From 8164c2be3b4849a9c77e44b9659a3de8af010bd7 Mon Sep 17 00:00:00 2001
From: Norbert Preining <norbert@preining.info>
Date: Mon, 13 Dec 2021 03:02:58 +0000
Subject: CTAN sync 202112130302

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+%% LyX 2.4.0-alpha3 created this file.  For more info, see https://www.lyx.org/.
+%% Do not edit unless you really know what you are doing.
+\documentclass[english,tableposition=top]{report}
+\usepackage{lmodern}
+\renewcommand{\sfdefault}{lmss}
+\renewcommand{\ttdefault}{lmtt}
+\usepackage[T1]{fontenc}
+\usepackage{textcomp}
+\usepackage[utf8]{inputenc}
+\setcounter{secnumdepth}{3}
+\setcounter{tocdepth}{3}
+\usepackage{color}
+\definecolor{shadecolor}{rgb}{0.667969, 1, 1}
+\usepackage{babel}
+\usepackage{array}
+\usepackage{cprotect}
+\usepackage{wrapfig}
+\usepackage{booktabs}
+\usepackage{framed}
+\usepackage{url}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage[unicode=true,pdfusetitle,
+ bookmarks=true,bookmarksnumbered=true,bookmarksopen=true,bookmarksopenlevel=2,
+ breaklinks=true,pdfborder={0 0 1},backref=section,colorlinks=true]
+ {hyperref}
+
+\makeatletter
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LyX specific LaTeX commands.
+\providecommand{\LyX}{\texorpdfstring{\ensureascii{%
+  L\kern-.1667em\lower.25em\hbox{Y}\kern-.125emX\@}}{LyX}}
+\DeclareRobustCommand*{\lyxarrow}{%
+\@ifstar
+{\leavevmode\,$\triangleleft$\,\allowbreak}
+{\leavevmode\,$\triangleright$\,\allowbreak}}
+%% Because html converters don't know tabularnewline
+\providecommand{\tabularnewline}{\\}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Textclass specific LaTeX commands.
+\newenvironment{centred}%
+  {\begin{center}\baselineskip=13pt\parskip=1pt}{\end{center}}
+\newenvironment{lyxcode}
+	{\par\begin{list}{}{
+		\setlength{\rightmargin}{\leftmargin}
+		\setlength{\listparindent}{0pt}% needed for AMS classes
+		\raggedright
+		\setlength{\itemsep}{0pt}
+		\setlength{\parsep}{0pt}
+		\normalfont\ttfamily}%
+	 \item[]}
+	{\end{list}}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% User specified LaTeX commands.
+\usepackage{numerica}
+
+\usepackage{upquote}
+\newcommand\rel{\,\varrho\;}
+\DeclareMathOperator{\erf}{erf}
+\DeclareMathOperator{\gd}{gd}
+
+\reuse{}
+
+\makeatother
+
+\begin{document}
+\title{\texttt{numerica}}
+\author{Andrew Parsloe\\
+(\url{ajparsloe@gmail.com})\\
+}
+\maketitle
+\begin{abstract}
+The \verb`numerica` package defines a command to numerically evaluate
+mathematical expressions in the LaTeX form in which they are typeset.
+For programs like LyX with a preview facility, or compile-as-you-go
+systems, interactive back-of-envelope calculations and numerical exploration
+are possible within the document being worked on. The package requires
+the bundles \verb`l3kernel` and \verb`l3packages`, and the \verb`amsmath`
+and \verb`mathtools` packages. \\
+\\
+\noindent\begin{minipage}[t]{1\columnwidth}%
+\begin{shaded}%
+
+\paragraph*{Note:}
+\begin{itemize}
+\item {\normalsize This document applies to version 2.0.0 of }{\normalsize\texttt{numerica.sty}}{\normalsize .}{\small\par}
+\item {\normalsize Reasonably recent versions of the \LaTeX 3 bundles }{\normalsize\texttt{l3kernel}}{\normalsize{}
+and }{\normalsize\texttt{l3packages}}{\normalsize{} are required (although
+much of }{\normalsize\verb`l3kernel`}{\normalsize{} has
+been incorporated into \LaTeXe{} since February 2020).}{\small\par}
+\item {\normalsize The package requires }{\normalsize\texttt{amsmath}}{\normalsize{}
+and }{\normalsize\texttt{mathtools}}{\normalsize .}{\small\par}
+\item {\normalsize I refer many times in this document (especially §\ref{sec:Argument-parsing})
+to }{\normalsize\emph{Handbook of Mathematical Functions}}{\normalsize ,
+edited by Milton Abramowitz and Irene A. Segun, Dover, 1965. This
+is abbreviated to }{\normalsize\emph{HMF}}{\normalsize , often followed
+by a number like 1.2.3 to locate the actual expression referenced.}{\small\par}
+\item {\normalsize Version 2.0.0 of }{\normalsize\texttt{numerica}}{\small\par}
+\begin{itemize}
+\item {\normalsize splits into distinct packages the additional functionality
+previously available with the }{\normalsize\texttt{plus}}{\normalsize{}
+and }{\normalsize\texttt{tables}}{\normalsize{} package options of version
+1;}{\small\par}
+\item {\normalsize allows for user-defined macros and constants (with the
+}{\normalsize\texttt{\textbackslash nmcMacros }}{\normalsize and }{\normalsize\texttt{\textbackslash nmcConstants}}{\normalsize{}
+commands) to be used in expressions to be evaluated;}{\small\par}
+\item {\normalsize rewrites the code and changes the behaviour of }{\normalsize\texttt{\textbackslash nmcReuse}}{\normalsize{}
+to maintain uniformity across all commands (}{\normalsize\texttt{\textbackslash nmcEvaluate}}{\normalsize ,
+}{\normalsize\texttt{\textbackslash nmcInfo}}{\normalsize , }{\normalsize\texttt{\textbackslash nmcMacros}}{\normalsize ,
+}{\normalsize\texttt{\textbackslash nmcConstants}}{\normalsize , }{\normalsize\texttt{\textbackslash nmcReuse}}{\normalsize );
+this command is no longer compatible with its use in v.1;}{\small\par}
+\item {\normalsize changes the behaviour of }{\normalsize\texttt{\textbackslash text}}{\normalsize{}
+and }{\normalsize\texttt{\textbackslash mbox}}{\normalsize{} in the
+}{\normalsize\texttt{\textbackslash eval}}{\normalsize{} command; adds
+}{\normalsize\texttt{\textbackslash textrm}}{\normalsize , }{\normalsize\texttt{\textbackslash textsf}}{\normalsize ,
+and }{\normalsize\texttt{\textbackslash texttt}}{\normalsize{} to compensate;}{\small\par}
+\item {\normalsize includes many adjustments to the code, including around
+nesting of commands;}{\small\par}
+\item {\normalsize adds to and amends documentation.}{\small\par}
+\end{itemize}
+\end{itemize}
+\end{shaded}%
+\end{minipage}
+\end{abstract}
+\begin{center}
+\tableofcontents{}
+\par\end{center}
+
+\chapter{Introduction}
+
+\texttt{numerica} is a \LaTeX{} package offering the ability to numerically
+evaluate mathematical expressions in the \LaTeX{} form in which they
+are typeset.\footnote{\texttt{numerica} evolved from the author's \texttt{calculyx} package
+that was designed for use with the document processor \LyX{} (and available
+for download from a link on the \LyX{} wiki website but not CTAN). }
+
+There are a number of packages which can do calculations in \LaTeX ,\footnote{A simple search finds the venerable \texttt{calc} in the \LaTeX{} base,
+\texttt{calculator }(including an associated \texttt{calculus} package),
+\texttt{fltpoint}, \texttt{fp} (\emph{fixed} rather than floating
+point), \texttt{spreadtab} (using either \texttt{fp} or \texttt{l3fp}
+as its calculational engine) if you want simple spreadsheeting with
+your calculations, the elaborate \texttt{xint}, \texttt{pst-calculate}
+(a limited interface to \texttt{l3fp}), \texttt{l3fp} in the \LaTeX 3
+kernel, and \texttt{xfp}, the \LaTeX 3 interface to \texttt{l3fp}.
+Other packages include a calculational element but are restricted
+in their scope. (\texttt{longdivision} for instance is elegant, but
+limited only to long division.) } but those I am aware of all require the mathematical expressions
+they operate on to be changed to an appropriate syntax. Of these packages
+\texttt{xfp} comes closest to my objective with \texttt{numerica}.
+For instance, given a formula
+\begin{centred}
+\verb`\frac{\sin (3.5)}{2} + 2\cdot 10^{-3}`
+\end{centred}
+(in a math environment), this can be evaluated using \texttt{xfp}
+by transforming the expression to \verb`sin(3.5)/2 + 2e-3` and wrapping
+this in the command \verb`\fpeval`. In \texttt{numerica} you don't
+need to transform the formula, just wrap it in an \verb`\eval` command:
+\begin{centred}
+\verb`\eval{ \frac{\sin (3.5)}{2} + 2\cdot 10^{-3} }`.
+\end{centred}
+(for the acutal calculation see §\ref{subsec:introSimple-examples}).\texttt{ }
+
+\texttt{numerica}, like \texttt{xfp} and a number of other packages,
+uses \texttt{l3fp} (the \LaTeX 3 floating point module in \texttt{l3kernel})
+as its calculational engine. To some extent the main command, \verb`\nmcEvaluate`,
+short-name form \verb`\eval`, is a pre-processor to \texttt{l3fp},
+converting mathematical expressions written in the \LaTeX\texttt{
+}form in which they will be typeset into an `fp-ified' form that
+is digestible by \texttt{l3fp}. The aim is to make the command act
+as a wrapper around such formulas. Ideally, one should not have to
+make \emph{any} adjustment to them, although any text on Fourier series
+suggests that hope in full generality is delusional. Surprisingly
+often however it \emph{is} possible. We shall see shortly that even
+complicated formulas like
+
+\[
+\cos\tfrac{m}{n}\pi-(1-4\sin^{2}\tfrac{m}{3n}\pi)\frac{\sin\tfrac{1}{n}\pi\sin\tfrac{m-1}{n}\pi}{2\sin^{2}\tfrac{m}{3n}\pi},
+\]
+and
+\[
+\left(\frac{1-4\sin^{2}\tfrac{m}{3n}\pi}{2\sin^{2}\tfrac{m}{3n}\pi}\right)\sin\tfrac{2m-3}{3n}\pi\sin\tfrac{m-3}{3n}\pi,
+\]
+
+\noindent can be evaluated `as is' (see below, §\ref{subsec:introReassurance}).
+There is no need to shift the position of the superscript $2$ on
+the sines, no need to parenthesize the arguments of $\sin$ and $\cos$,
+no need to insert asterisks to indicate multiplication, no need to
+change the \verb`\frac` and \verb`\tfrac`-s to slashes, \texttt{/},
+no need to delete the \verb`\left` and \verb`\right` that qualify
+the big parentheses (in the underlying \LaTeX ) in the second expression.
+Of course, if there are variables in an expression, as in these examples,
+they will need to be assigned values. And how the result of the evaluation
+is presented also requires specifying, but the aim is always: to evaluate
+mathematical expressions in \LaTeX{} with as little adjustment as possible
+to the form in which they are typeset.
+
+\texttt{numerica} is written in \texttt{expl3}, the programming language
+of the \LaTeX 3 project. It uses the \LaTeX 3 module \texttt{l3fp}
+(part of \texttt{l3kernel}) as its calculational engine. This enables
+floating point operations to 16 significant figures, with exponents
+ranging between $-10000$ and $+10000$. Many functions and operations
+are built into \texttt{l3fp} – arithmetic operations, trigonometric,
+exponential and logarithm functions, factorials, absolute value, max
+and min. Others have been constructed for \texttt{numerica }from \texttt{l3fp}
+ingredients – binomial coefficients, hyperbolic functions, sums and
+products – but to the user there should be no discernible difference. 
+
+Associated packages provide for additional operations: iteration of
+functions, finding zeros of functions, recurrence relations, mathematical
+table building; others are planned (e.g. calculus).
+
+\section{How to use \texttt{numerica}}
+
+The package is invoked in the usual way: put
+\begin{lyxcode}
+\textbackslash usepackage\{numerica\}
+\end{lyxcode}
+\noindent in the \LaTeX{} preamble. \texttt{numerica} requires the
+\texttt{amsmath} and \texttt{mathtools} packages and loads these automatically.
+\texttt{numerica} will also accept use of some relational symbols
+from the \texttt{amssymb} package provided that package is loaded
+by the user; see §\ref{subsec:evalBoolean-output}. 
+
+\subsection{Package options}
+
+\label{subsec:introPackagesOptions}Currently there are none. With
+version 2.0.0 of \verb`numerica` a change has been made to how additional
+functionality for the package is invoked; see §\ref{subsec:Related-packages}
+below. This means that the options available in version 1 have been
+discontinued.
+
+\subsection{Associated packages}
+
+\label{subsec:Related-packages}In version 1 of \verb`numerica` some
+additional functionality for the package could be gained by specifying
+package options – for instance the ability to create tables of function
+values or to iterate or find fixed points of functions. However this
+manner of invoking the addtional functionality makes the maintaining
+of semantic version numbering across the whole \verb`numerica` package
+difficult. With version 2.0.0 of the package, the addtional functionality
+has been separated into separate \LaTeX{} packages. Currently there
+are two of these, \verb`numerica-plus` and \verb`numerica-tables`.
+They are loaded with the familiar \verb`\usepackage` command in the
+document preamble and require the availability of the \verb`numerica`
+package in your \TeX{} distribution. Neither package requires a \verb`\usepackage{numerica}`
+statement; they take care of that themselves. So, if you enter
+\begin{lyxcode}
+\textbackslash usepackage\{numerica-plus\}
+\end{lyxcode}
+in the preamble of your document you gain access not only to the commands
+in the \verb`numerica` package but also to the commands \verb`\nmcIterate`,
+\verb`\nmcSolve`, and \verb`\nmcRecur`. \verb`\nmcIterate` enables
+the iteration of functions of a single variable, including finding
+fixed points. \verb`\nmcSolve` enables the solving of equations of
+the form $f(x)=0$ (i.e. finding the zeros of $f$), or the finding
+of local maxima or minima of a function of one variable. \verb`\nmcRecur`
+enables the calculation of terms in recurrence relations, like the
+terms of the Fibonacci series, or othogonal polynomials defined recurrently.
+See the associated document \verb`numerica-plus.pdf` for details.
+
+If you enter
+\begin{lyxcode}
+\textbackslash usepackage\{numerica-tables\}
+\end{lyxcode}
+in the preamble of your document you gain access not only to the commands
+in the \verb`numerica` package but also to the command \verb`\nmcTabulate`
+which enables the creation of (possibly multi-column) tables of function
+values and makes available most of the table formats evident in \emph{HMF}.
+See the associated document \verb`numerica-tables.pdf` for details. 
+
+A package \verb`numerica-calculus` is currently being developed.
+
+\subsection{Simple examples of use}
+
+\label{subsec:introSimple-examples}A simple example of use is provided
+by the document
+\begin{verbatim}
+    \documentclass{article} 
+    \usepackage{numerica}
+    \begin{document}
+
+        \eval{$ mc^2 $}[m=70,c=299 792 458][8x]
+
+    \end{document}
+
+\end{verbatim}
+\noindent We have a formula between math delimiters: \verb`$ mc^2 $`.
+We have wrapped a command \verb`\eval` around the lot, added an optional
+argument in parentheses specifying numericaal values for the quantities
+\texttt{m} and \texttt{c}, and concluded it all with a trailing optional
+argument specifying that the result should be presented to $8$ places
+of decimals and in scientific notation (the \verb`x`). Running \texttt{pdflatex}
+on this document generates a pdf displaying
+\begin{centred}
+\eval{$ mc^2 $}[m=70,c=299 792 458][8x]
+\end{centred}
+\noindent where the formula ($mc^{2})$ is equated to the numerical
+value resulting from substituting the given values of $m$ and $c$.
+Those values are displayed in a list following the result. The calculation
+is presented to $8$ decimal places in scientific notation. (According
+to Einstein's famous equation $E=mc^{2}$ this is the enormous energy
+content, in joules, of what was once considered an average adult Caucasian
+male. Only a minute fraction is ever available.) 
+
+A second example is provided by the formula in earlier remarks:
+\begin{verbatim}
+    \documentclass{article} 
+    \usepackage{numerica}
+    \begin{document}
+
+        \eval{\[ \frac{\sin(3.5)}{2} + 2\cdot 10^{-3} \]}
+
+    \end{document}
+
+\end{verbatim}
+Running \texttt{pdflatex} on this document produces the result
+
+\eval{\[ \frac{\sin(3.5)}{2} + 2\cdot 10^{-3} \]}
+
+The \verb`\eval` command used in these examples is the main command
+of the \texttt{numerica} package. It is discussed in full in the next
+chapter, but first some preliminaries.
+
+\subsection{Display of the result}
+
+\label{subsec:introResultDisplay}In what follows I shall write things
+like (but generally more complicated than)
+\begin{centred}
+\verb`$ \eval{ 1+1 } $` $\Longrightarrow \eval{ 1 + 1 } $
+\end{centred}
+to mean: run \texttt{pdflatex} on a document containing \verb`\eval{1+1}`
+in the document body to generate a pdf containing the calculated result
+($2$ in this instance). The reader will note that I have used dollar
+signs to delimit the math environment. I could (and perhaps should)
+have used the more \LaTeX -pure \verb`\( \)`, which will do equally
+well, but habit has won out. In the example the \verb`\eval` command
+is used \emph{within} a math environment (delimited by the dollar
+signs). It is not limited to this behaviour. The command can also
+wrap \emph{around} the math delimiters (as we saw in the previous
+examples):
+\begin{centred}
+\verb`\eval{$ 1+1 $}` $\Longrightarrow$ \eval{$ 1+1 $}.
+\end{centred}
+As you can see, the display that results is different. 
+\begin{itemize}
+\item When the \verb`\eval` command is used\emph{ within} a math environment,
+only the \emph{result,} followed possibly by the \emph{variable~=~value
+list} (see §\ref{subsec:evalVv-list}) is displayed.
+\end{itemize}
+Environments may include the various AMS environments as well as the
+standard \LaTeX{} inline ( \verb`$ $` or \verb`\( \)` ), \texttt{equation}
+( \verb`\[ \]` ) and \texttt{eqnarray} environments. For an example
+of \verb`\eval` within an \verb`align*` environment see §\ref{subsec:introExploring}
+below. 
+\begin{itemize}
+\item When the \verb`\eval` command is wrapped\emph{ around} a math environment,
+the result is displayed in the form, \emph{formula = result} (followed
+possibly by the \emph{variable~=~value list}) within that environment,
+\begin{itemize}
+\item If the formula is long or contains many variables then it may be desirable
+to split the display over two lines; see §\ref{subsec:evalChanging-display-format}
+and §\ref{subsec:settings New-line-display},
+\end{itemize}
+\end{itemize}
+the whole presented as an inline expression if \verb`$` delimiters
+are used, or as a display-style expression otherwise. (See the $mc^{2}$
+example for an illustration.) 
+
+It is not clear to me that wrapping \verb`\eval` \emph{around} the
+AMS environments, except for \texttt{multline}, makes much sense,
+although it can be done. Here is an example of \verb`\eval` wrapped
+around a \texttt{multline{*}} environment (the phantom is there so
+that the hanging $+$ sign spaces correctly),\medskip{}
+
+\begin{minipage}[t]{0.8\columnwidth}%
+\begin{verbatim}
+\eval{ \begin{multline*}
+          1+2+3+4+5+6+7+8+9+10+\phantom{0}\\
+            11+12+13+14+15+16+17+18+19 
+       \end{multline*} }
+\end{verbatim}
+%
+\end{minipage}
+
+\medskip{}$\Longrightarrow$ %
+\begin{minipage}[t]{0.8\columnwidth}%
+\vspace{-2ex}\eval{ \begin{multline*}       
+		1+2+3+4+5+6+7+8+9+10+\phantom{0}\\       		  11+12+13+14+15+16+17+18+19      
+\end{multline*} }%
+\end{minipage}
+\begin{itemize}
+\item It is also possible to dispense with math delimiters entirely, neither
+wrapped within nor wrapped around the \verb`\eval` command, but in
+that case \texttt{numerica} acts as if \verb`\eval` had been used
+within \verb`\[` and \verb`\]` and displays the result accordingly.
+\end{itemize}
+
+\subsection{Checking}
+
+A question I found on the internet that caught my attention was to
+simplify $\sqrt{220-30\sqrt{35}}$. I found myself intrigued. After
+some bumbling and fumbling, I let 
+\[
+x=\sqrt{220-30\sqrt{35}},\qquad y=\sqrt{220+30\sqrt{35}},
+\]
+(which seems an obvious thiing to do). Then 
+\[
+xy=10\sqrt{484-315}=10\sqrt{13^{2}}=130.
+\]
+Since $x^{2}+y^{2}=440$ it was then easy to form both $(x+y)^{2}$
+and $(x-y)^{2}$ and by separating the resulting numbers into their
+prime factors, to work out that $x=5\sqrt{7}-3\sqrt{5}$. Was I right? 
+\begin{centred}
+\verb`\eval{$ \sqrt{220-30\sqrt{35}} $}` $\Longrightarrow$ \eval{$ \sqrt{220-30\sqrt{35}} $},
+
+\verb`\eval{$ 5\sqrt{7}-3\sqrt{5} $}` $\Longrightarrow$ \eval{$ 5\sqrt{7}-3\sqrt{5} $}.
+\end{centred}
+Yes, the simplification was correct. And indeed $y=5\sqrt{7}+3\sqrt{5}$:
+\begin{centred}
+\verb`\eval{$ \sqrt{220+30\sqrt{35}} $}` $\Longrightarrow$ \eval{$ \sqrt{220+30\sqrt{35}} $},
+
+\verb`\eval{$ 5\sqrt{7}+3\sqrt{5} $}` $\Longrightarrow$ \eval{$ 5\sqrt{7}+3\sqrt{5} $}.
+\end{centred}
+As a final flourish,
+\begin{verbatim}
+    \eval{$ xy $}
+       [ x=5\sqrt{7}-3\sqrt{5}, 
+         y=5\sqrt{7}+3\sqrt{5} ]
+\end{verbatim}
+$\Longrightarrow$     \eval{$ xy $}
+       [x=5\sqrt{7}-3\sqrt{5}, 
+        y=5\sqrt{7}+3\sqrt{5}].
+
+\subsection{Exploring}
+
+\label{subsec:introExploring}When working on \texttt{numerica}'s
+predecessor package, I constantly tested it against known results
+to check for coding errors. One test was to ensure that 
+\[
+\left(1+\frac{1}{n}\right)^{n}
+\]
+did indeed converge to the number $e$ as $n$ increased.\texttt{ }Let's
+do that here. Try first $n=10$:
+\begin{center}
+\verb`\eval{$ e-(1+1/n)^n $}[n=10][x]` $\Longrightarrow$ \eval{$ e-(1+1/n)^n $}[n=10][x].
+\par\end{center}
+
+\noindent (The default number of decimal places displayed is $6$.)
+The difference between $e$ and $(1+1/n)^{n}$ is about an eighth
+($0.125$) when $n=10$, which is encouraging but hardly decisive.
+The obvious thing to do is increase the value of $n$. I'll use an
+\verb`align*` environment to `prettify' the presentation of the
+results:
+\begin{verbatim}
+  \begin{align*}
+    e-(1+1/n)^{n} & =\eval{e-(1+1/n)^n}[n=1\times10^5][*x],\\
+    e-(1+1/n)^{n} & =\eval{e-(1+1/n)^n}[n=1\times10^6][*x],\\
+    e-(1+1/n)^{n} & =\eval{e-(1+1/n)^n}[n=1\times10^7][*x],\\
+    e-(1+1/n)^{n} & =\eval{e-(1+1/n)^n}[n=1\times10^8][*x].
+  \end{align*}
+\end{verbatim}
+(most of which was written using copy and paste) which produces
+\begin{align*}
+e-(1+1/n)^{n} & =\eval{e-(1+1/n)^{n}}[n=1\times10^{5}][x],\\
+e-(1+1/n)^{n} & =\eval{e-(1+1/n)^{n}}[n=1\times10^{6}][*x],\\
+e-(1+1/n)^{n} & =\eval{e-(1+1/n)^{n}}[n=1\times10^{7}][x],\\
+e-(1+1/n)^{n} & =\eval{e-(1+1/n)^{n}}[n=1\times10^{8}][x].
+\end{align*}
+Clearly $(1+1/n)^{n}$ converges to $e$, the difference between them
+being of order $1/n$, but that is not what catches the eye. There
+is an unanticipated regularity here. 1.35914? Double the number: \verb`$\eval{2\times 1.35914}[5]$`\texttt{
+}$\Longrightarrow\eval{2\times1.35914}[5]$ which is close enough
+to $e$ to suggest a relationship, namely,
+\[
+\lim_{n\to\infty}n\left(e-\left(1+\frac{1}{n}\right)^{n}\right)=\tfrac{1}{2}e.
+\]
+
+This was new to me. Is it true? From the familiar expansion of the
+logarithm
+
+\begin{align*}
+\ln\left(1+\frac{1}{n}\right)^{n} & =n\ln\left(1+\frac{1}{n}\right)\\
+ & =n\left(\frac{1}{n}-\frac{1}{2}\frac{1}{n^{2}}+\frac{1}{3}\frac{1}{n^{3}}-\ldots\right)\\
+ & =1-\frac{1}{2n}\left(1-\frac{2}{3}\frac{1}{n}+\frac{2}{4}\frac{1}{n^{2}}-\right)\\
+ & \equiv1-\frac{1}{2n}E_{n},
+\end{align*}
+say. Since $E_{n}$ is an alternating series and the magnitudes of
+the terms of the series tend to $0$ monotonically, $1>E_{n}>1-2/3n$.
+From this and the inequalities $1/(1-x)>e^{x}>1+x$ when $x<1$ it
+proved a straightforward matter to verify the proposed limit.
+
+\subsection{Reassuring}
+
+\label{subsec:introReassurance}In the course of some hobbyist investigations
+in plane hyperbolic geometry I derived the formula
+\[
+\Phi_{1}(m,n)=\cos\tfrac{m}{n}\pi-(1-4\sin^{2}\tfrac{m}{3n}\pi)\frac{\sin\tfrac{1}{n}\pi\sin\tfrac{m-1}{n}\pi}{2\sin^{2}\tfrac{m}{3n}\pi},
+\]
+for $m=2,3,\ldots$ and integral $n\ge2m+1$. A key concern was: when
+is $\Phi_{1}$ positive? After an embarrassingly laborious struggle,
+I managed to work this expression into the form
+\[
+\Phi_{2}(m,n)=\left(\frac{1-4\sin^{2}\tfrac{m}{3n}\pi}{2\sin^{2}\tfrac{m}{3n}\pi}\right)\sin\tfrac{2m-3}{3n}\pi\sin\tfrac{m-3}{3n}\pi,
+\]
+in which the conditions for positivity are clear: with $n\ge2m+1$,
+so that $m\pi/3n<\pi/6$, the first factor is always positive, the
+second is positive for $m\ge2$, and the third is positive for $m\ge4$.
+All well and good, but given the struggle to derive $\Phi_{2}$, was
+I confident that $\Phi_{1}$ and $\Phi_{2}$ really\emph{ }are equal?
+It felt all too likely that I had made a mistake. 
+
+The simplest way to check was to see if the two expressions gave the
+same numerical answers for a number of $m,\thinspace n$ values. I
+wrote \verb`\eval{\[ \]}[m=2,n=5]` twice and between the delimiters
+pasted the already composed expressions for $\Phi_{1}$ and $\Phi_{2}$,
+namely: 
+\begin{verbatim}
+    \eval{\[ 
+            \cos\tfrac{m}{n}\pi-(1-4\sin^{2}\tfrac{m}{3n}\pi)
+            \frac{\sin\tfrac{1}{n}\pi\sin\tfrac{m-1}{n}\pi}
+            {2\sin^{2}\tfrac{m}{3n}\pi} 
+          \]}[m=2,n=5]
+    \eval{\[ 
+            \left(
+              \frac{1-4\sin^{2}\tfrac{m}{3n}\pi}
+              {2\sin^{2}\tfrac{m}{3n}\pi}
+            \right)
+            \sin\tfrac{2m-3}{3n}\pi\sin\tfrac{m-3}{3n}\pi
+          \]}[m=2,n=5]
+\end{verbatim}
+I have added some formatting – indenting, line breaks – to make the
+formulas more readable for the present document but otherwise left
+them unaltered. The \verb`\eval` command can be used for even quite
+complicated expressions without needing to tinker with their \LaTeX{}
+form, but you may wish – as here – to adjust white space to clarify
+the component parts of the formula. Running \texttt{pdflatex} on these
+expressions, the results were
+
+\eval{\[ 
+        \cos\tfrac{m}{n}\pi-(1-4\sin^{2}\tfrac{m}{3n}\pi)
+        \frac{\sin\tfrac{1}{n}\pi\sin\tfrac{m-1}{n}\pi}
+        {2\sin^{2}\tfrac{m}{3n}\pi} 
+      \]}[m=2,n=5]
+
+\eval{\[ 
+        \left(
+          \frac{1-4\sin^{2}\tfrac{m}{3n}\pi}
+          {2\sin^{2}\tfrac{m}{3n}\pi}
+        \right)
+       \sin\tfrac{2m-3}{3n}\pi\sin\tfrac{m-3}{3n}\pi
+      \]}[m=2,n=5]
+
+\noindent which was reassuring. Doing it again but with different
+values of $m$ and $n$, again the results coincided:
+
+\eval{\[ 
+        \cos\tfrac{m}{n}\pi-(1-4\sin^{2}\tfrac{m}{3n}\pi)
+        \frac{\sin\tfrac{1}{n}\pi\sin\tfrac{m-1}{n}\pi}
+        {2\sin^{2}\tfrac{m}{3n}\pi} 
+      \]}[m=5,n=13]
+
+\eval{\[ 
+        \left(
+          \frac{1-4\sin^{2}\tfrac{m}{3n}\pi}
+          {2\sin^{2}\tfrac{m}{3n}\pi}
+        \right)
+       \sin\tfrac{2m-3}{3n}\pi\sin\tfrac{m-3}{3n}\pi
+      \]}[m=5,n=13]
+
+\noindent Thus reassured that there was \emph{not }an error in my
+laborious derivation of $\Phi_{2}$ from $\Phi_{1}$, it was not difficult
+to work back from $\Phi_{2}$ to $\Phi_{1}$ then reverse the argument
+to find a straightforward derivation. 
+
+\chapter{\texttt{\textbackslash nmcEvaluate} (\texttt{\textbackslash eval)}}
+
+The main calculational command in \texttt{numerica} is \verb`\nmcEvaluate`.
+Unlike some other commands which are loaded optionally, \verb`\nmcEvaluate`
+is \emph{always} loaded, and therefore always available. Because \verb`\nmcEvaluate`
+would be tiresome to write too frequently,\texttt{ }particularly for
+back-of-envelope calculations, there is an equivalent short-name form,
+\verb`\eval`, used almost exclusively in the following. But note:
+wherever you see the command \verb`\eval`, you can substitute \verb`\nmcEvaluate`
+and obtain the same result.
+
+\verb`\eval` (like other short-name forms of other commands in the
+\texttt{numerica} suite) is defined using \verb`\ProvideDocumentCommand`
+from the \texttt{xparse} package. Hence if \verb`\eval` has already
+been defined in some other package already loaded, it will not be
+redefined by \texttt{numerica}. It will retain its meaning in the
+other package. Its consequent absence from \texttt{numerica} may be
+an irritant, but only that; \verb`\nmcEvaluate` is defined using
+\texttt{xparse}'s \verb`\DeclareDocumentCommand` which would override
+any (freakishly unlikely) previous definition of \verb`\nmcEvaluate`
+in another package and would therefore still be available.
+
+\section{Syntax of \texttt{\textbackslash nmcEvaluate (\textbackslash eval)}}
+
+There are five arguments to the \verb`\nmcEvaluate` (or \verb`\eval`)
+command, of which only one, the third, is mandatory. All others are
+optional. If all are deployed the command looks like
+\begin{centred}
+\noindent \verb`\nmcEvaluate*[settings]{expr.}[vv-list][num. format]`
+\end{centred}
+I discuss the various arguments in the referenced sections.
+\begin{enumerate}
+\item \verb`*` optional switch; if present ensures display of only the
+numerical result (suppresses display of the formula and vv-list);
+see §\ref{subsec:evalVvSuppresList}
+\item \verb`[settings]` optional comma-separated list of \emph{key=value
+}settings for this particular calculation; see §\ref{sec:settingsOption}
+\item \verb`{expr.}` the only mandatory argument; the mathematical expression/formula
+in \LaTeX{} form that is to be evaluated 
+\item \verb`[vv-list]` optional comma-separated list of \emph{variable=value
+}items; see §\ref{subsec:evalVv-list}
+\item \verb`[num. format]` optional format specification for presentation
+of the numerical result (rounding, padding with zeros, scientific
+notation, boolean output); see~§\ref{subsec:evalRoundingEtc}
+\end{enumerate}
+Note that arguments 4 and 5 are both square-bracket delimited optional
+arguments. Should only one such argument be used, \texttt{numerica}
+determines which is intended by looking for an equals sign within
+the argument. Its presence indicates the argument is the vv-list;
+its absence indicates the argument is the number format specification. 
+
+The vv-list and number-format specification are \emph{trailing} optional
+arguments. They do not need to be hard against their preceding arguments;
+intervening spaces are allowed. This means there is a possibility
+that should the \verb`\eval` command be followed by a square-bracketed
+mathematical expression \texttt{numerica} might confuse it with one
+of its trailing arguments. Experience using \texttt{numerica} suggests
+that this will be a (very) rare occurrence and is easily prevented
+by inserting an empty brace pair (\verb`{}`) before the offending
+square-bracketed expression. Allowing spaces between the arguments
+enables complicated expressions and large vv-lists to be formatted
+with new lines and white space to aid clarity – without requiring
+the insertion of comment characters (\verb`%`).
+
+Recommended practice is to minimise the number of optional arguments
+used in \LaTeX{} commands by consolidating such arguments into a single
+\emph{key=value} list. Although \texttt{numerica} uses such an argument,
+the vv-list does not fit naturally into that scheme. And practice
+suggests that separating out the elements of the number format specification
+(rounding value, padding with zeros, scientific notation, boolean
+output) and placing them in a trailing argument is both convenient
+and intuitive for the kind of back-of-envelope calculations envisaged
+for \texttt{numerica}. 
+
+\section{The variable=value list}
+
+\label{subsec:evalVv-list}To evaluate algebraic, trigonometric and
+other formulas that involve \emph{variables} we need to give those
+variables values. This is done in the \emph{variable=value list} –
+or \emph{vv-list} for short. This is the fourth argument of the \texttt{\textbackslash nmcEvaluate}
+command and is a square-bracket delimited optional argument (optional
+because an expression may depend only on constants and numbers).
+
+\subsection{Variable names}
+
+\label{subsec:evalVariableNames}In mathematical practice, variable
+names are generally single letters of the Roman or Greek alphabets,
+sometimes also from other alphabets, in a variety of fonts, and often
+with subscripts or primes or other decorations. In \texttt{numerica}
+a variable name is \emph{what lies to the left of the equals sign
+in an item} of the vv-list. Thus variables can be multi-token affairs:
+$x',x'',x^{iv},x_{n},x'_{n},x''_{mn}$, $^{k}C_{n},var,\mathrm{var},Fred,\mathbf{Fred},\mathcal{FRED}\ldots$
+(This criterion for what makes a variable name means a name may contain
+spaces – for instance \verb`x x` should not cause a \verb`numerica`
+error – but such names are not part of mathematical practice.) Usually,
+for the kind of back-of-envelope calculations envisaged for \verb`numerica`,
+and for ease of typing, most variables will be single letters from
+the Roman or Greek alphabets. 
+
+Because equals signs and commas give structure to the vv-list, it
+should also be clear that a variable name should not contain a \emph{naked}
+equals sign or a \emph{naked} comma. They can be incorporated in a
+variable name but only when decently wrapped in braces, like \verb`R_{=}`
+displaying as $R_{=}$ or \verb`X_{,i}` displaying as $X_{,i}$. 
+
+Note that $x$ and $\mathrm{x}$ will be treated by \verb`numerica`
+as \emph{different} variables since, in the underlying \LaTeX , one
+is \texttt{x} and the other \verb`\mathrm{x}`. Even names that look
+identical in the pdf may well be distinct in \LaTeX . This is true
+particularly of superscripts and subscripts: \verb`x_0` and \verb`x_{0}`
+appear identical in the pdf but in the underlying \LaTeX{} they are
+distinct, and will be treated as distinct variables by \verb`numerica`. 
+
+Although multi-token variables are perfectly acceptable, \emph{internally}
+\verb`numerica` works with single tokens. Variable names can be so
+different in structure, one from another, that to ease the parsing
+of formulas, all \emph{internal} variable names are assumed to be
+single tokens. Hence a necessary initial step for the package is to
+map all multi-token variable names in the vv-list and the formula
+to single tokens. \verb`numerica` does this by turning the multi-token
+variable names into control sequences with names in the sequence \verb`\nmc_a`,
+\verb`\nmc_b`, \verb`\nmc_c`, etc., then searches through the vv-list
+and the formula for every occurrence of the multi-token names and
+replaces them with the relevant control sequences. It does this in
+order of decreasing size of name, working from the names that contain
+most tokens down to names containing only two tokens. (Doing the replacing
+in this order prevents \emph{parts} of longer names possibly being
+mistaken for shorter variable names.) 
+
+The conversion process uses computer resources. Even if there are
+no multi-token variables present, \verb`numerica` still needs to
+check that this is so – unless the user alerts the program to the
+fact. This can be done by making a brief entry \texttt{xx=0 }in the
+settings option (the second optional argument of \verb`\nmcEvaluate`);
+see §\ref{subsec:settingsMultitokSwitch}. If the user never (or hardly
+ever) uses multi-token variables, then a more permanent solution is
+to create a file \verb`numerica.cfg`\texttt{ }with the line \texttt{multitoken-variables
+= false}; see §\ref{sec:settingsDefaults} for this.
+
+\subsection{The vv-list and its use}
+
+A vv-list is a comma-separated list where each item is of the form
+\emph{variable=value}. It might be something simple like
+\begin{lyxcode}
+{[}g=9.81,t=2{]}
+\end{lyxcode}
+or something more complicated like
+\begin{lyxcode}
+{[}V\_S=\textbackslash tfrac43\textbackslash pi~r\textasciicircum 3,V\_C=2\textbackslash pi~r\textasciicircum 2h,h=3/2,r=2{]}.
+\end{lyxcode}
+Spaces around the equals signs or the commas are stripped away during
+processing so that 
+\begin{lyxcode}
+{[}g=9.81,t=2{]}~\textrm{and}~{[}~g~=~9.81~,~t~=~2{]}
+\end{lyxcode}
+are the \emph{same} variable=value list.
+
+\subsubsection{Evaluation from right to left}
+
+In these examples, with variables depending on other variables, there
+is an implication: that the list is evaluated \emph{from the right}.
+Recall how a function of a function is evaluated, say\emph{ }$y=f(g(h(x)))$.
+To evaluate $y$, first $x$ is assigned a value then $h(x)$ is calculated,
+then $g(h(x))$ then $f(g(h(x)))=y$. We work from right to left,
+from the innermost to the outermost element. Or consider an example
+like calculating the area of a triangle by means of the formula
+\[
+A=\sqrt{s(s-a)(s-b)(s-c)}.
+\]
+First we write the formula; then we state how $s$ depends on $a,b,c$,
+namely $s=\frac{1}{2}(a+b+c)$, then we give values to $a,b,c$. In
+\texttt{numerica} this is mirrored in the layout of the \verb`\eval`
+command:
+\begin{verbatim}
+    \eval{$ \sqrt{s(s-a)(s-b)(s-c)} $}
+                 [s=\tfrac12(a+b+c),a=3,b=4,c=5]
+\end{verbatim}
+The formula in a sense is the leftmost extension of the vv-list. The
+entire evaluation occurs from right to left.
+
+This means that the rightmost variable in the vv-list can depend only
+on constants and numbers – although it may be a complicated expression
+of those elements. Other variables in the vv-list can depend on variables
+\emph{to their right} but not to their left.
+
+\subsubsection{Expressions in the variable=value list}
+
+Suppose our expression is $\tfrac{4}{3}\pi r^{3}$, the volume $V_{S}$
+of a sphere in terms of its radius $r$, and we want to calculate
+the volume for different values of $r$ to get a sense of how rapidly
+volume increases with radius.
