HP HP-15C - Page 183

Appendix: Accuracy of Numerical Calculations 181
i/n = 0.000000003567351598
1 + i/n = 1.000000004
when rounded to 10 significant digits. There is the rounding error
that hurts. Subsequently attempting to calculate (1 + i/n)n, Susan
must get instead (1.000000004)31'536'000 = 1.134445516, which is
wrong in its second decimal place.
How can the correct value be calculated? Only by not throwing
away so many digits of i/n. Observe that
so we might try to calculate the logarithm in some way that does
not discard those precious digits. An easy way to do so on the
HP-15C does exist.
To calculate X(x) = ln(l + x) accurately for all x > — 1, even if | x
very small:
1. Calculate u = 1 + x rounded.
2. Then ,
I x
X(*) = <
I \n(u)x/(u-l)
The following program calculates \(x) = ln(l + x).
Display Keystrokes
fglfp/Rl
[Tl CLEAR fPRGMl
rr||LBL|[Al
| ENTER |
| ENTER |
000-
001-42,21,11 Assumes x is in X-register.
002- 36
003- 36
004- 26 Places 1 in X-register.
005- 40 Calculates u — l+x
rounded.
006- 4312 Calculates ln(u) (zero for
u = l).
007- 34 Restores x to X-register.
008- 4336 Recalls u.