How to subtract both very large numbers and numbers smaller than one?

Question

In the context of Scientific Notation Only For Large Numbers have we been looking for a way to handle large numbers, as pgfmathparse can't cope with these. The only package we could find was bigintcalc, with the problem that it only works with integers.

MWE:

\documentclass{article}
\usepackage{bigintcalc}
\begin{document}
\bigintcalcSub{1}{100}
\end{document}

works fine, but

\documentclass{article}
\usepackage{bigintcalc}
\begin{document}
\bigintcalcSub{0.1}{100}
\end{document}

gives an error:

! Missing number, treated as zero.
<to be read again> 
                   .
l.4 \bigintcalcSub{0.1}{100}

So apparently \bigintcalcSub (or bigintcalc at all) is not the way to go. How can one handle large numbers that are not integers?

Edit: of course one can use an if statement to check if a number is big, and then only use it in that case, but that would be a bit hacky, wouldn't it?

Edit 2: to be more precise, we're trying to make a command that tests whether any number is greater than a certain threshold (and then do something if it is). There are both numbers as 0.01 and numbers as 1e75 fed to this command. pgfmathparse can do the first, bigintcalc can do the second, but is there something that can do both, so that the workarounds with if statements can be avoided and also large numbers that are not integers can be processed?

Answer 1

The l3fp package can compute expressions involving floating points, with 16 significant digits, with a large range of exponents (currently positive numbers range in [10^-10000, 10^10000], but this may change a bit). If 16 digits is enough for your purposes, you could use that. Some day, I'll code an arbitrary precision floating point unit, but that takes time.

\documentclass{article}

\usepackage{expl3}
\ExplSyntaxOn
  \cs_new_eq:NN \fpeval \fp_eval:n
  \cs_new_eq:NN \fpcmpTF \fp_compare:nTF
\ExplSyntaxOff

\begin{document}
  $0.1-100$ is $\fpeval{0.1-100}$, and it is
  \fpcmpTF{0.1-100 < -99}{greater}{smaller} than $-99$.
\end{document}

Answer 2

run with lualatex

\documentclass{article}
\begin{document}
$0.1-100$ is $\directlua{tex.print(0.1-100)}$
\end{document}

Answer 3

The package xintfrac may be used for such computations. The package is primarily concerned with computations, not typesetting, so I have added here a little macro to transmit the typesetting task to other packages such as siunitx or numprint.

Update: I wondered in the comments to the code below if wrapping across lines was possible with siunitx, or with numprint in mathmode. From very brief examination of some log a test document using \showoutput it seems that

1. siunitx puts the whole thing in a \hbox, whether in math mode or in text, and additionally also encloses the thousand separator in a \hbox,
2. numprint, when in math mode, typesets the thousand separator in a \hbox, but when in text mode, it does not: hence with numprint+text mode wrapping is possible (the entire thing is fortunately not enclosed in a \hbox), but in math mode it does not seem to be.

I apologize for these details perhaps only indirectly related to the OP's question: but if we are to subtract power of tens like 10^{100}-1, we have to think about how to print the resulting one hundred nine's.

\documentclass{article}
\usepackage{color} % for testing an option of siunitx

% formatting numbers with macro \np of numprint
% http://www.ctan.org/tex-archive/macros/latex/contrib/numprint
\usepackage[np]{numprint}
\npthousandsep{\allowbreak\,} 
% The \allowbreak allows very long numbers to wrap at endlines
% (but does not work in math mode, I didn't investigate why, perhaps numprint
% uses an \hbox) 
\npdecimalsign{.}

% formatting with macro \num of siunitx
% http://www.ctan.org/tex-archive/macros/latex/exptl/siunitx
% I don't know if very long numbers
% wrapping at endlines are possible with it. I tried the following option but it
% doesn't do what I hoped for:
\usepackage[group-separator={\allowbreak\,}]{siunitx}

% computes expandably with long fractions and decimal numbers
% http://www.ctan.org/tex-archive/macros/generic/xint
\usepackage{xintfrac} 

% The next macro prepares a decimal fraction as output by macros of the xintfrac
% package for printing by specialized software.

% Another macro could be written to directly prepare the number in scientific
% notation, but here we simply prepare the number in a.b form with b the
% possibly empty strings of digits after the decimal mark (which is a dot), and
% the transformation into scientific notation can be done at the level of the 
% typesetting package. 

% The first parameter to \decprint is optional and is the command to be applied.
% The default is \np from the numprint package. But one may use for example
% [{\num[key=value]}] to employ \num from the siunitx package with some options.

% The second argument is mandatory and is the decimal number to be typeset. On
% input, xintfrac understands the following format: a, or a.b, or a[c], or
% a.b[c], where [c] means 10 to the power c, and a and b are strings of digits.

% Example: \decprint {\xintAdd {1[-10]}{1[10]}}, which first computes the sum of
% 1e-10 and 1e10 (xintfrac does not understand the scientific notation, it uses 
% brackets rather than e or E) and then
% prints the result using \np command from numprint package.

\makeatletter

\newcommand{\decprint}[2][\np]
   {\expandafter\expandafter\expandafter\dec@print\xintREZ{#2}{#1}}

\def\dec@print #1/#2[#3]#4{%
    \ifnum #3<0
        \expandafter\@firstoftwo
    \else\expandafter\@secondoftwo
    \fi
    {#4{\xintTrunc {-#3}{#1[#3]}}}% non-zero digits after decimal point
    {#4{\xintDSH {-#3}{#1}}}% number is an integer, no decimal point needed
    \ifcase\XINT@isOne{#2} 
       \expandafter\@firstofone % unexpected, denominator wasn't power of ten
    \or\expandafter\@gobble
    \fi {/#4{#2}}}

\makeatother


\begin{document}

  $0.1-100$ is $\decprint{\xintSub{0.1}{100}}$, and it is
  \ifcase\xintCmp{\xintSub{0.1}{100}}{-99}
  equal \or greater \else smaller \fi than $-99$.

  $0.01-10^{10}$ is \decprint{\xintSub{0.01}{1[10]}}.

  $0.01-10^{10}$ is \decprint[\num]{\xintSub{0.01}{1[10]}}.

  $0.01-10^{75}$ is \decprint{\xintSub{0.01}{1[75]}}.

  $0.01-10^{75}$ is \decprint[{\num[mode=text]}]{\xintSub{0.01}{1[75]}}.

  $3.14-\np{5e-15}+\np{3e15}$ is \decprint{\xintSumExpr
    {3.14}{-5[-15]}{3[15]}\relax }

  $3.14-\np{5e-15}+\np{3e15}$ is \decprint[{\num[mode=text]}]{\xintSumExpr
    {3.14}{-5[-15]}{3[15]}\relax }

  $3.14-\np{5e-25}+\np{3e25}$ is \decprint{\xintSumExpr
    {3.14}{-5[-25]}{3[25]}\relax } and we could make it wrap accross lines using
  \textbackslash np.

  $3.14-\np{5e-25}+\np{3e25}$ is \decprint[{\num[color=blue]}]{\xintSumExpr
    {3.14}{-5[-25]}{3[25]}\relax } and we tried but failed to let it wrap
  accross lines using \textbackslash num.


\end{document}