+\begin{centred}
+\verb`$ V_S=\eval{ \tfrac43\pi r^3 }[r=1] $` $\Longrightarrow$ $ V_S=\eval{ \tfrac43\pi r^3 }[r=1] $.
+\end{centred}
+Having set up this calculation it is now an easy matter to change
+the value of $r$ in the vv-list:
+\begin{centred}
+\verb`$ V_S=\eval{ \tfrac43\pi r^3 }[r=1.5] $` $\Longrightarrow$
+$ V_S= \eval{ \tfrac43\pi r^3 }[r=1.5] $.
+
+\verb`$ V_S=\eval{ \tfrac43\pi r^3 }[r=2] $` $  \Longrightarrow $ $V_S= \eval{ \tfrac43\pi r^3 }[r=2] $.
+\end{centred}
+To compute the volume $V_{C}=\pi r^{2}h$ of a cylinder, we have two
+variables to assign values to:
+\begin{centred}
+\verb`$ V_C=\eval{ \pi r^2h }[h=4/3,r=1] $` $\Longrightarrow$ $ V_C=\eval{ \pi r^2h }[h=4/3,r=1] $.
+\end{centred}
+Although values in the vv-list are generally either numbers or simple
+expressions (like \texttt{4/3}), that is not essential. A little more
+complicated is
+\begin{centred}
+\verb`$ V_C=\eval{ hA_C }[A_C=\pi r^2,h=4/3,r=1] $` $\Longrightarrow$
+$ V_C=\eval{ hA_C }[A_C=\pi r^2,h=4/3,r=1] $.
+\end{centred}
+where calculation of the volume of the cylinder has been split into
+two: first calculate the area $A_{C}$ of its circular base and then,
+once that has been effected, calculate the volume.
+
+A second example is provided by Brahmagupta's formula for the area
+of a triangle in terms of its semi-perimeter. In a triangle ABC, the
+sides are $a=3$, $b=4$ and $c=5$. (Of course we know this is a
+right-angled triangle with area $\tfrac{1}{2}ab=6$.) The semi-perimeter
+$s=\tfrac{1}{2}(a+b+c)$ and the area of ABC is \medskip{}
+\begin{verbatim}
+    \eval{$ \sqrt{s(s-a)(s-b)(s-c) $}
+                 [s=\tfrac12(a+b+c),a=3,b=4,c=5] 
+\end{verbatim}
+$\Longrightarrow$ \eval{$ \sqrt{s(s-a)(s-b)(s-c)} $}
+    [s=\tfrac12(a+b+c),a=3,b=4,c=5].
+
+\subsubsection{Constants}
+
+\label{subsec:Built-in-Constants}\verb`numerica` has five built-in
+constants and can also accept user-defined constants. For the latter,
+see §\ref{sec:supplConstants}. The five built-in constants known
+to \texttt{numerica} are \verb`\pi`, the ratio of circumference to
+diameter of a circle; \verb`e`, the base of natural logarithms; Euler's
+constant \verb`\gamma`, the limit of $\left(\sum_{1}^{N}1/n\right)-\ln N$
+as $N\to\infty$; the golden ratio \verb`\phi`, equal to $\tfrac{1}{2}(1+\surd5)$;
+and the utilitarian constant \verb`\deg`, the number of radians in
+a degree.
+\begin{centred}
+\verb`\eval{$ \pi $}` $\Longrightarrow$ \eval{$ \pi $},
+
+\verb`\eval{$ e $}` $\Longrightarrow$ \eval{$ e $},
+
+\verb`\eval{$ \gamma $}` $\Longrightarrow$ \eval{$ \gamma $},
+
+\verb`\eval{$ \phi $}` $\Longrightarrow$ \eval{$ \phi $},
+
+\verb`\eval{$ \deg $}` $\Longrightarrow$ \eval{$ \deg $},
+\end{centred}
+so that \verb`\eval{$ 180\deg $}` $\Longrightarrow$ \eval{$ 180\deg $}
+(as it should).
+
+Let's combine two of these in a formula:
+\begin{centred}
+\verb`\eval{$ e^\pi-\pi^e $}` $\Longrightarrow$ \eval{$ e^\pi-\pi^e $},
+\end{centred}
+which is close-ish to $\tfrac{1}{4}e$: \verb`\eval{$ \tfrac14e $}`
+$\Longrightarrow$ \eval{$ \tfrac14e $}.
+
+\noindent\begin{minipage}[t]{1\columnwidth}%
+\begin{shaded}%
+In some contexts it may feel natural to use any or all of \verb`\pi`,
+\verb`e`, \verb`\gamma` and \verb`\phi` as variables by assigning
+values to them in the vv-list. \cprotect\texttt{numerica} does not
+object. The values assigned in this way override the constants' values.
+For example, if the triangle we labelled ABC previously was instead
+labelled CDE then it has sides $c=3,d=4$ and (note!) $e=5$. It's
+area therefore is\medskip{}
+\begin{verbatim}
+    \eval{$ \sqrt{s(s-c)(s-d)(s-e)} $}
+              [s=\tfrac12(c+d+e),c=3,d=4,e=5]       
+\end{verbatim}
+$\Longrightarrow$ 
+
+\eval{$ \sqrt{s(s-c)(s-d)(s-e)} $}
+    [s=\tfrac12(c+d+e),c=3,d=4,e=5].\medskip{}
+
+\noindent Since this is the correct area we see that \cprotect\texttt{e}
+has been treated as a variable with the assigned value $5$, not as
+the constant. But if \cprotect\texttt{e} (or \verb`\pi` or \verb`\gamma`
+or \verb`\phi`) is not assigned a value in the vv-list then it has,
+by default, the value of the constant. In the case of \cprotect\texttt{e},
+if you wish to use it as a variable, the constant is always available
+as \verb`\exp(1)`. There is no similar alternative available for
+\verb`\pi`, \verb`\gamma` or \verb`\phi`. \end{shaded}%
+\end{minipage}
+
+\subsection{Display of the vv-list}
+
+By default, the vv-list is displayed with (in fact following) the
+numerical result. That and the format of the display can both be changed.
+
+\subsubsection{Star option: suppressing display of the vv-list}
+
+\label{subsec:evalVvSuppresList}If display of the vv-list is not
+wanted at all, only the numerical result, it suffices to attach an
+asterisk (star) to the \texttt{\textbackslash eval} command:
+\begin{centred}
+\verb`$ V_C=\eval*{ hA_C }[A_C=\pi r^2,h=4/3,r=1] $` $\Longrightarrow$
+$ V_C=\eval*{ hA_C }[A_C=\pi r^2,h=4/3,r=1] $,
+\end{centred}
+or simply the naked result:
+\begin{centred}
+\verb`\eval*{$ hA_C $}[A_C=\pi r^2,h=4/3,r=1]` $\Longrightarrow$
+\eval*{$ hA_C $}[A_C=\pi r^2,h=4/3,r=1].
+\end{centred}
+In the latter case, note that a negative result will display with
+a hyphen for the minus sign unless you, the user, explicitly write
+math delimiters around the \verb`\eval*` command as a whole. Wrapping
+them around the formula has no effect:
+\begin{centred}
+\verb`\eval*{$ y $}[y=ax+b,x=2,a=-2,b=2]` $\Longrightarrow$ \eval*{$ y $}[y=ax+b,x=2,a=-2,b=2],
+
+\verb`$ \eval*{ y }[y=ax+b,x=2,a=-2,b=2] $` $\Longrightarrow$ $ \eval*{ y }[y=ax+b,x=2,a=-2,b=2] $.
+\end{centred}
+The star option delivers a number as result, pure and simple.
+
+\subsubsection{Suppressing display of items}
+
+\label{subsec:evalVvSuppressVars}You may wish to retain some variables
+in the vv-list display, but not all. For those variables you wish
+omitted from the display, wrap each variable (but not the equals sign
+or value) in braces. When calculating the volume of a cylinder in
+the previous examples, the base area $A_{C}$ has a different status
+from the `fundamental' variables $r$ and $h$. It is an intermediate
+value, one that we pass through on the way to the final result. To
+suppress it from display enclose the variable in braces:
+\begin{centred}
+\verb`$ V_C=\eval{ hA_C }[{A_C}=\pi r^2,h=4/3,r=1] $` $\Longrightarrow$
+$ V_C=\eval{ hA_C }[{A_C}=\pi r^2,h=4/3,r=1] $.
+\end{centred}
+As you can see, $A_{C}$ no longer appears in the displayed vv-list.
+Of course the name and its value are still recorded `behind the scenes'
+and can still be used in calculations.
+
+\subsubsection{Empty vv-list suppressed}
+
+Should the vv-list be empty, or display of \emph{all} variables is
+suppressed by wrapping each in braces, then \emph{nothing} is displayed
+where the vv-list would normally be, not even any punctuation:
+\begin{centred}
+\verb`$ V_C=\eval{ hA_C }[{A_C}=\pi r^2,{h}=4/3,{r}=1] $` $\Longrightarrow$
+$ V_C=\eval{ hA_C }[{A_C}=\pi r^2,{h}=4/3,{r}=1] $
+\end{centred}
+If you want a full stop after the result then you will need to add
+it by hand or use the \verb`p=.` setting of §\ref{subsec:settingsPunctuation}.
+
+\subsubsection{Changing the display format}
+
+\label{subsec:evalChanging-display-format}In two examples above,
+we have calculated the area of a triangle using Brahmagupta's formula.
+Display of the result is crowded. Two remedies have just been suggested,
+but a third one and preferable in this case would be to force display
+of the vv-list and result to a new line. This can be done through
+the settings option to the \verb`\eval` command, discussed in §\ref{subsec:settings New-line-display}.
+However, if \verb`\eval` is wrapped around an \emph{appropriate}
+environment (like \verb`multline`, but not \verb`equation`) it can
+also be done simply by including \texttt{\textbackslash\textbackslash}
+at the end of the formula. 
+
+In the following example I use Brahmagupta's formula for calculating
+the area of a cyclic quadrilateral (of which his formula for a triangle
+is a special case). The cyclic quadrilateral in the example is formed
+by a 45-45-90 triangle of hypotenuse 2 joined along the hypotenuse
+to a 30-60-90 triangle. The sides are therefore $\surd2,\surd2,\surd3,1$.
+Adding the areas of the two triangles, the area of the quadrilateral
+is $A=1+\tfrac{1}{2}\surd3$, or in decimal form, \verb`$\eval{1+\tfrac12\surd3}$`
+$\Longrightarrow$ $\eval{1+\tfrac12\surd3}$. Let's check with Brahmagupta's
+formula:
+\begin{verbatim}
+    \eval{
+      \begin{multline*}
+        \sqrt{(s-a)(s-b)(s-c)(s-d)}\\
+      \end{multline*} 
+         }[s=\tfrac12(a+b+c+d),
+           a=\surd2,b=\surd2,c=\surd3,d=1] 
+\end{verbatim}
+$\Longrightarrow$     \eval{
+      \begin{multline*}
+        \sqrt{(s-a)(s-b)(s-c)(s-d)}\\
+      \end{multline*} 
+         }[s=\tfrac12(a+b+c+d),
+           a=\surd2,b=\surd2,c=\surd3,d=1]
+
+\noindent\begin{minipage}[t]{1\columnwidth}%
+\begin{shaded}%
+
+\subsubsection{Abusing multi-token variable names}
+
+\label{subsec:evalDon't-do-this!}A variable name is what lies to
+the left of the equals sign of an item in the vv-list. Since multi-token
+variables are converted to single tokens (like \verb`\nmc_a`) before
+any calculating is done, it is possible to sin. Thus :
+\begin{centred}
+\verb`\eval{$ \sin\pi $}[{\sin\pi}=1]` $\Longrightarrow$ \eval{$ \sin\pi $}[{\sin\pi}=1];
+\end{centred}
+and (more?) egregiously,
+\begin{centred}
+\verb`\eval{$ 10 $}[{10}=20]` $\Longrightarrow$ \eval{$ 10 $}[{10}=20].
+\end{centred}
+What is happening here is that the multi-token `variables' \verb`\sin\pi`
+and \verb`10` are being converted, right at the start of proceedings,
+to single tokens like \verb`\nmc_a`, which in \TeX -speak are macros
+containing their respective multiple tokens. For display purposes
+they expand to those multiple tokens, but for calculating within \verb`numerica`
+the single token is used. By this means one can easily create further
+grotesqueries:
+\begin{centred}
+\verb`\eval{$ ++ + ++ $}[{++}=1]` $\Longrightarrow$ \eval{$ ++ + ++ $}[{++}=1],
+
+\verb`\eval{$ 2(1+1) $}[{2(1}=3,{+1)}=5]` $\Longrightarrow$ \eval{$ 2(1+1) $}[{2(1}=3,{+1)}=5],
+
+\verb`\eval{$ 1!! $}[{!!}=42]` $\Longrightarrow$ \eval{$ 1!! $}[{!!}=42].
+\end{centred}
+Should \verb`numerica` try to check variable names to avoid consequences
+like this? I don't see any reasonable way of doing that. Symbols like
+\verb`(` and \verb`+` can easily be part of valid variable names
+– $k^{+},\,k^{-}$, $C_{n}^{(0)}$ and so on. It is left to the user,
+in any \emph{public} document, to avoid such sins. (And they could
+easily construct the displayed expressions in \LaTeX{} if they so wished
+without recourse to \verb`\eval` at all.) See also §\ref{subsec:supplMacrosDisplay}
+where a similar issue arises with user-defined macros.\end{shaded}%
+\end{minipage}
+
+\section{Formatting the numerical result}
+
+\label{subsec:evalRoundingEtc}Internally, values are stored to $16$
+significant figures (if available), calculations are carried out to
+$16$ significant figures, but only rarely do we want to view the
+result to $16$ figures. Generally, we round to some smaller number
+of figures. The default rounding value is $6$, meaning by default
+at most $6$ decimal places are shown. So far, all results have been
+rounded to this figure, although not all digits are always displayed
+– for instance if the sixth one is $0$, or the result is an integer. 
+
+Like other elements of the display, both rounding value and the (dis)appearance
+of trailing zeros can be customized, in this case by means of an optional
+argument following the vv-list (or the formula if there is no vv-list).
+This optional argument may contain up to four juxtaposed items from
+six possibilities:
+\begin{itemize}
+\item a question mark ?, which gives boolean output, or 
+\item an integer, the \emph{rounding value}, positive, negative or zero,
+specifying how many decimal places to display the result to, or
+\item an asterisk {*}, which pads the result with zeros should it not have
+as many decimal places as the rounding value specifies, or
+\item the character \texttt{x} (lower case!) which presents the result in
+`proper' scientific notation (a form like $1.2345\times10^{5}$
+for 123450), or
+\item the character \texttt{t} (lower case!) which presents the result in
+a bastardized scientific notation useful in tables (a form like $(5)1.2345$
+for 123450), or
+\item a character other than \texttt{?}, \texttt{{*}}, \texttt{x}, \texttt{t}
+or an integer, usually  one of the letters\texttt{ e d} \texttt{E
+D}, which presents the result in scientific notation with that character
+as the exponent mark (a form like $1.2345\text{e}5$ for $123450$).
+\end{itemize}
+If you use \texttt{?} in the same specification as some other character,
+the \texttt{?} prevails; if you use \texttt{x} in the same specification
+as some other character except for \texttt{?}, the \texttt{x} prevails;
+if you use \texttt{t} in the same specification as some other character
+except for \texttt{?} or \texttt{x}, the \texttt{t} prevails.
+
+If you repeat the character serving as the exponent mark in scientific
+notation – say \verb`xx` or \verb`dd` – then scientific notation
+extends to numbers in the interval \verb`[1,10)`.
+
+If you repeat a question mark specifying boolean output, then the
+formatting of that output is changed from \verb`1/0` to $T/F$ or
+\verb`T/F` depending as there are two or three question marks used.
+
+\subsection{Rounding value }
+
+\label{subsec:evalRounding-value}If the number is displayed as a
+decimal, the rounding value specifies the number of decimal places
+displayed. If a number is displayed in scientific notation (see below
+§\ref{subsec:evalScientificNotation}) that is still true, but it
+can mean differences in the overall number of digits displayed. For
+the moment, I show the effect of rounding in a purely decimal display:
+\begin{centred}
+\verb`$ \eval{ 1/3 }[4] $` $\Longrightarrow$ $ \eval{ 1/3 }[4] $
+\end{centred}
+In this case \verb`4` was entered in the number-format option and
+the result is displayed to four decimal places. The default rounding
+value is $6$:
+\begin{centred}
+\verb`$ \eval{ 35/3 } $` $\Longrightarrow$ $ \eval{ 35/3 } $
+\end{centred}
+Following the default behaviour in \verb`l3fp`, the calculational
+engine which \verb`numerica` uses, `ties' are rounded to the nearest
+\emph{even} digit. Thus a number ending $55$ with a `choice' of
+rounding to $5$ or $6$ rounds up to the even digit $6$, and a number
+ending $65$ with a `choice' of rounding to $6$ or $7$ rounds
+down to the even digit $6$:
+\begin{centred}
+\verb`$ \eval{ 0.1234555 } $`  $\Longrightarrow\eval{0.1234555}$ 
+
+\verb`$ \eval{ 0.1234565 } $`  $\Longrightarrow\eval{0.1234565}$ 
+\end{centred}
+\verb`l3fp` works to 16 significant figures and never displays more
+than that number (and often fewer). 
+\begin{itemize}
+\item In the first of the following although I have specified a rounding
+value of $19$ only $16$ decimal places are displayed, with the final
+digit rounded up to $7$; 
+\item in the second I have added $10$ zeros after the decimal point, meaning
+that all $19$ decimal places specified by the rounding value can
+be displayed since the 10 initial zeros do not contribute to the significant
+figures; 
+\item in the third I have changed the figure \emph{before} the decimal point
+to $1$ so that the $10$ added zeros are now included among the significant
+figures; 
+\item and in the fourth, I have added $9$ digits before the decimal point:
+\end{itemize}
+\begin{centred}
+\verb`$ \eval{ 0.1234567890123456789 }[19] $` $\Longrightarrow$
+$\eval{ 0.1234567890123456789 }[19]$
+
+\verb`$ \eval{ 0.00000000001234567890123456789 }[19] $` $\Longrightarrow$
+$\eval{ 0.00000000001234567890123456789 }[19]$
+
+\verb`$ \eval{ 1.00000000001234567890123456789 }[19] $` $\Longrightarrow$
+$\eval{ 1.00000000001234567890123456789 }[19]$
+
+\verb`$ \eval{ 987654321.1234567890123456789 }[19] $` $\Longrightarrow$
+$\eval{ 987654321.1234567890123456789 }[19]$
+\end{centred}
+In all cases, no more than $16$ \emph{significant} figures are displayed,
+although the number of decimal places displayed may exceed $16$ as
+in the second example.
+
+It is possible to use \emph{negative} rounding values. Such a value
+zeroes the specified number of digits \emph{before} the decimal point.
+\begin{centred}
+\verb`$ \eval{ 987654321.123456789 }[-4] $` $\Longrightarrow$ $\eval{ 987654321.123456789 }[-4]$
+\end{centred}
+A rounding value of $0$ rounds to the nearest integer:
+\begin{centred}
+\verb`$ \eval{ 987654321.123456789 }[0] $` $\Longrightarrow$ $\eval{ 987654321.123456789 }[0]$
+\end{centred}
+If you wish to change the \emph{default} rounding value from $6$
+to some other value, this can be done by creating or editing a file
+\texttt{numerica.cfg} in a text editor; see §\ref{sec:settingsDefaults}.
+
+\subsection{Padding with zeros }
+
+\label{subsec:evalPadding-with-zeros}A result may contain fewer decimal
+places than the rounding value specifies, the trailing zeros being
+suppressed by default (this is how \texttt{l3fp} does it). Sometimes,
+perhaps for reasons of presentation like aligning columns of figures,
+it may be desirable to pad results with zeros. This is achieved by
+inserting an asterisk, {*}, into the final optional argument of the
+\verb`\eval` command:
+\begin{centred}
+\verb`$ \eval{ 1/4 }[4] $` $\Longrightarrow$ $ \eval{ 1/4 }[4] $,
+
+\verb`$ \eval{ 1/4 }[4*] $` $\Longrightarrow$ $ \eval{ 1/4 }[4*] $.
+\end{centred}
+
+\subsection{Scientific notation }
+
+\label{subsec:evalScientificNotation} \verb`l3fp` can output numbers
+in scientific notation. For example, $1234$ is rendered as $\eval{1234}[e]$,
+denoting $1.234\times10^{3}$ , and $0.008$ as $\eval{0.008}[e]$,
+denoting $8\times10^{-3}$. The `e' here, the \emph{exponent mark},
+separates the \emph{significand} ($1.234$) from the \emph{exponent}
+($3$). In scientific notation, the significand always has one \emph{non-zero}
+digit before the decimal point.\footnote{Except for $0$ itself.}
+
+For scientific notation rounding still means the number of decimal
+places displayed, but it can result in very different numbers of digits
+being shown from the number shown in decimal form. To switch on output
+in scientific notation in \verb`numerica` enter \verb`e` in the
+trailing optional argument: 
+\begin{centred}
+\verb`$ \eval{ 123.456789 }[e] $` $\Longrightarrow$ $ \eval{ 123.456789 }[e] $. 
+\end{centred}
+The default rounding value $6$ is in play here, with seven digits
+of the significand displayed overall, one preceding the decimal point,
+six following it. Compare this with the same number rounded in decimal
+form:
+\begin{centred}
+\verb`$ \eval{ 123.456789012345 } $` $\Longrightarrow$ $ \eval{ 123.456789012345 } $. 
+\end{centred}
+In this instance, nine digits are displayed, three before the decimal
+point and six after. Similarly compare
+\begin{centred}
+\verb`$ \eval{ 0.0123456789 }[e] $` $\Longrightarrow$ $ \eval{ 0.0123456789 }[e] $ 
+\end{centred}
+with
+\begin{centred}
+\verb`$ \eval{ 0.0123456789 } $` $\Longrightarrow$ $ \eval{ 0.0123456789 } $.
+\end{centred}
+This time scientific notation has gained two extra decimal digits
+to display.
+
+Negative rounding values are pointless for scientific notation. A
+zero might on occasion be relevant:
+\begin{centred}
+\verb`$ \eval{ 987654321 }[0e] $` $\Longrightarrow$ $ \eval{ 987654321 }[0e] $.
+\end{centred}
+Sometimes letters other than `e' are used to indicate scientific
+notation, like `E' or `d' or `D'. With a few exceptions, \texttt{numerica}
+allows any letter or text character to be used as the exponent marker:
+\begin{centred}
+\verb`\eval{$ 1/23456789 $}[4d]`\texttt{ $\Longrightarrow$} \eval{$ 1/23456789 $}[4d].
+\end{centred}
+But when \texttt{x} is inserted in the trailing optional argument,
+the output is in the form $d_{0}.d_{1}\ldots d_{m}\times10^{n}$ (except
+when $n=0$), where each $d_{i}$ denotes a digit.
+\begin{centred}
+\verb`\eval{$ 1/23456789 $}[4x]`\texttt{ $\Longrightarrow$ }\eval{$ 1/23456789 $}[4x] .
+\end{centred}
+The requirements of tables leads to another form of scientific notation.
+Placing \texttt{t} in the trailing argument turns on this table-ready
+form of notation:
+\begin{centred}
+\verb`\eval{$ 1/23456789 $}[4t]`\texttt{ $\Longrightarrow$ }\eval{$ 1/23456789 $}[4t].
+\end{centred}
+This is discussed more fully in the documentation for the \texttt{numerica-tables}
+package.
+
+In the next example three options are used in the trailing argument.
+The order in which the items are entered does not matter: 
+\begin{centred}
+\verb`\eval{$ 1/125 $}[*e4]`  $\Longrightarrow$ \eval{$ 1/125 $}[*e4].
+\end{centred}
+Finally, to illustrate that `any' text character\footnote{Be sensible! An equals sign for instance might confuse \texttt{numerica}
+into thinking the number-format option is the vv-list, and will certainly
+confuse the reader.} save for \texttt{x} or \texttt{t} can be used to distinguish the
+exponent, I use an @ character:
+\begin{centred}
+\verb`\eval{$ 1/125 $}[@4]` $\Longrightarrow$ \eval{$ 1/125 $}[@4].
+\end{centred}
+
+\subsubsection{Numbers in the interval \texttt{{[}1,10)}}
+
+Usually when scientific notation is being used, numbers with magnitude
+in the interval $[1,10)$ are rendered in their normal decimal form,
+$3.14159$ and the like. Occasionally it may be desired to present
+numbers in this range in scientific notation (this can be the case
+in tables where the alignment of a column of figures might be affected).
+\texttt{numerica} offers a means of extending scientific notation
+to numbers in this range by repeating the letter chosen as the exponent
+mark in the trailing optional argument.
+\begin{centred}
+\verb`\eval{$ \pi $}[4tt]` $\Longrightarrow$ \eval{$ \pi $}[4tt] 
+\end{centred}
+
+\subsubsection{\textbackslash eval{*} and scientific notation}
+
+Scientific notation can be used for the numerical result output by
+\verb`\eval*`:
+\begin{centred}
+\verb`\eval*{ \pi }[ee]` $\Longrightarrow$ \eval*{ \pi }[ee] 
+\end{centred}
+There is one catch: if you substitute \texttt{x} for \texttt{e} here,
+\LaTeX{} will complain about a missing \verb`$`. An \texttt{x} in
+the number-format option produces a \verb`\times` in the output which
+requires a math environment. It is up to you, as the user, to provide
+the necessary delimiters outside the \verb`\eval*` command. (This
+applies even when \verb`\eval*` wraps around math delimiters.)
+
+\subsection{Boolean output}
+
+\label{subsec:evalBoolean-output}\verb`l3fp` can evaluate comparisons,
+outputting $0$ if the comparison is false, $1$ if it is true. By
+entering a question mark, \texttt{?}, in the trailing optional argument,
+you can force \verb`numerica` to do the same depending as the result
+of a calculation is zero or not. The expression being evaluated does
+not need to be a comparison, \verb`$ \eval{\pi}[?] $` $\Longrightarrow$ $ \eval{\pi}[?]$,
+but comparisons are what this is designed for.
+
+Possible comparison relations are \verb`=`, \verb`<`, \verb`>`,
+\verb`\ne`, \verb`\neq`, \verb`\ge`, \verb`\geq`, \verb`\le`,
+\verb`\leq`. Although programming languages use combinations like
+\verb`<=` or \verb`>=`, \texttt{numerica} does \emph{not} accept
+these (they are not part of standard \emph{mathematical} usage) and
+will generate an error. An example where the relation is equality
+exhibits a numerological curiosity:\footnote{The \texttt{{[}p=.{]}} of this and the next example ensures a full
+stop appears in the correct place; see §\ref{subsec:settingsPunctuation}.}
+\begin{centred}
+\verb`\eval[p=.]{\[ \frac1{0.0123456789}=81 \]}[5?]` $\Longrightarrow$
+\eval[p=.]{\[ \frac1{0.0123456789}=81 \]}[5?]\smallskip{}
+\end{centred}
+Notice the $5$ alongside the question mark in the trailing argument.
+That is critical. Change the $5$ to a $6$ (or omit it since the
+default rounding value is $6$) and the outcome is different: 
+\begin{centred}
+\verb`\eval[p=.]{\[ \frac1{0.0123456789}=81 \]}[6?]` $\Longrightarrow$
+\eval[p=.]{\[ \frac1{0.0123456789}=81 \]}[6?]
+\end{centred}
+Now the relation is false. Evaluating the fraction to more than $6$
+places, say to $9$, we can see what is going on:
+\begin{centred}
+\verb`\eval{$ 1/0.0123456789 $}[9]` $\Longrightarrow$ \eval{$ 1/0.0123456789 $}[9]. 
+\end{centred}
+
+\subsubsection{Outputting \texttt{T} or \texttt{F}}
+
+To my eye, outputting $0$ or $1$ in response to a `question' like
+$1/0.0123456789=81$ is confusing. It is easy to change the boolean
+output from $0,1$ to a more appropriate $F,T$, or \texttt{$\texttt{F,\texttt{T}}$}
+by adding one or two more question marks respectively in the number-format
+option.
+\begin{centred}
+\verb`\eval[p=.]{\[ \frac1{0.0123456789}=81 \]}[6???]` $\Longrightarrow$
+\eval[p=.]{\[ \frac1{0.0123456789}=81 \]}[6???]
+\end{centred}
+The default boolean output format is chosen to be $0,1$ in case an
+\verb`\eval` command is used within another \verb`\eval` command
+(`nesting'– see Chapter~\ref{chap:Nesting}~). The inner command
+needs to output a \emph{numerical} answer.
+
+\subsubsection{Rounding error tolerance}
+
+\label{subsec:evalToleranceRounding}If at least one of the terms
+in a comparison is the result of a calculation, then it's value is
+likely to contain rounding errors. What level of rounding error can
+we tolerate before such errors interfere with the comparison being
+made? \texttt{l3fp} tolerates none. It decides the truth or falsity
+of a comparison to all $16$ significant figures: 1.000 0000 0000
+0000 and 1.000 0000 0000 0001 are \emph{not }equal in \texttt{l3fp}.
+But for most purposes this will be far too severe a criterion.
+
+Suppose our comparison relation is $\varrho$, denoting one of =,
+<, >, \verb`\le`, etc. If $X\rel Y$ then $X-Y\rel Y-Y$, i.e. $X-Y\rel0$.
+This is what \texttt{numerica} does. It takes the right-hand side
+of the relation from the left-hand side and then compares the \emph{rounded}
+difference under $\varrho$ to $0$. The rounding value used is the
+number specified with the question mark in the trailing argument of
+the \verb`\eval` command or, if no number is present, the default
+rounding value (`out of the box' this is $6$). Thus, in a recent
+example, $1/0.0123456789-81$ when rounded to $5$ decimal places
+is $0.00000$, indistinguishable from zero at this rounding value;
+hence the equality $1/0.0123456789=81$ is true. But when rounded
+to $6$ places it is $0.000001$ which \emph{is} distinguishable from
+zero and so the equality is false. Truth or falsity depends on the
+rounding value. 
+
+When dealing with numbers generated purely mathematically, rounding
+values of $5$ or $6$ are likely to be too small. More useful would
+be rounding values closer to \texttt{l3fp}'s $16$ – perhaps $14$?
+– depending on how severe the calculations are that generate the numbers.
+However if the numbers we are dealing with come from outside mathematics,
+from practical experiments perhaps, then even a rounding value of
+$5$ or $6$ may be too large. 
+
+\noindent\begin{minipage}[t]{1\columnwidth}%
+\begin{shaded}%
+Mathematically, the claim that $X=Y$ at a rounding value $n$ is
+the claim that 
+\[
+\abs{X-Y}\le5\times10^{-(n+1)}.
+\]
+since this rounds \emph{down} to zero at $n$ places of decimals.
+This gives a more accurate test of equality than doing things in the
+opposite order – rounding each number first and then taking the difference.
+One might, for instance, have numbers like $X=0.12345$, $Y=0.12335$.
+Rounding to $n=4$ places, both round to $0.1234$ and yet the difference
+between them is $0.0001$ – they are distinguishable numbers to $4$
+places of decimals. This is why \texttt{numerica} forms the difference
+\emph{before }doing the rounding.\end{shaded}%
+\end{minipage}
+
+\subsubsection{And, Or, Not}
+
+For logical And \LaTeX{} provides the symbols \verb`\wedge` and \verb`\land`,
+both displaying as $\land$, but \texttt{numerica} adds thin spaces
+( \verb`\,` ) around the symbol for \verb`\land` (copying the package
+\texttt{gn-logic14.sty}). For logical Or \LaTeX{} provides the symbols
+\verb`\vee` and \verb`\lor`, both displaying as $\lor$, but again
+\texttt{numerica} adds thin spaces around the symbol for \verb`\lor`. 
+\begin{centred}
+\verb`\eval{$ 1<2 \wedge 2<3 $}[??]` $\Longrightarrow$ \eval{$ 1<2 \wedge 2<3 $}[??],
+
+\verb`\eval{$ 1<2 \land 2<3 $}[???]` $\Longrightarrow$ \eval{$ 1<2 \land 2<3 $}[???].
+\end{centred}
+To my eye the second of these with its increased space around the
+wedge symbol displays the meaning of the overall expression better
+than the first. Both And and Or have equal precedence; in cases of
+ambiguity the user needs to parenthesize as necessary to clarify what
+is intended.
+
+\LaTeX{} provides two commands for logical Not, \verb`\neg` and \verb`\lnot`,
+both displaying as $\lnot$ . Not binds tightly to its argument:
+\begin{centred}
+\verb`\eval{$ \lnot A \land B $}[A=0,B=0]` $\Longrightarrow$ \eval{$ \lnot A \land B $}[A=0,B=0].
+\end{centred}
+Here \verb`\lnot` acts only on the $A$; if it had acted on $A\land B$
+as a whole the result would have been different:
+\begin{centred}
+\verb`\eval{$ \lnot(A \land B) $}[A=0,B=0]` $\Longrightarrow$ \eval{$ \lnot(A \land B) $}[A=0,B=0].
+\end{centred}
+For a little flourish, I evaluate a more complicated logical statement:\footnote{Quoting from an article in \emph{Quanta Magazine} (August 2020) by
+Kevin Hartnett: `Let’s say you and two friends are planning a party.
+The three of you are trying to put together the guest list, but you
+have somewhat competing interests. Maybe you want to either invite
+Avery or exclude Kemba. One of your co-planners wants to invite Kemba
+or Brad or both of them. Your other co-planner, with an ax to grind,
+wants to leave off Avery or Brad or both of them. Given these constraints,
+you could ask: Is there a guest list that satisfies all three party
+planners?' I have written $C$ for Kemba, $A$ and $B$ for Avery
+and Brad.}
+\begin{verbatim}
+    \eval{$(A\lor\lnot C)\land(C\lor B)\land
+           (\lnot A\lor\lnot B)$}[A=1,B=0,C=1][???]
+\end{verbatim}
+$\Longrightarrow$ \eval{$(A\lor\lnot C)\land(C\lor B)\land
+                (\lnot A\lor\lnot B)$}[A=1,B=0,C=1][???].
+
+\subsubsection{Chains of comparisons}
+
+\texttt{numerica} can handle chains of comparisons like $1<2<1+2<5-1$.
+`Behind the scenes' it inserts logical And-s into the chain, $1<2\land2<1+2\land1+2<5-1$,
+and evaluates the modified expression:
+\begin{centred}
+\verb`\eval{$ 1<2<1+2<5-1 $}[?'']` $\Longrightarrow$ \eval{$ 1<2<1+2<5-1 $}[?''].
+\end{centred}
+
+\subsubsection{\texttt{amssymb} comparison symbols}
+
+\label{subsec:evalAmssymb-comparisons}\texttt{numerica} accepts some
+alternative symbols for the basic comparison relations from the \texttt{amssymb}
+package provided that package is loaded, i.e. the preamble of your
+document includes the statement
+\begin{lyxcode}
+\textbackslash usepackage\{amssymb\}
+\end{lyxcode}
+The variants from this package are: \verb`\leqq` ( $\leqq$ ), \verb`\leqslant`
+( $\leqslant$ ), \verb`\geqq` (~$\geqq$~), and \verb`\geqslant`
+( $\geqslant$ ).\footnote{No, that is not \texttt{eggplant}.} There
+are also negations: \verb`\nless` ( $\nless$ ), \verb`\nleq` (~$\nleq$~),
+\verb`\nleqq` ( $\nleqq$ ), \verb`\nleqslant` ( $\nleqslant$ ),
+\verb`\ngtr` ( $\ngtr$ ), \verb`\ngeq` ( $\ngeq$ ), \verb`\ngeqq`
+( $\ngeqq$ ), \verb`\ngeqslant` ( $\ngeqslant$ ).
+
+\section{Calculational details}
+
+\subsection{Arithmetic}
+
+\label{subsec:evalArithmetic}Addition, subtraction, multiplication,
+division, square roots, \emph{$n$}th roots, and exponentiating (raising
+to a power) are all available. 
+
+Multiplication can be rendered explicitly with an asterisk, 
+\begin{centred}
+\verb`\eval{$ 9*9 $}` $\Longrightarrow$ \eval{$ 9*9 $},
+\end{centred}
+but that's ugly. More elegant is to use \verb`\times`:
+\begin{centred}
+\verb`\eval{$ 9\times9 $}` $\Longrightarrow$ \eval{$ 9\times9 $}.
+\end{centred}
+\verb`\cdot` is also available and in many cases juxtaposition alone
+suffices: 
+\begin{centred}
+\verb`\eval{$ \surd2\surd2 $}` $\Longrightarrow$ \eval{$ \surd2\surd2 $},
+
+\verb`\eval{$ ab $}[a=123,b=1/123]` $\Longrightarrow$ \eval{$ ab $}[a=123,b=1/123].
+\end{centred}
+Division can be rendered in multiple ways too:
+\begin{centred}
+\verb`\eval{$ 42/6 $}` $\Longrightarrow$ \eval{$ 42/6 $},
+
+\verb`\eval{$ 42\div6 $}` $\Longrightarrow$ \eval{$ 42\div6 $},
+\end{centred}
+or by using \verb`\frac` or \verb`\tfrac` or \verb`\dfrac` as in
+\begin{centred}
+\verb`\eval{$ \frac{42}6 $}` $\Longrightarrow$ \eval{$ \frac{42}6 $}.
+\end{centred}
+But note that since juxtaposition means multiplication, it is also
+true that $42\tfrac{1}{6}$\texttt{ }evaluates to $7$ inside an \verb`\eval`
+command rather than denoting `forty two and a sixth'. Hence if you
+want to use `two and a half' and similar values in \texttt{numerica},
+they need to be entered as improper fractions like $\tfrac{5}{2}$
+or in decimal form, $2.5$ (as one does automatically in mathematical
+expressions anyway because of the ambiguity in a form like $2\tfrac{1}{2}$).
+
+Powers are indicated with the superscript symbol \verb`^`:
+\begin{centred}
+\verb` \eval{$ 3^{2^2} $}` $\Longrightarrow$ \eval{$ 3^{2^2} $} .
+\end{centred}
+
+\subsubsection{Square roots and $n$th roots}
+
+\label{subsec:evalSquareRootsEtc}Let us check that 3, 4, 5 and 5,
+12, 13 really are Pythagorean triples (I use \verb`\sqrt` in the
+first, \verb`\surd` in the second):
+\begin{centred}
+\verb`\eval{$ \sqrt{3^2+4^2} $}` $\Longrightarrow$ \eval{$\sqrt{3^{2}+4^{2}}$},
+
+\verb`\eval{$ \surd(5^2+12^2) $}` $\Longrightarrow$ \eval{$ \surd(5^2+12^2) $}.
+\end{centred}
+The \verb`\sqrt` command has an optional argument which can be used
+for extracting $n$th roots of a number. This notation is generally
+used when $n$ is a small positive integer like $3$ or $4$. This
+practice is followed in \texttt{numerica}: $n$ must be a (not necessarily
+small) \emph{positive integer}:
+\begin{centred}
+\verb`\eval{$ \sqrt[4]{81} $}` $\Longrightarrow$  \eval{$ \sqrt[4]{81} $},
+
+\verb`\eval{$ \sqrt[n]{125} $}[n=\floor{\pi}]` $\Longrightarrow$
+\eval{$ \sqrt[n]{125} $}[n=\floor{\pi}].
+\end{centred}
+If $n$ should not be a positive integer, an error message is generated;
+see §\ref{sec:evalErrors}.
+
+For display-style expressions, the \verb`\sqrt` command grows to
+accommodate the extra vertical height; the surd doesn't. Here is an
+example which anticipates a number of matters not discussed yet. It
+shows \verb`\eval` wrapping around a square root containing various
+formatting commands (negative spaces, \verb`\left` and \verb`\right`
+nested within \verb`\bigg` commands), all digested without complaint
+(see §\ref{subsec:evalFormatting-commands}; and see §\ref{subsec:settingsPunctuation}
+for the \verb`[p=.]`): \medskip{}
+\begin{verbatim}
+    \eval[p=.]{\[ \sqrt[3]{\!
+      \biggl(\!\left.\frac AD\right/\!\frac BC\biggr)
+        }\]}[A=729,B=81,C=9,D=3]
+\end{verbatim}
+$\Longrightarrow$\eval[p=.]
