HP HP-15C Advanced Functions Handbook

Appendix: Accuracy of Numerical Calculations 177
Regarding the first misconception, example 1 would behave in the
same perverse way if it suffered only one rounding error, the one
that produces R(x) — I or 0.9999999999, in error by less than one
unit in its last (10th) significant digit.
Regarding the second misconception, cancellation is what happens
when two nearly equal numbers are subtracted. For example,
calculating
c(x) — (1 — cosx)/x2
in Radians mode for small values of x is hazardous because of
cancellation. Using x — l.2X 10~5 and rounding results to 10 digits,
cos x = 0.9999999999
and
1 - cos x = 0.0000000001
with cancellation leaving maybe one significant digit in the
numerator. Also
x2 = 1.44X10-10.
Then
C( x) = 0.6944.
This calculated value is wrong because 0 ^ c(x) < l/i for all x ¥^ 0. To
avoid numerical cancellation, exploit the trigonometric identity
cos x = 1 — 2 sin2(x/2) to cancel the 1 exactly and obtain a better
formula
1 / sin (x/2) \
2 \ )
When this latter expression is evaluated (in Radians mode) at
x — 1.2 X 10~5, the computed result C(x) — Q.5 is correct to 10
significant digits. This example, while explaining the meaning of
the word "cancellation," suggests that it is always a bad thing.
That is another misconception to be dispatched later. For the

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Brand	HP
Model	HP-15C
Category	Calculator
Language	English