Monte Carlo simulation using modified rejection technique

Question

Plot the probability density function

$\qquad w(x)=(1/(1+x^2))(1/x^{0.5})$ for $(0 < x < 1)$

Make up the subroutine flowchart to get the random numbers $x$, using the modified rejection technique with first factor used as the comparison function.

Use the flowchart to get a few random numbers in Wolfram Mathematica

Trial answer:

a = 0; b = 1;
f[x_] := (1/(1 + x^2)) (1/x^0.5)
h = 1; Nt = 1000000;
gr1 = Plot[f[x], {x, a, b}]

S := (
  x = y = b h; 
  While[y > f[x], 
    x = a + (b - a) RandomReal[]; 
    y = h RandomReal[]]; 
  Return[x];
)

rez = Table[S, {i, Nt}];
gr2 = Histogram[rez, Automatic, "PDF"]
Show[gr1, gr2]

Answer 1

Taking @mjw's correction into account and rewriting your code like so

f[x_] := Sqrt[2]/π/((1 + x^2) Sqrt[x])

a = .0001; b = 100;
gr1 = Plot[f[x], {x, a, 5}, PlotRange -> All];
S :=
  Module[{x, y, h = 1.},
    x = y = b h;
    While[y > f[x],
      x = RandomReal[{a, b}];
      y = h RandomReal[]];
    x]
gr2 = Histogram[Table[S, 1000], Automatic, "PDF"];
Show[gr2, gr1, PlotRange -> {{0, 5}, {0, Sqrt[2]}}]

produces

which seems fairly satisfactory.