HP -15C Advanced Functions Handbook page 182

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180
Appendix: Accuracy of Numerical Calculations
The foregoing statements about errors can be summarized for all
functions in Level 1 in a way that will prove convenient later:
Attempts to calculate a function / in Level 1 produce
instead a computed value F — (1 + e)/ whose relative error
e, though unknown, is very small:
I 5 X 10"
10
if Fis correctly rounded
I 1 X 10~
9
for all other functions F in Level 1.
This simple characterization of all the functions in Level 1 fails to
convey many other important properties they all possess,
properties like
• Exact integer values: mentioned in Level 0.
• Sign symmetry: sinh(-^) = -sinh(x), cosh(-x) = cosh(jc),
ln(l/x) = — ln(x) (if l/x is computed exactly).
• Monotonicity: if f(x) ^ f(y), then computedF(x) ^ F ( y ) .
These additional properties have powerful implications; for
instance, TAN(20°) = TAN(200°) = TAN(2,000°) = ... =
TAN(2 X 10" °) = 0.3639702343 correctly. But the simple character-
ization conveys most of what is worth knowing, and that can be
worth money.
Example 2 Explained. Susan tried to calculate
total = payment X
i/n
where
payment = $0.01,
/ = 0.1125, and
re = 60 X 60 X 24 X 365 = 31,536,000.
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:

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