PGFmathprint 分数精度问题

Question

我尝试使用pgf数字排版制作此表，但存在精度问题。摘自手册：

如果您仍然遇到稳定性问题，请在序言中使用 \usepackage{fp}。然后，frac 样式将自动采用 fp 的更高绝对精度来计算 1/r。

但是还是提醒一个问题。

\documentclass[french]{article}

\usepackage{tikz,array,tabularx,luacode,xparse,ragged2e}
\usepackage{amsmath,amsfonts,amssymb,calc}
\usepackage{fp}

\usepackage{siunitx} % \num{} ...

\sisetup{%
    unit-mode = text,%
    locale=FR,%
    detect-all,% Problème avec euro, utiliser \text{€} en mode math
    inter-unit-product = \ensuremath{{}\cdot{}},%
    group-minimum-digits=4,
    text-angstrom={Å},math-angstrom={\text{Å}}
} 

\usetikzlibrary{calc,fpu}

\directlua{require ("Maths")}

\NewDocumentCommand{\Frac}{sm}{%
    \IfBooleanTF{#1}{%
    \directlua{Frac(#2,1)}
    }{%
    \directlua{Frac(#2)}}
}

\NewDocumentCommand{\Around}{mm}{%
    \directlua{Around(#1,#2)
    }
}

\NewDocumentCommand{\pgfAround}{mm}{%
    {\pgfkeys{/pgf/number format/.cd,fixed,precision=#2,use comma}
    \pgfmathprintnumber{#1}}
}

\NewDocumentCommand{\pgfFrac}{sm}{%
    \IfBooleanTF{#1}{%
    {\pgfkeys{/pgf/number format/.cd,frac,frac whole = false}
    \pgfmathprintnumber{#2}}
    }{%
    {\pgfkeys{/pgf/number format/.cd,%
        frac,frac whole = false,frac TeX={\dfrac}}
    \pgfmathprintnumber{#2}}
    }
}

\newcommand{\vtab}[1][1.2]{\rule[-0.9em*\real{#1}]{0pt}{2.3em*\real{#1}}}

\begin{document}


Lua -- Version

\medskip

{\small\setlength{\tabcolsep}{0pt}
\begin{tabularx}{\linewidth}{|*{13}{>{\Centering}X|}}\hline
Do$_1$&
Ré$_1$&
Mi$_1$&
Fa$_1$&
Sol$_1$&
La$_1$&
Si$_1$&
Do$_2$&
Ré$_2$&
Mi$_2$&
Fa$_2$&
Sol$_2$&
La$_2$\\\hline
\num{20}&
\Around{17.7777777778}{-1}&
\Around{15.8024691358}{-1}&
\Around{14.0466392318}{-1}&
\Around{13.3333333333}{-1}&
\Around{11.8518518519}{-1}&
\Around{10.5349794239}{-1}&
\num{10}&
\Around{8.8888888889}{-1}&
\Around{7.9012345679}{-1}&
\Around{7.0233196159}{-1}&
\Around{6.6666666667}{-1}&
\Around{5.9259259259}{-1}\\\hline
\vtab\num{20}&
\Frac{17.7777777778}&
\Frac*{15.8024691358}&
\Frac*{14.0466392318}&
\Frac{13.3333333333}&
\Frac{11.8518518519}&
\Frac*{10.5349794239}&
\Frac{10}&
\Frac{8.8888888889}&
\Frac{7.9012345679}&
\Frac*{7.0233196159}&
\Frac{6.6666666667}&
\Frac{5.9259259259}\\\hline
\end{tabularx}
}

\bigskip


PGF -- Version

\medskip

{\small\setlength{\tabcolsep}{0pt}
\begin{tabularx}{\linewidth}{|*{13}{>{\Centering}X|}}\hline
Do$_1$&
Ré$_1$&
Mi$_1$&
Fa$_1$&
Sol$_1$&
La$_1$&
Si$_1$&
Do$_2$&
Ré$_2$&
Mi$_2$&
Fa$_2$&
Sol$_2$&
La$_2$\\\hline
\num{20}&
\pgfAround{17.7777777778}{1}&
\pgfAround{15.8024691358}{1}&
\pgfAround{14.0466392318}{1}&
\pgfAround{13.3333333333}{1}&
\pgfAround{11.8518518519}{1}&
\pgfAround{10.5349794239}{1}&
\num{10}&
\pgfAround{8.8888888889}{1}&
\pgfAround{7.9012345679}{1}&
\pgfAround{7.0233196159}{1}&
\pgfAround{6.6666666667}{1}&
\pgfAround{5.9259259259}{1}\\\hline
\vtab\num{20}&
\pgfFrac*{17.7777777778}&
\pgfFrac{15.8024691358}&
\pgfFrac*{14.0466392318}&
\pgfFrac{13.3333333333}&
\pgfFrac*{11.8518518519}&
\pgfFrac*{10.5349794239}&
\pgfFrac{10}&
\pgfFrac{8.8888888889}&
\pgfFrac{7.9012345679}&
\pgfFrac*{7.0233196159}&
\pgfFrac{6.6666666667}&
\pgfFrac*{5.9259259259}\\\hline
\end{tabularx}
}
\end{document}

和Maths.lua

-- ####################################
--  Transforme un décimal en fraction #
-- ####################################

function Frac( n , sf)

local s = ""

if n<0 then n,s = -n,"-" end

local tolerance = 1.e-8
local h1=1
local h2=0
local k1=0
local k2=1
local b = 1/n
local aux
local a

    while math.abs(n-h1/k1) > n*tolerance
    do
        b = 1/b
        a = math.floor(b)
        aux = h1
        h1 = a*h1+h2
        h2 = aux
        aux = k1
        k1 = a*k1+k2
        k2 = aux
        b = b-a
    end

    if k1 == 1
    then 
        tex.print ("\\num{"..s..h1.."}")
    else
        if sf == 1
        then
        tex.print ("\\ensuremath{\\frac{\\num{"
                ..s..h1.."}}{\\num{"..k1.."}}}")
        else
        tex.print ("\\ensuremath{\\dfrac{\\num{"
                ..s..h1.."}}{\\num{"..k1.."}}}")
        end
    end
end


function Around ( n , prec )
    tex.print("\\num{"..math.floor(n/10^prec+0.5)*10^prec.."}")
end