SD of a function of signals Say $a_x$, $a_y$, $b_x$, $b_y$ are signals with values in $\mathbb{R}$. Say they are independednt, normally distributed variables with given SDs $\sigma_{a_x}$, $\sigma_{a_y}$, $\sigma_{b_x}$, $\sigma_{b_y}$.

Say $f: \mathbb{R}^4\longrightarrow\mathbb{R}$ is a continuous function of these signals (i.e. $f(a_x, a_y, b_x, b_y)\in \mathbb{R}$ for all $(a_x, a_y, b_x, b_y)\in\mathbb{R}^4$.


General question

What is the SD of $f(a_x, a_y, b_x, b_y)$?
Example/Question


Say $a$ and $b$ are vectors (say in $\mathbb{R}^2$) and $\sphericalangle$ is the angle between $a$ and $b$, say defined as
$$\sphericalangle(a,b) := \mathrm{arcos}\left(\frac{ab}{\vert a\rvert \lvert b\rvert}\right)\quad\text{where}\quad a=(a_x, a_y)\quad\text{and}\quad b=(b_x, b_y).$$ 

Say $a_x, a_y, b_x, b_y$ are independent and normally distributed, have SDs $\sigma_{a_x}$, $\sigma_{a_y}$, $\sigma_{b_x}$, $\sigma_{b_y}$ respectively.

Then Gauss error proparation yields 
$$\sigma_\sphericalangle:= \sqrt{ { \left(\frac{\delta \sphericalangle}{\delta a_x} (a,b) |\mu \ \cdot \sigma_{a_x}\right) }^2 + { \left(\frac{\delta \sphericalangle}{\delta a_y} (a,b) |\mu \ \cdot \sigma_{a_y}\right) }^2 + { \left(\frac{\delta \sphericalangle}{\delta b_x} (a,b) |\mu \ \cdot \sigma_{b_x}\right) }^2 + { \left(\frac{\delta \sphericalangle}{\delta b_y} (a,b) |\mu \ \cdot \sigma_{b_y}\right) }^2}$$

as the error of $\sphericalangle$ (where $..|\mu$ shall note evaluation at average $\mu$).


Question


Is that $\sigma_\sphericalangle$ the SD of $\sphericalangle$?
It is an estimation of the angle SD.