+  {\[ 
+    \sqrt[3]
+      {\!\biggl(\!\left.\frac AD\right/\!\frac BC\biggr)}
+  \]}[A=729,B=81,C=9,D=3]
+
+As implemented in \texttt{numerica}, $n$th roots found using \verb`\sqrt[n]`
+are \verb`n=<integer>`\emph{ }roots. This raises an interesting question:
+if the `$n$' of an $n$th root is the result of a calculation,
+what happens with rounding errors? The calculation may not produce
+an \emph{exact} integer. (This problem also arises with factorials;
+see §\ref{subsec:evalFactorialsBinom}.) The solution employed in
+\texttt{numerica} is to make what is considered an integer depend
+on a rounding value. Most calculations will produce rounding errors
+in distant decimal places. For `int-ifying' calculations, \texttt{numerica}
+uses a rounding value of $14$: a calculation produces an integer
+if, when rounded to $14$ figures, the result is an integer. Since
+\texttt{l3fp} works to $16$ significant figures, a rounding value
+of $14$ allows ample `elbowroom' for rounding errors to be accommodated
+when judging what is an integer and what is not. As a practical matter
+problems should not arise. 
+
+\subsubsection{\emph{n}th roots of negative numbers}
+
+Odd (in the sense of `not even') integral roots of \emph{negative}
+numbers are available with \verb`\sqrt`,
+\begin{centred}
+\verb`\eval{$ \sqrt[3]{-125} $}` $\Longrightarrow$ \eval{$ \sqrt[3]{-125} $},
+
+\verb`\eval{$ \sqrt[3]{-1.25} $}` $\Longrightarrow$ \eval{$ \sqrt[3]{-0.125} $}.
+\end{centred}
+
+\subsubsection{Inverse integer powers }
+
+Of course to find an $n$th root we can also raise to the inverse
+power,
+\begin{centred}
+\verb`\eval{$ 81^{1/4} $}` $\Longrightarrow$ \eval{$ 81^{1/4} $}.
+\end{centred}
+However, raising a \emph{negative} number to an inverse power generates
+an error even when, mathematically, it should not. This matter, which
+is a product of floating point representation of numbers, is discussed
+below in §\ref{subsec:errorsInverse-powers}.
+
+\subsection{Precedence, parentheses}
+
+The usual precedence rules apply: multiplication and division bind
+equally strongly and more strongly than addition and subtraction which
+bind equally stongly. Exponentiating binds most strongly. Evaluation
+occurs from the left.
+\begin{centred}
+\verb`\eval{$ 4+5\times6+3 $}` $\Longrightarrow$ \eval{$ 4+5\times6+3 $},
+
+\verb`\eval{$ 6\times10^3/2\times10^2 $}` $\Longrightarrow$ \eval{$ 6\times10^3/2\times10^2 $},
+\end{centred}
+which may not be what was intended. Parentheses (or brackets or braces)
+retrieve the situation:
+\begin{centred}
+\verb`\eval{$ (4+5)(6+3) $}` $\Longrightarrow$ \eval{$ (4+5)(6+3) $},
+
+\verb`\eval{$ (6\times10^3)/(2\times10^2) $}` $\Longrightarrow$
+\eval{$ (6\times10^3)/(2\times10^2) $}.
+\end{centred}
+Because exponentiating binds most strongly, negative values must be
+parenthesized when raised to a power. If not,
+\begin{centred}
+\verb`\eval{$ -4^2 $}` $\Longrightarrow$ \eval{$ -4^2 $},
+\end{centred}
+which is clearly not $(-4)^{2}$. But
+\begin{centred}
+\verb`\eval{$ (-4)^2 $}` $\Longrightarrow$ \eval{$ (-4)^2 $}.
+\end{centred}
+
+\subsubsection{Command-form brackets}
+
+\label{subsec:evalCommandBrackets}Note that brackets of all three
+kinds are available also in command form: \verb`\lparen \rparen`
+(from \verb`mathtools`) for \verb`( )`, \verb`\lbrack \rbrack`
+for \verb`[ ]`, and \verb`\lbrace \rbrace` for \verb`\{ \}`.
+
+\subsection{Modifiers\texttt{ (\textbackslash left \textbackslash right}, etc.)}
+
+The  \verb`\left` and \texttt{\textbackslash right} modifiers and
+also the series of \verb`\big...` modifiers\texttt{ }(\texttt{\textbackslash}\verb`bigl \bigr`,
+\verb`\Bigl \Bigr`, \verb`\biggl \biggr`, \verb`\Biggl \Biggr`)
+are available for use with all brackets (parentheses, square brackets,
+braces):
+\begin{verbatim}
+    \eval[p=.]{\[ \exp\left(
+        \dfrac{\ln2}{4}+\dfrac{\ln8}{4} 
+      \right) \]}
+\end{verbatim}
+$\Longrightarrow$ \eval[p=.]{\[ \exp\left( \dfrac{\ln2}{4}+\dfrac{\ln8}{4} \right) \]}
+
+\texttt{numerica} also accepts their use with \texttt{.} (dot) and
+with \texttt{/} (as noted earlier, the \verb`[p]` and \verb`[p=.]`
+are explained at §\ref{subsec:settingsPunctuation}):
+\begin{centred}
+\verb`\eval[p]{\[ \left.\dfrac{3+4}{2+1}\right/\!\dfrac{1+2}{4+5} \]}`
+$\Longrightarrow$ \eval[p=.]{\[ \left. \dfrac{3+4}{2+1} \right/\!\dfrac{1+2}{4+5} \]}
+\end{centred}
+They can be nested.
+
+\subsection{Trigonometric \& hyperbolic functions}
+
+\LaTeX{} provides all six trignometric functions, \verb`\sin`, \verb`\cos`,
+\verb`\tan`, \verb`\csc`, \verb`\sec`, \verb`\cot` and the three
+principal inverses \verb`\arcsin`, \verb`\arccos`, \verb`\arctan`.
+It also provides four of the six hyperbolic functions: \verb`\sinh`,
+\verb`\cosh`, \verb`\tanh`, \verb`\coth`, and \emph{no} inverses.
+\texttt{numerica} provides the missing hyperbolic functions, \verb`\csch`
+and \verb`\sech`, and all missing inverses, the three trigonometric
+and all six hyperbolic: \verb`\arccsc`, \verb`\arcsec`, \verb`\arccot`,
+and \verb`\asinh`, \verb`\acosh`, \verb`\atanh`, \verb`\acsch`,
+\verb`\asech`, \verb`\acoth`. (\emph{HMF} writes $\text{arcsinh}$,
+$\text{arccosh}$, etc. and ISO recommends $\text{arsinh}$, $\text{arcosh}$,
+etc. The first seems ill-advised, the second not widely adopted. At
+present neither is catered for in \texttt{numerica}.)\emph{ }
+\begin{centred}
+\verb`\eval{$ \arctan1/1\deg $}` $\Longrightarrow$ \eval{$ \arctan 1/1\deg $} ,
+
+\verb`\eval{$ \atanh\tanh3 $}` $\Longrightarrow$ \eval{$ \atanh\tanh3 $} .
+\end{centred}
+Inverses can also be constructed using the `$-1$' superscript notation.
+Thus
+\begin{centred}
+\verb`\eval{$ \sin^{-1}(1/\surd2)/1\deg $}` $\Longrightarrow$ \eval{$ \sin^{-1}(1/\surd2)/1\deg $} ,
+
+\verb`\eval{$ \tanh\tanh^{-1}0.5 $}` $\Longrightarrow$ \eval{$ \tanh\tanh^{-1}0.5 $} .
+\end{centred}
+\noindent\begin{minipage}[t]{1\columnwidth}%
+\begin{shaded}%
+
+\subsubsection*{Hyperbolic functions}
+
+Please note that \texttt{l3fp} does not (as yet) provide \emph{any}
+hyperbolic functions natively. The values \texttt{numerica} provides
+for these functions are \emph{calculated} values using familiar formulas
+involving exponentials (for the direct functions) and natural logarithms
+and square roots for the inverses. Rounding errors mean the values
+calculated may not have $16$-figure accuracy. The worst `offenders'
+are likely to be the least used, \verb`\acsch` and \verb`\asech`.
+For instance,
+\[
+\acsch x=\ln\left[\frac{1}{x}+\left(\frac{1}{x^{2}}+1\right)^{1/2}\right],
+\]
+
+\begin{centred}
+\verb`\eval{$ \csch \acsch 7 $}[16]` $\Longrightarrow$ \eval{$ \csch \acsch 7 $}[16].
+\end{centred}
+\end{shaded}%
+\end{minipage}
+
+\subsection{Logarithms}
+
+The natural logarithm \verb`\ln`, base $10$ logarithm \verb`\lg`,
+and binary or base $2$ logarithm \verb`\lb` are all recognized,
+as is \verb`\log`, preferably with a subscripted base:
+\begin{centred}
+\verb`\eval{$ \log_{12}1728 $}` $\Longrightarrow$ \eval{$ \log_{12}1728 $}
+\end{centred}
+If there is no base indicated, base $10$ is assumed. (The notations
+\verb`\ln`, \verb`\lg`, and \verb`\lb` follow ISO 80000-2 recommendation,
+which frowns upon the use of the unsubscripted \verb`\log` although
+only \verb`\ln` appears widely used.) The base need not be explicitly
+entered as a number. It could be entered as an expression or be specified
+in the vv-list:
+\begin{centred}
+\verb`\eval*{$ \log_b c $}[b=2,c=1024]` $\Longrightarrow$ \eval*{$ \log_b c $}[b=2,c=1024],
+\end{centred}
+the log to base $2$ in this case. It is possible to use the unadorned
+\verb`\log` with a base different from $10$; if you wish to do this
+only for a particular calculation see §\ref{subsec:settingsLogBase},
+or see §\ref{sec:settingsDefaults} if you want to make this default
+behaviour.
+
+\subsection{Other unary functions}
+
+Other unary functions supported are the exponential function \verb`\exp`
+and signature function \verb`\sgn` (equal to $-1$, $0$, or $1$
+depending as its argument is $<0$, $=0$, or $>0$). 
+
+\subsection{Squaring, cubing, \ldots unary functions}
+
+\texttt{numerica} has no difficulty reading a familiar but `incorrectly
+formed' expression like 
+\[
+\sin^{2}1.234+\cos^{2}1.234.
+\]
+You do not have to render it $(\sin1.234)^{2}+(\cos1.234)^{2}$ or
+(heaven forbid) $(\sin(1.234))^{2}+(\cos(1.234))^{2}$. The everyday
+usage is fine:
+\begin{centred}
+\verb`\eval{$ \sin^2\theta+\cos^2\theta $}[\theta=1.234]` $\Longrightarrow$
+\eval{$ \sin^2\theta+\cos^2\theta $}[\theta=1.234] .
+\end{centred}
+Equally \texttt{numerica} has no difficulty reading the `correct'
+but pedantic form
+\begin{centred}
+\verb`\eval{$ (\sin(\theta))^2+(\cos(\theta))^2 $}[\theta=1.234]`
+$\Longrightarrow$ \eval{$ (\sin(\theta))^2+(\cos(\theta))^2 $}[\theta=1.234] .
+\end{centred}
+A hyperbolic identity is corroborated in this example:
+\begin{centred}
+\verb`\eval{$ \sinh 3x $}[x=1]` $\Longrightarrow$ \eval{$ \sinh 3x $}[x=1],\medskip{}
+
+\verb`\eval{$ 3\sinh x+4\sinh^3x $}[x=1]` $\Longrightarrow$ \eval{$ 3\sinh x+4\sinh^3x $}[x=1].
+\end{centred}
+In fact all named unary functions in \texttt{numerica} can be squared,
+cubed, etc., in this `incorrect' but familiar way, although the
+practice outside the trigonometric and hyperbolic context seems (vanishingly?)
+rare.
+
+When the argument of the function is parenthesized and raised to a
+power – like $\sin(\pi)^{2}$ – it is read by \texttt{numerica} as
+the `sine of the square of pi', $\sin(\pi^{2})$, and \emph{not
+}as the `square of the sine of pi', $(\sin\pi)^{2}$:
+\begin{centred}
+\verb`\eval{$ \sin(\pi)^2 $}` $\Longrightarrow$ \eval{$ \sin(\pi)^2  $} .
+\end{centred}
+Things are done like this in \texttt{numerica} above all to handle
+the logarithm in a natural way. Surely $\ln x^{n}=n\ln x$, i.e. $\ln x^{n}=\ln(x^{n})$
+rather than $(\ln x)^{n}$? And if we wish to write (as we do) $\ln(1+1/n)^{n}=n\ln(1+1/n)=1-1/2n+1/3n^{2}-\ldots$
+to study the limiting behaviour of $(1+1/n)^{n}$, then we cannot
+avoid $\ln(x)^{n}=n\ln(x)=\ln(x^{n})$ too.
+
+\subsection{\emph{n}-ary functions}
+
+The functions of more than one variable ($n$-ary functions) that
+\texttt{numerica} supports are \verb`\max`, \verb`\min` and \verb`\gcd`,
+greatest common divisor. The comma list of arguments to \verb`\max`,
+\verb`\min` or \verb`\gcd` can be of arbitrary length. The arguments
+themselves can be expressions or numbers. For \verb`\gcd`, \emph{non-integer
+arguments are truncated to integers}. Hence both $y$ and $3y$ are
+independently truncated in the following example – to $81$ and $243$
+respectively:
+\begin{centred}
+\verb`\eval{$ \gcd(12,10x^2,3y,y,63) $}[y=1/0.0123456789,x=3]` $\Longrightarrow$
+\eval{$ \gcd(12,10x^2,3y,y,63) $}[y=1/0.0123456789,x=3] .
+\end{centred}
+(The truncation occurs in the argument of \verb`\gcd`, not in the
+vv-list.) 
+
+For $n$-ary functions, squaring, cubing, etc. follows a different
+pattern from that for unary functions. For \verb`\max`, \verb`\min`,
+\verb`\gcd` the argument of the function is a comma list. Squaring
+the argument makes no sense. We understand the superscript as applying
+to the function as a whole. (Consistency is not the point here; it
+is what mathematicians do that \texttt{numerica} tries to accommodate.)
+\begin{centred}
+\verb`\eval{$ \gcd(3x,x,\arcsin 1/\deg)^2 $}[x=24]` $\Longrightarrow$
+\eval{$ \gcd(3x,x,\arcsin 1/\deg)^2 $}[x=24] .
+\end{centred}
+
+\subsection{Delimiting arguments with brackets \& modifiers }
+
+Arguments of unary and $n$-ary functions can be delimited not only
+with parentheses, but also with square brackets and braces, both in
+explicit character form and also in the command form of §\ref{subsec:evalCommandBrackets}.
+The brackets, of whatever kind, can be qualified with \verb`\left \right`,
+\verb`\bigl \bigr`, etc.\footnote{See §\ref{subsec:settingsPunctuation} for the \texttt{{[}p=.{]}}
+(which ensures the concluding full stop appears in the correct place.} 
+\begin{centred}
+\verb`\eval[p=.]{\[ \sin\left\lbrack \dfrac\pi{1+2+3}\right\rbrack \]}`
+$\Longrightarrow$\eval[p=.]{\[ \sin\left\lbrack\dfrac\pi{1+2+3}\right\rbrack \]}
+\end{centred}
+
+\subsection{Absolute value, floor \& ceiling functions}
+
+It is tempting to use the \texttt{|} key on the keyboard for inserting
+an absolute value sign. \texttt{numerica} accepts this usage, but
+it is deprecated. The spacing is incorrect – compare $|-l|$ using
+\texttt{|} against $\lvert-l\rvert$ using \verb`\lvert \rvert`.
+Also, the identity of the left and right delimiters makes nested absolute
+values difficult to parse. \texttt{numerica} does not attempt to do
+so. Placing an absolute value constructed with \texttt{|} within another
+absolute value constructed in the same way is likely to produce a
+compilation error or a spurious result. \verb`\lvert \rvert` are
+better in every way except ease of writing.\texttt{ }To aid such ease
+\texttt{numerica }provides the \verb`\abs` function (using the \texttt{\textbackslash DeclarePairedDelimiter}
+command of the \texttt{mathtools} package). This takes a mutually
+exclusive star (asterisk) or square bracketed optional argument, and
+a mandatory braced argument. The starred form expands to \verb`\left\lvert #1 \right\rvert`
+where \verb`#1` is the mandatory argument:
+\begin{centred}
+\verb`\eval[p=.]{\[ 3\abs*{\frac{\abs{n}}{21}-1} \]}[n=-7]` $\Longrightarrow$
+\eval[p=.]{\[ 3\abs*{\frac{\abs{n}}{21}-1} \]}[n=-7]
+\end{centred}
+The optional argument provides access to the \verb`\big...` modifiers:
+\begin{verbatim}
+    \eval[p=.]{\[
+      \abs[\Big]{\abs{a-c}-\abs[\big]{A-C}} 
+    \]}[A=12,a=-10,C=7,c=-5] 
+\end{verbatim}
+$\Longrightarrow$     \eval[p=.]{\[ 
+      \abs[\Big]{\abs{a-c}-\abs[\big]{A-C}} 
+    \]}[A=12,a=-10,C=7,c=-5] 
+
+The form without either star or square bracket option dispenses with
+the modifiers altogether:
+\begin{centred}
+\verb`\eval{$ \tfrac12(x+y)+\tfrac12\abs{x-y} $}[x=-3,y=7].` $\Longrightarrow$
+\eval{$ \tfrac12(x+y)+\tfrac12\abs{x-y} $}[x=-3,y=7].
+\end{centred}
+As noted, the star and square bracketed option are mutually exclusive
+arguments.
+
+\texttt{numerica} also provides the functions \verb`\floor` and \verb`\ceil`,
+defined in the same way, taking a mutually exclusive star or square
+bracketed optional argument and for the starred forms expanding to
+\verb`\left\lfloor #1 \right\rfloor` and \verb`\left\lceil #1 \right\rceil`
+where \verb`#1` is the mandatory argument, and for the square bracket
+option forms replacing the \verb`\left` and \verb`\right` with the
+corresponding \verb`\big` commands. The form without star or square-bracket
+option dispenses with any modifier at all.
+\begin{centred}
+\verb`\eval{$ \floor{-\pi} $}` $\Longrightarrow$ \eval{$ \floor{-\pi} $},
+
+\verb`\eval{$ \ceil{\pi} $}` $\Longrightarrow$ \eval{$ \ceil{\pi} $}.
+\end{centred}
+The floor function, $\lfloor x\rfloor$, is the greatest integer $\le x$;
+the ceiling function, $\lceil x\rceil$ is the smallest integer $\ge x$.
+Like the absolute value, the floor and ceiling functions, can be nested: 
+\begin{centred}
+\verb`\eval{$ \floor{-\pi+\ceil{e}} $}` $\Longrightarrow$ \eval{$ \floor{-\pi+\ceil{e}} $}.
+\end{centred}
+
+\subsubsection{Squaring, cubing, \ldots{} absolute values, etc.}
+
+These three functions can be raised to a power \emph{without} extra
+parentheses:
+\begin{centred}
+\verb`\eval{$ \ceil{e}^2 $},` $\Longrightarrow$ \eval{$ \ceil{e}^2 $},
+
+\verb`\eval{$ \abs{-4}^2 $}.` $\Longrightarrow$ \eval{$ \abs{-4}^2 $}.
+\end{centred}
+
+\subsection{Factorials, binomial coefficients}
+
+\label{subsec:evalFactorialsBinom}Factorials use the familiar trailing
+\texttt{!} notation: 
+\begin{centred}
+\verb`\eval{$ 7! $}` $\Longrightarrow$ \eval{$ 7! $},
+
+\verb`\eval{$ (\alpha+\beta)!-\alpha!-\beta! $}[\alpha=2,\beta=3]`
+$\Longrightarrow$ \eval{$ (\alpha+\beta)!-\alpha!-\beta! $}[\alpha=2,\beta=3].
+\end{centred}
+The examples illustrate how \texttt{numerica} interprets the argument
+of the factorial symbol:\texttt{ }it `digests' 
+\begin{itemize}
+\item a preceding (possibly multi-digit) integer, or
+\item a preceding variable token, or
+\item a bracketed expression, or
+\item a bracket-like expression.
+\end{itemize}
+A bracket-like expression is an absolute value, floor or ceiling function,
+since they delimit arguments in a bracket-like way:
+\begin{centred}
+\verb`\eval{$ \abs{-4}!+\floor{\pi}!+\ceil{e}! $}` $\Longrightarrow$
+\eval{$ \abs{-4}!+\floor{\pi}!+\ceil{e}! $}.
+\end{centred}
+The result of feeding the factorial an expression different in kind
+from one of these four cases may give an error message or an unexpected
+result. Use parentheses around such an expression; for example write
+$(3^{2})!$, rather than $3^{2}!$.
+
+Nesting of brackets for factorials is accepted:
+\begin{centred}
+\verb`\eval{$ ((5-2)!+1)! $}` $\Longrightarrow$ \eval{$ ((5-2)!+1)! $}.
+\end{centred}
+The factorials of negative integers or of non-integers are not defined
+in \texttt{numerica}, and again there is the problem met in relation
+to $n$th roots of what happens if the argument of a factorial is
+the result of a calculation and rounding errors mean it is not an
+exact integer. This problem is unlikely to be of practical concern
+since \texttt{numerica} rounds the result of such a calculation by
+default to $14$ significant figures before offering it to the factorial.
+Since \texttt{l3fp} works to $16$ significant figures, there is ample
+`elbowroom' to accommodate rounding errors before the result of
+a calculation ceases to round to an integer. 
+
+\subsubsection{Double factorials}
+
+The double factorial, written $n!!,$ is the product $n(n-2)(n-4)\ldots\times4\times2$
+when $n$ is even and the product $n(n-2)(n-4)\ldots\times3\times1$
+when $n$ is odd. 
+\begin{centred}
+\verb`\eval{$ 6!! $}` $\Longrightarrow$ \eval{$ 6!! $},
+
+\verb`\eval{$ n!! $}[n=\sqrt{49}]` $\Longrightarrow$ \eval{$ n!! $}[n=\sqrt{49}],
+\end{centred}
+Since $n!=n!!(n-1)!!$ it follows that
+\[
+n!!=\frac{n!}{(n-1)!!}=\frac{(n+1)!}{(n+1)!!}.
+\]
+Putting $n=0$ in the outer equality shows that $0!!=1$. Now putting
+$n=0$ in the left equality gives $(-1)!!=1$. Double factorials therefore
+are defined for integers $\ge-1$.
+
+\subsubsection{Binomial coefficients}
+
+Binomial coefficients are entered in \LaTeX{} with the \verb`\binom`\textbf{
+}command. It takes two arguments and has a text-style version \verb`\tbinom`
+and a display-style version \verb`\dbinom`. As implemented in {\ttfamily\verb`numerica`},
+these are \emph{generalised} binomial coefficients:
+\[
+\binom{x}{k}=\frac{x(x-1)\dots(x-k+1)}{k(k-1)\dots1},\quad(x\in\mathbb{R},~k\in\mathbb{N}),
+\]
+where $x$ need not be a non-negative integer, and where $\binom{x}{0}=1$
+by definition. Although the first (or upper) argument can be any real
+number, the lower argument \emph{must} be a non-negative integer.
+Thus, \verb`\eval{$ \tbinom53 $}` $\Longrightarrow$ \eval{$\tbinom53$},
+\verb`\eval{$ \tbinom70 $}` $\Longrightarrow$ \eval{$\tbinom70$},
+\verb`\eval{$ \tbinom{4.2}3 $}` $\Longrightarrow$ \eval{$\tbinom{4.2}3$},
+but if the second (or lower) argument of \verb`\binom` is \emph{not}
+a non-negative integer, {\ttfamily\verb`numerica`} displays
+a message; see §\ref{subsec:errorsIntegerArgs}. 
+
+\subsection{Sums and products}
+
+\verb`numerica` recognizes sums (\verb`\sum` displaying as $\sum$)
+and products (\verb`\prod` displaying as $\prod$), and expects both
+symbols to have lower and upper summation/product limits specified.
+The lower limit must be given in the form \emph{sum/prod variable
+= initial value}; the upper limit requires only the final value to
+be specified (although it can also be given in the form \emph{sum/prod
+variable = final value}). The values may be expressions depending
+on other variables and values but must evaluate to integers (or infinity
+– see §\ref{sec:settingsInfiniteSumsProds}). Evaluating to an integer
+means that they \emph{round} to an integer, using a rounding value
+that is set by default to $14$; (recall that \texttt{l3fp} works
+to $16$ significant figures). If a limit evaluates to a non-integer
+at this `int-ifying' rounding value, an error message results. (To
+change this `int-ifying' rounding value, see §\ref{subsec:defaultsIntifyingRounding}.)
+
+As an example of expressions in the limits, this example uses the
+floor and ceiling functions to convert combinations of constants to
+integers (the \verb`[p]` is explained in §\ref{subsec:settingsPunctuation}),
+\begin{centred}
+\verb`\eval[p]{\[ \sum_{n=\floor{\pi/e}}^{\ceil{\pi e}}n \]}` $\Longrightarrow$\eval[p]{\[ \sum_{n=\floor{\pi/e}}^{\ceil{\pi e}}n \]}
+\end{centred}
+\noindent (which is $\sum_{n=1}^{9}n$).\emph{ }If the upper limit
+is less than the lower limit the result is zero. Notice that there
+is no vv-list. The summation variable does not need to be included
+there unless there are other variables that depend on it. However,
+in the case
+\begin{centred}
+\verb`\eval[p]{\[ \sum_{k=1}^N\frac1{k^3} \]}[N=100][4]` $\Longrightarrow$
+\eval[p]{\[ \sum_{k=1}^N\frac1{k^3} \]}[N=100][4]
+\end{centred}
+the upper limit $N$ is necessarily assigned a value in the vv-list. 
+
+To the author it seems natural to enter the lower limit first, immediately
+after the \verb`\sum` command (the sum is \emph{from }something \emph{to}
+something), but no problem will accrue if the upper limit is placed
+first (after all, the appearance of the formula in the pdf is the
+same):
+\begin{centred}
+\verb`\eval[p=.]{\[ \sum^N_{k=1}\frac1{k^3} \]}[N=100][4]` $\Longrightarrow$
+\eval[p=.]{\[ \sum^N_{k=1}\frac1{k^3} \]}[N=100][4]
+\end{centred}
+Another example of a sum, using binomial coefficients this time, is 
+\begin{centred}
+\verb`\eval[p]{\[ \sum_{m=0}^5\binom{5}{m}x^m y^{5-m} \]}[x=0.75,y=2.25]`
+$\Longrightarrow$ \eval[p]{\[ \sum_{m=0}^5\binom{5}{m}x^m y^{5-m} \]}[x=0.75,y=2.25]
+\end{centred}
+which is just \verb`\eval{$(x+y)^5$}[x=0.75,y=2.25]` $\Longrightarrow$
+\eval{$ (x+y)^5 $}[x=0.75,y=2.25], or $3^{5}$.
+
+Now let's calculate a product:
+\begin{verbatim}
+    \eval[p]{\[
+      \prod_{k=1}^{100} 
+        \biggl(\frac{x^2}{k^2\pi^2} +1\biggr)
+          \]}[x=1][3]
+\end{verbatim}
+$\Longrightarrow$ \eval[p]{\[\prod_{k=1}^{100} \biggl(\frac{x^2}{k^2\pi^2} +1\biggr)\]}[x=1][3]
+
+\noindent to be compared with \verb`\eval{$ \sinh 1 $}[3]` $\Longrightarrow$
+\eval{$ \sinh1 $}[3]. Obviously more terms than $100$ are required
+in the product to achieve 3-figure accuracy. 
+
+\subsubsection{Infinite sums and products}
+
+How many more? Let's `go the whole hog' and put $\infty$ in the
+upper limit of this product:
+\begin{verbatim}
+    \eval[p=.]{\[
+      \prod_{k=1}^{\infty} 
+        \biggl(\frac{x^2}{k^2\pi^2} +1\biggr)
+          \]}[x=1][3] 
+\end{verbatim}
+$\Longrightarrow$ \eval[p=.]
+  {\[
+    \prod_{k=1}^{\infty} 
+      \biggl(\frac{x^2}{k^2\pi^2} +1\biggr)
+  \]}[x=1][3] 
+
+\noindent Disappointingly, we still get the same result, deficient
+by $1$ in the third decimal place. Obviously \texttt{numerica} has
+not multiplied an infinite number of terms and, just as obviously,
+the finite number of terms it \emph{has} multiplied are too few. How
+\texttt{numerica} decides when to stop evaluating additional terms
+in an infinite sum or product is discussed later, §\ref{sec:settingsInfiniteSumsProds}.
+
+For this particular product the problem is that it converges slowly.
+Any criterion for when to stop multiplying terms or, for an infinite
+sum adding terms, seems bound to fail for some product or series.
+Presumably any stopping criterion must measure smallness in some way.
+But terms of, for example, the divergent harmonic series $\sum(1/n)$
+can always be found smaller than any value we care to specify. It
+is not surprising that a sufficiently slowly converging product or
+series falls foul of a given criterion.
+
+The default criterion however can be changed. Because this involves
+values assigned in the settings option of the \verb`\eval` command,
+I discuss infinite sums and products in the section discussing that
+optional argument; see §\ref{sec:settingsInfiniteSumsProds}.
+
+Other infinite sums converge more rapidly, and the default settings
+work admirably. For example \verb`\eval{$ (1+0.1234)^{4.321} $}`
+$\Longrightarrow$ \eval{$ (1+0.1234)^{4.321} $}. Using binomial
+coefficients we can express this as an infinite sum:\medskip{}
+\begin{verbatim}
+    \eval[p=.]{\[
+        \sum_{n=0}^{\infty}\binom{\alpha}{n}x^{n}
+           \]}[\alpha=4.321,x=0.1234]
+\end{verbatim}
+$\Longrightarrow$ \eval[p=.]
+  {\[
+    \sum_{n=0}^{\infty}\binom{\alpha}{n}x^{n}
+  \]}[\alpha=4.321,x=0.1234]
+
+\subsection{Formatting commands}
+
+\label{subsec:evalFormatting-commands}There are many formatting commands
+which change the layout of a formula on the page but do not alter
+its calculational content. \verb`numerica` copes with a great many
+of these formatting commands, although there will surely be some that
+it has overlooked and which will trigger an `Unknown token' message;
+see §\ref{sec:evalErrors}. \footnote{Please contact the author in that case: ajparsloe@gmail.com}
+
+\subsubsection{Spaces, phantoms, struts}
+
+These include cryptic forms like \verb`\,` \verb`\:` and \verb`\>`,
+\verb`\;` and the corresponding `verbose' forms, \verb`\thinspace`,
+\verb`\medspace` and \verb`\thickspace` and their negative equivalents
+\verb`\!` or \verb`\negthinspace`, \verb`\negmedspace` and \verb`\negthickspace`:
+\begin{centred}
+\verb`\eval{$ 1\negthickspace+\negthickspace 1 $}` $\Longrightarrow$
+\eval{$ 1\negthickspace+\negthickspace 1 $}
+\end{centred}
+which gives the text spacing of 1+1 as against the usual math spacing
+$1+1$ but doesn't affect the result of the calculation. 
+
+Other spacing commands are \verb`\quad` and \verb`\qquad`, and \verb`\hspace{arg}`
+and \verb`\mspace{arg}`. For \verb`\hspace` there is also a starred
+form, \verb`\hspace*{arg}`. Phantoms similarly take an argument:
+\verb`\phantom{arg}`, \verb`\hphantom{arg}` and \verb`\vphantom{arg}`.
+\begin{centred}
+\verb`\eval{$ 1\hphantom{mmm}+\hphantom{mmm}1 $}` $\Longrightarrow$
+\eval{$ 1\hphantom{mmm}+ \hphantom{mmm}1 $}.
+\end{centred}
+Like \verb`\vphantom`, struts allow vertical spacing adjustments.
+\verb`numerica` should digest both \verb`\xmathstrut[optarg]{arg}`
+from \verb`mathtools` and its `baby cousin' \verb`\mathstrut`
+from \TeX . An example from \emph{The \TeX{} book} demonstrating the
+use of \verb`\mathstrut` is
+\begin{verbatim}
+     \eval{$\sqrt{\mathstrut a}+\sqrt{\mathstrut d}+
+           \sqrt{\mathstrut y}$}[a=4,d=9,y=16] 
+\end{verbatim}
+$\Longrightarrow$ \eval{$\sqrt{\mathstrut a}+\sqrt{\mathstrut d}+\sqrt{\mathstrut y}$}[a=4,d=9,y=16], 
+
+And here is an evaluation of an expression from the \verb`mathtools`
+documentation using \verb`\xmathstrut`:
+\begin{verbatim}
+    \eval{\[ \frac{ \frac{ \xmathstrut{0.1} x-1 }
+           { \xmathstrut{0.25} x-\sin{ x} } }
+             {\xmathstrut{0.4} \sqrt{ 10-x } } \]}
+               [x=\pi/6]
+\end{verbatim}
+$\Longrightarrow$ \eval{\[ \frac{ \frac{ \xmathstrut{0.1} x-1 }
+           { \xmathstrut{0.25} x-\sin{ x} } }
+             {\xmathstrut{0.4} \sqrt{ 1-x } } \]}
+               [x=\pi/6]
+
+\subsubsection{\texttt{\textbackslash splitfrac}}
+
+The \verb`mathtools` package provides \verb`\splitfrac` and \verb`\splitdfrac`
+to aid handling of clumsy fractions. The documentation gives an (artificial)
+example of use. I've mangled it to produce an even more ridiculous
+illustration, adding to the mess an enormous square root, the modifiers
+\verb`\left` and \verb`\right`, and the command-form alternatives
+to parentheses, \verb`\lparen` and \verb`\rparen`; also the use
+of \verb`\dfrac`. A little mental arithmetic will convince that we
+are evaluating the square root of $(9\times7)^{2}$ which indeed is
+what we get:\footnote{For the \texttt{{[}p=.,vvd={]}} see §\ref{subsec:settingsPunctuation}
+and §\ref{subsec:settingsVvDisplayChangeLocal}. The first puts the
+concluding full stop in the right place; the second suppresses the
+vv-list.} \medskip{}
+\begin{verbatim}
+	\eval[p=.,vvd=]{\[
+    \sqrt{\left\lparen
+      \frac{ \splitfrac{xy + xy + xy + xy + xy}
+              {+ xy + xy + xy + xy}
+           }
+           { \dfrac z7}
+          \right\rparen \left\lparen
+            \frac{ \splitdfrac{xy + xy + xy + xy + xy}
+                     {+ xy + xy + xy + xy}
+                 }
+                 {\dfrac z7}\right\rparen}
+      \]}[x=2,y=5,z=10]
+\end{verbatim}
+$\Longrightarrow$ \eval[p=.,vvd=]{\[
+    \sqrt{\left\lparen
+      \frac{ \splitfrac{xy + xy + xy + xy + xy}
+              {+ xy + xy + xy + xy}
+           }
+           { \dfrac z7}
+          \right\rparen \left\lparen
+            \frac{ \splitdfrac{xy + xy + xy + xy + xy}
+                     {+ xy + xy + xy + xy}
+                 }
+                 {\dfrac z7}\right\rparen}
+      \]}[x=2,y=5,z=10]
+
+\subsubsection{Colour}
+
+\label{subsec:Colour}(Anglicised spelling at least for the heading!)
+If you add to the preamble of your document the line
+\begin{lyxcode}
+\textbackslash usepackage\{color\}
+\end{lyxcode}
+two commands become available, \verb`\textcolor[optarg]{arg1}{arg2}`
+and the declaration form of command, \verb`\color[optarg]{arg}`.
+\verb`numerica` readily accepts the former in a formula to be evaluated:
+\begin{centred}
+\verb`\eval{$ \sin \tfrac\pi6n\textcolor{red}{T}+1 $}[T=9,n=3]` $\Longrightarrow$
+\eval{$ \sin \tfrac\pi6n\textcolor{red}{T}+1  $}[T=9,n=3]
+\end{centred}
+(assuming you had some wish to highlight the time $T$). 
+
+However there are restrictions on the use of \verb`\color` in \verb`\eval`
+commands. \verb`\color` is a \emph{declaration} form of command.
+It has effect until the end of the current group or environment. If
+you want to restrict it to only part of that group you need to em-brace
+the command and what it is to apply to,
+\begin{lyxcode}
+<pre-stuff>\{\textbackslash color\{red\}<red-stuff>\}<post-stuff\}
+\end{lyxcode}
+but that is where the problem arises. \verb`numerica` does not check
+for `unannounced' brace groups. It expects a brace group to be introduced
+by a preceding instruction like \verb`\sqrt` or \verb`\frac` or
+\verb`^`. When announced in this way, \verb`numerica` can handle
+the brace group appropriately. But the brace group \verb`{\color{red}<red-stuff>}`
+is not so announced. \verb`numerica`'s parsing routine will not recognize
+what it has just swallowed and a \LaTeX{} error will result. So, \verb`\color`
+cannot be used in a formula in a `naked' or unannounced brace group.
+Writing \verb`\eval{$ \color{red} \sin \tfrac\pi6nT+1 $}[T=9,n=3]`
+is fine, as is 
+\begin{centred}
+\verb`\eval{$  \sin \tfrac\pi6nT+1 \color{red} $}[T=9,n=3]` $\Longrightarrow$
+\eval{$ \sin \tfrac\pi6nT+1 \color{red} $}[T=9,n=3].
+\end{centred}
+So too, because the \verb`\frac` introduces the confining brace group,
+is
+\begin{centred}
+\verb`\eval{$ \frac{\color[gray]{0.5}A}B $}[A=12,b=4]` $\Longrightarrow$\eval{$ \frac{\color[gray]{0.5}A}B $}[A=12,B=4],
+\end{centred}
+where both arguments of the \verb`\color` command are used for grayscale
+output. 
+
+But trying something like \verb`\eval{$ 3{\color[gray]{0.5}x}+1 $}[x=2]`
+will cause a \LaTeX{} error and halt compilation since there is no
+command announcing the brace group confining the \verb`\color` command.
+
+\subsubsection{\texttt{\textbackslash text}, \texttt{\textbackslash mbox}, font
+commands}
+
+\label{subsec:Text-mbox-fonts}Following a rethink of the behaviour
+of a number of font and formatting commands, in version 2 of \verb`numerica`
+the content of a \verb`\text` or \verb`\mbox` command is \emph{invisible}
+to the \verb`\eval` command. \emph{This behaviour is different from
+that of version 1.} Now the content is ignored in a calculation,
+\begin{centred}
+\verb`\eval*{ 1/0.0123456789 \mbox{approx.} }[5]` $\Longrightarrow$
+\eval*{ 1/0.0123456789 \mbox{approx.}}[5],
+\end{centred}
+even when the \verb`\text` or \verb`\mbox` contains mathematical
+content.
+
+Conversely, the content of font commands (like \verb`\mathbf` or
+\verb`\mathcal`) is \emph{visible} to \verb`\eval`. This becomes
+useful should numbers be input in scientific notation (see §\ref{subsec:settingsInputtingSciNotation}).
+As well as the math font commands, \verb`\eval` also accepts \verb`\textrm`,
+\verb`\textsf` and \verb`\texttt`. Thus a number in scientific notation
+like \verb`2e-1` appearing in the formula or the vv-list can display
+correctly by wrapping it in a \verb`\textrm` or \verb`\texttt` command,
+rather than displaying inappropriately as the algebraic expression
+$2e-1$. 
+
+\subsubsection{\texttt{\textbackslash ensuremath},\texttt{ \$},\texttt{ \textbackslash (},\texttt{
+\textbackslash )},\texttt{ \textbackslash{[}},\texttt{ \textbackslash{]}}}
+
+Should \verb`\ensuremath` be included in a formula for evaluation
+(but why?) it is digested without demur, irrespective of whether explicit
+math delimiters are present or not. More generally, should math delimiters
+(through some momentary oversight) be used both within and outside
+an \verb`\eval` command, the command is processed as if only the
+outside environment is involved; the inner delimiters are ignored:
+\begin{centred}
+\verb`$ \eval{\[ -4^2 \]} $` $\Longrightarrow$ $ \eval{\[ -4^2 \]} $.
+\end{centred}
+
+\section{Error messages }
+
+\label{sec:evalErrors}There are two kinds of error in \texttt{numerica}:
+those in the underlying \LaTeX{} which are reported in the \LaTeX{}
+log, shown on the terminal, and generally halt compilation, and specifically
+\texttt{numerica}-related errors which do not halt compilation and
+produce messages displayed in the pdf where one would expect the result
+of the calculation to be. The original reason for doing things this
+way was to enable \texttt{numerica} to be used effectively with the
+instant preview facility of the document processor \LyX . More philosophically,
+one might view such errors as similar to errors of grammar or spelling
+mistakes in text. It is not clear that they should halt compilation.
+Hence strictly \texttt{numerica}-related errors leave brief messages
+in the pdf at the offending places.
+
+Before discussing specific error messages, note that there is a debug
+facility (of a sort) discussed below in §\ref{subsec:settingsDebug}. 
+
+Error messages are in two parts: a \emph{what} part and a \emph{where}
+part.
