Here, I preserve the fraction in its native \textstyle
, but add a (default) 1pt buffer above and below the numerator and denominator of each fraction, which can be changed with an optional argument. I call it \qfrac[]{}{}
. The MWE shows before and after.
\documentclass{article}
\usepackage{stackengine,scalerel}
\stackMath
\newcommand\qfrac[3][1pt]{\frac{%
\ThisStyle{\addstackgap[#1]{\SavedStyle#2}}}{%
\ThisStyle{\addstackgap[#1]{\SavedStyle#3}}%
}}
\usepackage{amsmath}
\begin{document}
Combining two Gaussians with mean $\mu_1, \mu_2$ and variance $\sigma_1^2,
\sigma_2^2$ yields a new Gaussian with mean $\mu = \frac{\sigma_2^2 \mu_1 +
\sigma_1^2 \mu_2}{\sigma_1^2 + \sigma_2^2}$ and variance $\sigma^2 =
\frac{1}{\frac{1}{\sigma_1^2} + \frac{1}{\sigma_2^2}}$
Combining two Gaussians with mean $\mu_1, \mu_2$ and variance $\sigma_1^2,
\sigma_2^2$ yields a new Gaussian with mean $\mu = \qfrac{\sigma_2^2 \mu_1 +
\sigma_1^2 \mu_2}{\sigma_1^2 + \sigma_2^2}$ and variance $\sigma^2 =
\qfrac[.5pt]{1}{\qfrac{1}{\sigma_1^2} + \qfrac{1}{\sigma_2^2}}$
\end{document}

\dfrac
instead of\frac
.\vphantom
command (for examle\vphantom{A^{A^A}}
) in the beginning of each problematic dominator, and keep the style as inline math.