HP 15c - Manual (Page 156)

156 Appendix: Accuracy of Numerical Calculations
156
[(2x) = −0.00001100815000
2[(x) \(x) = −0.00001100815000.
Note the close agreement even though for this x, sin(2x) = 2sin(x)cos(x) =
−0.0000110150176 ... disagrees with [(2x) in its fourth significant digit. The same
identities are satisfied by Æ(x) values as by trig(x) values even though Æ(x) and
trig(x) may disagree.

Despite the two kinds of errors in Æ, its computed values preserve familiar
relationships wherever possible:

Sign symmetry:
\(−x) = \(x)
[(−x) = − [(x)

Monotonicity: if trig(x) ≥ trig(y),
then Æ(x) ≥
Æ(y)
(provided |x − y| < 3)

Limiting inequalities:
[(x) / x ≤ 1 for all x ≠ 0
](x) / x ≥ 1 for 0 < |x| < π/2
−1 ≤ [(x) and \(x) ≤ 1
for all x
What do these properties imply for engineering calculations? You don't have to remember
them!
In general, engineering calculations will not be affected by the difference between p and π,
because the consequences of that difference in the formula defining Æ(x) above are
swamped by the difference between $
and π and by ordinary unavoidable roundoff in x or
in trig(x). For engineering purposes, the ratio π / p = 0.9999999999999342... could be
replaced by 1 without visible effect upon the behavior of Æ.
Example 5: Lunar Phases. If the distance between our Earth and its moon were known
accurately, we could calculate the phase difference between radar signals transmitted to and
reflected from the moon. In this calculation the phase shift introduced by p ≠ π has less effect
than changing the distance between Earth and moon by as little as the thickness of this page.
Moreover, the calculation of the strength, direction, and rate of change of radiated signals
near the moon or reflected signals near the Earth, calculations that depend upon the
trigonometric identities' continuing validity, are unaffected by the fact that p ≠ π; they rely
instead upon the fact that p is a constant (independent of x in the formula for Æ(x)), and
that constant is very near π.
The HP·15C's keyboard functions that involve p are the trigonometric functions [,
\, and ]for real and complex arguments; hyperbolic functions P[,
P\, and P] for complex arguments; complex operations ', @, and
Y; and real and complex ;.