A Hierarchy Of Errors; Level 0: No Error - HP -15C Advanced Functions Handbook

178 Appendix: Accuracy of Numerical Calculations
present, recall that example 1 contains no subtraction, therefore no
cancellation, and is still devastated by its rounding error. In this
respect example 1 is counterintuitive, a little bit scary. Nowhere in
it can we find one or two arithmetic operations to blame for the
catastrophe; no small rearrangement will set everything right as
happened for c(x). Alas, example 1 is not an isolated example. As
computers and calculators grow in power, so do instances of
insidious error growth become more common.
To help you recognize error growth and cope with it is the ultimate
goal of this appendix. We shall start with the simplest kinds of
errors and work our way up gradually to the subtle errors that can
afflict the sophisticated computations possible on the HP-15C.
A Hierarchy of Errors
Some errors are easier to explain and to tolerate than others.
Therefore, the functions delivered by single keystrokes on the
HP-15C have been categorized, for the purposes of easier
exposition, according to how difficult their errors are to estimate.
The estimates should be regarded as goals set by the calculator's
designers rather than as specifications that guarantee some stated
level of accuracy. On the other hand, the designers believe they can
prove mathematically that their accuracy goals have been
achieved, and extensive testing has produced no indication so far
that they might be mistaken.
Level 0: No Error
Functions which should map small integers (smaller than 10
10
) to
small integers do so exactly, without error, as you might expect.
Examples:
v/I=2 -2
3
= -8 3
20
= 3,486,784,401
log(10
9
) = 9 6! = 720
cos'^O) = 90 (in Degrees mode)
ABS(4,684,660 + 4,684,6590 = 6,625,109 (in Complex mode)
Also exact for real argments are I ABS|. | FRAC|, | I N T | , | R N D | , and
comparisons (such as |x^y|). But the matrix functions S, 0 > 1
1 /
* l ,
| MATRIX|6, and | MATRIX]9 (determinant) are exceptions (refer to
page 192).