+
+\subsection{Mismatched brackets}
+
+\label{subsec:errorsMismatched-brackets}An unmatched left parenthesis
+or other left bracket (in this case a missing right parenthesis) usually
+results in a \texttt{numerica} error:
+\begin{centred}
+\verb`$\eval{\sin(\pi/(1+x)}[x=1]$` $\Longrightarrow$ $\eval{\sin(\pi/(1+x)}[x=1]$
+\end{centred}
+For the same error in the vv-list, the what-part remains unchanged
+but the where-part is altered:
+\begin{centred}
+\verb`$\eval{ 1+y }[x=1,y=\sin(\pi/(1+x)]$` $\Longrightarrow$ $\eval{ 1+y }[y=\sin(\pi/(1+x),x=1]$
+\end{centred}
+The \emph{what} message is the same; the \emph{where} is different.
+
+An unmatched right parenthesis or other right bracket (in this case
+a missing \emph{left} parenthesis) usually results in a similar \texttt{numerica}
+error:
+\begin{centred}
+\verb`$\eval{2((x+y)/(y+z)))^2}[x=1,y=2,z=3]$` $\Longrightarrow$
+\eval{2((x+y)/(y+z)))^{2}}[x=1,y=2,z=3]
+\end{centred}
+But note that an unmatched modifier like \verb`\left` or \verb`\right`
+is a \LaTeX{} error and is caught by \LaTeX{} before \texttt{numerica}
+can respond and so results in a terminal and logfile message. 
+
+\subsection{Unknown tokens}
+
+An `Unknown token' message can arise in a number of ways. If an
+expression involves a number of variables, some of which depend on
+others, their order in the vv-list matters: 
+\noindent \begin{center}
+\verb`$\eval{\tfrac12 vt}[t=2,v=gt,g=9.8]$` $\Longrightarrow$ \eval{\tfrac{1}{2}vt}[t=2,v=gt,g=9.8] 
+\par\end{center}
+
+The vv-list is evaluated from the \emph{right} so that in this example
+the variable \texttt{v} depends on a quantity \texttt{t} that is not
+yet defined. Hence the message. The remedy is to move \texttt{t} to
+the right of \texttt{v} in the vv-list.
+
+Similarly, if we use a variable in the formula that has not been assigned
+a value in the vv-list, we again get the `Unknown token' message,
+but this time the location is the formula:
+\begin{centred}
+\verb`$\eval{\pi r^2h}[r=3]$` $\Longrightarrow$ \eval{\pi r^{2}h}[r=3]
+\end{centred}
+The remedy obviously is to assign a value to \texttt{h} in the vv-list\texttt{.}
+
+The same message will result if a mathematical operation or function
+is used that has not been implemented in \texttt{numerica}:
+\begin{centred}
+\verb`$\eval{u \bmod v }[v=7,u=3]$` $\Longrightarrow$ \eval{u\bmod v}[v=7,u=3] 
+\end{centred}
+A missing comma in the vv-list will generally result in an unknown
+token message:
+\begin{centred}
+\verb`$\eval{axy}[a=3 y=2,x=1]$` $\Longrightarrow$ \eval{axy}[a=3y=2,x=1] 
+\end{centred}
+Because of the missing comma, \verb`numerica` assumes \verb`a` has
+the `value' \verb`3y=2`, an expression which it then tries to evaluate,
+but the variable \verb`y` in this expression has not been assigned
+a value, which generates the message.
+
+\emph{Extra} commas in the vv-list should cause no problems:
+\begin{centred}
+\verb`$\eval{axy}[,a=3,,y=2,x=1,]$` $\Longrightarrow$ $\eval{axy}[,a=3,,y=2,x=1,]$ 
+\end{centred}
+The presence of multi-token variables can also cause an unknown token
+message if the check for such variables is turned off; see §\ref{subsec:settingsMultitokSwitch}.
+
+\subsection{Overlooked value assignments}
+
+Perhaps if one is evaluating a formula with a number of variables
+and assigning different experimental values to them to see the effect,
+a variable might be overlooked:
+\begin{centred}
+\verb`$\eval{axy}[a=3,y=,x=1]$` $\Longrightarrow$ \eval{axy}[a=3,y=,x=1] 
+\end{centred}
+In the example the variable \verb`y` has been overlooked. The remedy
+is obvious – assign a value to it.
+
+\subsection{Integer argument errors}
+
+\label{subsec:errorsIntegerArgs}Some functions require integer arguments
+– factorials, the second argument of a binomial coefficient, and (in
+\texttt{numerica}) $n$th roots using the optional argument of \texttt{\textbackslash sqrt};
+also summation and product variables. If integers are explicitly entered
+for these arguments there is no problem, but if the value of the argument
+is the result of a calculation, rounding errors require thinking about.
+What accumulation of rounding errors is \emph{too} much so that the
+result of the calculation \emph{cannot} be considered an integer?
+\texttt{numerica} is generous: in the default setup, if a calculation
+rounds to an integer at rounding value $14$ the result of the calculation
+is considered an integer (obviously, the value resulting from the
+rounding). Since \texttt{l3fp} works to $16$ significant figures
+that gives ample room for rounding errors to `get lost in' and be
+ignored, while still ruling out such things as (recall the example
+in §\ref{subsec:evalBoolean-output}),
+\begin{centred}
+\verb`\eval{\[ \sum_{n=1}^N n \]}[N=1/0.0123456789]` $\Longrightarrow$
+\eval{\[ \sum_{n=1}^N n \]}[N=1/0.0123456789]
+\end{centred}
+where $N$ differs from $81$ not until the seventh decimal place. 
+
+The default rounding value of $14$ for `int-ifying' calculations
+can be changed: see §\ref{subsec:defaultsIntifyingRounding}.
+
+\subsection{Comparison errors}
+
+Should a user try to make a comparison using a combination like \verb`>=`
+rather than \verb`\geq`, \texttt{numerica} admonishes like this: 
+\begin{centred}
+\verb`$\eval{ e^\pi >= \pi^e }[?]$` $\Longrightarrow$ $\eval{ e^\pi >= \pi^e }[?]$
+\end{centred}
+(The relation is true by the way.) The same error is generated by
+other multi-token comparisons. They are used in programming languages,
+yes, but \emph{not} in mathematics.
+
+\subsection{Invalid base for \texttt{\textbackslash log}}
+
+ISO recommends using \verb`\log` only with a subscripted base specified.
+Otherwise how is one to know whether the base is \verb`e` or $10$
+or $2$ or whatever? Nonetheless \texttt{numerica} assumes that when
+\verb`\log` is used unsubscripted, the base is 10. Suppose you want
+to make $12$ the base, but forget to put braces around the $12$:
+\begin{centred}
+\verb`$\eval{ \log_12 1728 }$` $\Longrightarrow$ $\eval{ \log_12 1728 } $
+\end{centred}
+Here, \texttt{numerica} has taken \texttt{1} as the base (and $21728$
+as the argument) of the logarithm and responds accordingly.
+
+\subsection{\texttt{l3fp} errors}
+
+Some errors arising at the \texttt{l3fp} level are trapped and a message
+displayed. 
+
+\subsubsection{Dividing by zero}
+\begin{centred}
+\verb`$\eval{1/\sin x}[x=0]$` $\Longrightarrow$ \eval{1/\sin x}[x=0] 
+\end{centred}
+Note however that \verb`$\eval{1/\sin x}[x=\pi]$` $\Longrightarrow\,\eval{1/\sin x}[x=\pi]$,
+because of rounding errors in distant decimal places. No doubt this
+is true for other functions as well.
+
+\subsubsection{Invalid operation}
+
+\label{subsec:errorsInverse-powers}Finding inverse integer powers
+of \emph{positive} numbers should always be possible, but raising
+a \emph{negative} number to an inverse power generates an error even
+when – mathematically – it should not: 
+\begin{centred}
+\verb`\eval{$ (-125)^{1/3} $}` $\Longrightarrow$ \eval{$ (-125)^{1/3} $}
+\end{centred}
+This is a feature of floating point arithmetic. When a number is raised
+to a rational power, say $p/q$ where $p$ and $q$ are non-zero integers,
+then the result is the $p$th power of the $q$th root of the number.
+Can a $q$th root be taken? If our floating point system used (for
+ease of illustration) only $4$ significant digits, $p/q=1/3$ would
+be the fraction $3333/10^{4}$, an odd numerator over an even denominator.
+But a negative number does not possess an even ($10^{4}$th) root. 
+
+Trying to evaluate a function like a factorial or square root or inverse
+trig. function outside its domain of definition also produces this
+error:
+\begin{centred}
+\verb`$\eval{\arccos x}[x=2]$` $\Longrightarrow$ \eval{\arccos x}[x=2]
+\end{centred}
+In this case the inverse cosine, which is defined only on the interval
+$[-1,1]$, has been fed the value $2$. 
+
+Trying to evaluate an expression that resolves to $0/0$ also produces
+this message:
+\begin{centred}
+\verb`$\eval{\frac{1-y}{x-2}}[x=2,y=1]$` $\Longrightarrow$ \eval{\frac{1-y}{x-2}}[x=2,y=1]
+\end{centred}
+
+\subsubsection{Overflow/underflow}
+
+The factorial (discussed in §\ref{subsec:evalFactorialsBinom}) provides
+an example of overflow:
+\begin{centred}
+\verb`$\eval{3249!}$`\texttt{ }$\Longrightarrow$ \eval{3249!} 
+\end{centred}
+This is hardly surprising since
+\begin{centred}
+\verb`$\eval{3248!}[x]$` $\Longrightarrow$ $\eval{3248!}[x]$.
+\end{centred}
+There is a limit on the size of exponents that \texttt{l3fp} can handle.
+A number in the form $a\times10^{b}$ must have $-10001\le b<10000$.
+If this is not the case an overflow or underflow condition occurs.
+As the examples show, an overflow condition generates a \texttt{numerica}
+error. 
+
+For underflow, where the number is closer to $0$ than $10^{-10001}$,
+\texttt{l3fp} assigns a zero value to the quantity. \texttt{numerica}
+accepts the zero value and the error is ignored.
+
+\chapter{Settings}
+
+\label{chap:Settings}A calculation is effected against a background
+of default values for various quantities. For a particular calculation,
+these values may not be appropriate; or you may have different preferences.
+The way to change settings for a particular calculation is through
+the settings option of \verb`\nmcEvaluate` discussed next. The way
+to change a \emph{default} setting is by creating a configuration
+file \texttt{numerica.cfg} discussed in §\ref{sec:settingsDefaults}.
+
+\section{Settings option}
+
+\label{sec:settingsOption}The second argument of the \verb`\nmcEvaluate`
+command is the settings option, delimited by square brackets. This
+option is a \emph{key=value} list, hence comma-separated. \emph{Key=value
+}lists tend to be wordy. For back-of-envelope calculations one wants
+to be able to `dash off' the calculation, hence the short, cryptic
+nature of the keys. Most settings are generic, applicable not only
+to \verb`\nmcEvaluate` but also to other commands that are available
+if the packages \texttt{numerica-plus} or \texttt{numerica-tables}
+are loaded; see §\ref{subsec:Related-packages}. 
+
+\subsection{\textquoteleft Debug\textquoteright{} facility}
+
+\label{subsec:settingsDebug}It is rather grandiose to call this a
+debug facility, but if a calculation goes wrong or produces a surprising
+result, \texttt{numerica} offers a means of examining various quantities
+at some intermediate stages on the way to the final result. To use
+the facility, enter 
+\begin{lyxcode}
+dbg~=~<integer>
+\end{lyxcode}
+into the settings option. (White space around the equals sign is optional.) 
+\begin{itemize}
+\item \texttt{dbg=0 }turns off the debug function, displays the result or
+error message (this is the default);
+\item \texttt{dbg=1 }equivalent to \texttt{dbg=2{*}3{*}5{*}7};
+\end{itemize}
+\begin{table}[t]
+\centering
+\noindent \centering{}\caption{Settings options}
+\noindent \begin{center}
+\begin{tabular}{ll>{\raggedright}p{4cm}>{\raggedright}p{4cm}}
+\toprule 
+{\small key} & {\small type} & {\small meaning} & {\small default}\tabularnewline
+\midrule
+{\small\texttt{dbg}} & {\small int} & {\small debug `magic' integer} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{view}} &  & {\small equivalent to }{\small\texttt{dbg=1}} & \tabularnewline
+{\small\texttt{\textasciicircum}} & {\small char} & {\small exponent mark for sci. notation input} & {\small\texttt{e}}\tabularnewline
+{\small\texttt{xx}} & {\small int (0/1)} & {\small multi-token variable switch} & {\small\texttt{1}}\tabularnewline
+{\small\texttt{()}} & {\small int (0/1/2)} & {\small trig. arg. parsing} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{o}} & {\small int (0/1)} & {\small degree switch for trig. functions} & {\small\texttt{1}}\tabularnewline
+{\small\texttt{log}} & {\small num} & {\small base of logarithms for }{\small{\small\verb`\log`}} & {\small\texttt{10}}\tabularnewline
+{\small\texttt{vv@}} & {\small int (0/1)} & {\small vv-list calculation mode} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{vvd}} & {\small token(s)} & {\small vv-list display-style spec.} & {\small\texttt{\{,\}\textbackslash mskip 12mu  6mu minus 9mu(vv)}}\tabularnewline
+{\small\texttt{vvi}} & {\small token(s)} & {\small vv-list text-style spec.} & {\small\texttt{\{,\}\textbackslash mskip 36mu minus 24mu(vv)}}\tabularnewline
+{*} &  & {\small suppress equation numbering if }{\small\texttt{\textbackslash\textbackslash}}{\small{}
+in }{\small\texttt{vvd}} & \tabularnewline
+{\small\texttt{p}} & token(s) & {\small punctuation (esp. in display-style)} & {\small\texttt{,}}\tabularnewline
+{\small\texttt{reuse}} & {\small int} & {\small form of result saved with }{\small{\small\verb`\nmcReuse`}} & {\small\texttt{0}}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+\end{table}
+
+The `magic' integers are the following primes and their products
+\begin{itemize}
+\item \texttt{dbg=2} displays the vv-list after multi-token variables have
+been converted to their single token form, \texttt{\textbackslash nmc\_a},
+\texttt{\textbackslash nmc\_b}, etc.;
+\item \texttt{dbg=3} displays the formula after multi-token variables have
+been converted to their single token form;
+\item \texttt{dbg=5} displays the stored variables and their \emph{evaluated}
+values (\texttt{dbg=2} lists the values as expressions);
+\item \texttt{dbg=7} displays the formula after it has been fp-ified but
+before it has been fed to \texttt{l3fp} to evaluate; 
+\begin{itemize}
+\item should the formula successfully evaluate, the result of the evaluation
+is also displayed (but without any formatting).
+\end{itemize}
+\end{itemize}
+To display two or more of the debug elements simultaneously, use the
+product of their debug numbers for the magic integer. This can be
+entered either as the multiplied-out product, or as the `waiting
+to be evaluated' product with asterisks (stars) between the factors.
+Thus \texttt{dbg=6} or \verb`dbg=2*3` display both the vv-list and
+formula after multi-token variables have been converted to single
+token form; \texttt{dbg=10} or \verb`dbg=2*5` display both the vv-list
+after multi-token variables have been converted to single token form
+and the recorded variables with their evaluated values. And similarly
+for the other magic integers listed. For other integers, if they are
+divisible by $2$ or $3$ or $5$ or $7$, they will display the corresponding
+component. Both \texttt{dbg=210} and \verb`dbg=2*3*5*7` display all
+four elements, but rather than remembering this product, it suffices
+to put \texttt{dbg=1}. This is equivalent and displays all elements.
+
+The debug option uses an \verb`aligned`\texttt{ }or \verb`align*`
+environment to display its wares, depending on the presence or absence
+of math delimiters around the \verb`\eval` command. The following
+uses \verb`align*` and shows how multi-token variables are handled,
+how a chain of comparisons is evaluated (§\ref{subsec:evalBoolean-output})
+and how formatting instructions in the number-format option are ignored
+in the debug display:
+\begin{verbatim}
+    \eval[dbg=1]{ a < 2a' < 3a'' }
+      [a=\pi,a'=\phi,a''=e\gamma][4???]
+\end{verbatim}
+$\Longrightarrow$     \eval[dbg=1]{ a < 2a' < 3a'' }
+      [a=\pi,a'=\phi,a''=e\gamma][4???]
+
+\noindent The various items are displayed in chronological order.
+First comes the vv-list after conversion of multi-token to single-token
+variables, then the formula in those single-token variables; these
+are created essentially at the same time. Next the stored values of
+the variables are displayed. These are the values \emph{after }evaluation.
+The fourth element both in the display and chronologically is the
+fp-ified formula. Often this can be a thicket of parentheses, especially
+if unary functions or fractions are involved. The final element of
+both the display and chronologically is the result from evaluating
+the formula. This is shown only if $7$ is a factor of the \texttt{dbg}
+integer, and there is no error. Despite the appearance of \verb`???`
+in the number-format option, the result displays as 1. Results are
+never rounded or formatted in the debug display, although as is apparent
+here, the rounding number \verb`4` is used in the comparisons.
+
+When interpreting the fp-form, differences in the ways \texttt{numerica}
+and \texttt{l3fp} read formulas can lead to more or less parentheses
+than seem strictly necessary. In particular be aware that in \texttt{l3fp}
+function calls bind most tightly so that, for example, \verb`sin 2pi`
+evaluates not to zero but to $(\sin2)\times\pi$, and \verb`sin x^2`
+evaluates to $(\sin x)^{2}$. \verb`numerica` takes care of the former
+by inserting extra parentheses and exploits the latter by not inserting
+parentheses:
+\begin{verbatim}
+    \eval[dbg=1]{ \sin 2x \cos^2 y }
+         [x=\pi/12,y=\pi/4]
+\end{verbatim}
+$\Longrightarrow$ \eval[dbg=1]{ \sin 2x \cos^2 y }[x=\pi/12,y=\pi/4]Finally,
+note that those mathematical operations that have no direct representation
+in \verb`l3fp` contribute only their value to the fp-form. This applies
+to sums and products, double factorials, partly to binomial coefficients,
+and also to \verb`\eval` and other commands when nested one within
+another (see Chapter~\ref{chap:Nesting}). The following (ridiculous)
+example illustrates the matter:
+\begin{verbatim}
+    \eval[dbg=1]{\[ 
+      \sum_{n=1}^k n + \binom{2k}{m} - \frac1{4k} + 
+        \prod_{n=2}^k (1-1/n) + m!! \]}[m=6,k=5]
+\end{verbatim}
+$\Longrightarrow$     \eval[dbg=1]{\[ 
+      \sum_{n=1}^k n + \binom{2k}{m} - \frac1{4k} + 
+        \prod_{n=2}^k (1-1/n) + m!! \]}[m=6,k=5]
+
+\noindent ($0$\textdegree{}~C in kelvin!) In the \verb`fp-form`
+line, the various contributions to the overall result are displayed
+simply as numbers because \verb`l3fp` does not (at least as yet)
+handle these elements natively.
+
+\subsection{Negative \texttt{dbg} values}
+
+Negative \texttt{dbg} values are possible: \texttt{dbg=-2}, \texttt{dbg=-3},
+etc. (and \texttt{dbg=-1} meaning \texttt{dbg=-210}) have exactly
+the same effects as the corresponding positive values except for some
+details of display. The display for positive \texttt{dbg} values is
+the one evident in the examples above. Lines wrap, the left margin
+is not indented and the display occupies the page width. For negative
+\texttt{dbg} values, lines do not wrap, the left margin is indented
+and the display occupies the text width. An example is presented in
+§\ref{subsec:nestDebugging} below where the display for a nested
+\verb`\eval` is significantly improved with a negative \texttt{dbg}
+value.
+
+\subsection{\texttt{view} setting}
+
+Putting \texttt{dbg=1} may seem a little obscure in order to view
+internal values of \verb`numerica`. In that case, simply writing
+\verb`view` in the settings option will produce the same effect as
+entering \verb`dbg=1`.
+
+\subsection{Inputting numbers in scientific notation}
+
+\label{subsec:settingsInputtingSciNotation}\emph{Outputting} numbers
+in scientific notation is controlled by the final trailing argument
+of the \texttt{\textbackslash eval} command. Such output is turned
+off by default and needs to be explicitly ordered. Similarly, \emph{inputting}
+numbers in scientific notation is turned off by default and needs
+to be explicitly ordered. To turn it on, write
+\begin{lyxcode}
+\textasciicircum ~=~<char>
+\end{lyxcode}
+\noindent in the settings option, where \texttt{<char>} is any single
+character, usually \texttt{e} or \texttt{d} or their upper-casings,
+but not restricted to them: \texttt{\textasciicircum =@} for instance
+is perfectly possible, and has the advantage over \texttt{e} or \texttt{d}
+that it doesn't conflict with the use of the character as a variable
+or constant. 
+\begin{centred}
+\verb`$ \eval[^=@]{ 1.23@-1 } $` $\Longrightarrow$ $ \eval[^=@]{ 1.23@-1 } $.
+\end{centred}
+With letters for the exponent mark – say \verb`d` or \verb`e` –
+the problem is interpreting forms like \texttt{8d-3} or \texttt{2e-1}.
+Does such a form denote a number in scientific notation or an algebraic
+expression? In \texttt{numerica}, if the settings option shows \texttt{\textasciicircum =d},
+then a form like \texttt{8d-3} is treated as a number in scientific
+notation. Similarly for \texttt{e} or any other letter used as the
+exponent marker for the input of scientific numbers. (But only one
+character can be so used at a time.) Note that the number \emph{must}
+start with a digit: \verb`e-1` for instance does not, and will be
+treated as an algebraic expression involving the exponential constant:
+\begin{centred}
+\verb`$ \eval[^=e]{ x+e-1 }[x=1] $` $\Longrightarrow$ $ \eval[^=e]{ x+e-1 }[x=1] $ 
+\end{centred}
+but
+\begin{centred}
+\verb`$ \eval[^=e]{ x+1e-1 }[x=1] $` $\Longrightarrow$ $ \eval[^=e]{ x+1e-1 }[x=1] $. 
+\end{centred}
+A problem of appearance arises if scientific numbers appear in the
+vv-list or formula and either is displayed in the result. A number
+like \verb`2e-1` will display as $2e-1$, as if it were an algebraic
+expression. In version 1 of \verb`numerica` the cure was to wrap
+\verb`2e-1` in a \verb`\text` or \verb`\mbox` command. In version
+2 of \verb`numerica` the behaviour of \verb`\text` and \verb`\mbox`
+has been re-thought; see §\ref{subsec:Text-mbox-fonts}. Their contents
+are now invisible to the \verb`\eval` command. The solution is to
+wrap \verb`2e-1` in a \verb`\textrm` or \verb`\textsf` or \verb`\texttt`
+command. These commands were not recognized by \verb`\eval` in version
+1 but \emph{are} in version 2:
+\begin{centred}
+\verb`\eval[^=e]{$  5x $ }[x=\texttt{2e-1}]` $\Longrightarrow$ \eval[^=e]{$  5x $ }[x=\texttt{2e-1}]
+,
+
+\verb`\eval[^=e]{$ 5\texttt{2e-1} $ }` $\Longrightarrow$ \eval[^=e]{$  5(\texttt{2e-1}) $ }
+.
+\end{centred}
+If you use a particular character as the exponent marker for inputting
+numbers in scientific notation, it is good practice \emph{not} to
+use that character as a variable, not because it will cause an error
+but because it makes expressions harder to read. 
+
+\subsection{Multi-token variables}
+
+\label{subsec:settingsMultitokSwitch}Variables need not consist of
+a single character or token (like $x$ or $\alpha$). Multi-token
+symbols like $x'$ or $t_{i}$ or $var$ are perfectly acceptable.
+For its internal operations, \texttt{numerica} converts such multi-token
+names to single tokens (as discussed in §\ref{subsec:evalVariableNames}).
+This conversion takes time. Even if there are no multi-token variables
+used at all, \texttt{numerica} still needs to check that that is so.
+There is a setting that allows a user to turn off or turn on the check
+for such variables by entering
+\begin{lyxcode}
+xx~=~<integer>
+\end{lyxcode}
+into the settings option. If \texttt{<integer>} is \texttt{0}, the
+check for (and conversion of) multi-token variables is turned off;
+if \texttt{<integer>} is \texttt{1} (or any other\emph{ non-zero}
+integer), the check, and conversion if needed, goes ahead. By default,
+checking for multi-token variables and converting them if found is
+turned \emph{on}. (The name for the key, \texttt{xx}, is chosen because
+\texttt{x} is the most familiar variable of all, introduced in elementary
+algebra, and doubling it like this suggests multi-token-ness.)
+
+If checking is turned off when a multi-token variable is present,
+an error results. We don't need to enter \texttt{xx=1} in the first
+of the following examples because the check for multi-token variables
+is on by default. Explicitly turning it off in the second produces
+an error.
+\begin{centred}
+\verb`\eval{$ x_0^{\,2} $}[x_0=5]` $\Longrightarrow$ \eval{$ x_0^{\,2} $}[x_0=5],\medskip{}
+\verb`\eval[xx=0]{$ x_0^{\,2} $}[x_0=5]` $\Longrightarrow$ \eval[xx=0]{$ x_0^{\,2} $}[x_0=5]
+\end{centred}
+
+\subsection{Parsing arguments of trigonometric functions}
+
+This setting allows a wider range of arguments to trigonometric  functions
+to be parsed (think Fourier series) without needing to insert extra
+parentheses in order for them to be read correctly by \verb`\eval`;
+see §\ref{subsec:parseTrigFns}.
+
+\subsection{Using degrees rather than radians}
+
+\label{subsec:settingsDegrees}You may find it more convenient to
+use degrees rather than radians with trigonometric functions. This
+can be switched on simply by entering a lowercase \texttt{o} in the
+settings option. (The author hopes the charitable eye sees a degree
+symbol in the  \texttt{o}.) Thus
+\begin{centred}
+\verb`\eval[o]{$ \sin 30 $}` $\Longrightarrow$ \eval[o]{$ \sin 30 $},
+
+\verb`\eval[o]{$ \arcsin 0.5 $}` $\Longrightarrow$ \eval[o]{$ \arcsin 0.5 $}.
+\end{centred}
+This is a \verb`0/1` switch, \verb`0` signifying \verb`off` or
+`don't use degrees', \verb`1` signifying \verb`on` or `do use
+degrees'. Although the \verb`o` default is \verb`1`, out-of-the-box
+\verb`numerica` assumes radians are being used. Thus if \verb`o`
+is absent from the settings option of an \verb`\eval` command, the
+out-of-the-box setting prevails and radians are used, but if \verb`o`
+is present, it is equivalent to \verb`o=1`. To explicitly turn off
+the use of degrees requires the full setting, \verb`o=0`. 
+
+If you want to change the out-of-the-box setting you need to put the
+line \verb`use-degrees = 1` into a configuration file; see §\ref{sec:settingsDefaults}.
+
+\subsection{Specifying a logarithm base}
+
+\label{subsec:settingsLogBase}If you wish to use \verb`\log` without
+a subscripted base in a particular calculation, then add an entry
+like 
+\begin{lyxcode}
+log~=~<positive~number>
+\end{lyxcode}
+where \verb`<positive number>`~$\ne1$ to the settings option of
+the \verb`\eval` command. The \verb`<positive number>` does not
+need to be an integer. It could be \verb`e` (if you object to writing
+\verb`\ln`) but is more likely to be $2$ or another small integer;
+$10$ is the default. If you want to use this changed base not for
+one but most calculations, then add an entry with your choice of base
+to a configuration file; see §\ref{sec:settingsDefaults}.
+
+\subsection{Calculation mode}
+
+\label{subsec:settingsRecalcMode}A variable may change in the course
+of a calculation. This is certainly true of sums and products. If
+a parameter in the vv-list depends on the variable then that parameter
+will need to be recalculated, perhaps repeatedly, in the course of
+a calculation. By entering either
+\begin{lyxcode}
+vv@~=~<integer>
+\end{lyxcode}
+or (as in version 1 of \verb`numerica`),
+\begin{lyxcode}
+vvmode~=~<integer>
+\end{lyxcode}
+in the settings option it is possible to turn on or off the ability
+to repeatedly evaluate the vv-list; \verb`<integer>` here takes two
+possible values, \texttt{0} or \texttt{1}. \texttt{vv@=0} (or \texttt{vvmode=0})
+means the vv-list is evaluated once at the start of the calculation;
+\texttt{vv@=1} (or\texttt{ vvmode=1}) means the vv-list is recalculated
+every time the relevant variable changes.\footnote{In version 1 of \texttt{numerica} only the \texttt{vvmode} name for
+this setting was available. To the author's eye, the \texttt{@} sign
+seems sufficiently close to a symbol like $\circlearrowleft$, suggesting\texttt{
+}redo or recalculate, that \texttt{vv@} is now preferred. The \texttt{@}
+symbol is – universally? – available on keyboards and \texttt{vv@}
+is only half as many keypresses as \texttt{vvmode}.}
+
+For example, in a sum it may be desirable to place the summand, or
+some part of it, in the vv-list. Since the summation variable obviously
+changes during the course of the calculation, we need to enter \texttt{vv@=1}
+in the settings option. Repeating an earlier sum (the seting \verb`p=.`
+is discussed in §\ref{subsec:settingsPunctuation}), \medskip{}
+\begin{verbatim}
+    \eval[p=.,vv@=1]{\[ \sum_{k=1}^N f(k) \]}
+      [N=100,f(k)=1/k^3,{k}=1][4] 
+\end{verbatim}
+$\Longrightarrow$      \eval[p=.,vv@=1]{\[ \sum_{k=1}^N f(k) \]}
+        [N=100,f(k)=1/k^3,{k}=1][4]
+
+As you can see, the summand \texttt{f(k)} has been given explicit
+form in the vv-list – equated to \texttt{1/k\textasciicircum 3}.
+That means we need to give a preceding value to \texttt{k} in the
+vv-list to avoid an unknown token message, hence the rightmost entry.
+But we don't want \texttt{k=1} appearing in the final display, so
+we wrap \texttt{k} in braces (see §\ref{subsec:evalVvSuppressVars}).
+Since the value \texttt{k=1} applies only to the first term in the
+sum, to ensure it is not used for all terms, we enter \texttt{vv@=1}
+in the settings option. This turns vv-recalculation mode on and ensures
+\texttt{k=1} is overwritten by \texttt{k=2}, \texttt{k=3} and so on,
+and the vv-list recalculated each time. The final result is the same
+as before, although recalculating the vv-list at each step is a more
+resource-hungry process. The difference may not be marked for this
+example; with more complicated expressions it noticeably takes longer.
+
+Because it is necessary to activate this switch when using \emph{implicit}
+notations – like $f(k)$ in the example – rather than the explicit
+form of the function in the main argument, it seems natural to call
+\texttt{vv@=1} \emph{implicit }mode and \texttt{vv@=0} (the default)
+\emph{explicit }mode. Most calculations are explicit mode – the vv-list
+is evaluated only once.\emph{ }
+
+\subsection{Changing the vv-list display format}
+
+\label{subsec:settingsVvDisplayChangeLocal}In previous formulas with
+variables the vv-list has been displayed following the result. It
+is wrapped in parentheses following a comma followed by a space. These
+formatting elements – comma, space, parentheses – can all be changed
+with the settings option.
+
+The default format specification is 
+\begin{lyxcode}
+\{,\}\textbackslash mskip~12mu~plus~6mu~minus~9mu(vv)
+\end{lyxcode}
+for a text-style display (an inline formula) and
+\begin{lyxcode}
+\{,\}\textbackslash mskip~36mu~minus~24mu(vv)
+\end{lyxcode}
+in a display-style context. The commas are wrapped in braces because
+these are items in a comma-separated list. Both entries exhibit the
+elements: punctuation (comma), preceding a variable space, preceding
+the parenthesized vv-list (the \texttt{vv} placeholder). No full stop
+is inserted after the closing parentheses because the \texttt{\textbackslash eval}
+command may occur in the middle of a sentence (even in display style).
+For inline use, the elasticity of the space becomes relevant when
+\TeX{} is adjusting individual lines to fit sentences into paragraphs
+and paragraphs into pages. The largest spacing that can be stretched
+to is a quad, $18$~mu (mu~=~math unit), and the smallest that
+can be shrunk to is a thin space, $3$~mu. In display style,\texttt{
+}the largest spacing specified is the double quad, in line with the
+recommendation in \emph{The \TeX{} Book}, Chapter~18, but this can
+shrink to a single quad, for instance if the vv-list is heavily populated
+with variables so that the evaluated result is pushed well to the
+left by the vv-list. (But see below, §\ref{subsec:settings New-line-display}.)
+
+If you want to change these defaults, enter in the settings option
+\begin{lyxcode}
+vvi~=~<new~specification>
+\end{lyxcode}
+to change the inline display and
+\begin{lyxcode}
+vvd~=~<new~specification>
+\end{lyxcode}
+to change the display-style display For example the settings
+\begin{lyxcode}
+vvi~=~\{,\}\textbackslash quad(vv)
+
+vvd~=~\{,\}\textbackslash qquad(vv)
+\end{lyxcode}
+would give a comma (in braces since the settings option is a comma-separated
+list) and a fixed space (of one or two quads) between the result and
+the parenthesized vv-list. 
+
+The vv-list itself in the display specification is represented by
+the placeholder \texttt{vv}. If the \texttt{vv} is omitted from the
+specification, then the vv-list will not appear at all:
+\begin{centred}
+\verb`\eval[vvi=?!]{$ \pi $}[\pi=3]` $\Longrightarrow$  \eval[vvi=?!]{$ \pi $}[\pi=3]
+\end{centred}
+More relevantly, it may well be the case that all variables in the
+vv-list are suppressed (wrapped in braces). In that case nothing is
+displayed. Compare the last example with
+\begin{centred}
+\verb`\eval[vvi=?!]{$ \pi $}[{\pi}=3]` $\Longrightarrow$ \eval[vvi=?!]{$ \pi $}[{\pi}=3]
+\end{centred}
+and
+\begin{centred}
+\verb`\eval[vvi=?!]{$ \pi $}` $\Longrightarrow$ \eval[vvi=?!]{$ \pi $}
+\end{centred}
+See also the punctuation setting below, §\ref{subsec:settingsPunctuation}.
+
+\subsection{Displaying the vv-list on a new line }
+
+\label{subsec:settings New-line-display}Display of a long formula
+with many variables, hence a full vv-list, may not fit comfortably
+on a line. In an earlier example I used Brahmagupta's formula to calculate
+the area of a triangle. It squeezed onto a line. I shall now use his
+formula for the area of a cyclic quadrilateral:
+\[
+A=\sqrt{(s-a)(s-b)(s-c)(s-d)}.
+\]
+The extra side (quadrilateral as against triangle) means there is
+a further variable to accommodate, not only in the formula but also
+in the vv-list. In the following example, the cyclic quadrilateral
+is formed by a 45-45-90 triangle of hypotenuse 2 joined along the
+hypotenuse to a 30-60-90 triangle. The sides are therefore $\surd2,\surd2,\surd3,1$.
+Adding the areas of the two triangles, the area of the quadrilateral
+is $A=1+\tfrac{1}{2}\surd3$, or in decimal form, \verb`$\eval{1+\tfrac12\surd3}$`
+$\Longrightarrow$ $\eval{1+\tfrac12\surd3}$. Let's check with Brahmagupta's
+formula:
+\begin{verbatim}
+    \eval[p=.,vvd={,}\\(vv),*]
+      {\[ \sqrt{(s-a)(s-b)(s-c)(s-d)} \]}
+        [s=\tfrac12(a+b+c+d),
+           a=\surd2,b=\surd2,c=\surd3,d=1] 
+\end{verbatim}
+$\Longrightarrow$     \eval[p=.,vvd={,}\\(vv),*]
+      {\[ \sqrt{(s-a)(s-b)(s-c)(s-d)} \]}
+        [s=\tfrac12(a+b+c+d), 
+           a=\surd2,b=\surd2,c=\surd3,d=1] 
+
+\noindent The values agree. The point to note here is the\texttt{
+vvd=\{,\}\textbackslash\textbackslash (vv)} and the \texttt{{*}}
+in the settings option. The \texttt{\textbackslash\textbackslash}
+in a specification for \texttt{vvd} acts as a trigger for \texttt{numerica}
+to replace whatever math delimiters are enclosed by the \verb`\eval`
+command with a \verb`multline` environment. As you can see, the specification
+inserts a comma after the formula and places the parenthesized vv-list
+on a new line. The star \texttt{{*}} if present suppresses equation
+numbering by turning the \verb`multline` into a \verb`multline*`
+environment. 
+
+Things to note in the use of\texttt{ \textbackslash\textbackslash}
+in a \texttt{vvd} specification are that
+\begin{itemize}
+\item it applies only to the \texttt{vvd} specification, not the \texttt{vvi}
+spec.;
+\item it applies only when\emph{ }\verb`\eval`\emph{ wraps around }a math
+environment of some kind; 
+\item it has no effect when the \verb`\eval` command is used \emph{within}
+a math environment when the presentation of the result is of the form
+\emph{result, vv-list}. The formula is not displayed and so the pressure
+on space is less and the `ordinary' vv-list specification is used.
+\end{itemize}
+
+\subsection{Punctuation}
+
+\label{subsec:settingsPunctuation}The \verb`\eval` command can be
+used within mathematical delimiters or it can be wrapped around mathematical
+delimiters. The latter gives a \emph{formula=result} style of display
+automatically, which is convenient. One doesn't need to write the
+\emph{formula=} part of the expression, but it causes a problem when
+\verb`\eval` wraps around a display-style or similar environment:
+how to display a following punctuation mark? For an inline display
+we can simply follow the \verb`\eval` command with the appropriate
+punctuation, for instance: \verb`\eval{$ 1+1 $}.` $\Longrightarrow$
+\eval{$ 1+1 $}. But with \verb`\[ \]` delimiters used \emph{within}
+the \verb`\eval` command a trailing fullstop will slide off to the
+start of the next line, since it is beyond the closing delimiter.
+We want it to display as if it were the last element \emph{before}
+the closing delimiter.
+
+Explicitly putting it there – \verb`\eval{\[ 1+1. \]}` – means the
+punctuation mark becomes part of the formula. Potentially \texttt{numerica}
+then needs to check not just for a fullstop but also other possible
+punctuation marks like comma, semicolon, perhaps even exclamation
+and question marks. All these marks have roles in mathematics or \texttt{l3fp}.
+Including them in the formula means distinguishing their punctuation
+role from their mathematical role and can only cause difficulties
+and slow evaluation.
+
+Instead, \texttt{numerica} uses the setting 
+\begin{lyxcode}
+p~=~<char(s)>~
+\end{lyxcode}
+to place the {\ttfamily\verb`<char(s)>`} after the result
+but within the environment delimiters. The default punctuation mark
+is the comma so that simply entering \texttt{p} will produce a comma
+in the appropriate place. This saves having to write \texttt{p=\{,\}}
+as would otherwise be required, since the settings option is a \emph{comma}-separated
+list.
+
+Nor is one limited to a single punctuation mark: 
+\begin{centred}
+\verb`\eval[p=\ (but no 8!)]{\[ \frac{1}{81} \]}[9]` $\Longrightarrow$
+\eval[p=\ \text{(but no 8!)}]{\[ \frac{1}{81} \]}[9]
+\end{centred}
+
+\subsection{Reuse setting}
+
+This setting determines whether the entire display or only the numerical
+result is saved to file with the \verb`\nmcReuse` command. See below,
+§\ref{subsec:supplReuseEvalSetting}.
+
+\section{Infinite sums and products}
+
+\label{sec:settingsInfiniteSumsProds}There are ways of tweaking various
+default settings to nudge infinite sums and products to a correct
+limit. These tweaks are applied via the settings option of the \verb`\eval`
+command. 
+
+The normal convergence criterion used by \texttt{numerica} to determine
+when to stop adding/multiplying terms in an infinite sum/product is
+\emph{when the next term added/multiplied leaves the total unaltered
+when rounded to 2 more digits than the specified rounding value.}
+Suppose $T_{k}$ is the sum/product after the inclusion of $k$ terms,
+and $r$ is the rounding value. Denote $T_{k}$ rounded to $r$ figures
+by $\left(T_{k}\right)_{r}$. \emph{The infinite sum or product stops
+at the $(k+1)$th term (and the value is attained at the $k$th term)
+when }$\left(T_{k+1}\right)_{r+2}=\left(T_{k}\right)_{r+2}$. The
+hope is that if this is true at rounding value $r+2$ then at rounding
+value $r$ the series or product will have attained a stable value
+at that level of rounding.
+
+For a series of monotonic terms converging quickly to a limit, this
+stopping criterion works well, less so if convergence is slower, as
+seen earlier with the infinite product for $\sinh1$. The criterion
+can fail completely when terms behave in a non-monotonic manner. Terms
+of a Fourier series, for example, may take zero values; the criterion
+is necessarily satisfied but the series may still be far from its
+limit. In a product the equivalent would be a term taking unit value.
