[x-δ/2,x+δ/2]

IdPA[x_]:={x,{Precision[x],Accuracy[x]}}
IPA[x_]:={InputForm[x],{Precision[x],Accuracy[x]}}
Error[x_]:=10^-Accuracy[x]

test=IdPA/@{1,Pi,1.,345.,3`11,123`10,4``13,321``5};
TableForm[test]

test=IPA/@{1,Pi,1.,345.,3`11,123`10,4``13,321``5};
TableForm[test]

TableForm[{InputForm[{3.,3.`10},NumberMarks->#],#}&/@{True,False,Automatic}]

$NumberMarks=True;

test2=IdPA/@{22.22`4,0.11`2};
TableForm[test2]

Acc[x_]:=Precision[x]-Log[10,x]
Prec[x_]:=Accuracy[x]+Log[10,x]

Through[{Precision,Prec,Accuracy,Acc}[3.`10]]

RealDigits[1`10,10,12]

N[Pi]//IPA

N[Pi,10]//IPA

N[Pi,{10,10}]//IdPA
N[Pi,{11,10}]//IdPA

N[Pi x,10]//IPA

1`10x+1`5y//IPA

$MaxExtraPrecision

IdPA[N[Sin[2^1000],10]]

Block[{$MaxExtraPrecision=Infinity},IdPA[N[Sin[2^1000],10]]]

Timing[NestList[#^2&,1`10000,10000];]
Timing[NestList[#^2&,1.,10000];]

Precision[0.`10]

TableForm[{#,(Precision[Nest[#,1`200,200]])}&/@{Sin,Cos,#^2&,Sqrt,Exp[#-1]&}]
TableForm[{#,(Accuracy[Nest[#,1.,200]])}&/@{Sin,Cos,#^2&,Sqrt,Exp[#-1]&}]

Nest[Sqrt,Nest[#^2&,2.,20],20]-2.

Series[NestList[Sqrt,1+ϵ,10],{ϵ,0,2}]//TableForm

Series[NestList[#^2&,1+ϵ,10],{ϵ,0,2}]//TableForm

[1-δ/2,1+δ/2]
Accuracy[x] = -(Subscript[log, 10])(δ/x)

Series[f[x+ϵ],{ϵ,0,1}]

err[f_,x0_,n_]:=-Log[10,f'[x0]^n 10^-Accuracy[x0]/x0]
err[#^2&,1.`200,200]

Clear[lp,lpp,lpa,lpany]
lp[f_,s_,n_,max_]:=ListPlot[NestList[f,N[s,#],n]&/@(MachinePrecision{max,1}(*2^Reverse@Range[max,max]*)),Joined->True,PlotStyle->{Directive[Thickness[0.01],Lighter[Red]],Directive[{Thickness[0.005],Black}]},Frame->True,PlotLabel->(max MachinePrecision)]
lpga[f_,s_,n_,max_]:=GraphicsArray[Partition[lp[f,s,n,#]&/@ Range@@max,2]]
lpp[f_,s_,n_,max_]:=lpany[f,s,n,max,Precision]
lpa[f_,s_,n_,max_]:=lpany[f,s,n,max,Accuracy]
lpany[f_,s_,n_,max_,a_]:=ListLogPlot[(a/@NestList[f,N[s,#],n])&/@(MachinePrecision Range[1,max]),Joined->True,PlotStyle->Thick,PlotRange->All,PlotLabel->a]
lppa[f_,s_,n_,max_]:=GraphicsArray[{Through[{lpp,lpa}[f,s,n,max]]}]

q[x_]:=4x(1-x)

lpga[q,1/3,100,{1,8,2}]

lppa[q,1/3,100,6]

s20=Nest[4#(1-#)&,1/3,20];

Precision/@(Nest[30/10#(1-#)&,N[1/2,2^#],20]&/@Range[10])

m7[x_]:=Mod[10x,7]

lpga[m7,1/3,100,{1,200,49}]

lppa[m7,1/3,100,200/$MachinePrecision]

Take[NestList[Mod[10#,7]&,N[7/9],100],-10]

NestList[Mod[10#,7]&,6,10]

lpga[m7,(3983/999),100,{5,16,10}]

m7[(3983/999)]

7 1/9+1/99999
NestList[m7,%,10]

m7@m7[12.9]
Mod[100 12.9,7]

Reduce[7>10a/9>0,a]

First/@Select[{#,#/99,m7[m7[#/99]]}&/@Range[1,999],Equal@@#[[2;;3]]&]

686/7

Reduce[Nest[Mod[10#,7]&,x,#]==x,Reals]&/@Range[3]

TableForm[IPA/@Take[NestList[Mod[10#,7]&,N[7/9,20],20],-10]]

$MaxExtraPrecision

m11[α_,x_]:=Mod[α x,7]

lpga[m11[107/100,#]&,1/3,1000,{1,12,10}]

Manipulate[Plot[{Nest[m11[z,#]&,x,10],x},{x,0,7}],{z,0,10}]

tent[μ_]:=Piecewise[{{μ #,#<1/2},{μ(1-#),#>=1/2}}]&

ArrayPlot[First/@RealDigits[NestList[tent[2],N[Pi/4,2000],100],2,100,-1]]