HP 15c - Manual (Page 150)

150 Appendix: Accuracy of Numerical Calculations
150
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:
2 4

-2
3
= -8 3
20
= 3,486,784,401
log(10
9
) = 9 6! = 720
cos
-1
(0) = 90 (in Degrees mode)
ABS(4,686,660 + 4,684,659 i) = 6,625,109 (in Complex mode)
Also exact for real arguments are a,
q, ‘, &, and comparisons (such as
£). But the matrix functions *, ÷, ⁄, >6, and >9 (determinant)
are exceptions (refer to page 160).
Level ∞: Overflow/Underflow
Results which would lie closer to zero than 10
-99
underflow quietly to zero. Any result that
would lie beyond the overflow thresholds ±9.999999999×10
99
is replaced by the nearest
threshold, and then flag 9 is set and the display blinks. (Pressing == or "9 or −
will clear flag 9 and stop the blinking.) Most functions that result in more than one
component can tolerate overflow/underflow in one component without contaminating the
other; examples are ;, :, complex arithmetic, and most matrix operations. The
exceptions are matrix inversion (⁄ and ÷), > 9 (determinant), and L.
Level 1: Correctly Rounded, or Nearly So
Operations that deliver "correctly rounded" results whose error cannot exceed ½ unit in their
last (10th) significant digit include the real algebraic operations +, -, *, ÷, x,
¤, ⁄, and k, the complex and matrix operations + and -, matrix by scalar
operations *and ÷(excluding division by a matrix), and h. These results are the
best that 10 significant digits can represent, as are familiar constants $, 1', 2N,
10
N, 1r, and many more. Operations that can suffer a slightly larger error, but still