+Sometimes the initial terms of series or products are `irregular'
+and take these `stopping' values meaning sum or product would stop
+after only one or two additions/multiplications and far from any limit. 
+
+\begin{table}[t]
+\centering
+\noindent \centering{}\caption{Settings for infinite sums \& products}\label{tab:settingsSumsProducts}
+\noindent \begin{center}
+{\small{}%
+\begin{tabular}{ll>{\raggedright}p{4cm}l}
+\toprule 
+key & type & meaning & default\tabularnewline
+\midrule 
+\texttt{S+} & int & extra rounding for stopping criterion & \texttt{2}\tabularnewline
+\texttt{S?} & $\text{int}\ge0$ & stopping criterion query terms for sums & \texttt{0}\tabularnewline
+\texttt{P+} & int & extra rounding for stopping criterion & \texttt{2}\tabularnewline
+\texttt{P?} & $\text{int}\ge0$ & stopping criterion query terms for products & \texttt{0}\tabularnewline
+\bottomrule
+\end{tabular}}
+\par\end{center}
+\end{table}
+To cope with these possibilities, \texttt{numerica} offers two settings
+for sums, two for products, summarized in Table~\ref{tab:settingsSumsProducts}.
+These are entered in the settings option of the \verb`\eval` command.
+\begin{itemize}
+\item \texttt{S+=<integer> }or \texttt{P+=<integer>} additional rounding
+on top of the specified (or default) rounding for the calculation;
+default = $2$
+\begin{itemize}
+\item the larger the additional \texttt{<integer>} is, the more likely that
+sum or product has attained a stable value at the specified rounding
+$r$ 
+\end{itemize}
+\item \texttt{S?=<integer${}\,\mathtt{\ge0}$> }or \texttt{P?=<integer${}\,\mathtt{\ge0}$>}
+the number of final terms to query after the stopping criterion has
+been achieved to confirm that it is not an `accident' of particular
+values; default = $0$
+\begin{itemize}
+\item a final few terms to be summed/multiplied and the rounded result after
+each such operation to be compared with the rounded result at the
+time the stopping criterion was achieved. Suppose the additional rounding
+(\texttt{S+} or \texttt{P+}) is $n$ on top of the specified rounding
+$r$ and let the number of final checking terms be $m$. Suppose $T_{k_{0}}$
+is the first term at which the stopping criterion is achieved: $\left(T_{k_{0}}\right)_{r+n}=\left(T_{k_{0}+1}\right)_{r+n}$.
+What we require of the final query terms is that $\left(T_{k_{0}}\right)_{r+n}=\left(T_{k_{0}+1+j}\right)_{r+n}$
+for $j=0,1,\ldots,m$.
+\end{itemize}
+\end{itemize}
+Previously we found that the infinite product for $\sinh1$ with the
+default settings gave the wrong value, $0.174$, deficient by $1$
+in the last digit. We now have the means to tweak the stopping criterion
+by increasing the additional rounding: 
+\begin{verbatim}
+    \eval[p,P+=3]{\[
+      \prod_{k=1}^{\infty}
+      \biggl(\frac{x^2}{k^2\pi^2} +1\biggr) 
+    \]}[x=1][3] \nmcInfo{prod}.
+\end{verbatim}
+\noindent $\Longrightarrow$ \noindent \eval[p,P+=3]{\[
+  \prod_{k=1}^{\infty}
+  \biggl(\frac{x^2}{k^2\pi^2} +1\biggr) 
+\]}[x=1][3]  \nmcInfo{prod}.
+
+\noindent To obtain that last item of information (350 factors), I've
+anticipated a little and used the command \verb`\nmcInfo` with the
+argument \verb`prod`; see §\ref{sec:supplInfo}. The product now
+produces the correct three-figure value, but it takes $350$ factors
+to do so. 
+
+Knowing how many terms or factors have been needed helps assess how
+trustworthy the result from an infinite sum or product is. For example,
+for the exponential series,
+\begin{verbatim}
+    \eval[p]{\[
+      \sum_{k=0}^\infty \frac1{k!} 
+    \]}[9] \nmcInfo{sum}.
+\end{verbatim}
+$\Longrightarrow$ \eval[p]{\[\sum_{k=0}^\infty \frac1{k!} \]}[9] \nmcInfo{sum}. 
+
+To $9$ places of decimals, using the default value \texttt{S+=2},
+the exponential series arrives at the right sum after only $15$ terms.
+Convergence is rapid. We can trust this result (and it is in fact
+the correct nine-figure value). By contrast, if we didn't know the
+value of $\sinh1$ beforehand, noting the number of factors required
+would make us justly cautious about accepting the result of the infinite
+product calculation. 
+
+\noindent\begin{minipage}[t]{1\columnwidth}%
+\begin{shaded}%
+One way to gain confidence in a result is to choose a possibly unrealistic
+rounding value – say, the default $6$ for the infinite product –
+then use \emph{negative} values for the extra rounding, \texttt{S+=-5},
+\texttt{S+=-4}, \ldots{} , so that the stopping criterion applies at
+rounding values $s$ of $6+(-5)=1$, one decimal place, $6+(-4)=2$,
+two decimal places, and so on, but the result is always presented
+to $6$ decimal places. You can then see how the $6$-figure results
+behave relative to the number of terms it takes to meet the stopping
+criterion. A little experimenting shows that for the infinite product
+for $\sinh1$ the number of factors $N_{s}$ required at a stopping
+rounding value $s$ increases in geometric proportion with a scale
+factor of about $3$: $N_{s}\approx\text{const}\times3^{s}$. This
+rapidly becomes large ($3^{4}=81,3^{5}=243\dots$). For the exponential
+series on the other hand $N_{s}=4+s$, the number of terms increases
+only slowly, in direct proportion to the stopping rounding value.
+Similar experiments with the sums of inverse fourth, third and second
+powers of the integers using \verb`\nmcInfo` to find how many terms
+are required at each stopping rounding value, show that at least over
+the rounding value range $1$ to $8$, for inverse fourth powers $N_{s}\approx\text{const}\times1.7^{s}$,
+for inverse third powers $N_{s}\approx\text{const}\times2^{s}$ and
+for inverse squares $N_{s}\approx\text{const}\times3^{s}$. All are
+geometric rather than arithmetic progressions, but for inverse fourth
+powers the scale factor ($\approx1.7$) is sufficiently small that
+for these low values of $s$ the number of terms required does not
+grow too quickly (e.g. $1.7^6\approx\eval{1.7^6}[0]$). It is a standard
+result (Euler) that the series sums to $\pi^{4}/90$: \verb`$ \eval{ \pi^4/90 } $`
+$\Longrightarrow$ $ \eval{ \pi^4/90 } $ to six places, and indeed,
+with the default \texttt{S+=2},
+\begin{centred}
+\verb`\eval[p]{\[ \sum_{k=1}^\infty \frac1{k^4} \]}` $\Longrightarrow$
+\eval[p=.]{\[ \sum_{k=1}^\infty \frac1{k^4} \]}
+\end{centred}
+\end{shaded}%
+\end{minipage}
+
+\subsection{Premature ending of infinite sums}
+
+All the series considered so far have been monotonic. Trigonometric
+series will generally not be so, nor even single-signed. 
+
+Trigonometric sums are computationally intensive and so, for the following
+example, I have specified a rounding value of 2. The series 
+\[
+\sum_{n=1}^{\infty}\frac{4}{n^{2}\pi^{2}}(1-\cos n\pi)\cos2\pi nt
+\]
+is the Fourier series for the triangular wave function \textbackslash\!/\!\textbackslash\!/\!\textbackslash\!/\!\textbackslash{}
+\ldots{} of period 1, symmetric about the origin where it takes its
+maximum value 1, crossing the axis at $t=0.25$ and descending to
+its minimum $-1$ at $t=0.5$, before ascending to a second maximum
+at $t=1$ (and so on). In the interval $[0,0.5)$ the series should
+sum to $1-4t$. The problem is that the summand $\frac{4}{n^{2}\pi^{2}}(1-\cos n\pi)\cos2\pi nt$
+vanishes both when $n$ is even and when $4nt$ is an odd integer.
+If $t=0.1$ then $4nt$ is never an odd integer so the summand vanishes
+only for $n$ even, every second term. We expect the result to be
+$1-4\times0.1=0.6$.
+\begin{verbatim}
+    \eval[p]{\[
+      \sum_{n=1}^{\infty}
+        \frac{4}{n^{2}\pi^{2}}
+        (1-\cos n\pi)\cos2\pi nt 
+    \]}[t=0.1][2] \nmcInfo{sum}.
+\end{verbatim}
+$\Longrightarrow$ \eval[p]{\[
+  \sum_{n=1}^{\infty}
+    \frac{4}{n^{2}\pi^{2}}
+    (1-\cos n\pi)\cos2\pi nt 
+\]}[t=0.1][2]  \info{sum}.
+
+\noindent Only one term? Of course – since for the second term $n$
+is even, the term vanishes and the stopping criterion is satisfied.
+The way around this problem is to query terms \emph{beyond} the one
+where the stopping criterion is achieved, i.e., to set \texttt{S?}
+to a nonzero value. We try \texttt{S?=1}: 
+\begin{verbatim}
+    \eval[p,S?=1]{\[
+      \sum_{n=1}^{\infty}
+        \frac{4}{n^{2}\pi^{2}}
+        (1-\cos n\pi)\cos2\pi nt 
+    \]}[t=0.1][2] \nmcInfo{sum}.
+\end{verbatim}
+$\Longrightarrow$ \eval[p,S?=1]{\[
+  \sum_{n=1}^{\infty}
+    \frac{4}{n^{2}\pi^{2}}
+    (1-\cos n\pi)\cos2\pi nt 
+\]}[t=0.1][2] \info{sum}.
+
+\noindent\begin{minipage}[t]{1\columnwidth}%
+\begin{shaded}%
+\begin{wraptable}{o}{0.35\columnwidth}%
+\centering{}\vspace{-5ex}
+\caption{Finite sums}\label{tab:settingsFinite-sums}
+\abovetopsep=2ex %
+\begin{tabular}{cc}
+\toprule 
+$N$ & $\Sigma$\tabularnewline
+\midrule
+$63$ & $0.6001$\tabularnewline
+$64$ & $0.6001$\tabularnewline
+$65$ & $0.5999$\tabularnewline
+$66$ & $0.5999$\tabularnewline
+$67$ & $0.5999$\tabularnewline
+\bottomrule
+\end{tabular}\end{wraptable}%
+Table~\ref{tab:settingsFinite-sums} lists the results of evaluating
+the \emph{finite }sums from $n=1$ to $N$ for values of $N$ around
+$65$. Since the specified rounding value is $2$ for the calculation,
+the stopping criterion\emph{ }applies at a rounding value of $2$
+more than that, $4$. Since $N=64$ is even, the summand for the $64$th
+term is zero and the sum takes the same value as for $N=63$. The
+$65$th term is the query term and the sum differs, so the summation
+continues. The $66$th term vanishes, so the stopping criterion is
+met. This time for the query term, the $67$th, the sum retains the
+same $4$-figure value, and the summation stops. The result was attained
+at the $65$th term. Should we be confident in the result? Increase
+the number of query terms to $3$ (there is no point in increasing
+\texttt{S?} to $2$ because of the vanishing of the even terms), the
+sum stops after $113$ terms, with the same $0.6$ result.  \end{shaded}%
+\end{minipage}
+
+For a final example, consider the error function 
+\[
+\erf z=\dfrac{2}{\sqrt{\pi}}\int_{0}^{z}e^{-t^{2}}dt
+\]
+which can also be rendered as an infinite sum (\emph{HMF }7.1.5):
+\[
+\erf z=\sum_{n=0}^{\infty}(-1)^{n}\frac{z^{2n+1}}{n!(2n+1)}.
+\]
+(\verb`\erf` expanding to \verb`erf` has been defined in the preamble
+to this document using \verb`\DeclareMathOperator`.) We calculate
+this sum for $z=2$ to $10$ places of decimals. Although this is
+an alternating series, it is obvious that the summand never vanishes
+when $z\ne0$ as here. Hence there seems no need to change the default
+value \texttt{S?=0}.
+\begin{verbatim}
+    \eval[p]{\[ 
+      \frac2{\sqrt{\pi}}
+        \sum_{n=0}^\infty(-1)^n
+          \frac{z^{2n+1}}{n!(2n+1)}   
+    \]}[z=2][10*] \nmcInfo{sum}.
+\end{verbatim}
+$\Longrightarrow$ \eval[p]{\[ 
+  \frac2{\sqrt{\pi}}
+    \sum_{n=0}^\infty(-1)^n
+      \frac{z^{2n+1}}{n!(2n+1)}   
+\]}[z=2][10*] \nmcInfo{sum}.
+
+According to \emph{HMF }Table 7.1, this calculated value of $\erf2$
+is correct to all $10$ places. But beyond $z=2$ errors will begin
+to interfere with the result. Note that $26$ terms means $n=26$
+was the last value of $n$ for which the summand was evaluated. (The
+sum stops at the $26$th term, $n=25$, but the next term $n=26$
+needs to be calculated for the stopping criterion.) Fortuitously,
+$2^{2\times26+1}=2^{53}$ is the greatest power of $2$ that can be
+\emph{exactly} rendered to the $16$ significant figures that \texttt{l3fp}
+uses. But $n!$ exceeds the $16$-significant figure limit of \texttt{l3fp}
+when $n>21$, so despite the 10-figure result, errors have already
+begun to occur in the denominator of the summand and accrue in the
+sum when $z=2$. For larger $z$ values the errors can only get worse
+and at some point will render the calculated value worthless at any
+meaningful rounding value. For example, when $z=7$ the sum apparently
+`evaluates' to over $929$ whereas we know that 
+\[
+\erf z<\dfrac{2}{\sqrt{\pi}}\int_{0}^{\infty}e^{-t^{2}}dt=1.
+\]
+
+
+\subsection{Double sums or products}
+
+Sums or products can be iterate d. For instance, the exponential function
+can be calculated this way:
+\begin{verbatim}
+    \eval[p]
+      {\[ \sum_{k=0}^{\infty}
+        \prod_{m=1}^{k}\frac{x}{m} \]}[x=2]
+\end{verbatim}
+$\Longrightarrow$     \eval[p]
+     {\[ \sum_{k=0}^{\infty}
+       \prod_{m=1}^{k}\frac{x}{m} \]}[x=2]
+
+\noindent which is \verb`\eval{$ e^2 $}` $\Longrightarrow\eval{\ensuremath{e^{2}}}$.
+
+A second example is afforded by Euler's transformation of series (\emph{HMF~}3.6.27).
+To calculate $e^{-1}$ we use 
+\begin{verbatim}
+    \eval[p]
+      {\[ \sum_{n=0}^{\infty}
+              \frac{(-1)^{n}}{n!} \]}[3] \info{sum}.
+\end{verbatim}
+$\Longrightarrow$ \eval[p]{\[ \sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!} \]}[3] \info{sum}.
+
+Following Euler, this series can be transformed to the form
+\begin{verbatim}
+    \eval[p,S?=1]{\[
+      \sum_{k=0}^\infty \frac{(-1)^k}{2^{k+1}}
+      \sum_{n=0}^k(-1)^n\binom kn \frac1{(k-n)!} 
+    \]}[3] \nmcInfo{sum}.
+\end{verbatim}
+$\Longrightarrow$ \eval[p,S?=1]{\[ \sum_{k=0}^\infty \frac{(-1)^k}{2^{k+1}}\sum_{n=0}^k(-1)^n\binom kn \frac1{(k-n)!} \]}[3] \nmcInfo{sum}.
+
+\noindent Note the setting \verb`S?=1`. Without it, the summation
+stops after $1$ term, the $k=0$ term, because the $k=1$ term vanishes.
+With \verb`S?=1` it takes $16$ terms of the \emph{outer }sum to
+reach the stopping criterion. Since that sum starts at $0$, that
+means that changing the upper limit from $\infty$ to $15$ should
+give the same result – which it does – but it takes $\tfrac{1}{2}\times16\times17=136$
+terms in total to get there, to be compared with the $9$ terms of
+the earlier simpler sum, and the terms are more complicated. Obviously
+such double sums are computationally intensive.
+
+\section{Changing default values}
+
+\label{sec:settingsDefaults}The settings option enables various settings
+to be changed for an individual calculation. You may find yourself
+wanting to make such changes sufficiently often that a change of default
+value is a better plan than encumbering each calculation with a list
+of settings.
+
+The way to do that is to create a \emph{configuration file} named
+\texttt{numerica.cfg} in a text editor. Its entries are of the form
+\emph{key=value} followed by a comma, and for clarity preferably one
+entry per line (although this is not essential).The key names are
+noticeably more verbose than the corresponding keys of the settings
+option.\emph{ }The possible keys are listed in Table~\ref{tab:settingsDefaults},
+together with their current default values. 
+
+\begin{table}[t]
+\centering
+\noindent \centering{}\caption{Default values, \texttt{\textbackslash eval} command}\label{tab:settingsDefaults}
+\noindent \begin{center}
+\begin{tabular}{ll}
+\toprule 
+{\small key} & {\small value}\tabularnewline
+\midrule
+{\small rounding} & {\small\texttt{6}}\tabularnewline
+{\small pad} & {\small\texttt{0}}\tabularnewline
+{\small output-sci-notation} & {\small\texttt{0}}\tabularnewline
+{\small output-exponent-char} & {\small\texttt{e}}\tabularnewline
+{\small\%} & \tabularnewline
+{\small input-sci-notation} & {\small\texttt{0}}\tabularnewline
+{\small input-exponent-char} & {\small\texttt{e}}\tabularnewline
+{\small multitoken-variables} & {\small\texttt{1}}\tabularnewline
+{\small use-degrees} & {\small\texttt{0}}\tabularnewline
+{\small logarithm-base} & {\small\texttt{10}}\tabularnewline
+{\small vv-display} & {\small\texttt{\{,\}\textbackslash mskip 36mu minus 24mu(vv)}}\tabularnewline
+{\small vv-inline} & {\small\texttt{\{,\}\textbackslash mskip 12mu 6mu minus 9mu(vv)}}\tabularnewline
+{\small\%} & \tabularnewline
+{\small sum-extra-rounding} & {\small\texttt{2}}\tabularnewline
+{\small sum-query-terms} & {\small\texttt{0}}\tabularnewline
+{\small prod-extra-rounding} & {\small\texttt{2}}\tabularnewline
+{\small prod-query-terms} & {\small\texttt{0}}\tabularnewline
+\% & \tabularnewline
+{\small intify-rounding} & {\small\texttt{14}}\tabularnewline
+{\small eval-reuse} & {\small\texttt{0}}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+\end{table}
+Keys taking one of two possible values, \verb`0` (for \verb`false/off`)
+or \verb`1` (for \verb`true/on`), are \verb`pad` (the result with
+zeros), \verb`output-sci-notation`, \verb`input-sci-notation`, (check
+for) \verb`multitoken-variables`, and \verb`use-degrees` (for trig.
+functions).
+
+The table is divided into four parts. 
+\begin{itemize}
+\item The top four rows concern elements that can be changed for individual
+calculations with the trailing optional argument of \verb`\eval`:
+rounding, padding with zeros, and outputting in scientific notation;
+see §\ref{subsec:evalRoundingEtc}. 
+\begin{itemize}
+\item Note that to output the result always in scientific notation requires
+two settings, first setting \texttt{output-sci-notation} to \texttt{1},
+and then choosing a character to act as the exponent marker. Because
+\texttt{l3fp} uses \texttt{e} for this character, \texttt{numerica}
+has made \texttt{e} its default. But this option is turned off by
+default (hence the \texttt{0} against this key).
+\end{itemize}
+\item The next block of rows concern general elements that can be changed
+for individual calculations with the settings option of \verb`\eval`;
+see §\ref{sec:settingsOption}. The key names are more expansive here
+but the effect is the same. 
+\begin{itemize}
+\item Note that to input numbers in scientific notation requires two settings,
+first setting \texttt{input-sci-notation} to \texttt{1}, and then
+choosing a character to act as the exponent marker. Because \texttt{l3fp}
+uses \texttt{e} for this character, \texttt{numerica} has made \texttt{e}
+its default. The option is turned off by default (hence the \texttt{0}
+against this key). 
+\end{itemize}
+\item The third block of rows concern default settings for infinite sums
+and products. These correspond to the keys \texttt{S+}, \texttt{S?}
+and \texttt{P+}, \texttt{P?} of the settings option that can be used
+to tweak the behaviour of the stopping criterion for such sums or
+products; see §\ref{sec:settingsInfiniteSumsProds}.
+\item The last block is for `left-overs': specifying at what rounding
+value a floating point should be considered an integer (see §\ref{subsec:defaultsIntifyingRounding}
+below), and specifying what kind of result is saved to file when the
+\verb`\nmcReuse` command is used (see §\ref{subsec:supplReuseEvalSetting}).
+\end{itemize}
+If you are dissatisfied with any of the default values listed, then
+in a text editor create a new file called \texttt{numerica.cfg} and
+assign \emph{your} values to the relevant keys. For instance, if you
+find yourself working to $4$ figures, that rounding to $6$ is too
+many, then make the entry \texttt{rounding~=}~4. If also you want
+results always presented in proper scientific notation,\emph{ $d.d_{1}d_{2}d_{3}d_{4}\times10^{n}$},
+then add a comma after \texttt{4} and enter on a new line (recommended
+but not strictly necessary; the comma is the crucial thing), \texttt{output-sci-notation~=~1,}
+(note the comma) and on another new line, \texttt{output-exponent-char~=~x}. 
+
+Perhaps you also want a non-zero setting for the final query terms
+for infinite sums and products. This makes sense if you are largely
+dealing with non-monotonic series – like Fourier series. Even the
+Euler transformation of the exponential series for $e^{-1}$ discussed
+above required a non-zero \texttt{S?}. If you wish to make this change
+then add a comma and on a new line add (for instance) \texttt{sum-query-terms~=~1,}
+and again on a new line, \texttt{prod-query-terms~=}~1. If this
+is all you wish to change, then no comma is necessary after this final
+entry. Your newly created file should look something like
+\begin{lyxcode}
+rounding~~~~~~~~~~~~~=~4,
+
+output-sci-notation~~=~1,
+
+output-exponent-char~=~x,
+
+sum-query-terms~~~~~~=~1,
+
+prod-query-terms~~~~~=~1
+\end{lyxcode}
+The white spacing may be different; white space is ignored by \texttt{numerica}
+when reading the file. Using it to align the equals signs helps \emph{us}
+read the file. Note that the last entry, because it is the last entry,
+lacks a comma. Now save the file with the name \texttt{numerica.cfg}.
+This file will be read by \texttt{numerica} near the end of its loading
+process. These settings will be \texttt{numerica}'s defaults for the
+relevant keys. 
+
+\subsection{Location of \texttt{numerica.cfg}}
+
+Save, yes, but where to? If the new settings are likely to apply only
+to your current document, then the document's directory is a sensible
+place to put it and \texttt{numerica} will certainly find it there
+since it is part of \LaTeX 3 file handling that file searches are
+not limited to the \TeX{} distribution (including your personal texmf
+tree) but also include the current document directory. But what happens
+when you start working on another document? Will you remember to copy
+\texttt{numerica.cfg} to its new location? That is why your \emph{personal
+texmf tree} is a better place. 
+
+\subsubsection{Personal texmf tree? }
+
+\label{subsec:settingsPersonal-texmf-tree}This is a directory for
+`waifs and strays' of the \TeX{} system that are not included in
+the standard distributions like MiK\TeX{} or \TeX Live. Here you place
+personal packages designed for your own particular circumstances.
+These may include your own \TeX{} or \LaTeX{} package, say \texttt{mypackage.sty},
+achieving some small or singular effect that doesn't warrant wider
+distribution on CTAN. Here you might place configuration files for
+other packages with your preferences (unless the package requires
+some specific location). Here you can put your personal bibliography
+files. 
+
+Your personal texmf tree is structured like the standard MiK\TeX{}
+or \TeX Live hierarchy but placed in another location so that there
+is no chance of its being overwritten when packages in MiK\TeX{} or
+\TeX Live are updated. But these distributions need to be alerted
+to its existence. 
+
+For example, in the MiK\TeX{} console, click on \textsf{Settings},
+and then on the \textsf{Directories} tab of the resulting dialog.
+Here you get to add your personal texmf hierarchy to the list of paths
+that MiK\TeX{} searches, by clicking on the \textsf{+} button, browsing
+to your texmf folder and selecting it. By using the up and down arrow
+keys that the MiK\TeX{} console provides, ensure that it lies \emph{above
+}the the entry for the main MiK\TeX{} tree. That way, files in your
+personal texmf tree will be found first and loaded. Now go to the
+\textsf{Tasks} menu and click on \textsf{Refresh the filename database}.
+This will let MiK\TeX{} know what is held in your personal texmf tree.
+Files there can then be used like standard \LaTeX{} packages.
+
+\subsection{Rounding in \textquoteleft int-ifying\textquoteright{} calculations}
+
+\label{subsec:defaultsIntifyingRounding}Factorials, binomial coefficients,
+summation and product variables, and (in \texttt{numerica}) $n$th
+roots from the \texttt{\textbackslash sqrt} command, all require
+integer arguments. These integers may indeed be entered explicitly
+as integers, but they can also be determined as the result of a calculation.
+Rounding errors may mean the result is not an exact integer. How much
+leeway should be allowed before it is clear that the calculation did
+not give an integer result? In the default setup, \texttt{numerica}
+is generous. A number is considered an integer if it rounds to an
+integer when the rounding value is $14$. Since \texttt{l3fp} works
+to $16$ significant figures this provides more than enough `elbowroom'
+for innocuous rounding errors to be accommodated. If a calculation
+does not round to an integer at a rounding value of $14$ then it
+seems reasonable to conclude that it has \emph{really }not given an
+integer answer, not just that rounding errors have accumulated. If
+you want to change this `int-ifying' value for a particular calculation,
+then add a line to \texttt{numerica.cfg} like
+\begin{lyxcode}
+intify-rounding~=~<integer>
+\end{lyxcode}
+Since \texttt{l3fp} works to $16$ significant figures, values of
+{\ttfamily\verb`<integer>`} greater than $16$ are pointless.
+Generally int-ifying rounding values will be less than but close to
+$16$ (although when testing the code I used some ridiculous values
+like $3$ or $4$). If other entries follow this one in the file,
+then conclude the line with a comma.
+
+\section{Parsing mathematical arguments}
+
+\label{sec:Argument-parsing} A main aim of the \texttt{numerica}
+package is to require minimal, preferably no, adjustment to the \LaTeX{}
+form in which an expression is typeset in order to evaluate it. But
+mathematicians do not follow codified rules of the kind programming
+languages insist on when writing formulas – like parenthesizing the
+arguments of functions, or inserting explicit multiplication signs
+({*}) between juxtaposed terms. Hence the question of where the arguments
+of mathematical functions end is acute. For a few functions \LaTeX{}
+delimits the argument: think of \verb`\sqrt`, \verb`\frac`, \verb`\binom`;
+also \verb`^`. But for functions like \verb`\sin` or \verb`\tanh`
+or \verb`\ln`, unary functions, this is not so; nor is it for sums
+and products, and comparisons.
+
+Before discussing the parsing rules for different groups of functions,
+I discuss the means \texttt{numerica} provides to handle exceptions
+to those rules, when one \emph{does} need to make some adjustment
+to a formula.
+
+\subsection{The cleave commands \texttt{\textbackslash q} and \texttt{\textbackslash Q}}
+
+\label{subsec:parseCleave}The word \emph{cleave} has two opposed
+meanings: to adhere or cling to, and to split apart or separate. \texttt{numerica}
+defines two commands, \verb`\q` and \verb`\Q` to achieve these opposite
+effects. When a mathematical argument is being parsed, the \verb`\q`
+command joins the next token to the argument (\emph{cleaves to});
+the \verb`\Q` command severs the next token from the argument (\emph{cleaves
+apart}). Neither command is added to the argument nor leaves a visible
+trace in the output. 
+
+Thus, without \verb`\q`, 
+\begin{centred}
+\verb`\eval{$ \sin(n+\tfrac12)(x-t) $}[n=3,x=t+\pi,t=1.234]` $\Longrightarrow$
+\eval{$ \sin(n+\tfrac12)(x-t) $}[n=3,x=t+\pi,t=1.234],
+\end{centred}
+which is $(\sin\tfrac{7}{2})\times\pi$. With \verb`\q` between the
+bracketed factors,
+\begin{centred}
+\verb`\eval{$ \sin(n+\tfrac12)\q(x-t) $}[n=3,x=t+\pi,t=1.234]` $\Longrightarrow$
+\eval{$ \sin(n+\tfrac12)\q(x-t) $}[n=3,x=t+\pi,t=1.234],
+\end{centred}
+which is $\sin(\tfrac{7}{2}\pi)$. Similarly, without \verb`\q`,
+\begin{centred}
+\verb`\eval[p]{\[ \cos\frac{2\pi}{T}n(t+\tfrac12T) \]}[T=2,t=1,n=3]`
+$\Longrightarrow$ \eval[p]{\[ \cos\frac{2\pi}{T}n(t+\tfrac12T) \]}[T=2,t=1,n=3]
+\end{centred}
+which is $(\cos\pi)\times3\times(1+\tfrac{1}{2}\times2)$. With \verb`\q`
+used twice, once after the fraction and once before the left parenthesis,
+\begin{verbatim}
+    \eval[p]{\[ 
+      \cos\frac{2\pi}{T}\q n\q(t+\tfrac12T)
+        \]}[T=2,t=1,n=3]
+\end{verbatim}
+$\Longrightarrow$ \eval[p]{\[ \cos\frac{2\pi}{T}\q n\q(t+\tfrac12T) \]}[T=2,t=1,n=3]
+
+\noindent which is $\cos(\pi\times3\times2)$.
+
+It should be noted that for \emph{trigonometric }functions, because
+of their use in Fourier series especially, there is another way of
+handling arguments that contain parentheses (and fractions). This
+is discussed in §\ref{subsec:parseTrigFns} below.
+
+For the \verb`\Q` command which splits an argument we have, without
+it,
+\begin{centred}
+\verb`\eval{$ 1/2e $}` $\Longrightarrow$ \eval{$ 1/2e $}, 
+\end{centred}
+which is the reciprocal of $2e$, whereas with the \verb`\Q` command
+inserted before \verb`e`,
+\begin{centred}
+\verb`\eval{$ 1/2\Q e $}` $\Longrightarrow$ \eval{$ 1/2\Q e $},
+\end{centred}
+which is one half of $e$, although it is unlikely to be read that
+way. If one half of $e$ is intended then parenthesize the $1/2$
+or present as a \verb`\tfrac`.
+
+\subsubsection{Mnemonic}
+
+As mnemonic, best seen in sans serif for the Latin Modern fonts used
+in this document, think of the letter \textsf{q} as a circle \emph{cleaving
+to} a vertical descender; think of the letter \textsf{Q} as a circle
+\emph{cleaved apart} by the diagonal stroke.
+
+\subsection{Parsing groups}
+
+The arguments of different groups of functions are handled in different
+ways. The criterion used for deciding when an argument ends for one
+group will not be that used for others. Table §\ref{subsec:defaultsIntifyingRounding}
+lists the different groups that \texttt{numerica} takes account of.
+At the top are functions or operations that have the smallest reach
+when determining where their arguments end; at the bottom are operations
+that have the greatest reach. The denominator of a slash fraction
+is treated as a unary function and is assigned to group II. By default
+trigonometric functions are treated the same as other unary functions
+but there is a setting which enables the direct (rather than inverse)
+trigonometric functions to accept a wider range of arguments, as occurs
+in Fourier series. Hence they are separated into their own group.
+
+\begin{table}
+\centering
+
+\caption{ Parsing groups}\label{tab:settingsParsing-groups}
+{\ttfamily{}%
+\begin{tabular}{ll}
+\toprule 
+{\small\textrm{group}} & {\small\textrm{function/operation}}\tabularnewline
+\midrule
+{\small\textrm{I}} & {\small\textrm{surd, logical Not}}\tabularnewline
+{\small\textrm{II}} & {\small\textrm{unary functions (direct trig. functions default), /}}\tabularnewline
+{\small\textrm{III}} & {\small\textrm{direct trig. functions with special setting}}\tabularnewline
+{\small\textrm{IV}} & {\small\textrm{sums, products}}\tabularnewline
+{\small\textrm{V}} & {\small\textrm{comparisons}}\tabularnewline
+{\small\textrm{VI}} & {\small\textrm{logical And, logical Or}}\tabularnewline
+\bottomrule
+\end{tabular}}
+
+\end{table}
+ A formula is a sequence of tokens and brace groups. All parsing occurs
+from the left, \LaTeX{} argument by \LaTeX{} argument, where \emph{argument}
+means either a token (an N-type argument in \verb`expl3`-speak) or
+a brace group (an n-type argument). To distinguish \LaTeX{} arguments
+from mathematical arguments I shall when necessary refer to L-args
+and M-args. A mathematical argument may end \emph{at} an L-arg, meaning
+immediately before the L-arg, or end \emph{with} the L-arg, meaning
+immediately after the L-arg. Ending or not will in general depend
+on whether the argument is in \emph{first position} – the position
+immediately following a function token like \verb`\sin` or \verb`\log`
+– or in \emph{general position} – any later position (although for
+trigonometric functions we will also need to consider \emph{second}
+and even \emph{third }positions). 
+
+For counting position, we need to allow for formatting elements and
+multi-token numbers – in both decimal and scientific formats. Formatting
+elements do not change the position count. This applies to things
+like thin spaces or phantoms (and their arguments) or modifiers like
+\verb`\left` or \verb`\biggl`. Multi-token numbers (in decimal or
+scientific formats) are treated as single items; they advance the
+position count by exactly one. \LaTeX{} functions – like \verb`\frac`
+– which take \LaTeX{} arguments again advance the position count only
+by one. Mathematically, the fraction is viewed as a single unit.
+
+I shall refer to a token or a token and its \LaTeX{} arguments – like
+\verb`\frac` and its arguments – as an \emph{item}. Similarly, a
+(possibly multi-token) number is an item. Also it will help to distinguish
+tokens within brackets where both brackets lie to the right of a function
+from those that do not. The former I call \emph{clothed}; the latter
+are \emph{naked}. Thus the plus sign in $(\sin x+y)$ is naked relative
+to the sine (one bracket to the left of the function), but is clothed
+in $\sin(x+y)$ (both brackets to the right of the function). 
+
+\subsubsection{Parsing group I}
+
+The only functions in this category are the surd and logical Not. 
+
+Why distinguish the surd from other unary functions? Surely we all
+agree that \verb`\sin2\pi`, displaying as $\sin2\pi$, vanishes?
+The argument of the sine extends beyond the $2$ to include the $\pi$.
+But \verb`\surd2\pi`, displaying as $\surd2\pi$, is understood to
+be the product $\surd2\times\pi$. The argument of the surd ends with
+the $2$. The surd binds more tightly to its argument than is true
+of unary functions generally. 
+
+For parsing group I
+\begin{enumerate}
+\item if a left bracket is in first position, the mathematical argument
+ends with the matching right bracket; otherwise
+\item the argument ends with the item in first position and any L- or M-args
+required by that item.
+\end{enumerate}
+If the factorial sign \verb`!` \emph{preceded} its argument, it too
+would belong to this parsing state, for it also binds tightly like
+the surd. This means that an expression like $\surd4!$ is intrinsically
+ambiguous. Is it the square root of $24$ or the factorial of $2$?
+In \texttt{numerica} it produces the (perhaps rather odd) error
+\begin{centred}
+\verb`\eval{$ \surd 4! $}` $\Longrightarrow$ \eval{$ \surd4! $}
+\end{centred}
+The surd has seized the argument; there is nothing for the factorial
+to operate on. The same error arises if the $4$ is parenthesized,
+but parenthesizing like either \verb`(\surd 4)!` or \verb`\surd(4!)`
+repairs the situation. Because other unary functions (like the sine
+or logarithm) do not bind as tightly, this ambiguity does not arise
+for them.
+
+Exponents cause no problem because taking square roots and raising
+to a power are commutative operations – the result is the same whichever
+is performed first. 
+\begin{centred}
+\verb`\eval{$ \surd 3^4 $}` $\Longrightarrow$ \eval{$ \surd 3^4 $}.
+\end{centred}
+
+\subsubsection{Parsing group II}
+
+In the default setup this category includes the trigonometric and
+hyperbolic functions, their inverses, the various logarithms and the
+exponential functions, the signum function \verb`\sgn`, and the denominators
+of slash fractions \verb`/`. Note however that there is a setting
+switch which enables trigonometric  functions to handle parentheses
+in arguments more generally; see §\ref{subsec:parseTrigFns}. 
+\begin{itemize}
+\item In parsing group II we wish to accommodate usages like $\ln z^{n}=n\ln z$
+(\emph{HMF} 4.1.11), or $\gd z=2\arctan e^{z}-\frac{1}{2}\pi$ (\emph{HMF}
+4.3.117), defining the Gudermannian. The exponent is included in the
+argument. Considering $\ln(1+1/n)^{n}$ exponents must also be part
+of parenthesized arguments. 
+\item An approximation to Stirling's formula for the factorial is often
+written $\ln N!\approx N\ln N-N$ (widely used in texts on statistical
+mechanics). Hence the factorial sign should also be considered part
+of the argument.
+\item $\ln xy=\ln x+\ln y$ means the argument must reach over a product
+of variables. Identities like $\sin2z=2\sin z\cos z$ mean the argument
+also reaches over numbers, and expressions like $\sin\tfrac{1}{2}\pi x$
+(\emph{HMF} 4.3.104) mean that it further reaches over \verb`\tfrac`-s
+and constants.
+\item Essentially \emph{anything }can be in first position, and without
+parentheses; e.g.
+\begin{itemize}
+\item unary functions: $\ln\ln z$ (\emph{HMF} 4.1.52), $\ln\tan\dfrac{z}{2}$
+(\emph{HMF} 4.3.116),
+\item fractions: $\ln\dfrac{z_{1}}{z_{2}}$ (\emph{HMF} 4.1.9), $\arcsin\dfrac{(2ax+b)}{(b^{2}-4ac)^{1/2}}$
+(\emph{HMF} 3.3.36), $\ln\dfrac{\tan z}{z}$ (\emph{HMF} 4.3.73),
+\item absolute values: $\ln\abs*{\dfrac{a+x}{a-x}}$ (\emph{HMF} 3.3.25),
+\item square roots: $\arctan\sqrt{\dfrac{\nu_{1}}{\nu_{2}}F}$ (\emph{HMF
+}26.6.8)
+\end{itemize}
+\end{itemize}
+With these examples in mind, for parsing group II
+\begin{enumerate}
+\item if a left bracket is in first position, the mathematical argument
+ends with the matching right bracket and any attached exponent, or
+factorial or double factorial sign; otherwise
+\item the mathematical argument includes the item in first position and
+any L- or M-args required by that item; 
+\begin{enumerate}
+\item if the item in first position is a number, variable, constant or \verb`\tfrac` 
+\begin{enumerate}
+\item the argument appends the next item if it is a number, variable, constant
+or \verb`\tfrac`, and so on recursively; or
+\item the argument appends the next item if it is an exponent, or facorial
+or double factorial sign, and ends there; otherwise
+\item the argument ends.
+\end{enumerate}
+\item if the item in first position is not a number, variable, constant
+or \verb`\tfrac`
+\begin{enumerate}
+\item the argument appends the next item if it is an exponent, or factorial
+or double factorial sign, and ends there; otherwise
+\item the argument ends.
+\end{enumerate}
+\end{enumerate}
+\end{enumerate}
+An argument may extend over (see 2(a)i) numbers, constants, variables
+and \verb`\tfrac`-s, as instanced with $\sin2\tfrac{p}{q}\pi x$
+which exhibits all elements.
+
+Illustrating 1, the exponent is included in the argument but not the
+following variable:
+\begin{centred}
+\verb`\eval{$ \log_{10}(1+2+3+4)^3n $}[n=5]` $\Longrightarrow$ \eval{$ \log_{10}(1+2+3+4)^3n $}[n=5].
+\end{centred}
+For the sake of the reader, and as one naturally does in any case
+to avoid ambiguity, the formula should be written with the variable
+$n$ preceding the logarithm: $n\log_{10}(1+2+3+4)^{3}$. The way
+the example is written suggests that the writer wished the $n$ to
+be considered part of the argument of the logarithm. If that is the
+case, inserting a \verb`\q` command before \verb`n` would achieve
+this, but that would still be confusing for the reader of the pdf.
+Inserting parentheses is the only sensible thing to do. 
