Btw, this is what I came with:

\documentclass[tikz,border=2mm]{article}
\usepackage{tikz}
\usetikzlibrary{intersections}
\usetikzlibrary{calc}
\usepackage{xfp}

% Define the new command for sign-related angle adjustment
\newcommand{\adjustangle}[1]{%
  \fpeval{(#1) < 0 ? 180 : 0}%
}

\begin{document}
\begin{tikzpicture}

% Define the line as a path to be reused
\coordinate (A) at (-4,0); 
\coordinate (B) at (4,1); 
\path[name path=line] (A) -- (B); 

% Place the point
\coordinate (P) at (1,1.5); 

% Circle with radius greater than distance to the line, choose arbitrary for this example
\def\radius{2cm}

% Define the circle path
\path[name path=circle] (P) circle (\radius);

% Calculate intersections
\path[name intersections={of=circle and line,by={C,D}}];

% Draw the straight line
\draw (A) -- (B); 

% Mark the point
\fill (P) circle (2pt) node[above] {$P$};
\fill (C) circle (2pt) node[above] {$C$};
\fill (D) circle (2pt) node[above] {$D$};

% Draw the part of the circle on the other half plane
\draw[very thick, blue] let \p1=(C), \p2=(D), \p3=(P) in (C)
arc[start angle={atan((\y1-\y3)/(\x1-\x3))-\adjustangle{\x1-\x3}},
end angle={atan((\y2-\y3)/(\x2-\x3))-\adjustangle{\x2-\x3}},
radius=\radius];

\end{tikzpicture}
\end{document}

I'm trying to have something like that working:

\documentclass{article}
\usepackage{tikz}

\newcommand{\adjustangle}[1]{%
  \pgfmathparse{ifthenelse(#1 < 0, 180, 0)}
  \pgfmathresult
}

\begin{document}

\begin{tikzpicture}
  \draw
  (0,1) arc[start angle={45-\adjustangle{45}},
    end angle={60},
    radius=1];

\end{tikzpicture}

\end{document}

But whichever way I put it I always have something like that:

\pgfmath@catcodes ...`\|=12 \catcode `\&=12 \let "
                                                  \pgfmath@char@quote

You can declare new pgfmath functions through /pgf/declare function key:

\tikzset{
  declare function={adjustangle(\a)=ifthenelse(\a < 0, 180, 0);}
}

You can declare new pgfmath functions through /pgf/declare function key:

\tikzset{
  declare function={adjustangle(\a)=ifthenelse(\a < 0, 180, 0);}
}

It's slower than \pgfmathdeclarefunction, but easier to use.

Full example

\documentclass{article}
\usepackage{tikz}

\tikzset{
  declare function={adjustangle(\a)=ifthenelse(\a < 0, 180, 0);}
}

\begin{document}

\begin{tikzpicture}
  \draw
  (0,1) arc[start angle={45-adjustangle{45}},
    end angle={60},
    radius=1];

\end{tikzpicture}

\end{document}

In reply to @inkbottle:matrix.org
I want to cut the plane in two parts through a straight line, then put a point in one of the half-plane, then consider a circle centered on this point with radius greater than the distance of the point to the straight line, and actually draw only the arc/part of the circle, that is on the half plane where the point (center of the circle) is not.
What are the best options to do that?

In reply to @muzzi:matrix.org
What's "inlining"?

In reply to @sourcecodedeleted:matrix.org
Is there any javascript interfaces with this?