Scientific Notation Only For Large Numbers

Question

I have a table full of numbers. Most of them are floating point numbers that are less than one, some are integers from 1-100, and a few are large (×10^75). I know how to set the precision of the small numbers using siunitx so that they all line up nicely, but I'd like the whole numbers to stay as they are and the large numbers (and only the large ones!) to be put in to scientific notation.

Is there a way to automatically set any number larger than 100 to be formatted in scientific notation, while leaving the smaller numbers in standard decimal notation and the intermediate numbers as integers? The intermediate numbers are less important to happen automatically, as they're always in a certain row, so I can simply treat them as text.

EDIT: Here is a MWE.

\documentclass[11pt,a4paper,twoside]{book}
\usepackage[top=30mm, bottom=30mm, left=40mm, right=20mm]{geometry} 

\usepackage{fancyhdr}
\pagestyle{fancy}
\setlength{\headheight}{15pt}

\usepackage{siunitx}
\sisetup{round-mode=places,round-precision=6}



\begin{document}


\begin{table}
%\begin{center}
\centering
\caption{Example table for Ask-Ubuntu.}

\begin{tabular}{rr|rrr}

          &       & Control Case & History & No History \\

    \hline & A & \num{0.143392788} & \num{9026780216140000000000} & \num{0.192293062} \\
    file & B    & \num{0.002162212} & \num{-725293214339000000000} & \num{0.004052388} \\
          & C  & 98    & 75    & 65 \\
    \end{tabular}%

\label{t:example_table}
\end{table}

\end{document}

Answer 1

UPDATE: Bruno le Floch gave a great suggestion in the comments on one of the answers to How to subtract both very large numbers and numbers smaller than one?. This now makes it possible to use very large numbers in the range of plus/minus [1e-10000,1e10000].

He uses the package expl3 to define a comparison (test which of the numbers is less or greater):

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

An MWE that only uses scientific notations for very small (now defined as 0.01) and very large (100 in this case) numbers is the following.

\documentclass{article}
\usepackage{expl3,siunitx}
\sisetup{scientific-notation=true}
\ExplSyntaxOn
    \cs_new_eq:NN \fpcmpTF \fp_compare:nTF
\ExplSyntaxOff

%Edit these as you wish:
\newcommand*{\ThresholdLow}{0.01}
\newcommand*{\ThresholdHigh}{100}

\let\OldNum\num%
\renewcommand*{\num}[2][]{%
    \fpcmpTF{abs(#2)<=\ThresholdLow}{%
        \OldNum[scientific-notation=true,#1]{#2}%
    }{%
        \fpcmpTF{abs(#2)>=\ThresholdHigh}{%
            \OldNum[scientific-notation=true,#1]{#2}%
        }{%
            \OldNum[scientific-notation=false,#1]{#2}%
        }%
    }%
}%
\begin{document}
\newcommand{\Row}[1]{#1 & \OldNum{#1} & \num{#1}}%
\begin{tabular}{l l l}
    Num & Old & New\\\hline\\[-0.7em]
    \Row{0.01}\\
    \Row{0.1}\\
    \Row{1}\\
    \Row{10}\\
    \Row{100}\\
\end{tabular}
\end{document}

---

A different way this solution can be used is by making a new .sty-file that you save as for example threshold.sty, in which you copy-paste the following:

\RequirePackage{expl3,kvoptions,siunitx}
\SetupKeyvalOptions{family=threshold,prefix=threshold@}
\DeclareStringOption[1]{low}[0.01]
\DeclareStringOption[1]{high}[100]
\ProcessKeyvalOptions*
\sisetup{scientific-notation=true}
\ExplSyntaxOn
    \cs_new_eq:NN \fpcmpTF \fp_compare:nTF
\ExplSyntaxOff
\let\OldNum\num%
\renewcommand*{\num}[2][]{%
    \fpcmpTF{abs(#2)<=\threshold@low}{%
        \OldNum[scientific-notation=true,#1]{#2}%
    }{%
        \fpcmpTF{abs(#2)>=\threshold@high}{%
            \OldNum[scientific-notation=true,#1]{#2}%
        }{%
            \OldNum[scientific-notation=false,#1]{#2}%
        }%
    }%
}

and which you then call using for example \usepackage[low=1e-2,high=1e2]{threshold}.

