# Uncertainty propagation in open equations

I don't know if this is the proper place for asking this kind of questions and I apologise in advance if it isn't, but anyways: is there a way to propagate linear uncertainties (i.e. through first-degree partial derivatives) for functions in an open form, that is, that cannot be reduced? I am particularly interested in finding the uncertainty of $y$, given the equation:

$y = A\cdot sinh(y)\cdot\sqrt{BC}$

where $A$, $B$, $C$ all have a given uncertainty. Does anybody have any insight? Thank you.

• Could you elaborate on what "open form" and "cannot be reduced" might mean? Would you perhaps mean an implicitly defined function, such as propagating the uncertainty from the vector $(A,B,C)$ to $y$ given that $F(y,A,B,C)=0$ for a known function $F$? – whuber May 20 '15 at 19:33
• Did you really intend for $y$ to be on both sides of that equation? Is there any time dependence in there, or are those both the same $y$? – Glen_b -Reinstate Monica May 21 '15 at 7:04

$f(y) = y/\sinh y = g(A,B,C) = A \sqrt{BC}$
\begin{align} \sigma_f = \sigma_y \left| \frac{df}{dy} \right|_{y = \overline{y}} = \sigma_y \mathop{\rm csch} \overline{y} \: (1 - \overline{y} \mathop{\rm \coth \overline{y}}) = \sigma_g \end{align}
where $\overline{y}$ is the expected value of $y$. You can work out $\sigma_g$ as a function of $\sigma_{A,B,C}$ by the usual error propagation.
Of course, this technique won't work if $\sigma_y$ is so large that $\frac{df}{dy}$ becomes nonlinear within $\overline{y} \pm \sigma_y$.