+
+Illustrating 2(a)ii, again the exponent is included in the argument
+but not the following variable:
+\begin{centred}
+\verb`\eval{$ \log_{10}m^3n $}[m=10,n=5]` $\Longrightarrow$ \eval{$ \log_{10}m^3n $}[m=10,n=5].
+\end{centred}
+Again, for the sake of the reader and as one naturally does to avoid
+ambiguity, the variable $n$ should precede the logarithm. If in fact
+the intention was for the $n$ to be included in the argument of the
+logarithm, then again the \verb`\q` command could be used or, better
+in this case, the $n$ could be shifted to precede the $m$, which
+illustrates 2(a)i:
+\begin{centred}
+\verb`\eval{$ \log_{10}nm^3 $}[m=10,n=5]` $\Longrightarrow$ \eval{$ \log_{10}nm^3 $}[m=10,n=5],
+\end{centred}
+the logarithm of $5000$, or better still, $m^{3}n$ could (and should)
+be parenthesized for the sake of the reader.
+
+Why the difference between $nm^{3}$ where $n$ and $m^{3}$ are included
+in the argument, and $m^{3}n$ where $n$ is not? Any criterion is
+going to miss some instances where a different outcome might be desirable.
+Where an argument ends is affected by visual appearance in the pdf.
+It is simple and easy to remember if it is understood that anything
+that breaks the `visual flow' of juxtaposed numbers, variables,
+constants and \verb`\tfrac`-s ends the argument. An exponent does
+just that. If you feel there is ambiguity, parenthesize to clarify.
+
+Illustrating 2(b)ii, the argument stops with the \verb`\dfrac` and
+its arguments and does not extend to the following constant:
+\begin{centred}
+\verb`\eval{$ \sin\dfrac12\pi $}` $\Longrightarrow$ \eval{$ \sin\dfrac12\pi $}.
+\end{centred}
+Obviously, someone writing an expression like this intends the $\pi$
+to be part of the argument. In that case, a \verb`\tfrac` should
+be used since the \verb`\dfrac` breaks the `visual flow' of the
+argument. 
+\begin{description}
+\item [{Fractions}]~
+
+But why not a plain \verb`\frac`? After all, for an inline expression
+it displays in the same way as a \verb`\tfrac`. I considered making
+the argument-behaviour of \verb`\frac` the same as \verb`\tfrac`
+for text-style contexts, and the same as \verb`\dfrac` for display-style
+contexts, but that would have meant the same expression evaluating
+to different results depending on whether it lay between \verb`$ $`
+or \verb`\[ \]` delimiters, which ruled it out. Because \verb`\frac`
+sometimes displays as \verb`\dfrac`, it is treated like \verb`\dfrac`
+(but see §\ref{subsec:parseTrigFns}, specifically \texttt{()=2}). 
+\item [{Slash~fractions}]~
+
+It is easy to write ambiguous expressions using the slash $/$ to
+indicate fractions or division. How should $\pi/2n$ be interpreted?
+With from-the-left evaluation and calculator precedence rules which
+give equal precedence to {*} (multiplication) and / (division), this
+would be interpreted as $(\pi/2)\times n$, but most people will instinctively
+interpret it as $\pi/(2n)$. By placing \verb`/` in parsing group
+II, this is what \texttt{numerica} does. It treats the right-hand
+argument of the slash \emph{as if it were the argument of a named
+function}. This means that $1/2\sin(\pi/6)$ is parsed as $(1/2)\sin(\pi/6)$
+rather than as $1/(2\sin(\pi/6))$. It also means that $1/2\exp(1)$
+and $1/2e$ give different results, which (in the author's view) is
+acceptable since they display differently and are not instinctively
+read in the same way. 
+\end{description}
+
+\subsubsection{Parsing group III}
+
+\label{subsec:parseTrigFns}By default trigonometric functions are
+set to parsing group II. This accommodates many instances of how arguments
+are used with these functions, but Fourier series in particular require
+more. For them we need to take account of how \emph{parentheses} are
+used in arguments. I find $\tan\tfrac{1}{2}(A+B)$ (\emph{HMF }4.3.148),
+$\sec\pi(\tfrac{1}{4}+\tfrac{1}{2}az)$ (\emph{HMF }19.3.3), $\cos(2m+p)z$
+(\emph{HMF }20.2.3), $\sin(2n+1)v$ (\emph{HMF }16.38.1). Looking
+through various texts discussing Fourier series it is easy to find
+examples like
+\[
+\cos\frac{2\pi}{T}nt,\quad\cos\frac{2\pi}{T}n(t+\tfrac{1}{2}T),
+\]
+and
+\[
+\cos(N+\tfrac{1}{2})\frac{2\pi\tau}{T},\quad\sin2\pi\left(\frac{x}{\lambda}-\frac{t}{T}\right).
+\]
+In the last of these \verb`\left` and \verb`\right` have been used
+to enlarge the parentheses. 
+
+All these usages can be accommodated by adjusting a setting in the
+settings option (§\ref{sec:settingsOption}) of the \verb`\eval`
+command:
+\begin{lyxcode}
+()~=~<integer>
+\end{lyxcode}
+where \texttt{<integer>} is one of \texttt{0, 1, 2}. 
+
+\noindent\begin{minipage}[t]{1\columnwidth}%
+\begin{shaded}%
+I remain unsure whether to persist with the \texttt{()} setting. A
+principal aim of \texttt{numerica} is to avoid having to modify a
+formula to bring it into a form that can be evaluated. The \texttt{()}
+setting arose in that context, but it complicates the code and may
+well confuse the user. Inserting cleave functions, \texttt{\textbackslash q}
+or \texttt{\textbackslash Q}, to achieve the same effects does mean
+modifying formulas, but is straightforward and easier to understand.
+(And \texttt{\textbackslash q} and \texttt{\textbackslash Q} have
+no effect on the visual appearance of formulas.)\end{shaded}%
+\end{minipage}
+
+For convenience of statement in what follows call parentheses, square
+brackets or braces \emph{brackets}. If preceded by a \verb`\left`
+or \verb`\right` or \verb`\biggl` or \verb`\biggr` etc. modifier,
+call them \emph{Brackets}, with an uppercase `B'. Modifiers do not
+contribute to the position count, so that a left Bracket in first
+position means the modifier and left bracket are both considered to
+be in first position. When it is immaterial whether it is a bracket
+or a Bracket I write b/Bracket. The rules that follow do not prescribe
+what mathematicians \emph{ought} to do but are intended to be descriptive
+of certain patterns of mathematical practice as discerned in \emph{HMF}
+and a number of texts (about half a dozen) on Fourier series.
+\begin{description}
+\item [{\texttt{()=0}}] is the \emph{default} setting, parsing group II
+behaviour; b/Brackets are included in the argument only if
+\begin{itemize}
+\item the left b/Bracket is in first position;
+\begin{itemize}
+\item if the first item beyond the matching right b/Bracket is an exponent,
+or factorial or double factorial sign, it is appended to the argument,
+which ends there, otherwise
+\item the argument ends with the right b/Bracket.
+\end{itemize}
+\end{itemize}
+\end{description}
+The default setting allows things like $\sin\tfrac{1}{2}a$, $\cos2\pi nt$
+and $\tan(A+B)$. It does \emph{not} encompass examples like $\tan\tfrac{1}{2}(A+B)$
+or $\cos2(n+\tfrac{1}{2})\pi$. For that we need the setting \verb`()=1`:
+\begin{description}
+\item [{\texttt{()=1}}] includes a b/Bracketed expression in the argument,
+provided 
+\begin{itemize}
+\item the left Bracket is in first position;
+\begin{itemize}
+\item if the first item beyond the matching right Bracket is an exponent,
+or factorial or double factorial sign, it is appended to the argument,
+which ends there, otherwise
+\item the argument ends with the right Bracket.
+\end{itemize}
+\item or the item in first position is a number, variable, constant or \verb`\tfrac`
+and the left bracket is in second position;
+\begin{itemize}
+\item if the first item beyond the matching right bracket is an exponent,
+or factorial or double factorial sign, it is appended to the argument,
+which ends there, or
+\item if the first item beyond the matching right bracket is a number, variable,
+constant, or \verb`\tfrac` it is appended to the argument, and so
+on recursively, until
+\begin{itemize}
+\item an exponent, or factorial or double factorial sign is met, which is
+appended to the argument which ends there, or
+\item an item is met which is \emph{not} an exponent, or factorial or double
+factorial sign, or a number, variable, constant or \verb`\tfrac`,
+at which point the argument ends, or
+\item the end of the formula is reached.
+\end{itemize}
+\end{itemize}
+\end{itemize}
+\end{description}
+With the \verb`()=1` setting, the arguments of $\tan\tfrac{1}{2}(A+B)$,
+$\sec\pi(\tfrac{1}{4}+\tfrac{1}{2}az)$, $\cos(2m+p)z$, $\sin(2n+1)v$
+are all accommodated, as is $\sin\tfrac{1}{2}(m+n)\pi$ with items
+on both sides of the parentheses. But, note, there must be at most
+\emph{one} item before the left parenthesis:
+\begin{centred}
+\verb`\eval[()=1]{$ \sin\tfrac16(m+n)\pi $}[m=1,n=2]`. $\Longrightarrow$
+\eval[()=1]{$ \sin\tfrac16(m+n)\pi $}[m=1,n=2],
+\end{centred}
+whereas, with two items before the left parenthesis,
+\begin{centred}
+\verb`\eval[()=1]{$ \sin2\tfrac1{12}(m+n)\pi $}[m=1,n=2]`. $\Longrightarrow$
+
+\eval[()=1]{$ \sin2\tfrac1{12}(m+n)\pi $}[m=1,n=2].
+\end{centred}
+Whatever the \verb`()` setting, \texttt{numerica} does not check
+what is included between the parentheses (or brackets generally) –
+it could be anything. However inserting \verb`\left`, \verb`\right`
+or other modifiers before the parentheses restricts the argument of
+the sine in this example, despite the \verb`()=1` setting, to the
+\verb`\tfrac`:
+\begin{centred}
+\verb`\eval[()=1]{$ \sin\tfrac16\left(m+n\right)\pi $}[m=1,n=2]`
+$\Longrightarrow$ \eval[()=1]{$ \sin\tfrac16\left(m+n\right)\pi $}[m=1,n=2].
+\end{centred}
+Although \verb`()=1` serves well for the kinds of expressions and
+identities involved in trigonometry, perusal of any text on Fourier
+series will show it does not cover the kinds of expressions met there.
+For that we need
+\begin{description}
+\item [{\texttt{()=2}}] includes a b/Bracketed expression in the argument
+provided
+\begin{itemize}
+\item the left b/Bracket is in first position, or the item in first position
+is a number, variable, constant, \verb`\dfrac`, \verb`\frac` or
+\verb`\tfrac` and the left b/Bracket is in second position, or the
+items in first and second positions are numbers, variables, constants,
+\verb`\dfrac`-s, \verb`\frac`-s or \verb`\tfrac`-s and the left
+b/Bracket is in third position;
+\begin{itemize}
+\item if the first item beyond the matching right b/Bracket is an exponent,
+or factorial or double factorial sign, it is appended to the argument,
+which ends there, or
+\item if the first item beyond the matching right b/Bracket is a number,
+variable, constant, \verb`\dfrac`, \verb`\frac` or \verb`\tfrac`
+it is appended to the argument, and so on recursively, until
+\begin{itemize}
+\item an exponent, or factorial or double factorial sign is met, which is
+appended to the argument which ends there, or
+\item an item is met which is \emph{not} an exponent, or factorial or double
+factorial sign, or a number, variable, constant, \verb`\dfrac`, \verb`\frac`
+or \verb`\tfrac`, at which point the argument ends, or
+\item the end of the formula is reached.
+\end{itemize}
+\end{itemize}
+\end{itemize}
+\end{description}
+{\ttfamily\verb`()=2`} draws no distinction between brackets
+and Brackets. It allows all \verb`()=1` possibilities but also \emph{two
+}items (of a suitable kind) before a left b/Bracket; it also treats
+\verb`\dfrac`-s and \verb`\frac`-s like \verb`\tfrac`-s for determining
+the scope of arguments. 
+
+The following examples are taken from different texts on Fourier series.
+The first shows a \verb`\frac` being included in the argument, the
+second shows \emph{two} items – including a \verb`\frac` – preceding
+the left parenthesis, the third shows a \verb`\frac` to the right
+of the parentheses, and the fourth shows parentheses using \verb`\left`-\verb`\right`
+modifiers with two items preceding them:
+\[
+\cos\frac{2\pi}{T}nt,\quad\cos\frac{2\pi}{T}n(t+\tfrac{1}{2}T),\quad\text{\ensuremath{\sin(N+\tfrac{1}{2})\frac{2\pi\tau}{T}}\ensuremath{\quad}and}\quad\sin2\pi\left(\frac{x}{\lambda}-\frac{t}{T}\right).
+\]
+All these usages are accommodated by the \verb`()=2` setting. For
+instance
+\begin{verbatim}
+    \eval[p,()=2]
+      { 
+        \[ \sin(N+\tfrac12)\frac{2\pi\tau}T \]
+      }[N=1,\tau=2,T=3]
+\end{verbatim}
+$\Longrightarrow$     \eval[p,()=2]
+      { 
+        \[ \sin(N+\tfrac12)\frac{2\pi\tau}T \]
+      }[N=1,\tau=2,T=3]which is the sine of $2\pi=(\tfrac{3}{2})\times(\tfrac{4}{3}\pi)$
+where a \verb`\frac` trailing the parentheses has been included in
+the argument, and \emph{not }$(\sin\tfrac{3}{2})(\tfrac{4}{3}\pi)$.
+Or consider
+\begin{verbatim}
+    \eval[p,()=2]
+      {\[ 
+         \sin2\pi\left(\frac{x}{\lambda}
+            -\frac{t}{T}\right) 
+      \]}[x=1,\lambda=2,t=3,T=4]
+\end{verbatim}
+$\Longrightarrow$     \eval[p,()=2]
+      {\[ 
+         \sin2\pi\left(\frac{x}{\lambda}
+            -\frac{t}{T}\right) 
+      \]}[x=1,\lambda=2,t=3,T=4]which is the sine of $-\tfrac{1}{2}\pi=2\pi\times(-\tfrac{1}{4})$
+where there are two items before the parentheses and \verb`\left`
+and \verb`\right` modifiers, and \emph{not} $\sin2\pi$ times the
+parenthesised expression.
+
+However a usage like $\sin(n+\tfrac{1}{2})(x-t)$, noted in two different
+texts, is not available without explicit use of the \verb`\q` command
+between the parenthesized groups. 
+
+\subsubsection{Parsing group IV}
+
+The only members of this group are \verb`\sum` and \verb`\prod`.
+
+For parsing group IV
+\begin{enumerate}
+\item the argument ends 
+\begin{enumerate}
+\item at the first naked plus or minus sign encountered, or 
+\item at the first comparison sign or comparison command encountered, or
+\item at the first logical And or logical Or sign encountered, or
+\item at the end of the formula.
+\end{enumerate}
+\end{enumerate}
+In practice this means mainly (a) and (d), and seems to be the instinctive
+practice. \emph{HMF} has multiple examples in multiple chapters of
+the argument to a sum ending at a naked plus sign: 7.3.12 \& 7.3.14,
+9.1.11 \& 9.1.77, 9.6.35 \& 9.6.43, 11.1.9, \ldots{} (at that point
+I stopped looking). They were all of the form
+\[
+\sum\text{argument}+\ldots
+\]
+ A minus sign serving the same purpose was harder to find but \emph{HMF}
+10.4.65 \& 10.4.67 are two instances. I considered whether a \verb`\times`
+or slash fraction sign \verb`/` might end the argument of a sum,
+but surely we need to allow things like $\sum1/n^{2}$ which rules
+out the slash and \emph{HMF} 9.9.11 provides two of a number of instances
+in \emph{HMF} of sum arguments continuing past explicit \verb`\times`
+signs (at line breaks when a summand spills onto a second line).
+
+Because they are evaluated using the same code as sums I (unthinkingly)
+placed products with sums but doubts later intruded. In \emph{HMF}
+products occur only occasionally and are almost all of the form
+\[
+\prod\left(\text{argument}\right)
+\]
+where the argument is bracketed (often with \verb`\left \right` modifiers)
+and the multiplicand ends with the right bracket. At least twice (\emph{HMF
+}6.1.25 and 24.2.2.1) an exponent ($-1$) is attached to the right
+bracket and the argument ends there. Looking further afield, a text
+on number theory has examples where the argument of the product extends
+to \emph{three} parenthesised factors, $\prod\left(\text{arg}1\right)\left(\text{arg2}\right)\left(\text{arg3}\right)$
+and a number of others where it extends to two. A text on theory of
+functions has 
+\[
+\prod_{n=1}^{\infty}\left(1+\frac{z}{n}\right)e^{z/n}
+\]
+although \emph{HMF}, for the same expression, encloses the two factors
+within (large) square brackets, as if some ambiguity existed as to
+how far the reach of the \verb`\prod` extended. 
+
+\emph{Tentatively} I retain products here in the same group as sums.
+
+\subsubsection{Parsing group V}
+
+Comparison symbols compose this group: \texttt{=}, \texttt{<}, \texttt{>},
+\verb`\ne`, \verb`\le`, \verb`\ge`, \verb`\leq`, \verb`\geq`,
+and the various comparison commands from the \texttt{amssymb} package
+listed in §\ref{subsec:evalAmssymb-comparisons}. Because of the way
+\texttt{numerica} handles comparisons, it is the argument on the right-hand
+side of the relation that needs determining.
+
+For parsing group V
+\begin{enumerate}
+\item the argument ends at
+\begin{enumerate}
+\item the first logical And or logical Or encountered, or
+\item the first comparison sign or command encountered, or
+\item the end of the formula.
+\end{enumerate}
+\end{enumerate}
+
+\subsubsection{Parsing group VI}
+
+Logical And and logical Or are the sole members of this group. It
+is the right-hand side of the And or Or command that needs determining.
+
+For parsing group VI
+\begin{enumerate}
+\item the argument ends at
+\begin{enumerate}
+\item the first logical And or logical Or encountered, or
+\item the end of the formula.
+\end{enumerate}
+\end{enumerate}
+
+\subsubsection{Disclaimer}
+
+The parsing rules of the different groups are not normative; they
+are not statements of how mathematical formulas should be written.
+Rather they are attempts to discern regularities in how mathematicians
+often do write formulas. It is \emph{how things look in the pdf},
+not \LaTeX , that is the guide. You are always free to parenthesize
+as you see fit and to insert cleave commands (\verb`\q` or \verb`\Q`)
+to force outcomes. 
+
+(But note that parenthesizing has its limits. For sums, writing 
+\[
+\sum\left(\mathtt{<stuff>}\right)\mathtt{<more\ stuff>}
+\]
+ does not necessarily end the summand at the right parenthesis: it
+ends at the first naked $+$ or $-$ sign, or \verb`\Q` command,
+encountered.) 
+
+The rule should always be to write expressions that are clear to the
+reader of the pdf. An expression that is ambiguous to the reader,
+even if it fits within the parsing rules, is to be deprecated. The
+\emph{intent} is that \verb`\eval` can parse unambiguous expressions
+correctly. 
+
+\chapter{Supplementary commands}
+
+\label{chap:Supplementary-commands}This chapter introduces four commands,
+\verb`\nmcInfo` (which we have already met), \verb`\nmcMacros`,
+\verb`\nmcConstants` and \verb`\nmcReuse`, supplementary to the
+principal command \verb`\nmcEvaluate`. They use the same machinery
+as \verb`\nmcEvaluate` and so have the same syntax. If all arguments
+are present it is
+\begin{centred}
+\noindent \verb`\nmc<cmd>*[settings]{main arg}[vv-list][rounding]`
+\end{centred}
+where \verb`<cmd>` is one of \verb`Info`, \verb`Macros`, \verb`Constants`
+and \verb`Reuse`. All four commands have short-name forms: \verb`\info`,
+\verb`\macros`, \verb`\constants`, \verb`\reuse`.
+
+Generally the final two optional arguments will not be used. The user
+should be aware of this if following a command with a square bracketed
+expression – the expression will be absorbed without trace unless
+it is preceded by, for example, an empty brace pair.
+
+Because the commands share the machinery of \verb`\nmcEvaluate`,
+the settings discussed earlier (Chapter~\ref{chap:Settings}) for
+the \verb`\eval` command are also available for these commands, although
+they will, in the main, be irrelevant. The `debug' code has been
+used by the \verb`view` setting of some of these supplementary commands
+to produce its effects. 
+
+The starred form of command is available in all four cases and in
+all cases produces a pure number. If both star and \verb`view` are
+used at the same time, the \verb`view` setting prevails over starring.
+
+\section{Feedback on \textquoteleft infinite\textquoteright{} processes:\texttt{ \textbackslash nmcInfo}}
+
+\label{sec:supplInfo}Used after the evaluation of an `infinite'
+process, the \verb`\nmcInfo` command, or its short-name form \verb`\info`
+will tell you how many terms or factors or other operations\footnote{It also applies to the commands \texttt{\textbackslash nmcIterate
+}and \texttt{\textbackslash nmcSolve} from the \texttt{numerica-plus
+}package and to derivatives and integrals from the \texttt{numerica-calculus}
+package.} were needed to arrive at the result.The main argument contains an
+identifier for the `infinite' process: 
+\begin{lyxcode}
+\textbackslash nmcInfo\{<arg>\}~
+\end{lyxcode}
+(or \verb`\info{<arg>}`) where, at this stage, \verb`<arg>` is either
+\verb`sum` or \verb`prod`. The display, as we have seen in earlier
+examples, is a number followed by a space then a descriptor. For \verb`sum`
+and \verb`prod` the descriptors are \verb`terms` and \verb`factors`.
+Starring \verb`\nmcInfo` – \verb`\nmcInfo*{arg}` or \verb`\info*{arg}`
+– suppresses the descriptor and leaves only the number. This allows
+the starred form to be nested in an \verb`\eval` command, which might
+sometimes be convenient. 
+
+As an example, let's test `the hard way' a standard identity, $\cosh^{2}x-\sinh^{2}x=1$.
+We know that $\cosh x=\sum_{n=0}^{\infty}\frac{x^{2n}}{(2n)!}$ and
+$\sinh x=x\prod_{k=1}^{\infty}\left(1+\frac{x^{2}}{k^{2}\pi^{2}}\right)$.
+The difference of their squares should be $1$:
+\begin{verbatim}
+    \eval{\[  
+      \left[\sum_{n=0}^{\infty}
+        \frac{x^{2n}}{(2n)!}
+      \right]^2-
+        \left[x\prod_{k=1}^{\infty}
+          \left(1+\frac{x^{2}}{k^{2}\pi^{2}}\right)
+        \right]^2
+      \]}[x=1][3] \info{sum},\quad \info{prod}
+\end{verbatim}
+$\Longrightarrow$     \eval{\[  
+      \left[\sum_{n=0}^{\infty}
+        \frac{x^{2n}}{(2n)!}\right]^2-
+      \left[x\prod_{k=1}^{\infty}
+        \left(1+\frac{x^{2}}{k^{2}\pi^{2}}\right)
+    \right]^2
+        \]}[x=1][3] \info{sum},\quad\info{prod}.
+
+Nearly right. Obviously the product converges only slowly which is
+where the error comes from (see the discussion in §\ref{sec:settingsInfiniteSumsProds},
+where we needed the extra rounding setting \texttt{P+=3} and $350$
+factors to get a correct 3-figure value). The point of the example
+is to show the information command being used for both sum and product
+in the one evaluation. One does not exclude the other. 
+
+\subsection{Suppressing the descriptor: \texttt{\textbackslash nmcInfo{*}}}
+
+The starred form of the \verb`\info` command suppresses the descriptor
+(`terms', `factors') and gives a purely numerical result: 
+\begin{verbatim}
+    \eval{$
+      \sum_{k=0}^{\infty}\binom \alpha k x^k
+        $}[x=1/2,\alpha=3], 
+    requiring $ \info*{sum}-1 $ additions. 
+\end{verbatim}
+$\Longrightarrow$     \eval{$
+      \sum_{k=0}^{\infty}\binom \alpha k x^k
+         $}[x=1/2,\alpha=3], 
+requiring $ \info*{sum}-1 $ additions. (Four terms added, therefore $3$ additions.)
+
+\subsection{Errors}
+
+Should the \emph{wrong} argument be used in the \verb`\nmcInfo` command,
+no harm is done:
+\begin{verbatim}
+    \eval{$
+      \sum_{k=0}^{\infty}\binom \alpha k x^k
+         $}[x=1/2,\alpha=3], \ \info{prod} 
+\end{verbatim}
+$\Longrightarrow$  \eval{$ \sum_{k=0}^{\infty}\binom \alpha k x^k $}[x=1/2,\alpha=3],\ \info{prod}.\medskip{}
+
+$119$ \emph{factors}? The information command is remembering a previous
+result, the last time \verb`prod` was used as its argument. Changing
+the argument from \verb`prod` to \verb`sum` reveals the correct
+number of \emph{terms}.
+
+Should a non-existent argument be used, an error message is generated:
+\begin{verbatim}
+    \eval{$ 
+      \sum_{k=0}^{\infty}\binom \alpha k x^k 
+         $}[x=1/2,\alpha=3], \\ \info{Fred} 
+\end{verbatim}
+$\Longrightarrow$  \eval{$ \sum_{k=0}^{\infty}\binom \alpha k x^k $}[x=1/2,\alpha=3],\\  \info{Fred}
+
+\subsection{\texttt{view} setting}
+
+As noted at the start of this chapter, \verb`\nmcInfo` uses the `machinery'
+of \verb`\nmcEvaluate`. Most of the settings available for \verb`\eval`
+are also available for \verb`\info` but of these only one seems relevant:
+the \verb`dbg` setting. However, rather than use the obscure \verb`dbg=<integer>`
+(which is possible), it suffices to enter \verb`view` in this argument:
+\begin{centred}
+\verb`\info[view]{}` $\Longrightarrow$ \info[view] {}
+\end{centred}
+The result is a display of all the current values of all the `infinite'
+processes available. All such values are initialized to $0$. (Further
+processes \verb`iter` and \verb`solve` become relevant if the \verb`numerica-plus`
+package is used; \verb`deriv` and \verb`integ` become relevant if
+the \verb`numerica-calculus` package, currently under development,
+is used.)
+
+\section{User-defined macros: \texttt{\textbackslash nmcMacros}}
+
+\label{sec:supplMacros}The \verb`\nmcMacros` and \verb`\nmcConstants`
+commands were prompted by a question on \TeX{} Stack Exchange.\cprotect\footnote{A question from Giacomo Petrillo on \TeX{} Stack Exchange and a response
+by `egreg' prompted the introduction of the \cprotect\texttt{\textbackslash nmcMacros}
+and \cprotect\texttt{\textbackslash nmcConstants} commands. See \url{https://tex.stackexchange.com/questions/602993/use-macros-in-numerica-vv-list/602998#602998}} Some time later the maintainer of the \verb`mandi` package\footnote{The maintainer is Joe Heafner, who explains that `mandi' is an abbreviation
+of `matter and interactions' after a physics textbook of that name
+(by different authors). Among other things, the package defines a
+long list of macros, each containing the value of a physical constant.} approached me with a similar problem. Suppose one has defined a macro
+to contain a value, say 
+\begin{itemize}
+\item \verb`\def\myvalue{0.35}`, or
+\item \verb`\newcommand\myvalue{0.35}`, or
+\item \verb`\NewDocumentCommand\myvalue{}{0.35}`, if you're using \verb`xparse`.
+\end{itemize}
+(If you're using the document processor \LyX{} then there is good reason
+to prefer \verb`\gdef` to define your macro, \verb`\gdef\myvalue{0.35}`;
+see Chapter~\ref{chap:LyX}). After one of these commands, \verb`\myvalue`
+is now known to \LaTeX , but it is not known to \verb`numerica`.
+The quantities \verb`numerica` \emph{does }know about are variables
+in the vv-list of an \verb`\eval` command, and those \LaTeX{} (and
+\verb`amsmath` and \verb`mathtools`) commands used for writing mathematical
+expressions. These quantities are stored in \verb`numerica` in structures
+called property lists. Since \verb`\myvalue` is not recorded in these
+lists yet, putting \verb`x=\myvalue` in the formula or vv-list of
+an \verb`\eval` command will produce an `Unknown token' error message:
+\begin{verbatim}
+    \NewDocumentCommand \myvalue {} {0.35}
+    \eval{ \myvalue }
+\end{verbatim}
+$\Longrightarrow$     \NewDocumentCommand \myvalue {} {0.35}
+    \eval{ \myvalue }
+
+With version 2 of \verb`numerica` a command is now available, \verb`\nmcMacros`,
+to register macros and their values with the property lists used internally
+by \verb`numerica`. (This command was not available in version 1.)
+The macro must have been defined earlier in the document or in a supporting
+package. 
+
+The basic usage is simple. If you have a list of macros you wish to
+make available to \verb`\nmcEvaluate`, enter them in a comma list
+in the mandatory argument of \verb`\nmcMacros`:
+\begin{lyxcode}
+\textbackslash nmcMacros\{~\textbackslash macro1,~\textbackslash macro2,~\ldots ~\}
+\end{lyxcode}
+There is an equivalent short-name form of the command, \verb`\macros`. 
+
+Multiple \verb`\nmcMacros` commands can be used in a document. If
+the command is placed in the preamble (\emph{after} the definition
+of the macros)\emph{ }then the user-defined macros and their values
+are available throughout the document, otherwise they are available
+from the position of the \verb`\macros` statement. However, macros
+do not need to be defined in your current document provided they are
+defined and accessible from elsewhere – for example from a loaded
+\LaTeX{} package. But always an \verb`\nmcMacros` command is required
+to `register' them with \verb`numerica` for use in an \verb`\eval`
+command. 
+
+\subsection{What can be stored in a macro?}
+
+Generally a user-defined macro will store a number. This macro might
+well be defined in an external package – for example the \verb`mandi`
+package defines a large number of macros containing the values of
+physical constants, some fundamental like the speed of light, others
+contingent like the earth–moon distance. If the \verb`mandi` package
+is loaded then writing, for instance, 
+\begin{verbatim}
+    \macros{ \electronmassprecisevalue,
+             \protonmassprecisevalue }
+\end{verbatim}
+will make these two macros available for use in \verb`numerica`.
+One could then write in the vv-list of an \verb`\eval` command
+\begin{verbatim}
+m_e=\electronmassprecisevalue,m_p=\protonmassprecisevalue
+\end{verbatim}
+which would allow (among other things) calculation of the mass ratio
+$m_{p}/m_{e}$ of proton to electron. (The length of name of some
+of the macros in the \verb`mandi` package has a pedagogical purpose,
+but makes them unwieldy for direct use in mathematical expressions.)
+
+\subsubsection{Macros containing formulas }
+
+Numbers are not the only quantities that can be stored in a macro
+for use in \verb`numerica`. In fact any mathematical expression that
+can be \verb`\eval`-uated can be stored in a macro:
+\begin{verbatim}
+    \NewDocumentCommand \mysumC {}
+        { \sum_{n=1}^{100}1/n - \ln 100 }
+    \macros{ \mysumC }
+    \eval{$ \mysumC $}[4]
+\end{verbatim}
+$\Longrightarrow$     \NewDocumentCommand \mysumC {}
+        { \sum_{n=1}^{100}1/n - \ln 100 }
+    \macros{ \mysumC }
+    \eval{$ \mysumC $}[4], \medskip{}
+
+\noindent (to be compared with Euler's constant \eval{$ \gamma $}[4]
+– obviously many more terms are needed). The \verb`\eval` command
+wraps around math delimiters in the example. Hence the result is presented
+in the form \emph{formula=result}. In that presentation, note how
+\verb`\mysumC` displays as the formula it contains.
+
+\paragraph{The essential space: }
+
+But the critical thing to notice in the example is \emph{the space
+preceding }\verb`\sum`\emph{ in the definition of }\verb`\mysumC`.
+When a formula starts with an expandable token, \emph{this space is
+essential}. For macros to register successfully with \verb`numerica`,
+the first character in their definition must be \emph{un}expandable.
+Thus a digit is fine: storing a number in a macro is straightforward
+and you don't need to fuss about such niceties. But a control sequence
+like \verb`\sum` does expand (to $\sum$ ). If it is the initial
+token of the formula, then it will cause a possibly obscure error
+– see §\ref{subsec:supplMacrosErrors} – unless preceded by an unexpandable
+token. Hence the space before \verb`\sum` in the \verb`\NewDocumentCommand`
+statement. (On the other hand the spacing in the \verb`\macros` statement
+is purely aesthetic.)
+
+When using macros from another package, this is a matter to be aware
+of. If the macros contain only numbers, there should be no problem,
+but if they contain more complicated expressions, the absence of an
+initial space could make them unusable in \verb`numerica`.
+
+\subsubsection{Vv-list}
+
+In the example it would be nice to be able to vary the number of terms
+summed. This is easily done by using a vv-list in the \verb`\macros`
+statement:
+\begin{verbatim}
+    \NewDocumentCommand \mysumN {}
+        { \sum_{n=1}^{N}1/n - \ln N }
+    \macros{ \mysumN }[N=150]
+    \eval{$ \mysumN $}
+\end{verbatim}
+$\Longrightarrow$     \NewDocumentCommand \mysumN {}
+        { \sum_{n=1}^{N}1/n - \ln N }
+    \macros{ \mysumN }[N=150]
+    \eval{$ \mysumN $}.\medskip{}
+
+\noindent \verb`numerica` needs a definite value to store; it does
+not store the formula as such. To give \verb`\mysumN` a definite
+value, give the variable \verb`N` a value. This is done in the vv-list
+added to the \verb`\macros` statement: \verb`N=150`. In this way
+a definite value is stored in \verb`numerica` against the macro \verb`\mysumN`.
+The definition of the macro is unaffected. If a new value is given
+to \verb`N` in the \verb`\macros` statement (which is the point
+of using a variable), the old value is overwritten and the new value
+is used in subsequent calculations.
+
+\subsection{Seeing what macros are available}
+
+Perhaps your document has a number of \verb`\nmcMacros` statements
+scattered through it and you want to remind yourself of what exactly
+has been stored. \verb`\nmcMacros` has the \verb`view` setting for
+this purpose. Writing
+\begin{centred}
+\verb`\macros[view]{}` $\Longrightarrow$ \macros[view]{}
+\end{centred}
+produces a list of all macros registered with \verb`numerica` and
+their values, as you can see. 
+
+If the braced argument is not empty, the display is slightly modified:
+\begin{verbatim}
+    \def\mydef{ \sin(m\pi/n) }
+    \newcommand\mynewcmd{ \cos(m\pi/n) }
+    \macros[view]{ \mydef,\mynewcmd }[m=3,n=18]
+\end{verbatim}
+$\Longrightarrow$     \def\mydef{ \sin(m\pi/n) }
+    \newcommand\mynewcmd{ \cos(m\pi/n) }
+    \macros[view]{ \mydef,\mynewcmd }[m=3,n=18]
+
+\noindent \verb`\mydef` and \verb`\mynewcmd` have been added to
+those available for use in \verb`numerica`. 
+
+\subsubsection{Freeing macros from storage}
+
+Rather than cluttering \verb`numerica`'s property lists with no-longer-needed
+macros, it is possible to remove them from there with the \verb`free`
+setting. This has no effect on the \LaTeX{} definition of the macro.
+It merely `de-registers' the macro with \verb`numerica`.
+\begin{centred}
+\verb`\macros[free,view]{ \mysumC }` $\Longrightarrow$ \macros[free,view]{ \mysumC }
+\end{centred}
+If you want to free \emph{all} macros registered with \verb`numerica`
+use an empty main argument with the \verb`free` setting. For an example,
+see just below.
+
+\subsubsection{Counting how many macros are available}
+
+You can count how many macros are currently registered with \verb`numerica`
+by starring the \verb`\nmcMacros`~command:
+\begin{centred}
+\verb`\macros*{}` $\Longrightarrow$ \macros*{}.
+\end{centred}
+If the braced argument is not empty, the list of macros it contains
+will be added to those registered with \verb`numerica` and included
+in the overall count.
+
+Note that the \verb`view` setting prevails over starring if both
+are used. 
+
+The star can also be used with the \verb`free` setting. As mentioned
+above, if the main argument is empty, then \emph{all} macros are freed:
+\begin{centred}
+\verb`\macros*[free]{}` $\Longrightarrow$ \macros*[free]{ }
+\end{centred}
+
+\subsection{Errors}
+
+\label{subsec:supplMacrosErrors}If a macro is used in a \verb`\macros`
+statement and the macro has not been defined in the document or a
+supporting package it will cause an error:
+\begin{verbatim}
+    \macros{ \mymacro }
+\end{verbatim}
+$\Longrightarrow$ \macros{\mymacro }
+
+\noindent As noted in the introduction to this section, an undefined
+macro used in an \verb`\eval`-uation will cause an `Unknown token'
+message in \verb`numerica`. The solution in this and the preceding
+case is (obviously) to define the macro.
+
+If a macro contains a formula which begins with an expandable token
+and a preceding space is omitted (see above), then entering that macro
+in a \verb`\macros` statement to register it with \verb`numerica`
+will generally cause a puzzling error:
+\begin{verbatim}
+    \newcommand\mysin{\sin(\pi/7)}
+    \macros{ \mysin }
+\end{verbatim}
+$\Longrightarrow$     \newcommand\mysin{\sin(\pi/7)}
+    \macros{ \mysin }
+
+\noindent The \verb`\protect` seems to be plucked from nowhere. In
+fact it comes from the expansion of \verb`\sin`. If \verb`\sum`
+had been the first token in the macro definition, again with no preceding
+space, then \verb`\protect` would have been replaced by the even
+more puzzling \verb`\DOTSB`. The solution is to insert a space as
+the first token in the macro definition.
+
+If a macro is defined but the \verb`\macros` statement is overlooked,
+and the macro is used in an \verb`\eval`-uation, it will generate
+an `Unknown token' message.
+
+If your macro stores a formula with variables, and you forget to give
+those variables values in the \verb`\macros` statement that will
+produce a message:
+\begin{verbatim}
+    \def\mysumk{ \sum_{n=1}^k n }
+    \macros{ \mysumk }
+\end{verbatim}
+$\Longrightarrow$     \def\mysumk{ \sum_{n=1}^k n } 
+    \macros{ \mysumk }
+
+\noindent The `where' part of the message is specific in this case,
+but is generally `\verb`\nmcMacros` command'. 
+
+And of course there can be `all the usual suspects' discussed at
+§\ref{sec:evalErrors} in the evaluation of the vv-list or the formula.
+
+\noindent\begin{minipage}[t]{1\columnwidth}%
+\begin{shaded}%
+
+\subsubsection{Display of macros}
+
+\label{subsec:supplMacrosDisplay}As shown in earlier examples, macros
+display as their content. Thus \verb`\mysumC` displayed as $\sum_{n=1}^{100}1/n-\ln100$.
+But once a macro is known to \LaTeX{} (not necessarily to \verb`numerica`)
+it can be used as a variable name. This has the same potential for
+abuse as noted earler for multi-token variables (§\ref{subsec:evalDon't-do-this!}).
+In the following example note that there is no \verb`\macros` statement.
+It suffices for the macro to be known to \LaTeX . 
+\begin{verbatim}
+    \def\mymac{1}
+    \eval[vvi=\,???]{$ \mymac+\mymac $}[\mymac=2]
+\end{verbatim}
+$\Longrightarrow$     \def\mymac{1}
+    \eval[vvi=\,???]{$ \mymac+\mymac $}[\mymac=2]
+
+The value assigned to a variable name – in this case \verb`\mymac`
+– by \verb`numerica` for \cprotect\emph{calculational} purposes and
+how that variable name \cprotect\emph{displays} in \LaTeX{} are two
+separate things. One relies on the user not to do something deliberately
+deceptive.\end{shaded}%
+\end{minipage}
+
+\subsection{Rounding value}
+
+\label{subsec:supplMacrosRounding}Values are stored to $16$ significant
+figures (if available). In most cases appending a rounding value to
+a \verb`\macros` statement has no effect on the value stored. In
+the following example note the \verb`o` setting, meaning the sine
+reads angles in degrees:
+\begin{verbatim}
+    \NewDocumentCommand\testi{}{ \sin 60 }
+    \NewDocumentCommand\testii{}{ \sin 60 }
+    \macros[o]{ \testi }[10]
+    \macros[o]{ \testii }[3]
+    \macros[view]{}
+\end{verbatim}
+$\Longrightarrow$     \NewDocumentCommand\testi{}{ \sin 60 }
+    \NewDocumentCommand\testii{}{ \sin 60 }
+    \macros[o]{ \testi }[10]
+    \macros[o]{ \testii }[3]
+    \macros[view]{}
+
+\noindent Despite the different rounding values the same $16$ figures
+are stored in both \verb`\testi` and \verb`\testii`. 
