and
rounded
to 10
significant digits,
the
error inherited
in its
10th significant digit
is
probably
orders
of
magnitude bigger
than
[TRIG
|'s
second perturbation
in x's
13th significant digit,
so
this
second perturbation
can be
ignored
unless
x is
regarded
as
known
or
calculated exactly.
•
Every trigonometric identity
that
does
not
explicitly involve
n
is
satisfied
to
within
roundoff
in the
10th significant digit
of
the
calculated values
in the
identity.
For
instance,
sin2(x)
+
cos2(x)
=
1,
so
(Q3IN](*))2
+
(fcOSlQc))2
= 1
sin(;c)/cos(:x;)
=
tan(x),
so|
SIN
\(x)/\COS\(x)
=
I
TANK*)
with each calculated result correct
to
nine significant
digits
for
all x.
Note
that
|COS\(x)
vanishes
for no
value
of x
representable exactly with just
10
significant digits.
And if 2x
can
be
calculated exactly given
x,
sin(2x)
=
2sin(*)cos(*),
so
[SJN](2x)
= 2
[SIN](x)[cOS](%)
to
nine significant digits.
Try the
last
identity
for x
=
52174
radians
on the
HP-15C:
[siN](2;c)
=
-0.00001100815000,
2[SII\n(x)rcOSl(a:)
=
-0.00001100815000.
Note
the
close agreement even though
for
this
x,
sin(2^;)
=
2sin(;t)cos(x)
=
-0.0000110150176...
disagrees
with
[§JN](2x)
in
its
fourth significant digit.
The
same identities
are
satisfied
by
|TRIG|(x)
values
as by
trig(jc)
values even though
[TRIG
\(x)
and
trig(x)
may
disagree.
•
Despite
the two
kinds
of
errors
in
[TRIG
|, its
computed values
preserve
familiar relationships wherever possible:
•
Sign symmetry:
|COS|(-*)
=
|COS|(x)
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