\(Needs /@ {"DerivativeBasedAdaptiveRejectionSampling`", "Graphics`Graphics`", "Statistics`ContinuousDistributions`", "ProbabilityPlot`"}\)

updf[x_] = -x^2/2;
derpdf[x_] = D[updf[x], x];
fun[x_] := Through[{updf, derpdf}[x]];
trueInvCumul[u_] = 
        x /. Solve[
              Integrate[
                  Exp[updf[y]]/Sqrt[2*Pi], {y, -Infinity, x}] == u, 
              x][[1]];
trueNC = Sqrt[2 Pi] // N;

repsCnt = 10^2;
lowerInit  =  -0.5;
upperInit  =  1.5;

ClearAll[data, NCs, LogPDFplot, invCumulPlot, probPlot, 
      NCplot]
data[0] = {};
NCs[0] = {};
LogPDFplot;
invCumulPlot;
probPlot;
NCplot;
\(\(Module[{obj, lowerLog, upperLog}, 
obj = 
          Make[DerivativeBasedAdaptiveRejectionSampling, \(-Infinity\), {lowerInit, upperInit}, Infinity, fun]; 
lowerLog = 
          ToExpression[
            Last[Names["*`lowerLog$*"]]]; 
upperLog = 
          ToExpression[
            Last[Names["*`upperLog$*"]]]; 
SeedRandom[
          0]; 
Do[data[0] = {data[0], Random[obj]}; 
          NCs[0] = {NCs[0], 
              NormalizingConst[obj]}, {repsCnt}]; 
data[
            0] = Flatten[data[0]]; 
NCs[0] = 
          Flatten[NCs[0]]; 
NCPlot[0] = 
          ListPlot[NCs[0], 
            PlotRange -> {Automatic, {trueNC, 
                  Max[NCs[0]]}}]; 
LogPDFplot[0] = 
          Plot[{\(lowerLog[]\)[x], updf[x], \(upperLog[]\)[x]}, {x, \(-1.9\), 
              1.9}, PlotStyle -> {Blue, Black, Red}, 
            DisplayFunction -> 
              Identity]; 
invCumulPlot[0] = 
          Plot[{trueInvCumul[u], InverseCDF[obj, u]}, {u, 0.001, 0.9999}, 
            DisplayFunction -> Identity]; 
probPlot[
            0] = ProbabilityPlot[data[0], 
            trueInvCumul];
];\)\)

data[1] = {};
Module[{repsCnt = 10^3, r, mean = 0, 
          var = 0}, 
Do[
Module[{obj}, 
obj = 
              Make[DerivativeBasedAdaptiveRejectionSampling, -Infinity, {lowerInit, upperInit}, Infinity, fun]; 
r = 
              Random[obj]; 
data[1] = {data[1], 
                r};
]; 
mean += 
            r; 
var += r^2; 
clear[
            obj];
, {repsCnt}
]; 
data[1] = 
          Flatten[data[1]]; 
probPlot[1] = 
          ProbabilityPlot[data[1], trueInvCumul]; 
mean /= 
          repsCnt; 
var -= 
          repsCnt mean^2; 
var /= (repsCnt - 
              1); 
Return[{mean, 
            var}];
];

Show[NCPlot[0], DisplayFunction -> $DisplayFunction, 
      PlotLabel -> "Sequence of normalizing constants"]
Show[LogPDFplot[0], DisplayFunction -> $DisplayFunction, 
      PlotLabel -> "PDF and lower/upper approximations"]
Show[invCumulPlot[0], DisplayFunction -> $DisplayFunction, 
      PlotLabel -> "Inverse cumulative"]
Show[probPlot[0], DisplayFunction -> $DisplayFunction, 
        PlotLabel -> "qqplot"];

\(\(\( (*\ INDEPENDENT\ DRAWS\ *) \)\(
\)\)\)

\(\(\(
\)\(Show[probPlot[1], DisplayFunction -> $DisplayFunction, 
        PlotLabel -> "qqplot"];\)\)\)