HP -15C Advanced Functions Handbook page 210

208 Appendix: Accuracy of Numerical Calculations
The last three cases show how severe are the results of perturbing
the 10th significant digit of any coefficient of any quadratic whose
roots are nearly coincident. The correct roots for these cases are
Case 4: 1 and 0.9999969434
Case 5: 62.77179203 ± J8.5375 X 10~
5
Case 6: 12.21711755 ±i'0.001374514
Despite errors in the fifth significant digit of the results, subroutine
"A" suffices for almost all engineering and scientific applications
of quadratic equations. Its results are correct to nine significant
digits for most data, including c, b, and a representable exactly
using only five significant digits; and the computed roots are
correct to at least five significant digits in any case because they
cannot be appreciably worse than if the data had been entered with
errors in the 10th significant digit. Nonetheless, some readers will
feel uneasy about results calculated to 10 significant digits but
correct to only 5. If only to simplify their understanding of the
relationship between input data and output results, they might still
prefer roots correct to nine significant digits in all cases.
Programs do exist which, while carrying only 10 significant digits
during arithmetic, will calculate the roots of any quadratic
correctly to at least nine significant digits regardless of how
nearly coincident those roots may be. All such programs calculate
d = b
2
— ac by some trick tantamount to carrying 20 significant
digits whenever b
2
and ac nearly cancel, so those programs are a
lot longer and slower than the simple subroutine "A" provided
above. Subroutine "B" below, which uses such a trick,* is a very
short program that guarantees nine correct significant digits on a
10-digit calculator. It uses labels "B", ".7", and ".8" and registers
R
0
through R
9
and the Index register. To use it, key in c | ENTER | b
I ENTER |a, run subroutine "B", and wait for results as before.
Keystrokes Display
000-
001-42,21,12
002- 44 25
003- 33
*Program "B" exploits a tricky property of the |_£-| and | S + | keys whereby certain
calculations can be carried out to 13 significant digits before being rounded back to 10.
'~-