HP 15c User Manual
Page 151
Appendix: Accuracy of Numerical Calculations
151
significantly smaller than one unit in the 10th significant digit of the result, include ∆,
À,r,d,p,and c; N,o, @,P]for real arguments;
:, ,, {, /,H[, H\and H]for real and complex
arguments; a, ¤, and⁄for complex arguments; matrix norms >7, >8;
and finally [, \, and ] for real arguments in Degrees and Grads modes (but not
in Radians mode − refer to Level 2, page 154)
A function that grows to ∞ or decays to 0 exponentially fast as its argument approaches ± ∞
may suffer an error larger than one unit in its 10th significant digit, but only if its magnitude
is smaller than 10
-20
or larger than 10
20
; and though the relative error gets worse as the result
gets more extreme (small or large), the error stays below three units in the last (10th)
significant digit. The reason for this error is explained later. Functions so affected,
are ',
Y, ! (for non-integer x), P[, and P\ for real arguments. The worst
case known is 3
201
, which calculated as 7.968419664×10
95
. The last digit 4 should be 6
instead, as is the case for 7.29
33.5
, calculated as 7.968419666×10
28
.
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 f in Level 1 produce instead a computed value F = (1 +ε) f
whose relative errorε, though unknown, is very small:
.
1
Level
in
functions
other
all
for
10
1
rounded
correctly
if
10
5
9
10
F
F
This simple characterization of all the functions in Level l fails to convey many other
important properties they all possess, properties like
Exact integer values: mentioned in Level 0.
Sign symmetry: sinh(−x) = −sinh(x), cosh(−x) = cosh(x), ln(1/x) = −ln(x) (if l/x is
computed exactly).
Monotonicity: if f(x) ≥ f(y), then computed F(x) ≥ F(y).
These additional properties have powerful implications; for instance, TAN(20°) =
TAN(200°) = TAN(2,000°) = ... = TAN(2× 10
99
°) = 0.3639702343 correctly. But the simple
characterization conveys most of what is worth knowing, and that can be worth money.
Example 2 Explained. Susan tried to calculate
n
i
n
i
n
1
1
payment
total
where
payment = $0.01,
i = 0.1125, and
n = 60 Ч 60 Ч 24 Ч 365 = 31,536,000.