Keystrokes display – HP 15c User Manual
Page 152
152
Appendix: Accuracy of Numerical Calculations
152
She calculated $376,877.67 on her HP-15C, but the bank's total was $333,783.35, and this
latter total agrees with the results calculated on good, modern financial calculators like the
HP-12C, HP-37E, HP-38E/38C, and HP-92. Where did Susan's calculation go awry? No
severe cancellation, no vast accumulation of errors; just one rounding error that grew
insidiously caused the damage:
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
n
i
n
e
n
n
i
1
ln
1
,
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 λ(x) = ln(1+x) accurately for all x>−1, even if |x| is very small:
1. Calculate u = 1 + x rounded.
2. Then
.
1
1
)
ln(
1
)
(
u
if
u
x
u
u
if
x
x
The following program calculates λ(x) = ln(1+x)
Keystrokes
Display
|¥
´CLEARM
000-
´bA
001-42,21,11
Assumes x is in X-register.
v
002- 36
v
003- 36
“
004- 26
Places 1 in X-register.
+
005- 40
Calculates u = 1 + x rounded.
|N
006- 43 12
Calculates ln(u) (zero for u = 1).
®
007- 34
Restores x to X-register.
|K
008- 43 36
Recalls u.
“
009- 26
Places 1 in X-register.
|T6
010-43,30, 6
Tests u≠1.