Janky pgfplot behaviour with arcsinh function

Question

I'm trying to draw a titration curve using pgfplot. Based off this paper, there is an equation for the titration curve:

I've attempted to implement this within pgfplots, and I get a very strange result:

I have no idea why this would occur; even attempting to recreate the exact formula I used within desmos did not work.

For reference, here is the code, and also a link to an overleaf project.

\documentclass{article}
\usepackage{pgfplots}

\begin{document}

\pgfkeys{
    /pgf/declare function={
        arcsinh(\x) = ln(\x + sqrt(\x^2+1));
    },
    /pgf/declare function={
        Va = 0.025;
        Ma = 0.1;
        Mb = 0.1;
        V(\x) = \x / 1000;
        Kw = 1*10^(-14);
        p(\o) = -ln(\o)/ln(10);
    }
}

\begin{center}
\begin{tikzpicture}
    \begin{axis}[
        xlabel = {Solution Added (mL)},
        ylabel = {pH},
        ymin=0,
        ymax=14,
        ytick distance=7,
        xtick distance=10,
    ]
    \addplot[%
        samples=100,
        color=red,
        domain=0:50,
    ]{%
    7 + 1/ln(10) * arcsinh( 1/(2*sqrt(Kw))  *  (Mb*V(x) - Ma*Va) / (Va + V(x)) )
    };
    \end{axis}
\end{tikzpicture}
\end{center}

\end{document}

endnote: if there's a way I can have variables e.g. V_a within the pgfplot function, that'd be much nicer than having constants throughout. thanks Torjorn

Answer 1

One gets this strange result because TeXs limits are reached -- as can also be seen by the dropped coordinates -- and thus the "zigzag" results from precision limits (red line). If you use either gnuplot (green dots) or Lua (blue line) as calculation engine, it works as expected. Of course for the Lua solution you have to use LuaLaTeX as TeX engine.

---

Sidenote:
If you want to avoid to use such a high number of samples, consider reformulating the eqution to make use of non-linear spacing. For that see e.g.

• https://tex.stackexchange.com/a/443731/95441
• https://tex.stackexchange.com/a/446727/95441

% used PGFPlots v1.17
\documentclass[border={5pt}]{standalone}
\usepackage{pgfplots}
    % use this `compat` level or higher to use LUA backend for calculation (if possible)
    \pgfplotsset{compat=1.12}
\begin{document}
    % for gnuplot solution
    \newcommand*{\Kw}{1e-14}
    \newcommand*{\Ma}{0.1}
    \newcommand*{\Mb}{0.1}
    \newcommand*{\Va}{0.025}
\begin{tikzpicture}[
    % from https://tex.stackexchange.com/q/144778
    /pgf/declare function={
        % for LUA solution
        arcsinh(\x) = ln(\x + sqrt(\x^2+1));
        Kw = 1e-14;
        Ma = 0.1;
        Mb = 0.1;
        Va = 0.025;
        V(\x) = \x / 1000;
        p(\o) = -0.5*ln(\o)/ln(10);
    },
]
    \begin{axis}[
        xlabel={Solution Added (mL)},
        ylabel={pH},
        domain=0:50,
        samples=201,
    ]
        % gnuplot
        \addplot+ [ultra thick,green,mark size=1pt,only marks,opacity=0.5] gnuplot
            {-0.5*log10(\Kw) + asinh(1/(2*sqrt(\Kw)) * (\Ma*\Va - \Mb*x/1000)/(\Va + x/1000) )/log(10)};
        % Lua(TeX)
        \addplot [thick,blue]
            {p(Kw) + arcsinh(1/(2*sqrt(Kw)) * (Ma*Va - Mb*V(x))/(Va + V(x)) )/ln(10)};
        % TeX
        \addplot [red,opacity=0.75]
            {p(\Kw) + arcsinh(1/(2*sqrt(Kw)) * (Ma*Va - Mb*V(x))/(Va + V(x)) )/ln(10)};
    \end{axis}
\end{tikzpicture}
\end{document}