Wide equations have too much vertical space above them

Question

It is the same question as How can I prevent wide equation having too large vertical space?, but the answers there don't address my case:

\documentclass[9pt]{beamer}
\usepackage[utf8]{inputenc}
\usefonttheme{serif}
\usepackage{tcolorbox}
\tcbuselibrary{magazine}
\tcbuselibrary{theorems}
\colorlet{greendark}{green!60!black}
\colorlet{bluelight}{blue!35!white}
\newtcbtheorem[number within=section]{mycomp}{Computation}%
{colframe=bluelight!60!greendark,enlarge top by=0.15cm,before
skip=3pt,after skip=6.5pt,fonttitle=\slshape,breakable}{comp}
\begin{document}
\begin{mycomp}{Mobius cancellation $G=\log \implies g = \frac 1t$}{mobiuscancellog}
   $$\frac{1}{1/x} = x\sum_{d\leq x} \mu(d) \frac {-\log(\frac dx)}d + O\left(\sum_{d\leq x} \mu(d) (\tfrac{1}{d/x}+1))\right).$$
  \end{mycomp}
\begin{mycomp}{Mobius cancellation $G=\log^2 \implies g = \frac {2\log t}{t}$}{mobiuscancellog2}
    $$\frac{2 \log(\frac 1x)}{1/x} = x\sum_{d\leq x} \mu(d) \frac {-\log^2(\frac dx)}d + O\Big(\sum_{d\leq x} \mu(d) \left[\tfrac{\log(\frac dx)}{d/x}+1\right]\Big)$$
   \end{mycomp}
\end{document}

Notice that the 2nd equation has more vertical space above the equation than below (compared distance from top of box to top of summation sign, and bottom of summation sign to bottom of box in both pictures).

I have determined that it must be solely based on width: by deleting enough of the 2nd equation, and it becomes shorter, it suddenly looks right again.

Can anyone suggest a fix?

Answer 1

Vertical space before an equation depends on its length and the length of the previous line. The corresponding skips are \abovedisplayskip and \abovedisplayshortskip.

In your case, setting

\setlength\abovedisplayshortskip{\abovedisplayskip}

gives the same spacing for the two equations.

The problems arises because you start a paragraph with a formula (first material in your boxes), and LaTeX doesn't support it. You can solve it with \noindent before your equations.

\documentclass[9pt]{beamer}
\usepackage[utf8]{inputenc}
\usefonttheme{serif}
\usepackage{tcolorbox}
\tcbuselibrary{magazine}
\tcbuselibrary{theorems}
\colorlet{greendark}{green!60!black}
\colorlet{bluelight}{blue!35!white}
\newtcbtheorem[number within=section]{mycomp}{Computation}%
{colframe=bluelight!60!greendark,enlarge top by=0.15cm,before
skip=3pt,after skip=6.5pt,fonttitle=\slshape,breakable}{comp}
\begin{document}
\begin{mycomp}{Mobius cancellation $G=\log \implies g = \frac 1t$}{mobiuscancellog}
  \noindent
  [\frac{1}{1/x} = x\sum_{d\leq x} \mu(d) \frac {-\log(\frac dx)}d + O\left(\sum_{d\leq x} \mu(d) (\tfrac{1}{d/x}+1))\right).]
  \end{mycomp}
\begin{mycomp}{Mobius cancellation $G=\log^2 \implies g = \frac {2\log t}{t}$}{mobiuscancellog2}
  \noindent
  [\frac{2 \log(\frac 1x)}{1/x} = x\sum_{d\leq x} \mu(d) \frac {-\log^2(\frac dx)}d + O\Big(\sum_{d\leq x} \mu(d) \left[\tfrac{\log(\frac dx)}{d/x}+1\right]\Big)]
   \end{mycomp}
\end{document}

I replaced $$ with \[...\] as suggested by Mico.

See also: https://tex.stackexchange.com/a/600078/316068

Answer 2

(La)TeX is not really at ease when a single equation out of context has to be displayed.

I suggest to define a specific environment for such cases, so you can force the “line before the display” (that doesn't really exist) to be short, by setting \predisplaysize=0pt.

\documentclass[9pt]{beamer}
\usefonttheme{serif}
\usepackage{tcolorbox}
\tcbuselibrary{magazine}
\tcbuselibrary{theorems}
\colorlet{greendark}{green!60!black}
\colorlet{bluelight}{blue!35!white}
\newtcbtheorem[number within=section]{mycomp}{Computation}{
  colframe=bluelight!60!greendark,
  enlarge top by=0.15cm,
  before skip=3pt,
  after skip=6.5pt,
  fonttitle=\slshape,
  breakable
}{comp}
\newenvironment{equation-}{[}{\predisplaysize=0pt]}
\begin{document}
\begin{mycomp}{Mobius cancellation $G=\log \implies g = \frac 1t$}{mobiuscancellog}
\begin{equation-}
\frac{1}{1/x} = x\sum_{d\leq x} \mu(d) \frac {-\log(\frac dx)}d

O\biggl(,\sum_{d\leq x} \mu(d) (\tfrac{1}{d/x}+1))\biggr).

\end{equation-}
\end{mycomp}
\begin{mycomp}{Mobius cancellation $G=\log^2 \implies g = \frac {2\log t}{t}$}{mobiuscancellog2}
\begin{equation-}
\frac{2 \log(\frac 1x)}{1/x} = x\sum_{d\leq x} \mu(d) \frac {-\log^2(\frac dx)}d

O\biggl(,\sum_{d\leq x} \mu(d) \Bigl[\tfrac{\log(\frac dx)}{d/x}+1\Bigr]\biggr)

\end{equation-}
\end{mycomp}
\end{document}

Note the small changes I made in order to avoid badly sized delimiters and clashes (they're not relevant as far as top and bottom spacing are concerned).

If you need to do an alignment as an out of context display, use aligned inside equation-.