HP HP-15C

Appendix: Accuracy of Numerical Calculations 195
for the figure shown below. Engineering and scientific calculations
often require that the angle 6 be calculated from given values p, q,
and r for the length of the triangle's sides. This calculation is
feasible provided 0 <p^q + r, 0<q^p + r, andO ^ r^p + q, and
then
cos-\((p2 + q2) -
otherwise, no triangle exists with those side lengths, or else Q = 0/0
is indeterminate.
The foregoing formula for 6 defines a function 6 — f(p,q,r) and also
in a natural way, a program F(p,q,r) intended to calculate the
function. That program is labeled "A" below, with results
FA(p,q,r) tabulated for certain inputs p, q, and r corresponding to
sliver-shaped triangles for which the formula suffers badly from
roundoff. The numerical unreliability of this formula is well known
as is that of the algebraically equivalent but more reliable formula
6 = f(p,q,r) — 2 ian~l\fab/(cs), where s = (p + q + r)/2, a — s — p,
b = s — q, and c = s — r. Another program F(p,q,r) based upon this
better formula is labeled "B" below, with results FB(p,q,r) for
selected inputs. Apparently FB is not much more reliable than FA.
Most of the poor results could be explained by backward error
analysis if we assume that the calculations yield F(p,q,r) —
f(p + 6p,q + 8q,r + 8r) for unknown but small perturbations
satisfying \5p\ 10~9|p| , etc. Even if this explanation were true, it
would have perplexing and disagreeable consequences, because the
angles in sliver-shaped triangles can change relatively drastically
when the sides are perturbed relatively slightly; f(p,q,r) is
relatively unstable for marginal inputs.
Actually the preceding explanation is false. No backward error
analysis could account for the results tabulated for FA and FB
under case 1 below unless perturbations 8p, 8q, and 8r were
allowed to corrupt the fifth significant digit of the input, changing
1 to 1.0001 or 0.9999 . That much is too much noise to tolerate in a
10-digit calculation. A better program by far is Fc, labeled "C" and
explained shortly afterwards.