The advantage of this is that you can use it more easily in other files, and that it doesn't take up so much space in the file you're editing. Also, it is more flexible, as you can just decide not to use any threshold and call the package without any argument (\usepackage{threshold}), which then essentially does the same as just using \usepackage{siunitx}. An other option is to use the arguments [low,high] when using the package, which then uses the default settings for the low and high thresholds (using scientific notation only for numbers outside the range ±[-0.01,100]).

Answer 2

I am looking at this for v3.2. A prototype implementation for a threshold: key names, etc. at present not fixed.

\documentclass{article}
\usepackage{siunitx}
\ExplSyntaxOn
\keys_define:nn { siunitx }
  {
    exponent-mode / threshold .code:n =
      { \tl_set:Nn \l__siunitx_number_exponent_mode_tl {#1} } ,
    exponent-threshold-range .code:n =
      { \__siunitx_number_set_thresholds:n {#1} }
  }
\cs_new_protected:Npx \__siunitx_number_set_thresholds:n #1
  {
    \exp_not:N \__siunitx_number_set_thresholds:w
      #1 \token_to_str:N : \token_to_str:N : \exp_not:N \q_stop
  }
\use:x
  {
    \cs_new_protected:Npn \exp_not:N \__siunitx_number_set_thresholds:w
      ##1 \token_to_str:N : ##2 \token_to_str:N : ##3 \exp_not:N \q_stop
  }
  {
    \__siunitx_number_set_threshold:nn { lower } {#1}
    \__siunitx_number_set_threshold:nn { upper } {#2}
  }
\cs_new_protected:Npn \__siunitx_number_set_threshold:nn #1#2
  { \int_set:cn { l__siunitx_number_ #1 _threshold_int } { 0 #2 } }
\int_new:N \l__siunitx_number_lower_threshold_int
\int_new:N \l__siunitx_number_upper_threshold_int
\cs_new:Npn \__siunitx_number_exponent_threshold:nnnnnnn #1#2#3#4#5#6#7
  {
    \exp_args:Ne \__siunitx_number_exponent_threshold:nn
      {
        \__siunitx_number_exponent_scientific:nnnnnnn
          {#1} {#2} {#3} {#4} {#5} {#6} {#7}
      }
      { {#1} {#2} {#3} {#4} {#5} {#6} {#7} }
  }
\cs_new:Npn  \__siunitx_number_exponent_threshold:nn #1#2
  { \__siunitx_number_exponent_threshold:nnnnnnnn #1 {#2} }
\cs_new:Npn  \__siunitx_number_exponent_threshold:nnnnnnnn #1#2#3#4#5#6#7#8
  {
    \bool_lazy_and:nnTF
      { \int_compare_p:nNn {#6#7} > \l__siunitx_number_lower_threshold_int }
      { \int_compare_p:nNn {#6#7} < \l__siunitx_number_upper_threshold_int }
      { \exp_not:n {#8} }
      { \exp_not:n { {#1} {#2} {#3} {#4} {#5} {#6} {#7} } }
  }
\ExplSyntaxOff
\sisetup{exponent-threshold-range = -2:2 }
\begin{document}
\newcommand{\Row}[1]{#1 & \num[exponent-mode = scientific]{#1} & \num[exponent-mode = threshold]{#1}}%
\begin{tabular}{l l l}
    Num & Old & New\\\hline\\[-0.7em]
    \Row{0.01}\\
    \Row{0.1}\\
    \Row{1}\\
    \Row{10}\\
    \Row{100}\\
    \Row{1234567890123}
\end{tabular}
\end{document}