+
+For the \verb`\eval` command, rounding values specify how results
+are \emph{displayed}. The rounding value matters only \emph{after},
+not during, the calculation. Only for infinite sums or products is
+this otherwise. There the rounding value is used to determine when
+to stop adding further terms or factors. The same is true of the \verb`\macros`
+command. Only if a macro contains an infinite sum or product does
+the rounding value become relevant. Sixteen figures are still stored,
+but most of them will be `wrong' since the infinite sum or product
+has stopped early, after only a finite number of terms or factors.
+Exactly how many of the first few figures are correct depends on the
+rounding value. An example may clarify the matter.
+\begin{verbatim}
+    \macros[free]{}
+    \def\zetaiii{ \sum_{n=1}^\infty 1/n^3 }
+    \macros[view]{ \zetaiii }[3]
+    \info{sum}
+    \macros[view]{ \zetaiii }[6]
+    \info{sum}
+\end{verbatim}
+$\Longrightarrow$     \macros[free]{} 
+    \def\zetaiii{ \sum_{n=1}^\infty 1/n^3 }
+    \macros[view]{ \zetaiii }[3]
+    \info{sum}
+    \macros[view]{ \zetaiii }[6]
+    \info{sum}
+
+\emph{HMF }Table 23.3 tells me that $\zeta(3)=1.202056903159594\dots$
+The different rounding numbers have restricted the infinite sums to
+the very finite $47$ and $468$ terms respectively. Although $16$
+figures are stored, only the first few are correct. Just how many
+depends on the number of terms summed which depends on when the stopping
+criterion is met which depends on the rounding value.
+
+\section{User-defined constants:\texttt{ \textbackslash nmcConstants}}
+
+\label{sec:supplConstants}As noted much earlier in this document
+(§\ref{subsec:Built-in-Constants}), there are five built-in constants:
+\verb`\pi`, \verb`e`, \verb`\phi`, \verb`\gamma` and \verb`\deg`,
+but a user may well want to define their own constant or constants.
+There are contexts where it would make sense to permanently record
+fundamental constants like the speed of light or Planck's constant,
+or more down-to-earth constants like the acceleration due to gravity
+or the viscosity of water, rather than having to enter them in the
+vv-list for each calculation. Or a parameter might be held constant
+for a particular problem or class of problems where other variables
+change – for example triangles of constant perimeter but varying sides.
+This is the purpose of the \verb`\nmcConstants` command.
+
+The symbols used to denote constants are subject to exactly the same
+constraints and freedoms as the symbols used to denote variables.
+They might be single latin letters like \verb`c` (e.g. $c=3\times10^{8}$),
+or greek letters like \verb`\alpha` (e.g. $\alpha=1/137)$, or multitoken
+combinations like the Rydberg constants \verb`R_\infty` or \verb`R_{\mathrm{H}}`
+from atomic physics, or \verb`\mu_0` and \verb`\epsilon_0` used
+to denote the permeability and permittivity of free space, or personal
+constants like \verb`total` of no wider significance. \verb`numerica`
+handles all these different forms of constant with the command \verb`\nmcConstants`:
+\begin{verbatim}
+    \nmcConstants{  const-n=value-n, ... ,
+                    const2=value2, const1=value1 }
+\end{verbatim}
+This is the simplest use – each constant is assigned a (numerical)
+value. But it is easy to envisage situations where it would be convenient
+to have a constant with value $1/\sqrt{2\pi}$ say, or another with
+value $e^{\tfrac{\pi}{2}}$, and so on. That is easy: simply put the
+expession for the value on the right:
+\begin{verbatim}
+    \constants{ a=1/\sqrt{2\pi},b=e^{\tfrac\pi2} } 
+\end{verbatim}
+Or the values could be expressions depending on parameters:
+\begin{verbatim}
+    \constants{ s=\tfrac12(a+b+c) }[a=3,b=5,c=7]
+\end{verbatim}
+Some constants might depend on earlier constants in the list:
+\begin{verbatim}
+    \constants{ A=\sqrt{s(s-a)(s-b)(s-c),
+                s=\tfrac12(a+b+c) }[a=3,b=5,c=7]
+\end{verbatim}
+Or the values could involve an `infinite' process, requiring a rounding
+number:
+\begin{verbatim}
+    \constants{ \zeta=\sum_{n=1}^\infty(1/n^k) }[k=4][5]
+\end{verbatim}
+In this, although $16$ figures will be stored, only the first few
+will be accurate, the precise number depending on the value of \verb`k`
+and the rounding number (\verb`5` in the example); see the discussion
+on this issue for user-defined macros, §\ref{subsec:supplMacrosRounding}.
+
+\subsection{New list replaces old by default}
+
+A particular group of constants may be relevant only to a particular
+part of a document. Another part of the document may use other constants.
+By default, a second list of constants \emph{replaces }the first list.
+Thus each of the \verb`\constants` statements above would replace
+the previous one.
+
+There is a technical reason why replacing rather than appending is
+the default. For each calculation all \emph{multi-token} constants
+(e.g. \verb`R_\infty`, \verb`N_0`, \ldots ) are added internally
+to the start of the vv-list of the \verb`\eval` command. Even if
+the vv-list is empty, this is still the case since the formula might
+well use constants. Like multi-token variables and for the same reason
+(see §\ref{subsec:evalVariableNames}), multi-token constants are
+mapped internally to single tokens. This occurs afresh for each calculation.
+If there are a lot of multi-token constants then each calculation
+is going to involve not only this mapping from multi- to single tokens
+but the evaluation of a long vv-list. In that case it seems better
+to make the default behaviour replacement of one constant list by
+another, rather than appending them. 
+
+\subsection{Adding constants to a list}
+
+Despite the default behaviour, there will be occasions when you want
+to add a new constant or constants to the current list. This is easily
+done with the \verb`add` setting. For instance,
+\begin{verbatim}
+    \nmcConstants[add]{ \sigma=5.67\times10^{-8},
+                          k_B = 1.381\times10^{-23} }
+\end{verbatim}
+would add \verb`\sigma` and \verb`k_B` to the current list. The
+presence of the \verb`add` setting triggers appending rather than
+replacement.
+
+\subsection{Examples of use}
+
+\subsubsection{Example 1: atomic constants}
+
+In the following example, the values of various atomic constants are
+taken from the \verb`mandi` package. I use two \verb`\constants`
+statements in order to show the use of the \verb`add` setting. I've
+also included a \verb`view` setting in the second \verb`\constants`
+statement.
+
+The constants are used to calculate the fine-structure constant \verb`\alpha`
+in the vv-list of the \verb`\eval` command, and its well-known reciprocal
+(close to $137)$ in the main argument. Note that the constants do
+not need to be entered in the vv-list of the \verb`\eval` command.
+Their values are available from the \verb`\constants` statements.
+\begin{verbatim}
+    \constants{ c=2.99792458\times10^{8},
+                h=6.62607015\times10^{-34},
+                e=1.602176634\times10^{-19} }
+    \constants[view,add]
+      { \epsilon_0=8.854187817\times10^{-12} }
+    \eval{$ 1/\alpha $}[\alpha=e^2/2\epsilon_0hc]
+\end{verbatim}
+$\Longrightarrow$     \constants{ c=2.99792458\times10^{8},
+                h=6.62607015\times10^{-34},
+                e=1.602176634\times10^{-19} }
+    \constants[view,add]{
+       \epsilon_0=8.854187817\times10^{-12} }
+    \eval{$ 1/\alpha $}[\alpha=e^2/2\epsilon_0hc].
+
+The \verb`view` setting produces a now familiar kind of display.
+It shows that the three-token \verb`\epsilon_0` (the control sequence
+\verb`\epsilon`, the underscore \verb`_` and the digit \verb`0`)
+has been replaced by \verb`\nmc_q` – which may look as if it is also
+three tokens but is in fact a single control sequence.
+
+\subsubsection{Example 2: local constants}
+
+Long ago, when there were such creatures as reference librarians,
+I was asked about a school physics project along these lines.
+
+A\emph{ car is travelling at 50 km/hr when it hits a lamppost. The
+bonnet crumples 1 metre and the car comes to an immediate halt. Although
+she herself is wearing a seat-belt, a woman in the passenger cabin
+is holding her 5 kg baby. Does the baby survive?}
+
+The enquirer was familiar with the equations describing constant acceleration,
+\[
+x=ut+\tfrac{1}{2}at^{2},\quad\text{and}\quad v^{2}-u^{2}=2ax,
+\]
+and Newton's second law, $F=ma$, force equals mass times acceleration.
+The question was really about understanding these laws and how to
+think with them. Here, $s$ is the distance travelled in time $t,$
+with initial speed $u$ at $t=0$, speed $v$ at time $t$, and constant
+acceleration $a$ – a deceleration in this case. 
+
+The given data provide our constants: distance $x=1$ metre, initial
+speed $u=1000*50/(60*60)=(10/36)*50$ metres per second, final speed
+$v=0$. To estimate whether the woman can hold on to her baby, we
+will need to make a comparison with forces we have personally experienced.
+Most of us have tried lifting someone else, so let's use a characteristic
+human weight as our test mass. Thus, we have the (baby's) mass $m=5$
+kilograms, and a test mass, $M$ say, which we will leave as a variable.
+But dealing with weight, we will need the acceleration due to gravity.
+For the kind of rough estimating we are doing, $g=10$ metres per
+second per second will be an adequate approximation.
+\begin{verbatim}
+    \constants{ x=1,v=0,u=(10/36)50,m=5,g=10 }
+\end{verbatim}
+The deceleration experienced by the woman is found from the second
+equation of constant acceleration, $a=(v^{2}-u^{2})/2x$. Even if
+the deceleration isn't constant this will give an estimate of its
+magnitude. (If some of the deceleration is less than this $a$, some
+must be greater.) This is also the deceleration experienced by the
+baby as long as the woman holds onto her. Hence the magnitude of the
+force exerted by the baby on the woman's arms is $ma=m(v^{2}-u^{2})/2x=-mu^{2}/2x$
+which we want to compare with our test force, say that required to
+lift $M=70$ kilograms, which was once considered the mass of an average
+western adult male (but is doubtless a considerable underestimate
+now). Hence the test force is $Mg$. Let's do the calculations. (I
+have altered the \verb`\constants` statement to allow for a later
+comparison with the effect of a small increase in speed.)
+\begin{verbatim}
+    \constants{ x=1,u=(10/36)U,m=5,g=10 }[U=50]
+    \eval{$ mu^2/2x $}[0], \par
+    \eval{$ Mg $}[M=70].
+\end{verbatim}
+$\Longrightarrow$     \constants{ x=1,u=(10/36)U,m=5,g=10 }[U=50]
+    \eval{$ mu^2/2x $}[0], \par
+    \eval{$ Mg $}[M=70].
+
+The force required to hold on to the baby is noticeably less than
+that required to lift a $70$~kg person – in fact about the same
+as that needed to lift a $50$~kg person. But we have ignored the
+force experienced by the mothers forearms – perhaps doubling $m$
+(baby plus forearms) would give a better estimate of the force she
+experiences. In that case $mu^{2}/2x$ obviously doubles and the total
+force required by the woman to retain her baby – now $964$ newtons
+– is significantly more than that required to lift a $70$~kg person.
+I think it almost certain that the baby is torn from her arms. 
+
+What difference does increasing the speed to 60 km/hr make? 
+\begin{verbatim}
+    \constants{ x=1,u=(10/36)U,m=5,g=10 }[U=60]
+    \eval{$ mu^2/2x $}[1], \par
+    \eval{$ Mg $}[M=70].
+\end{verbatim}
+$\Longrightarrow$     \constants{ x=1,u=(10/36)U,m=5,g=10 }[U=60]
+    \eval{$ mu^2/2x $}[0], \par
+    \eval{$ Mg $}[M=70].
+
+Now the force of baby alone is comparable to that required to lift
+a $70$ kg person. Including the woman's forearms in $m$, doubling
+$m$ say, will result in a force twice as great – like that required
+to lift two $70$~kg people or one $140$~kg person. There is no
+chance of the woman holding on to her baby. The force is too great. 
+
+\subsubsection{Example 3: macros and constants}
+
+Constants can depend on previously defined and registered user macros.
+Suppose I have defined two macros
+\begin{verbatim}
+    \NewDocumentCommand\electronmassprecisevalue {}
+                       {9.1093837015\times10^{-31}}
+    \NewDocumentCommand\protonmassprecisevalue {}
+                       {1.672621898\times10^{-27}}
+\end{verbatim}
+(I have taken both the names and the values from the \verb`mandi`
+package.) The long explicit names of the macros has a pedagogic purpose,
+but they are too cumbersome to use in calculations. For that purpose
+we need, first, a \verb`\macros` statement registering the two macros
+with \verb`numerica`, and then a \verb`\constants` statement like
+\begin{verbatim}
+    \nmcConstants{ m_e=\electronmassprecisevalue,
+                   m_p=\protonmassprecisevalue }
+\end{verbatim}
+With that \verb`m_e` and \verb`m_p` could be entered in formulas,
+taking the values contained in the macros. Let's do it:
+\begin{verbatim}
+    \NewDocumentCommand\electronmassprecisevalue {}
+                       {9.1093837015\times10^{-31}}
+    \NewDocumentCommand\protonmassprecisevalue {}
+                       {1.672621898\times10^{-27}}
+    \nmcMacros{ \electronmassprecisevalue,
+                \protonmassprecisevalue }
+    \nmcConstants{ m_e=\electronmassprecisevalue,
+                   m_p=\protonmassprecisevalue }
+    \eval{$ m_p/m_e $}
+\end{verbatim}
+$\Longrightarrow$ \NewDocumentCommand\electronmassprecisevalue {}
+                       {9.1093837015\times10^{-31}}
+\NewDocumentCommand\protonmassprecisevalue {}
+                       {1.672621898\times10^{-27}}
+\nmcMacros{ \electronmassprecisevalue,
+                \protonmassprecisevalue }
+\nmcConstants{ m_e=\electronmassprecisevalue,
+                   m_p=\protonmassprecisevalue }
+\eval{$ m_p/m_e $},
+
+\noindent the familiar mass ratio of proton and electron.
+
+\subsection{Viewing, counting constants}
+
+To see all constants currently `in play', use the \verb`view` setting
+in the \verb`\constants` command. The main argument can be empty,
+\begin{centred}
+\verb`\constants[view]{}` $\Longrightarrow$ \constants[view]{}
+\end{centred}
+or contain a list of constants. In the latter case, the display is
+of the above form but featuring the constants of the new list or,
+if the \verb`add` setting is used, featuring the joined lists, old
+and new:
+\begin{centred}
+\verb`\constants[view,add]{X=42}` $\Longrightarrow$ \constants[view,add]{X=42}
+\end{centred}
+To count how many constants are currently in play, star the \verb`\constants`
+command. The number will depend on whether the main argument is empty
+or not, and whether the \verb`add` setting is active:
+\begin{centred}
+\verb`\constants*{}` $\Longrightarrow$ \constants*{}.
+\end{centred}
+If the \verb`view` setting is being used at the same time as the
+star, the \verb`view` prevails.
+
+\subsection{Errors}
+
+When contemplating error messages from \verb`numerica` it needs to
+be remembered that \emph{multi-token} constants are added to the vv-list
+for every calculation. Hence an error may not be in the vv-list as
+indicated in the message but in the \verb`\constants` statement,
+specifically, the multi-token constants. 
+
+\section{Saving and reusing results: \texttt{\textbackslash nmcReuse}}
+
+\label{sec:supplReuse}You may want to use at some place in a document
+a result calculated earlier. It would be good to be able to do so
+without having to do the calculation again at the new location. \texttt{numerica}
+offers a command \verb`\nmcReuse` (short-name form, \verb`\reuse`)
+which saves a result to a control sequence that can then be used elsewhere
+in the document, expanding to the saved result. The control sequence
+and its content are also saved to file, allowing the possibility of
+using the result in other documents. 
+
+\noindent\begin{minipage}[t]{1\columnwidth}%
+\begin{shaded}%
+The \texttt{\textbackslash nmcReuse} command in version 2 of \texttt{numerica
+}has been completely rewritten. Its use is not compatible with how
+the command was used in version 1. I found that I could bring \texttt{\textbackslash nmcReuse}
+along with \texttt{\textbackslash nmcMacros} and \texttt{\textbackslash nmcConstants}
+into the coding scheme used for \texttt{\textbackslash nmcEvaluate}
+and the reasons for doing so were too compelling. \end{shaded}%
+\end{minipage}
+
+\subsection{Use of \texttt{\textbackslash nmcReuse}}
+
+As noted, all the supplementary commands share the syntax of the \verb`\eval`
+command, so that \verb`\nmcReuse` has an optional settings argument
+preceding a mandatory main argument, followed by two trailing optional
+arguments. \verb`\nmcReuse` does not use the last two. The command
+is used mainly in two ways:
+\begin{enumerate}
+\item {\small\verb`\nmcReuse{}`}{\small , which loads the saved control
+sequences from file, if not already loaded; and}{\small\par}
+\item {\small\verb`\nmcReuse{csname}`}{\small , which loads the saved
+control sequences from file, if not already loaded, assigns the latest
+result from }{\small\verb`\eval`}{\small{} to the control sequence
+}{\small\verb`\csname`}{\small , and saves }{\small\verb`\csname`}{\small{}
+to file.}{\small\par}
+\end{enumerate}
+You may wish to put \verb`\nmcReuse{}` in the preamble of your document
+(\emph{after} \verb`\usepackage{numerica}` of course). In that way,
+saved control sequences are available from the start. Indeed, control
+sequences saved from later in the document can be used in earlier
+sections in a later \LaTeX{} run.
+
+Note that only the \emph{name}, \verb`csname`, of the control sequence
+is supplied to \verb`\reuse`, not the control sequence (\verb`\csname`).
+The name should be composed of letters only. If the name has already
+been defined in \LaTeX{} a \verb`numerica` error is produced, see
+below §\ref{subsec:reuseDeletingOverwriting:}, although if you want
+to save a \emph{new} value in a previously saved control sequence,
+that can be done without invoking a message; see §\ref{subsec:reuseDeletingOverwriting:}.
+
+Once defined {\small with a }{\small\verb`\nmcReuse{csname}`}
+command, \verb`\csname` becomes available for use elsewhere in the
+document.
+
+\subsubsection{What is saved?}
+
+What is saved is the most recent result of an \verb`\eval`-uation.
+This is the \emph{full} result. It may include the vv-list; it may
+include formatting elements; it may include math delimiters. Thus,
+using \verb`\csname` in your document (after the command \verb`\nmcReuse{csname}`)
+may not be straightforward – simply writing \verb`\csname` where
+you want the value it expands to, may produce a \LaTeX{} error and
+halt compilation. You may have to write \verb`$ \csname $` or provide
+some other math environment in order for the control sequence to display
+correctly.
+
+It can be helpful to see \emph{exactly} what has been saved; to do
+that see §\ref{subsec:reuseSeeingSaved}.
+
+\paragraph{Use of \texttt{\textbackslash eval{*}}}
+
+Users will make life simpler for themselves if they make a habit of
+using the starred form \verb`\eval*` to produce the results to save.
+\verb`\eval*` produces solely a number with no formatting or delimiters;
+even a negative result uses a hyphen for the minus sign, just as one
+would type it. In this case \verb`\csname` can be used freely in
+both text and math environments.
+
+\subsubsection{The \texttt{.nmc} file}
+
+\label{subsec:suppleReuse.nmc-file}The file that control sequences
+are saved to has a filename composed of the document name with the
+extension \verb`.nmc`. If your document is \verb`mydoc.tex` (so
+that the \LaTeX{} command {\small\verb`\jobname`} expands to
+\verb`mydoc`) then the file to which results are saved is \verb`mydoc.nmc`,
+located in the document directory. 
+
+\verb`mydoc.nmc` is a comma list of pairs of the form \verb`\csname {value}`.
+Thus, the contents of \verb`mydoc.nmc` might be \texttt{\textbackslash csname1
+\{value1\},\textbackslash csname2 \{value2\},..., \textbackslash csname$n$
+\{value$n$\}}. If \verb`mydoc.nmc` does not already exist then it
+is created in the document directory, and \verb`\csname {value}`
+becomes its first element.
+
+\paragraph{Editing the \texttt{.nmc} file externally}
+
+The \verb`.nmc` file is a text file and can be edited in a text editor.
+Thus it is possible to externally add control sequences and values
+to it provided the structure of the file is strictly adhered to. It
+is also possible to delete items from it or rename control sequences
+or edit values by the same mechanism. Editing the file externally
+like this, or renaming it, or transferring items from one \verb`.nmc`
+file to another, provides a way of using saved values in multiple
+documents.
+
+\subsubsection{Messages}
+
+If a control sequence \verb`\csname` is already known to \LaTeX ,
+then writing \verb`\reuse` \verb`{csname}` will produce a \verb`numerica`
+message and the result of the latest \verb`\eval`-uation will \emph{not}
+be saved:
+\begin{verbatim}
+    \eval*{\sum_{n=1}^{10}n}\par
+    \reuse{sigma}
+\end{verbatim}
+$\Longrightarrow$ \eval*{\sum_{n=1}^{10}n} \par \reuse{sigma}
+
+If there is no result to save – perhaps an \verb`\eval`-uation produces
+an error message instead – then another message is generated:
+\begin{verbatim}
+    \eval*{1/0}\par
+    \reuse{oops}
+\end{verbatim}
+$\Longrightarrow$     \eval*{1/0}\par
+    \reuse{oops}
+
+\subsubsection{Deleting and renewing}
+
+\label{subsec:reuseDeletingOverwriting:}There may be occasions when
+you wish to change a previously saved value and yet, irritatingly,
+the control sequence name will now be known to \LaTeX{} and so will
+generate an `already known' message. If you choose a different name
+for the control sequence to save the new value to, do you want the
+old name cluttering the \verb`.nmc` file? Deleting and renewing the
+values of saved control sequences are controlled by the settings \verb`delete`
+and \verb`renew`.
+
+Entering \verb`delete` in the settings option \emph{deletes} a control
+sequence and its value from the \verb`.nmc` file and undefines it
+in \LaTeX{} terms. Thus \verb`\reuse[delete]` \verb`{csname}` would
+delete \verb`\csname` and its value from the \verb`.nmc` file and
+undefine \verb`\csname`. If \verb`\csname` is not present in the
+file, nothing happens. Entering \verb`renew` replaces the value of
+a saved control sequence with a new value. If there is no such \emph{saved}
+control sequence but the control sequence is otherwise known to \LaTeX{}
+the `already defined' message will still be generated. This prevents
+giving control sequences like \verb`\sin`~or \verb`\frac` new meanings
+with the \verb`renew` setting.
+\begin{itemize}
+\item \verb`\reuse[delete]{csname}` deletes \verb`\csname` and its value
+from the \verb`.nmc` file and from memory if present; otherwise has
+no effect;
+\item \verb`\reuse{csname}` (the default) saves the result of the latest
+\verb`\eval` command to \verb`\csname`, provided \verb`\csname`
+is not already defined; in that case a warning message is presented
+and the result is not saved;
+\item \verb`\reuse[renew]{csname}` behaves like the default mode unless
+\verb`\csname` is already a saved control sequence in the \verb`.nmc`
+file, in which case its previous value is replaced by the result of
+the latest \verb`\eval` command; 
+\item if \verb`delete` and \verb`renew` are used together, whichever occurs
+second prevails.
+\end{itemize}
+In the following example, the first \verb`\reuse` deletes \verb`\suma`
+should it be present in the \verb`.nmc` file, the second saves the
+result, $55$, of the latest \verb`\eval`-uation (in fact an \verb`\eval*`-uation)
+and the third overwrites that saved value with the new value, $210$. 
+\begin{verbatim}
+    \reuse[delete]{suma}
+    \eval*{\sum_{n=1}^{10}n} \par
+    \reuse{suma}
+    \eval*{\sum_{n=1}^{20}n} \par 
+    \reuse[renew]{suma}
+\end{verbatim}
+$\Longrightarrow$ \reuse[delete]{suma}
+\eval*{\sum_{n=1}^{10}n} \par 
+\reuse{suma}
+\eval*{\sum_{n=1}^{20}n} \par 
+\reuse[renew]{suma}
+
+\subsubsection{Viewing what has been saved}
+
+\label{subsec:reuseSeeingSaved}It would be good in this example to
+see that the new value $210$ has in fact been saved. That is easy.
+Simply enter \verb`view` in the settings option of \verb`\nmcReuse`
+(I've removed the now unnecessary \verb`\par` tokens from the example.)
+\begin{verbatim}
+    \reuse[delete]{suma}
+    \eval*{\sum_{n=1}^{10}n}
+    \reuse[view]{suma}
+    \eval*{\sum_{n=1}^{20}n} 
+    \reuse[renew,view]{suma}
+\end{verbatim}
+$\Longrightarrow$     \reuse[delete]{suma}
+    \eval*{\sum_{n=1}^{10}n}
+    \reuse[view]{suma}
+    \eval*{\sum_{n=1}^{20}n}
+    \reuse[renew,view]{suma}
+
+\noindent First the original value $55$ was saved to \verb`\suma`
+but then the value was overwritten by the new value $210$. 
+
+The \verb`view` setting allows us to see how formatting is stored
+if the \emph{un}starred form of the \verb`\eval` command is used.
+In the following example, \verb`\eval` wraps around math delimiters:
+\begin{centred}
+\verb`\eval{$ 1+1 $} \reuse[view,renew]{two}` $\Longrightarrow$
+\eval{$ 1+1 $} \reuse[view,renew]{two}
+\end{centred}
+The full \emph{formula=result} display has been captured in \verb`\two`
+along with the math delimiters. If a vv-list is also involved, things
+become messy (but informative):
+\begin{verbatim}
+    \eval{$ x+y $}[x=1,y=2]
+    \reuse[view,renew]{three}
+\end{verbatim}
+$\Longrightarrow$     \eval{$ x+y $}[x=1,y=2]
+    \reuse[view,renew]{three}
+
+You may want to see \emph{all} saved control sequences. In that case
+use an \emph{empty} main argument: \verb`\nmcReuse[view]{}`. We now
+have enough saved control sequences to make this worthwhile: 
+\begin{verbatim}
+    \reuse[view]{}
+\end{verbatim}
+$\Longrightarrow$ \reuse [view]{} 
+
+\noindent (The \verb`\seven` that appears here is defined shortly.
+Its appearance \emph{before} definition is presumably due to \LaTeX{}
+making a number of passes when compiling this document.)
+
+\subsubsection{Counting saved control sequences: \texttt{\textbackslash nmcReuse{*}}}
+
+Because \verb`\nmcReuse` uses the same machinery as \verb`\nmcEvaluate`,
+it has a starred form, \verb`\nmcReuse*`, which produces a purely
+numerical result (just like \verb`\eval*`, \verb`info*`, \verb`\macros*`
+and \verb`\constants*`). In this case, the number is the count of
+how many control sequences have been saved:
+\begin{centred}
+\verb`\reuse*{}` $\Longrightarrow$ \reuse*{ }.
+\end{centred}
+
+\subsection{\texttt{reuse} setting of \texttt{\textbackslash eval} command}
+
+\label{subsec:supplReuseEvalSetting}Using \verb`\eval*` for a calculation
+ensures a purely numerical result, with no vv-list or formatting in
+the display of the result. But sometimes we might want the full display
+yet wish to save only the numerical result. This is the point of the
+\verb`reuse` setting of the \verb`\eval` command. 
+
+For the \emph{starred} form of the \verb`\eval` command it is always
+\emph{only the numerical result} that is saved, whatever the value
+of the \texttt{reuse} key in the settings option of the \verb`\eval`
+command.
+
+For the \emph{unstarred} form of the \verb`\eval` command exactly
+what is saved with \verb`\nmcReuse` depends on the \texttt{reuse}
+setting:
+\begin{lyxcode}
+reuse~=~<integer>
+\end{lyxcode}
+where \verb`<integer>` can take one of two values,
+\begin{itemize}
+\item \texttt{reuse=0} (the default) saves\emph{ the form that is displayed}
+including the vv-list if there is one and possibly a formatting component
+(like math delimiters). Note that if the vv-list is empty, a formatting
+component (math delimiters) may still be present in the saved result; 
+\item \texttt{reuse=1} (or, indeed, any non-zero integer) saves only the
+numerical result with no other elements of the display (no vv-list,
+no formatting component, no math delimiters).
+\end{itemize}
+As we saw earlier, saving the result from \verb`\eval{$ x+y $}[x=1,y=2]`,
+corresponding to \verb`reuse=0`, means the full display is saved.
+Check by writing \verb`\three` $\Longrightarrow$ \three. The full
+display was saved (including math delimiters).
+
+On the other hand, with {\ttfamily\verb`reuse=1`} only
+the numerical value is saved:
+\begin{centred}
+\verb`\eval[reuse=1]{$ x + y $}[x=3,y=4] \reuse[renew]{seven}` $\Longrightarrow$
+\eval[reuse=1]{$ x + y $}[x=3,y=4] \reuse[renew]{seven}.
+\end{centred}
+The numerical result only of the calculation should be saved, although
+the formula and vv-list are displayed as the result of the \verb`\eval`-uation.
+We can easily check: \verb`\seven` $\Longrightarrow$ \seven. Indeed,
+only the numerical result was saved.
+
+\chapter{Nesting  commands}
+
+\label{chap:Nesting} The \verb`\eval` command and the supplementary
+commands of the previous chapter can be \emph{nested} –\emph{ }used
+within other \verb`\eval` or supplementary commands. Nesting may
+occur in the main argument, or the vv-list, or the settings option,
+or some combination of all three. With the commands currently introduced,
+nesting is unlikely to be a major concern, but it becomes significant
+for the commands defined in the associated package \texttt{numerica-plus}
+(see §\ref{subsec:Related-packages}). Since those additional commands
+are not available for this document, the examples below use the commands
+introduced earlier: \verb`\eval`, \verb`\info`, \verb`\macros`,
+\verb`\constants` and \verb`\reuse`.
+
+\section{Nesting in the formula}
+
+Consider a statement like \verb`\eval{...\eval...}`. There is an
+inner \verb`\eval` and an outer \verb`\eval`. The inner \verb`\eval`
+`digests' \emph{its} \LaTeX{} formula to produce an \verb`l3fp`-readable
+expression which is fed to \verb`l3fp` to evaluate. The result is
+then returned to (the inner) \verb`\eval` to display. In version
+1 of \verb`numerica` that meant the inner command \emph{had} to be
+starred, \verb`\eval*`, so that no display formatting was fed to
+the outer command to try to digest (and cause an error).\emph{ }In
+version 2 of \verb`numerica` this is no longer the case. \verb`numerica`
+detects whether a command is inner or outer, and if inner, suppresses
+all display formatting, producing only a number, as if the command
+had been starred: 
+\begin{centred}
+\verb`\eval{$ \sin(\eval{\sin x}[x=\pi/6]\pi) + 1 $}` $\Longrightarrow$
+\eval{$ \sin(\eval{\sin x}[x=\pi/6]\pi) + 1$}.
+\end{centred}
+In the presentation of the overall result, the inner \verb`\eval`
+command is evaluated, displaying as a number.
+
+In this example, the \verb`x=\pi/6` could be removed from the inner
+\verb`\eval` and placed in the vv-list of the outer command since
+outer variables are available to the inner command:
+\begin{centred}
+\verb`\eval{$ \sin(\eval{\sin x}\pi) + 1 $}[x=\pi/6]` $\Longrightarrow$
+\eval{$ \sin(\eval{\sin x}\pi) + 1$}[x=\pi/6].
+\end{centred}
+Just to show that it is possible, the next example shows \verb`\eval`
+being used in a \verb`\constants` command. The \verb`o` setting
+in the \verb`\constants` command pervades its argument; hence it
+needs to be explicitly turned off for the \verb`\eval` if \verb`\sin(\pi/6)`
+is to evaluate as expected.
+\begin{verbatim}
+    \constants[o]{ y=\sin 30,x=\eval[o=0]{\sin(\pi/6)} }
+    \eval{$ x+y $}
+\end{verbatim}
+$\Longrightarrow$     \constants[o]{ y=\sin 30,x=\eval[o=0]{\sin(\pi/6)} }
+    \eval{$ x+y $}.
+
+\subsection{Math delimiters and double evaluations}
+
+Any math delimiters in the inner \verb`\eval` are ignored. (This
+also differs from version 1 of \verb`numerica` where they caused
+an error.) Obviously it is simpler to omit them as I have done in
+the examples. 
+
+However, math delimiters in the \emph{outer} \verb`\eval` command
+still have their normal effect and produce a \emph{formula = result,
+(vv-list)} display. One consequence of such a display is that the
+formula in the \emph{inner} \verb`\eval` command is evaluated \emph{twice}
+– once when the overall result is being calculated (i.e. the formula
+of the \emph{outer} \verb`\eval`) and later when the overall display
+of the result is created. In the \emph{formula} part of the \emph{formula
+= result, {[}vv-list{]}} display, the tokens in the \emph{formula}
+are expanded to their display form. For example, \verb`\sin` is expanded
+to $\sin$, \verb`\pi` is expanded to $\pi$ – and the inner \verb`\eval`
+is expanded to the numerical result of its evaluation – a second evaluation.
+If the inner formula is simple, this will be of little moment, but
+should the inner formula contain, say, a slowly converging infinite
+series, then evaluating it twice is a bad idea and it would be better
+to remove the delimiters from the outer \verb`\eval`. That prevents
+the second evaluation. 
+
+The problem does not arise if the outer \verb`\eval` lies within
+a math environment (e.g. \verb`$ \eval{...} $`) since that produces
+a display of the form \emph{result, {[}vv-list{]}.} The formula is
+not displayed and so the second evaluation does not occur. The inner
+\verb`\eval` is evaluated once only  to calculate the result.
+
+\section{Nesting in the vv-list}
+
+The inner \verb`\eval` can be placed in the vv-list of the outer
+command. If the vv-list of the inner \verb`\eval` contains a comma
+(meaning there are at least two variables), then the entire inner
+\verb`\eval` and its \LaTeX{} arguments needs to be wrapped in braces
+to hide the comma or commas of its vv-list from the outer \verb`\eval`.
+To show the effect of not doing so, I have slightly complicated the
+previous example by adding a second (unnecessary) variable. The first
+example is with braces, the second without:
+\begin{centred}
+\verb`\eval{$ \sin k\pi + 1 $}[k={\eval{y\sin x}[x=\pi/6,y=1]}]`
+$\Longrightarrow$ \eval{$ \sin k\pi + 1 $} [k={\eval{y\sin x}[x=\pi/6,y=1]}].
+
+\verb`\eval{$ \sin k\pi + 1 $}[k=\eval{y\sin x}[x=\pi/6,y=1]]` $\Longrightarrow$
+\eval{$ \sin k\pi + 1 $}[k=\eval{y\sin x}[x=\pi/6,y=1]].
+\end{centred}
+The vv-list of the outer \verb`\eval` is parsed as containing two
+entries, \verb`k=\eval` \verb`{y\sin x}[x=\pi/6` and \verb`y=1]`.
+Both will cause errors but since the vv-list is evaluated from the
+right, it is \verb`y=1]` which actually does so. 
+
+\section{Nesting in the settings option}
+
+This will be rare, but commands can occur in the settings option of
+the outer command. The \verb`\info` command provides a good example.
+I have included it in the punctuation setting of an \verb`\eval`-uation. 
+\begin{verbatim}
+    \eval[p=\mbox{,\qquad\info{sum} terms.}]
+      {\[ \sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!} \]}[3] 
+\end{verbatim}
+$\Longrightarrow$ \eval[p=\mbox{,\qquad\info{sum} terms.}]{\[ \sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!} \]}[3]
+
+Because of the \verb`\[ \]` math delimiters, if the \verb`\info`
+command had been placed \emph{after} the \texttt{\textbackslash eval}
+command, it would have slid down to the next line. Used in the settings,
+as here, the display is \emph{inside} the \verb`\[ \]` delimiters,
+on the same line as the expression. This may be significant for adjusting
+vertical spacing of later parts of the document – widow and orphan
+control for instance.
+
+A point to note is the explicit writing of the `terms' descriptor.
+Normally \verb`\info{sum}` would automatically supply the descriptor,
+but as noted earlier, nesting of one command in another suppresses
+all elements of display of the inner command beyond the numerical
+result. It is as if the inner command is starred. Because the \verb`\info`
+command is nested in the \verb`\eval` command, the `terms' descriptor
+is suppressed and has had to be explicitly supplied by hand.
+
+\section{Rounding and display}
+
+In the display of the overall result, it is the result of the inner
+command which is shown, not the formula that the inner command acts
+on. How that number is displayed is determined by the number-format
+specification of the \emph{inner} command. Note however that this
+specification affects only how the result of the inner command is
+shown. Always 16 figures are passed from the inner command to the
+outer, as you can see in this example:
+\begin{verbatim}
+    \eval{$ \pi - \eval{ \pi }[4] $}[15]
+\end{verbatim}
+$\Longrightarrow$ \eval{$ \pi - \eval{ \pi }[4] $}[15].
+
+\noindent The outer result would not be zero to $15$ places of decimals
+if the inner result were restricted to $4$ decimal places. It is
+only the \emph{display} of the inner result which is so restricted.
+
+For infinite sums and products (and for \verb`\nmcIterate` and \verb`\nmcSolve`
+of the \verb`numerica-plus` package), the rounding value is not just
+for display purposes but is also used to determine the result. This
+may require judicious use of the extra rounding setting to get a sensible
+display. In the first instance below, the second sum stops at an effective
+rounding value of $5+2=7$, since the default extra rounding is $+2$,
+and the first sum (the inner one) also stops at $7=4+3$. No surprise
+then that the overall result is $0$. In the second instance, the
+inner sum stops at a rounding value of $4+2=6$. Although the left-hand
+side of the display is unaltered, the result is no longer $0$.
+\begin{verbatim}
+     \eval{$ \eval[S+=3]{\sum_{n=1}^\infty 1/n^3}[4*]
+       - \sum_{n=1}^\infty 1/n^3$}[5]
+\end{verbatim}
+$\Longrightarrow$      \eval{$ \eval[S+=3]{\sum_{n=1}^\infty 1/n^3}[4*]
+       - \sum_{n=1}^\infty 1/n^3$}[5]
+
+\noindent whereas
+\begin{verbatim}
+    \eval{$ \eval[S+=2]{\sum_{n=1}^\infty 1/n^3}[4*]
+       - \sum_{n=1}^\infty 1/n^3$}[5]
+\end{verbatim}
+$\Longrightarrow$      \eval{$ \eval[S+=2]{\sum_{n=1}^\infty 1/n^3}[4*]
+       - \sum_{n=1}^\infty 1/n^3$}[5].
+
+\section{Error messages}
+
+Errors in an inner command create a small change in error message
+display.
+\begin{centred}
+\verb`\eval{ 1 + \eval{ 1 + \eval{ k } } }` $\Longrightarrow$ \eval{ 1 + \eval{ 1 + \eval{ k } } }
+
+\verb`\eval{ x + \eval{ k }[k=\arcsin 2] }[x=1]` $\Longrightarrow$
+\eval{ x + \eval{ k }[k=\arcsin 2] }[x=1]
+\end{centred}
+An integer is added to the `where' part of the error message. The
+integer indicates the \emph{level of nesting} where the error occurs.
+
+If there is no nesting where the error occurs, the integer is suppressed,
+even though there may be nesting elsewhere in the overall expression.
+This is in the interests of straightforwardness when nesting is absent,
+which will be overwhelmingly the most common situation.
+\begin{centred}
+\verb`\eval{ k + \eval{ x }[x=1] }[k=\arcsin 2]` $\Longrightarrow$
+\eval{ k + \eval{ x }[x=1] }[k=\arcsin 2]
+\end{centred}
+
+\section{Debugging}
+
+\label{subsec:nestDebugging}It is worth looking at the debug display
+when \verb`\eval` commands are nested. For the outer \verb`\eval`
+command:
+\begin{centred}
+\verb`\eval[dbg=1]{$ \sin \eval*{\sin x}[x=\pi/6]\pi + 1 $}` $\Longrightarrow$
+\eval[dbg=210]{$ \sin \eval*{\sin x}[x=\pi/6]\pi + 1$}
+\end{centred}
+There is no vv-list for the outer command whence the two empty slots
+in the display but when the inner \verb`\eval` is in the vv-list,
+they are filled:
+\begin{centred}
+\verb`\eval[dbg=1]{$ \sin k\pi + 1 $}[k=\eval*{\sin x}[x=\pi/6]]`
+$\Longrightarrow$ \eval[dbg=1]{$ \sin k\pi + 1$} [k=\eval*{\sin x}[x=\pi/6]]
+\end{centred}
+For the inner \verb`\eval` command debugging may still work but in
+an idiosyncratic way. To clarify exactly what is going on I have added
+a \verb`\left( \right)` pair around the entire inner \verb`\eval`
+command. Note that I have also used a \emph{negative} \texttt{dbg}
+value. With a positive value, the right parenthesis is pressed toward
+the right margin of the page. The negative value limits the display
+to the text width and gives the much neater result shown.
