HP -15C Advanced Functions Handbook page 207

Appendix: Accuracy of Numerical Calculations 205
But backward error analysis is no panacea; it may explain errors
but not excuse them. Because it complicates computations
involving singularities, we have tried to eliminate the need for it
wherever we could. If we knew how to eliminate the need for
backward error analysis from every function built into the
calculator, and to do so at tolerable cost, we would do that and
simplify life for everyone. That simplicity would cost too much
speed and memory for today's technology. The next example will
illustrate the trade-offs involved.
Example 6 Continued. The program listed below solves the real
quadratic equation c — 2bz + az
2
= 0 for real or complex roots.
To use the program, key the real constants into the stack (c [ENTER |
b [ENTER] a) and run program "A".
The roots x and y will appear in the X- and Y-registers. If the roots
are complex, the C annunciator turns on, indicating that Complex
mode has been activated. The program uses labels "A" and ".9"
and the Index register (but none of the other registers 0 to .9);
therefore, the program may readily be called as a subroutine by
other programs. The calling programs (after clearing flag 8 if
necessary) can discover whether roots are real or complex by
testing flag 8, which gets set only if roots are complex.
The roots x and y are so ordered that \x\ |y| except possibly when
\x\d |y| agree to more than nine significant digits. The roots are
as accurate as if the coefficient c had first been perturbed in its 10th
significant digit, the perturbed equation had been solved exactly,
and its roots rounded to 10 significant digits. Consequently, the
computed roots match the given quadratic's roots to at least five
significant digits. More generally, if the roots x and y agree to n
significant digits for some positive n ^ 5, then they are correct to at
least 10 — n significant digits unless overflow or underflow occurs.
Keystrokes Display
[Tl CLEAR fPRGMl 000-
[Q[lBLl[Al 001-42,21,11
| ENTER | 002- 36
003-
43 33
004-
20
005- 43 se