Problem with curly brackets and \pgfmathsetmacro

Question

I want to create a loop over an false-position algorithm But I have two problems with \pgfmathsetmacro.

(1) The curly parentheses does not allow `\pgfmathsetmacro` to evaluate

In the code below, my \XM will not be defined if I insert the curly parenthesis there. This problem prevent me from using \foreach

(2) If the variable is redefined, it plots from the new position instead of the 
value where the plot was issued. 

This forces me to defined new variable which is not a good way.

What should I do ??

Here is the code

\documentclass[]{standalone}
\usepackage{pst-all}
\usepackage{graphicx}
\usepackage{tikz}
\usepackage{verbatim}
\usepackage{pgfplots}   
\usepackage{tikz-dimline}   
\usepackage{ifthen}
\usetikzlibrary{patterns}
\usetikzlibrary{arrows}
% Adapted from the 'patterns' library: enlarged the distance between the lines from 4pt to 10pt
\pgfdeclarepatternformonly{north east lines wide}{\pgfqpoint{-1pt}{-1pt}}{\pgfqpoint{10pt}{10pt}}{\pgfqpoint{9pt}{9pt}}%
{
\pgfsetlinewidth{0.4pt}
\pgfpathmoveto{\pgfqpoint{0pt}{0pt}}
\pgfpathlineto{\pgfqpoint{9.1pt}{9.1pt}}
\pgfusepath{stroke}
}

\tikzset{%
body/.style={inner sep=0pt,outer sep=0pt,shape=rectangle,draw,thick,pattern=north east lines wide},
dimen/.style={<->,>=latex,thin,every rectangle node/.style={fill=white,midway,font=\sffamily}},
symmetry/.style={dashed,thin},
}

\definecolor{gray3}{RGB}{50, 50, 50}
\definecolor{gray2}{RGB}{100, 100, 100}
\definecolor{gray1}{RGB}{150, 150, 150}

\definecolor{red3}{RGB}{255, 0, 0}
\definecolor{red2}{RGB}{220, 70, 91}
\definecolor{red1}{RGB}{255, 120, 120}

\begin{document}

    %the plotting function
\def\FunctionF(#1){(sin(3*#1*180.0/3.1415)-0.1)}%
\newcommand*{\FunctionFGnuplot}{ (sin(3*x)-0.1)}%
\newcommand\XL{0.1}
\newcommand\XR{2.0}
%----two beginning points leftmost and rightnost
\pgfmathsetmacro{\YL}{\FunctionF(\XL)}
\pgfmathsetmacro{\YR}{\FunctionF(\XR)}  

    \begin{tikzpicture}

\begin{axis}[
trig format plots=rad,
axis lines = middle,
enlargelimits,
, ymin=-1,ymax=1.0, restrict y to domain=-3:3,
xlabel=$x$,ylabel=$y$, 
every axis x label/.style={
at={(ticklabel* cs:1.025)},
anchor=north east,
}
,every axis y label/.style={
at={(ticklabel* cs:1.025)},
anchor=east,
}
]

\addplot[color=blue,domain=\XL:\XR,line width = 1.0,samples=64]{ \FunctionFGnuplot};
%---two beginning points
\addplot[mark=*,color=blue] coordinates {(\XL,\YL)};
\addplot[mark=*,color=blue] coordinates {(\XR,\YR)};


%---the straight line joining two points. 
%   The x-intersection will be the new xm   


% Line1
\draw [line width = 0.25] (axis cs:\XL,\YL)--(axis cs:\XR,\YR);


%%------------------- Problem with  {}   here -----------------
%% Problem 1:
%% If I open the curly bracket below here \XM will not be defined no matter
%% where I close it


\pgfmathsetmacro{\XM}{  \XL - \YL* (\XR-\XL ) / (\YR-\YL)  }    ;
    \pgfmathsetmacro{\YM}{\FunctionF(\XM)}  ;
      %\addplot[mark=*,color=gray1] coordinates {(\XM,0)};
      \node[gray,below] at (axis cs:\XM,0){$x_m^1$};
    %----------------------------------------------------------------


    \addplot[mark=*,color=red1] coordinates {(\XM,\YM   )}; 