+\begin{verbatim}
+    \eval[()=2]{$
+      \sin\left( 
+            \eval*[dbg=-1]{ \sin x }[x=\pi/6]
+          \right)\pi + 1 $}
+\end{verbatim}
+$\Longrightarrow$     \eval[()=2]{$
+      \sin\left( 
+            \eval*[dbg=-1]{ \sin x }[x=\pi/6]
+          \right)\pi + 1 $}
+
+\medskip{}
+The debug display from the inner \verb`\eval` command has been inserted
+into the formula of the outer \verb`\eval` in the position occupied
+by the inner \verb`\eval`. I did not deliberately code for this,
+but have decided to leave it as is despite the potential for some
+rather odd displays, since there can be no confusion about which \verb`\eval`
+command is being `debugged'. In this last example, in order to both
+use \verb`\left(...\right)` and have the calculation give the previous
+result I have employed the setting \verb`()=2` in the outer \verb`\eval`;
+see §\ref{subsec:parseTrigFns}.
+
+\chapter{Using \texttt{numerica} with \protect\LyX}
+
+\label{chap:LyX}The document processor \LyX{} has a facility that
+enables snippets from a document to be compiled separately and the
+results presented to the user without having to compile the entire
+document. The present document was written in \LyX . The demonstration
+calculations were evaluated using this \emph{instant preview} facility. 
+
+To use \texttt{numerica} in \LyX{} go to \textsf{Document \lyxarrow{}
+Settings \lyxarrow{} LaTeX Preamble} and enter
+\begin{lyxcode}
+\textbackslash usepackage\{numerica\}
+\end{lyxcode}
+then click \textsf{OK}. You may wish to follow the above line in the
+preamble with \verb`\nmcReuse{}`:
+\begin{lyxcode}
+\textbackslash usepackage{[}lyx{]}\{numerica\}
+
+\textbackslash nmcReuse\{\}
+\end{lyxcode}
+In that case, type the extra line and \emph{then} click \textsf{OK}.
+The additional line ensures all saved values are available in your
+document from the outset.
+
+\section{Instant~preview}
+
+The instant preview facility of \LyX{} performs mini-\LaTeX{} runs on
+selected parts of a document (for instance, the mathematical parts)
+and displays the results in \LyX{} while the user continues to work
+on the surrounding document.\texttt{ numerica} uses these mini-\LaTeX{}
+runs to do its evaluations and display their results. That means you
+get feedback on your calculations almost immediately.
+
+To use this facility first ensure that instant preview is turned on.
+This means selecting \textsf{Tools \lyxarrow Preferences \lyxarrow Look
+\& Feel \lyxarrow{} Display}, ensuring that the \textsf{Display graphics}
+checkbox is checked, and against \textsf{Instant preview} selecting
+\textsf{On}, then clicking \textsf{OK}.
+
+\subsection{Document location}
+
+It also matters where your document is located. You may have your
+own local or personal texmf tree (see §\ref{subsec:settingsPersonal-texmf-tree}).
+If your document is located there, perhaps in the \verb`doc` folder,
+then not all features of preview will work as expected. Presumably
+this is because both \LyX{} and your \LaTeX{} distribution (e.g. \TeX Live
+or MiK\TeX ) are interacting with the location and interfere. Move
+your document to another location which your \LaTeX{} distribution
+has no interest in, and open it in \LyX{} there.
+
+\subsection{Global vs local previewing}
+
+\label{subsec:LyXGlobal-vs-local}Compilation of previews occurs in
+two distinct modes. 
+
+\paragraph{Global preview generation:}
+
+When a document is opened (and preview is \emph{on}), all previews
+in the document are formed in sequence in the one \LaTeX{} run. This
+is the global mode. The mini-\LaTeX{} run may well be substantial.
+It compiles a \verb`.tex` file that begins with the document's preamble
+with some additions then comes \verb`\begin{document}`. That is followed
+by a sequence of preview environments,
+\begin{lyxcode}
+\textbackslash begin\{preview\}
+
+<stuff>
+
+\textbackslash end\{preview\}
+\end{lyxcode}
+one for each preview in the document. Finally there is an \verb`\end{document}`
+statement. The critical point is that all previews are between the
+same \verb`\begin{document}`, \verb`\end{document}` statements,
+and so earlier previews in the sequence can communicate with later
+ones.
+
+\paragraph{Local preview generation:}
+
+The other mode in which preview operates is local. Suppose you have
+your document open and want to add to it, for instance with a simple
+evaluation, \verb`\eval{x+y}[x=1,y=2]` in an ERT inset in a preview
+inset. The resulting mini-\LaTeX{} run is of the form
+\begin{lyxcode}
+<preamble>
+
+\textbackslash begin\{document\}
+
+\textbackslash begin\{preview\}
+
+\textbackslash eval\{x+y\}{[}x=1,y=2{]}
+
+\textbackslash end\{preview\}
+
+\textbackslash end\{document\}
+\end{lyxcode}
+The preamble is as before but there is only \emph{one} preview between
+the \verb`\begin{document}`, \verb`\end{document}` statements. That
+preview is isolated from all other, previous previews and will be
+isolated from all other, later previews. 
+
+This has implications for the supplementary commands of the previous
+chapter and means that if you want to transfer information (a macro,
+a constant, a result) from one preview to another, you need to do
+it through the preamble or by means of an external file or, in some
+cases, by forcing a global preview run in which all previews are recompiled
+between the same \verb`\begin{document}`, \verb`\end{document}`
+statements.
+
+\subsubsection{Forcing a global preview run}
+
+Closing then opening a document is one way to force a global preview
+compilation. Another is to change the zoom level. This causes \LyX{}
+to recompile all previews at the new zoom level. But you may not want
+to work at the new zoom level. Going back to the old zoom level will
+force a second recompilation of all previews. For a large document
+\emph{two} recompilations is too heavy a burden. The secret is to
+combine a zoom in and a zoom out into one command and attach it to
+a shortcut.
+
+If you go to \textsf{Tools \lyxarrow{} Preferences \lyxarrow{} Editing
+\lyxarrow{} Shortcuts}, click on the \textsf{New} button and enter
+\begin{lyxcode}
+command-sequence~buffer-zoom-in;~buffer-zoom-out
+\end{lyxcode}
+then assign a shortcut to it (\verb`Alt+Z` for zoom?) you will gain
+a simple means of forcing a global recompilation of previews. 
+
+\section{Mathed}
+
+(Mathed = the \LyX{} mathematics editor.) If you have instant preview
+\emph{on} then one way to use \texttt{numerica} in \LyX{} is to enter
+an \verb`\eval` command in mathed. Clicking the cursor outside the
+editor with the mouse or moving it outside with the arrow keys will
+then trigger formation of a preview of the editor's contents – a snippet
+of what will be shown in the pdf. This will be displayed in mathed's
+place after a generally short `pause for thought' as the mini-\LaTeX{}
+run progresses behind the scenes. 
+
+The original expression can be recovered by clicking on the preview.
+The content of mathed is immediately displayed and can be edited.
+
+\subsection{\protect\LaTeX{} braces~\{~~\}}
+
+\LyX{} does not support \texttt{numerica}'s \verb`\eval` command `out
+of the box' as it does, say, \verb`\frac` or \verb`\sqrt`. To use
+the \verb`\eval` command in mathed you will need to supply the braces
+used to delimit its mandatory argument. (For \verb`\frac` and \verb`\sqrt`
+by contrast, \LyX{} supplies these automatically in the form of blue-outlined
+boxes.) Unfortunately the \verb`{` key\footnote{\textsf{Shift+{[}} on my keyboard.}
+does not insert a left brace into the document but rather an escaped
+left brace \verb`\{` as you can see by looking at \textsf{View \lyxarrow{}
+Code Preview Pane}. Escaped braces like this are used for grouping
+terms in \emph{mathematics}; they are not the delimiters of a \LaTeX{}
+argument. 
+
+The brace delimiters for \LaTeX{} arguments are entered in mathed by
+typing a backslash \textsf{\textbackslash{} }then a left brace\textsf{
+\{} – two separate key presses rather than a single combined press.
+This enters a balanced pair of (unescaped) braces with the cursor
+sitting between them waiting for input. Alternatively, if you have
+already written an expression that you want to place between braces,
+select it, then type \textsf{\textbackslash{} }then\textsf{ \{}.
+
+\section{Preview insets}
+
+There are problems with using mathed for calculations.
+\begin{itemize}
+\item Expressions entered in mathed are necessarily of the form \verb`$ \eval... $`
+or more generally \verb`delimiter` \verb`\eval...` \verb`delimiter`.
+But you may wish to wrap the \verb`\eval` command \emph{around} the
+math delimiters to produce a \emph{formula=result} form of display.
+In mathed the only way to effect such a display is to write the \emph{formula=
+}part yourself – which may involve no more than copy and paste but
+is still additional mouse work/key pressing. 
+\item Mathed does not accept carriage returns. If you want to format a complicated
+expression for readability by breaking it into separate lines, you
+can't. The expression is jammed into the one line, along with the
+settings option content and the vv-list, often extending well beyond
+the edge of the screen. 
+\end{itemize}
+For these reasons I have come to prefer \emph{not} using mathed for
+calculations but instead to use preview insets wrapped around \TeX -code
+(ERT) insets. \LyX{} uses the shortcut \textsf{Ctrl+L} to insert an
+ERT inset. Since \LyX{} now does no printing itself, the shortcut \textsf{Ctrl+P}
+that was formerly used for printing is available for other purposes.
+On my keyboard, the \textsf{P} key lies diagonally up and to the right
+but adjacent to the \textsf{L} key. I suggest assigning \textsf{Ctrl+P}
+to inserting a preview inset. Then typing \textsf{Ctrl+P Ctrl+L} –
+which means holding the \textsf{Ctrl} key down and tapping two diagonally
+adjacent keys, \textsf{P} followed immediately by \textsf{L} – will
+insert an ERT inset inside a preview inset with the cursor sitting
+inside the ERT inset waiting for input. In the ERT inset you can enter
+carriage returns, and so format complicated expressions. You can place
+the vv-list on a separate line or onto consecutive lines. And when
+you have finished, clicking outside the preview inset will trigger
+preview into doing its thing and present the result `before your
+eyes'. 
+
+To assign the suggested shortcut, go to \textsf{Tools \lyxarrow{} Preferences
+\lyxarrow{} Editing \lyxarrow{} Shortcuts}. Under \textsf{Cursor, Mouse
+and Editing Functions} in the main window on the right, scroll down
+until you come to \textsf{preview-insert}, select it, then click \textsf{Modify}.
+Now press \textsf{Ctrl+P}. The shortcut will magically appear in the
+greyed, depressed key.\textsf{ }Click \textsf{OK} and then \textsf{OK}
+in the \textsf{Preferences} window to close it. (Most of the examples
+in this document have been evaluated in this way, using \textsf{Ctrl+P
+Ctrl+L.)}
+
+\section{Errors }
+
+Instant preview will display error messages generated by \verb`numerica`
+in \LyX{} just as it does the results of calculations. Clicking on
+the message will show the underlying expression which can then be
+edited. However \LaTeX{} errors will \emph{not} produce a preview;
+formation of the preview will stall. To find precisely what has gone
+wrong, you will need to look at the \LaTeX{} log, but not the log of
+the overall document; rather the \emph{preview} log. 
+
+\subsection{Temporary directory of \protect\LyX}
+
+Unfortunately this is tucked away in a temporary directory and is
+not immediately accessible in \LyX{} (unlike the main \LaTeX{} log from
+\textsf{Document \lyxarrow{} \LaTeX{} Log}). When \LyX{} is started,
+it sets up a temporary directory in which to perform various tasks.
+On Windows systems this will be located in \texttt{C:\textbackslash Users\textbackslash <your
+name>\textbackslash AppData\textbackslash Local\textbackslash Temp}
+and will have a name like \texttt{lyx\_tmpdir.XOsSGhBc1344}. 
+
+One of the tasks \LyX{} uses this temporary directory for is to create
+preview images when a document is opened. If you look inside \LyX 's
+temporary directory when a document is first loaded, you will see
+a subdirectory created, with a name like \texttt{lyx\_tmpbuf0}. There
+may already be such directories there, in which case the number on
+the end will be greater than \texttt{0} – it depends on whether other
+documents are or have been open in the current instance of \LyX .
+Inside the appropriate \texttt{lyx\_tmpbuf}\texttt{\emph{n}} folder
+will be the preview log with a name like \texttt{lyxpreviewZL1344.log}.
+It will usually be accompanied by other files with extensions like
+\texttt{.dvi}, \texttt{.tex}, and – depending on the number of previews
+in your document – a number, perhaps a lot, of image files with the
+extension \texttt{.png}, each one of which is a preview. For a document
+just loaded there will be only the one preview log, but if you have
+added preview insets or math insets to your document\textsf{ }in the
+current editing session there will be a number of such logs and you
+will need to determine the relevant one by the time stamp. 
+
+The log files are text files and can be opened in a text editor. The
+relevant part of the log is towards the end (just before the final
+statistical summary) where you will find a list of entries like \texttt{Preview: Snippet
+1 641947 163840 7864588}. If there is an error, it will be noted here
+among these snippets and will generally make clear what needs remedying.
+
+\subsection{CPU usage, \protect\LaTeX{} processes}
+
+It is possible when a preview stalls that the \LaTeX{} process associated
+with the preview will continue to run, using CPU cycles, slowing overall
+computer performance, and perhaps resulting in extra fan use giving
+a different sound to the computer. In Windows 10, the \textsf{Task
+Manager} (\textsf{Ctrl+Shift+esc}) under the \textsf{Details} tab
+shows the current executables running. The \textsf{CPU} column will
+show which processes are preoccupying the CPU. Check whether one or
+more of these processes looks \LaTeX -related (e.g. \texttt{latex.exe}
+or \texttt{pdflatex.exe}, or \texttt{miktex-pdftex.exe} if using MiK\TeX ).
+Click the \textsf{Name} column to sort the processes by name and look
+for the relevant name in the list, select it, and end the process
+(click the \textsf{End Task} button). 
+
+I am not familiar with the corresponding situation on Linux or Mac.
+
+\section{Hyperref support vs speed}
+
+If you want the pdf produced from your document to support hyperref
+links and show an outline window in your pdf viewer (generally placed
+on the left in the viewer) then you need to ensure the checkbox at
+\textsf{Document Settings \lyxarrow{} PDF Properties \lyxarrow{} Use
+Hyperref Support} is indeed checked. But you don't need to do this
+until the final compilation of the document. The advantage of leaving
+this until the last is that in a large document with many previews
+the time for preview generation is essentially halved. If hyperref
+support is enabled, preview generation not only creates all the individual
+image files that are the previews (files of extension \verb`.png`)
+but also requires the compilation of a single pdf document showing
+all the previews in sequence. (Like the previews, the pdf document
+`hides' in the termporary directory where \LyX{} does its work.)
+In other words, \emph{two} images are created for each preview, the
+\verb`.png` image which is the one \LyX{} displays, and another image
+buried inside the pdf of all images. That second step does not occur
+if hyperref support is disabled. In a small document, this is not
+going to matter; in a large document it becomes significant. It is
+well worth temporarily turning off hyperref support and then, when
+the time for final compilation comes, turning it back on.
+
+\section{Supplementary commands in \protect\LyX}
+
+There are some difficulties using the supplementary commands successfully
+with instant preview. 
+
+\subsection{Reuse of earlier previews}
+
+One is that whenever \LyX{} has generated a preview image for a particular
+\LaTeX{} expression, it will use that same image whenever it meets
+that same \LaTeX{} expression later. That means that a statement like
+\verb`\macros[view]{}` and the same statement later will display
+the same image, even though there may have been macros defined or
+freed in between. The same goes for all the other supplementary functions,
+including, for example, \verb`\info{sum}`. A second instance of \verb`\info{sum}`
+will display the image generated by the first instance even though
+further infinite sums may have been evaluated between the \verb`\info`
+statements. 
+
+The remedy is to make some small but insignificant difference to the
+\LaTeX{} expression  in the second instance – generally a change in
+white space will do. For example: first time \verb`\macros[view]{}`,
+second time \verb`\macros[view]{ }` where a space has been inserted
+between the braces; or: first time \verb`\info{sum}`, second time
+\verb`\info{ sum}` where a space has been inserted before \verb`sum`.
+This will ensure \LyX{} doesn't fall back on the previously generated
+image.
+
+\subsection{\textquoteleft Stalled\textquoteright{} previews}
+
+It is possible to put content into an ERT inset inside a preview inset
+(\textsf{Ctrl+P Ctrl+L}) and for nothing to happen. The preview has
+apparently stalled. Certainly this can be the case if there is an
+error in the input (e.g. a missing brace) but it also occurs if there
+is no output to display. For instance \verb`\constants{ c=300000000 }`
+does not produce any visual output. There is nothing for the preview
+to display and so the preview inset sits there, apparently stalled.
+This is a security measure for previews in \LyX{} to provide at least
+some guard against malicious code being run in the preview. If the
+preview resolved, it would disappear completely from view in the \LyX{}
+window. 
+
+If you find the visual appearance of such apparently stalled previews
+distracting, the addition of some displayable content to the preview
+will result in it resolving to that content; the content could be
+as small as a full stop.
+
+\subsection{Using \texttt{\textbackslash nmcMacros}}
+
+As noted earlier, previews are mini-\LaTeX{} runs, either local or
+global. Each local preview is of the form<preamble>
+\begin{lyxcode}
+\textbackslash begin\{document\}
+
+\textbackslash begin\{preview\}
+
+<stuff>
+
+\textbackslash end\{preview\}
+
+\textbackslash end\{document\}
+\end{lyxcode}
+Whatever goes into or comes out of the preview is isolated from any
+other local preview, unless it is through the preamble or an external
+file. Sometimes a global preview run can overcome this problem for
+then all the previews lie between the same \verb`\begin{document}`,
+\verb`end{document}` statements. However, this does not help with
+macro definitions. \verb`\def`, \verb`\newcommand`, \verb`\NewDocumentCommand`
+all provide \emph{local} definitions which remain trapped within their
+own \verb`\begin{preview}`, \verb`\end{preview}`) statements. Another
+preview, say containing an \verb`\eval` command, between a different
+pair of \verb`\begin{preview}`, \verb`\end{preview}`) statements,
+will not know about the macro definition.
+
+There are (at least) three ways out:
+\begin{enumerate}
+\item Confine everything to the same preview inset: the definition of a
+macro, the \verb`\macros` statement, and the use of the macro in
+an \verb`\eval` command. 
+\item Confine macro definitions to the preamble (\textsf{Document \lyxarrow{}
+Settings \lyxarrow{} \LaTeX{} Preamble}).
+\item Within previews use \verb`\gdef` (or \verb`\global\def`) exclusively
+for making your macro definitions; this makes the macro available
+to all later previews.
+\end{enumerate}
+
+\subsection{Using \texttt{\textbackslash nmcConstants}}
+
+Because \verb`\nmcConstants` doesn't use \verb`\def` or \verb`\newcommand`
+or \verb`\NewDocumentCommand` it is not subject to the same localisation
+problem as \verb`\nmcMacros`, but the reach of a \verb`\constants`
+command will still be confined to its own preview unless a \emph{global}
+preview run is forced; see above §\ref{subsec:LyXGlobal-vs-local}.
+
+\subsection{Using \texttt{\textbackslash nmcReuse}}
+
+As noted earlier, \LyX{} creates its previews in a temporary directory,
+not the document directory. If you want to save values from your current
+document – say, \texttt{mydoc.lyx} – to \texttt{mydoc.nmc} then you
+do so as described earlier (§\ref{sec:supplReuse}), but the file
+\texttt{mydoc.nmc} containing the saved results will be located in
+the temporary directory. When \LyX{} is closed the file will be deleted
+along with all the other contents of that directory.
+
+Fortunately \LyX{} has a copying mechanism for getting files out of
+the temporary directory and into the document directory. When a document
+is exported – say to pdf – it is possible to specify a \emph{copier}
+to automatically copy back to the document directory or subdirectory
+various files in the temporary directory. We want the \texttt{.nmc}
+file containing the saved values to be copied back. Go to \textsf{Tools
+\lyxarrow{} Preferences \lyxarrow{} File Handling \lyxarrow{} File Formats}
+and find \textsf{PDF (pdflatex)} (assuming export to \texttt{pdf}
+by this route) in the list of formats. In\textsf{ }the \textsf{Copier}
+slot of the dialogue insert the following line of code:
+\begin{lyxcode}
+{\small python~-tt~\$\$s/scripts/ext\_copy.py~-e~nmc,pdf~-d~\$\$i~\$\$o}{\small\par}
+\end{lyxcode}
+\verb`ext_copy.py` is a python script that is supplied with \LyX .
+The \texttt{-e nmc,pdf -d} part of the line tells \texttt{ext\_copy.py}
+that on export to \texttt{pdf} by the \texttt{pdflatex} route\texttt{
+}to copy any files with the extensions \texttt{.nmc} or \texttt{.pdf}
+from the temporary directory where \LyX{} does its work back to the
+document directory – the \verb`-d` option (which became available
+with \LyX{} 2.3.0). 
+
+But if you have a complex document, it may take too much time to want
+to export to pdf before closing \LyX , particularly if there are a
+lot of evaluations in the document. Much faster is to export to \emph{plain
+text}, not because you want a plain text version of your document
+but because it too can be used to trigger the copier mechanism. Go
+to \textsf{Tools \lyxarrow{} Preferences \lyxarrow{} File Handling \lyxarrow{}
+File Formats} and find \textsf{Plain text} in the list of formats.
+In the \textsf{Copier} slot enter
+\begin{lyxcode}
+{\small python~-tt~\$\$s/scripts/ext\_copy.py~-e~nmc~-d~\$\$i~\$\$o}{\small\par}
+\end{lyxcode}
+The only difference from the previous copier command is the absence
+of \texttt{pdf}.\footnote{I'm assuming that you don't actually want the plain text version of
+the file copied back. If you do, then change \texttt{-e nmc} to \texttt{-e
+nmc,txt}.} This will copy \texttt{mydoc.nmc} with its saved values from the
+temporary directory back to the document directory. To effect the
+export, go to \textsf{File \lyxarrow{} Export }and find \textsf{Plain
+text} in the list of formats and click on it. 
+
+A shortcut would be nice. For that go to \textsf{Tools \lyxarrow{}
+Preferences \lyxarrow{} Editing \lyxarrow{} Shortcuts}, click on \textsf{New},
+enter \texttt{buffer-export text} in the \textsf{Function:} slot,
+click on the blank key against \textsf{Shortcut:} and type your shortcut.
+You may have to try a number before you find one that hasn't already
+been assigned. (I'm using \textsf{Ctrl+}; for no particular reason
+beyond the fact that it fits under the fingers easily and saving values
+to the document directory has a punctuation-like feel to it, a pause
+in the process of writing.) It is now an easy matter to press the
+shortcut at the end of a \LyX{} session to copy all the values saved
+in \texttt{mydoc.nmc} back to a file of the same name in the document
+directory. And it is brisk, not least because plain text export ignores
+ERT insets (and hence preview insets wrapped around ERT insets), nor
+does it evaluate \verb`\eval` commands in math insets.
+
+\subsubsection{A final tweak?}
+
+But one still needs to \emph{remember} to press the shortcut. The
+thought arises: can \emph{closing} the current document trigger the
+copying process? \LyX{} provides a means of linking two commands and
+assigning a keyboard shortcut to them with its \texttt{command-sequence}
+\LyX{} function. I suggest assigning a shortcut\textsf{ }to
+\begin{lyxcode}
+command-sequence~buffer-export~text;~view-close
+\end{lyxcode}
+Indeed, why not reassign the current shortcut for \texttt{view-close},\texttt{
+}which is \textsf{Ctrl+W} on my system, to this command sequence?
+(I use the \texttt{cua} key bindings – check the \textsf{Bind file:}
+slot in \textsf{Tools \lyxarrow{} Preferences \lyxarrow{} Editing \lyxarrow{}
+Shortcuts}.)
+
+Please note, however, that \emph{this will work as intended only from
+\LyX{} 2.4.0}.\footnote{Due for release in 2021.} For \LyX{} 2.3 and
+earlier, the command sequence will generally fail because of `asynchronous'
+processing – \texttt{buffer-export }and \texttt{view-close} use different
+threads and the latter may well start before the former is complete.
+From \LyX{} 2.4.0 this defect has been fixed. You press your shortcut,
+the export to plain text occurs and the \texttt{.nmc} file is copied
+back to the document directory, then the current view is closed.
+
+Note that in the other direction, the \verb`.nmc` file in your document
+directory is \emph{automatically} copied to the temporary directory
+when needed. Nothing needs to be done by you, the user.
+
+\subsubsection{Use of \protect\LyX{} notes}
+
+The central fact about a \LyX{} note is that it does not contribute
+to the pdf. But instant preview still works there. This suggests a
+possibility: that a calculation be performed within a \LyX{} note and
+the result saved using \verb`\nmcReuse` within the same note. The
+saved value is now available \emph{from file} for use elsewhere in
+the document. In this way, some selected content from a LyX note \emph{can}
+find its way into the pdf when the document is compiled.
+
+\chapter{Reference summary}
+
+\section{Commands defined in \texttt{numerica}}
+\begin{enumerate}
+\item \texttt{\textbackslash nmcEvaluate, \textbackslash eval }
+\item \texttt{\textbackslash q, \textbackslash Q }(`cleave' commands)
+\item \texttt{\textbackslash nmcInfo, \textbackslash info}
+\item \texttt{\textbackslash nmcMacros, \textbackslash macros}
+\item \texttt{\textbackslash nmcConstants, \textbackslash constants}
+\item \texttt{\textbackslash nmcReuse, \textbackslash reuse}
+\end{enumerate}
+Provided they have not already been defined when \texttt{numerica}
+is loaded, the following commands are defined in \texttt{numerica}
+using \verb`\DeclareMathOperator` from \texttt{amsmath} :
+\begin{enumerate}
+\item \texttt{\textbackslash arccsc, \textbackslash arcsec, \textbackslash arccot}
+\item \texttt{\textbackslash csch, \textbackslash sech}
+\item \texttt{\textbackslash asinh, \textbackslash acosh, \textbackslash atanh,
+\textbackslash acsch, \textbackslash asech, \textbackslash acoth}
+\item \texttt{\textbackslash sgn, \textbackslash lb}
+\end{enumerate}
+Provided they have not already been defined, the following commands
+are defined in \texttt{numerica} using \verb`\DeclarePairedDelimiter`
+from \texttt{mathtools}:
+\begin{lyxcode}
+\textbackslash abs,~\textbackslash ceil,~\textbackslash floor
+\end{lyxcode}
+The following commands have been redefined in \texttt{numerica} to
+give more spacing around the underlying \verb`\wedge` and \verb`\vee`
+symbols:
+\begin{lyxcode}
+\textbackslash land,~\textbackslash lor
+\end{lyxcode}
+
+\section{\textquoteleft Digestible\textquoteright{} content}
+
+\texttt{numerica} knows how to deal with the following content, meaning
+that any of these elements occurring within an \verb`\eval` command
+should not of itself cause a \texttt{numerica} error. Not all formatting
+commands affect display of the output.
+\begin{enumerate}
+\item variable names (sequences of tokens given values in the variable~=~value
+list)
+\item digits, decimal point
+\begin{enumerate}
+\item \texttt{1, 2, 3, 4, 5, 6, 7, 8, 9, 0, .}
+\end{enumerate}
+\item constants
+\begin{enumerate}
+\item \texttt{e, \textbackslash pi, \textbackslash gamma, \textbackslash phi,
+\textbackslash deg, \textbackslash infty}
+\end{enumerate}
+\item arithmetic operators
+\begin{enumerate}
+\item \texttt{+, -, {*}, /, \textasciicircum , \textbackslash times, \textbackslash cdot,
+\textbackslash div}
+\end{enumerate}
+\item logical operators
+\begin{enumerate}
+\item \texttt{\textbackslash wedge, \textbackslash land, \textbackslash vee,
+\textbackslash lor, \textbackslash neg, \textbackslash lnot}
+\end{enumerate}
+\item comparisons
+\begin{enumerate}
+\item \texttt{=, <, >, \textbackslash ne, \textbackslash neq, \textbackslash le,
+\textbackslash leq, \textbackslash ge, \textbackslash geq}
+\item (if \texttt{amssymb} loaded) \texttt{\textbackslash nless, \textbackslash ngtr,
+\textbackslash geqq, \textbackslash geqslant, \textbackslash leqq,
+\textbackslash leqslant, \textbackslash ngeq, \textbackslash ngeqq,
+\textbackslash ngeqslant, \textbackslash nleq, \textbackslash nleqq,
+\textbackslash nleqslant}
+\end{enumerate}
+\item brackets, bracket-like elements, modifiers
+\begin{enumerate}
+\item \texttt{( ), {[} {]}, \textbackslash\{ \textbackslash\}}
+\item \texttt{\textbackslash lparen \textbackslash rparen} (from \texttt{mathtools})\texttt{,
+\textbackslash lbrack \textbackslash rbrack, \textbackslash lbrace
+\textbackslash rbrace}
+\item \texttt{\textbackslash lvert \textbackslash rvert, \textbackslash lfloor
+\textbackslash rfloor, \textbackslash lceil \textbackslash rceil}
+\item \texttt{| |} (no nesting, deprecated)
+\item \texttt{\textbackslash left \textbackslash right, \textbackslash bigl
+\textbackslash bigr, \textbackslash Bigl \textbackslash Bigr, \textbackslash biggl
+\textbackslash biggr, \textbackslash Biggl \textbackslash Biggr} 
+\item \texttt{.} \texttt{/ |} (used with a modifier)
+\item \texttt{\textbackslash abs{[}{]}\{\}, \textbackslash abs{*}\{\},
+\textbackslash floor{[}{]}\{\}, \textbackslash floor{*}\{\}, \textbackslash ceil{[}{]}\{\},
+\textbackslash ceil{*}\{\}} 
+\end{enumerate}
+\item unary functions (in the mathematical sense)
+\begin{enumerate}
+\item \texttt{\textbackslash sin, \textbackslash cos, \textbackslash tan,
+\textbackslash csc, \textbackslash sec, \textbackslash cot}
+\item \texttt{\textbackslash arcsin, \textbackslash arccos, \textbackslash arctan,
+arccsc, \textbackslash arcsec, \textbackslash arccot }
+\item \texttt{\textbackslash sin\textasciicircum\{-1\}, \textbackslash cos\textasciicircum\{-1\},
+\textbackslash tan\textasciicircum\{-1\}, \textbackslash csc\textasciicircum\{-1\},
+\textbackslash sec\textasciicircum\{-1\}, \textbackslash cot\textasciicircum\{-1\}}
+\item \texttt{\textbackslash sinh, \textbackslash cosh, \textbackslash tanh,
+\textbackslash csch, \textbackslash sech, \textbackslash coth }
+\item \texttt{\textbackslash asinh, \textbackslash acosh, \textbackslash atanh,
+\textbackslash csch, \textbackslash sech, \textbackslash acoth}
+\item \texttt{\textbackslash sinh\textasciicircum\{-1\}, \textbackslash cosh\textasciicircum\{-1\},
+\textbackslash tanh\textasciicircum\{-1\}, \textbackslash csch\textasciicircum\{-1\},
+\textbackslash sech\textasciicircum\{-1\}, \textbackslash acoth\textasciicircum\{-1\}}
+\item \texttt{\textbackslash exp, \textbackslash lb, \textbackslash lg,
+\textbackslash ln, \textbackslash log, \textbackslash log\_\{\},
+\textbackslash sgn, \textbackslash surd}
+\item \texttt{\textbackslash sqrt\{\}, \textbackslash abs{[}{]}\{\}, \textbackslash abs{*}\{\},
+\textbackslash floor{[}{]}\{\}, \textbackslash floor{*}\{\}, \textbackslash ceil{[}{]}\{\},
+\textbackslash ceil{*}\{\}}
+\item \texttt{!, !! }(prepended argument)
+\end{enumerate}
+\item binary functions 
+\begin{enumerate}
+\item \texttt{\textbackslash tfrac\{\}\{\}, \textbackslash frac\{\}\{\},
+\textbackslash dfrac\{\}\{\}}
+\item \texttt{\textbackslash tbinom\{\}\{\}, \textbackslash binom\{\}\{\},
+\textbackslash dbinom\{\}\{\}}
+\item \texttt{\textbackslash sqrt{[}{]}\{\}}
+\end{enumerate}
+\item $n$-ary functions
+\begin{enumerate}
+\item \texttt{\textbackslash min, \textbackslash max, \textbackslash gcd}
+\end{enumerate}
+\item sum, prod
+\begin{enumerate}
+\item \texttt{\textbackslash sum\_\{\}\textasciicircum , \textbackslash prod\_\{\}\textasciicircum{} }
+\end{enumerate}
+\item formatting commands
+\begin{enumerate}
+\item \texttt{,} (comma, in $n$-ary functions)
+\item \texttt{\{\}, \textbackslash\textbackslash , \&, \textbackslash to}
+\item \texttt{\textbackslash begin\{\}, \textbackslash end\{\}, \$, \textbackslash{[},
+\textbackslash{]}}
+\item \texttt{\textbackslash dots, \textbackslash ldots, \textbackslash cdots}
+\item \texttt{\textbackslash{} , \textbackslash ,{}, \textbackslash ;,
+\textbackslash :, \textbackslash !, \textbackslash >}
+\item \texttt{\textbackslash thinspace, \textbackslash medspace, \textbackslash thickspace,}
+\item \textbackslash\texttt{negthinspace, \textbackslash negmedspace,
+\textbackslash negthickspace,}
+\item \textbackslash\texttt{hspace{*}\{\}, \textbackslash mspace\{\},}
+\item \texttt{\textbackslash quad, \textbackslash qquad , \textbackslash hfill,
+\textbackslash hfil}
+\item \texttt{\textbackslash phantom\{\}, \textbackslash vphantom\{\},
+\textbackslash hphantom\{\}}
+\item \texttt{\textbackslash xmathstrut{[}{]}\{\}} \texttt{, \textbackslash splitfrac\{\}\{\},
+\textbackslash splitdfrac\{\}\{\} }(from \texttt{mathtools}), \texttt{\textbackslash mathstrut}
+\item \texttt{\textbackslash displaystyle, \textbackslash textstyle, \textbackslash scriptstyle,
+\textbackslash scriptscriptstyle}
+\item \texttt{\textbackslash label\{\}, \textbackslash ensuremath\{\},
+\textbackslash text\{\}, \textbackslash mbox\{\}, \textbackslash smash\{\}}
+\item \texttt{\textbackslash color{[}{]}\{\}, \textbackslash textcolor{[}{]}\{\}\{\}}
+\end{enumerate}
+\item font commands
+\begin{enumerate}
+\item \texttt{\textbackslash mathrm\{\}, \textbackslash mathit\{\}, \textbackslash mathcal\{\},
+\textbackslash mathtt\{\}, \textbackslash mathbf\{\}, \textbackslash mathbb\{\},
+\textbackslash mathsf\{\}, \textbackslash mathfrak\{\}, \textbackslash mathscr\{\}}
+\item \texttt{\textbackslash mathnormal\{\}, \textbackslash boldsymbol\{\}}
+\item \texttt{\textbackslash textrm, \textbackslash textsf, \textbackslash texttt}
+\end{enumerate}
+\end{enumerate}
+
+\section{Settings}
+
+\subsection{Available \texttt{\textbackslash nmcEvaluate} settings}
+\begin{center}
+\begin{tabular}[t]{>{\raggedright}p{1.5cm}l>{\raggedright}m{4cm}>{\raggedright}m{4cm}}
+\toprule 
+{\small key} & {\small type} & {\small meaning} & {\small default}\tabularnewline
+\midrule
+{\small\texttt{dbg}} & {\small int} & {\small debug `magic' integer} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{view}}  &  & {\small equivalent to }{\small\texttt{dbg=1}} & \tabularnewline
+{\small\texttt{\textasciicircum}} & {\small char} & {\small exponent mark for sci. notation input} & {\small\texttt{e}}\tabularnewline
+{\small\texttt{xx}} & {\small int (0/1)} & {\small multi-token variable switch} & {\small\texttt{1}}\tabularnewline
+{\small\texttt{()}} & {\small int (0/1/2)} & {\small trig. function arg. parsing} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{o}} & {\small int (0/1)} & {\small degree switch for trig. funcions} & {\small\texttt{1}}\tabularnewline
+{\small\texttt{log}} & {\small num} & {\small base of logarithms for }{\small{\small\verb`\log`}} & {\small\texttt{10}}\tabularnewline
+{\small\texttt{vv@}} & {\small int (0/1)} & {\small vv-list calculation mode} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{vvmode}} & {\small int (0/1)} & {\small equivalent to }{\small\texttt{vv@}} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{vvd}} & {\small token(s)} & {\small vv-list display-style spec.} & {\small\texttt{\{,\}\textbackslash mskip 12mu plus 6mu minus 9mu(vv)}}\tabularnewline
+{\small\texttt{vvi}} & {\small token(s)} & {\small vv-list text-style spec.} & {\small\texttt{\{,\}\textbackslash mskip 36mu minus 24mu(vv)}}\tabularnewline
+{*} &  & {\small switch to suppress equation numbering (if }{\small\texttt{\textbackslash\textbackslash}}{\small{}
+in }{\small\texttt{vvd}}{\small )} & \tabularnewline
+{\small\texttt{p}} & char(s) & {\small punctuation (esp. in display-style)} & {\small\texttt{,}}\tabularnewline
+{\small\texttt{S+}} & {\small int} & {\small extra rounding for stopping criterion, sums} & {\small\texttt{2}}\tabularnewline
+{\small\texttt{S?}} & {\small$\text{int}\ge0$} & {\small query stopping with these final terms, sums} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{P+}} & {\small int} & {\small extra rounding for stopping criterion, products} & {\small\texttt{2}}\tabularnewline
+{\small\texttt{P?}} & {\small$\text{int}\ge0$} & {\small query stopping with these final terms, products} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{reuse}} & {\small int} & {\small form of result saved with }{\small{\small\verb`\nmcReuse`}} & {\small\texttt{0}}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+
+\subsection{Available settings for supplementary commands}
+
+All settings for \verb`\nmcEvaluate`, the \verb`view` setting in
+particular (although most will be irrelevant), plus for
+\begin{itemize}
+\item \verb`\nmcMacros`
+\begin{itemize}
+\item \verb`free` `deregister' a macro from \verb`numerica`
+\end{itemize}
+\item \verb`\nmcConstants`
+\begin{itemize}
+\item \verb`add` add the new list of constants to the current one
+\end{itemize}
+\item \verb`\nmcReuse`
+\begin{itemize}
+\item \verb`delete` remove the listed control sequences from the \verb`.nmc`
+file
+\item \verb`renew` change the value of a control sequence in the \verb`.nmc`
+file
+\end{itemize}
+\end{itemize}
+
+\subsection{Available configuration file settings}
+\begin{center}
+\bigskip{}
+\begin{tabular}{ll}
+\toprule 
+key & default\tabularnewline
+\midrule
+{\small\texttt{rounding}} & {\small\texttt{6}}\tabularnewline
+{\small\texttt{pad}} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{output-sci-notation}} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{output-exponent-char}} & {\small\texttt{e}}\tabularnewline
+{\small\texttt{input-sci-notation}} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{input-exponent-char}} & {\small\texttt{e}}\tabularnewline
+{\small\texttt{multitoken-variables}} & {\small\texttt{1}}\tabularnewline
+{\small\texttt{use-degrees}} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{logarithm-base}} & {\small\texttt{10}}\tabularnewline
+{\small\texttt{intify-rounding}} & {\small\texttt{14}}\tabularnewline
+{\small\texttt{vv-display}} & {\small\texttt{\{,\}\textbackslash mskip 36mu minus 24mu(vv)}}\tabularnewline
+{\small\texttt{vv-inline}} & {\small\texttt{\{,\}\textbackslash mskip 12mu  6mu minus 9mu(vv)}}\tabularnewline
+{\small\texttt{sum-extra-rounding}} & {\small\texttt{2}}\tabularnewline
+{\small\texttt{sum-query-terms}} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{prod-extra-rounding}} & {\small\texttt{2}}\tabularnewline
+{\small\texttt{prod-query-terms}} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{eval-reuse}} & {\small\texttt{0}}\tabularnewline
+\bottomrule
+\end{tabular} 
+\par\end{center}
+\end{document}
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