For
$\left|x\right|>{E}_{1}$, the routine fails owing to danger of setting overflow in calculating
${e}^{x}$. The result returned for such calls is
$\mathrm{cosh}{E}_{1}$, i.e., it returns the result for the nearest valid argument. The value of machine-dependent constant
${E}_{1}$ may be given in the
Users' Note for your implementation.
If on entry
${\mathbf{ifail}}=0$ or
$-1$, explanatory error messages are output on the current error message unit (as defined by
x04aaf).
If
$\delta $ and
$\epsilon $ are the relative errors in the argument and result, respectively, then in principle
That is, the relative error in the argument,
$x$, is amplified by a factor, at least
$x\mathrm{tanh}x$. The equality should hold if
$\delta $ is greater than the
machine precision (
$\delta $ is due to data errors etc.) but if
$\delta $ is simply a result of round-off in the machine representation of
$x$ then it is possible that an extra figure may be lost in internal calculation round-off.
The behaviour of the error amplification factor is shown by the following graph:
Figure 1
It should be noted that near
$x=0$ where this amplification factor tends to zero the accuracy will be limited eventually by the
machine precision. Also for
$\left|x\right|\ge 2$
where
$\Delta $ is the absolute error in the argument
$x$.
None.