HP 35S: Hyperbolics Bug?

[Marcio]

(07-24-2015 11:27 AM)Paul Dale Wrote:  It would be surprising. The hyperbolics don't have the same nasty cases as the circular trigonometric functions. There are still concerns but they aren't so insidious as with the circular functions.

The näive formula for \( cosh(x) = \frac{e^x + e^{-x}}{2} \) is stable across the entire domain.

The näive formula for \( sinh(x) = \frac{e^x - e^{-x}}{2} \) isn't. It encounters cancellation issues as x -> 0. This would be the place I'd go hunting for accuracy loss both for sinh and tanh.

If I remember correctly, I played with these functions quite a bit a while ago and couldn't find any loss of accuracy inside the working limits of the 35s. It seems the loss of precision pops up when you use complex numbers to calculate SIN or COS:.

\[ cos(x) = \frac{e^{ix} + e^{-ix}}{2} \]

Even so, in order for this problem to appear, the argument must be really small (for example, 1E-7) and the result multiplied by its inverse (1E7).

Marcio