\draw [->,line width = 0.25] (axis cs:\XM,0) to   (axis cs:\XM,\YM/2);  
\draw [-, line width = 0.25] (axis cs:\XM,0) to   (axis cs:\XM,\YM);    

%--------------------------------------------------------------------------
%Problem 2: Line1 is not drawn immeadiately when it was defined above.
%           But it wait until \XL was redefined here.
%           Instead of having a line joining two blue dots,
%           I got something else. 
\pgfmathsetmacro{\XLL}{ ifthenelse( \YL*\YM <0, \XL, \XM)};
\pgfmathsetmacro{\XRR}{ ifthenelse( \YL*\YM <0, \XM, \XR)}; 
\pgfmathsetmacro{\YLL}{ ifthenelse( \YL*\YM <0, \YL, \YM)};
\pgfmathsetmacro{\YRR}{ ifthenelse( \YL*\YM <0, \YM, \YR)}; 


    %----------------second loop
\draw [line width = 0.25] (axis cs:\XLL,\YLL)--(axis cs:\XRR,\YRR);
\pgfmathsetmacro{\XMM}{  \XLL - \YLL* (\XRR-\XLL ) / (\YRR-\YLL)  } ;
    \pgfmathsetmacro{\YMM}{\FunctionF(\XMM)}    ;
    %\addplot[mark=*,color=gray2] coordinates {(\XMM,0)};
          \node[gray,above] at (axis cs:\XMM,0){$x_m^2$};

    \addplot[mark=*,color=red2] coordinates {(\XMM,\YMM )}; 
\draw [->,line width = 0.25] (axis cs:\XMM,0) to   (axis cs:\XMM,\YMM/2);   
\draw [-, line width = 0.25] (axis cs:\XMM,0) to   (axis cs:\XMM,\YMM); 

\pgfmathsetmacro{\XLLL}{ ifthenelse( \YLL*\YMM <0, \XLL, \XMM)};
\pgfmathsetmacro{\XRRR}{ ifthenelse( \YLL*\YMM <0, \XMM, \XRR)};    
\pgfmathsetmacro{\YLLL}{ ifthenelse( \YLL*\YMM <0, \YLL, \YMM)};
\pgfmathsetmacro{\YRRR}{ ifthenelse( \YLL*\YMM <0, \YMM, \YRR)};    

%----------------third loop 

\draw [line width = 0.25] (axis cs:\XLLL,\YLLL)--(axis cs:\XRRR,\YRRR);
\pgfmathsetmacro{\XMMM}{  \XLLL - \YLLL* (\XRRR-\XLLL ) / (\YRRR-\YLLL)  }  ;
    \pgfmathsetmacro{\YMMM}{\FunctionF(\XMMM)}  ;

    %\addplot[mark=*,color=gray3] coordinates {(\XMMM,0)};  
              \node[gray, above right, node distance = 1cm] at (axis cs:\XMMM,0){$x_m^3$};
    \addplot[mark=*,color=red3] coordinates {(\XMMM,\YMMM   )}; 

\draw [->,line width = 0.25] (axis cs:\XMMM,0) to   (axis cs:\XMMM,\YMMM/2);    
\draw [-, line width = 0.25] (axis cs:\XMMM,0) to   (axis cs:\XMMM,\YMMM);      

\draw [line width = 0.25] (axis cs:\XLLL,\YLLL)--(axis cs:\XRRR,\YRRR); 


\pgfmathsetmacro{\XLLLL}{ ifthenelse( \YLLL*\YMMM <0, \XLLL, \XMMM)};
\pgfmathsetmacro{\XRRRR}{ ifthenelse( \YLLL*\YMMM <0, \XMMM, \XRRR)};   
\pgfmathsetmacro{\YLLLL}{ ifthenelse( \YLLL*\YMMM <0, \YLLL, \YMMM)};
\pgfmathsetmacro{\YRRRR}{ ifthenelse( \YLLL*\YMMM <0, \YMMM, \YRRR)};   
%----------------fourth loop    
\pgfmathsetmacro{\XMMMM}{  \XLLLL - \YLLLL* (\XRRRR-\XLLLL ) / (\YRRRR-\YLLLL)  }   ;


     \addplot[mark=*,color=black] coordinates {(\XMMMM,0)};
\end{axis}

    \end{tikzpicture}
\end{document}