How to speedup evaluation of a random point from an implicit region?

Question

Consider three variables s2,t1,t2, and parameters ma,mb,m1,m2,m3,s.

The domain of definition for s2,t1,t2 is given by the region Region3Particle:

matrix\[CapitalDelta]4 = {{2*s2, s2 - t1 + mb^2, s + s2 - m1^2, 
    s2 - m2^2 + m3^2}, {s2 - t1 + mb^2, 2 mb^2, s - ma^2 + mb^2, 
    mb^2 + m3^2 - t2}, {s + s2 - m1^2, s - ma^2 + mb^2, 2 s, 
    s - s1 + m3^2}, {s2 - m2^2 + m3^2, mb^2 + m3^2 - t2, 
    s - s1 + m3^2, 2 m3^2}};
matrix\[CapitalDelta]4 // MatrixForm
\[CapitalDelta]4val[ma_, mb_, m1_, m2_, m3_, s_, s2_, t1_, t2_, s1_] =
   1/16 Det[matrix\[CapitalDelta]4] // Simplify;
s1sol[ma_, mb_, m1_, m2_, m3_, s_, s2_, t1_, t2_] = 
  s1 /. Solve[\[CapitalDelta]4val[ma, mb, m1, m2, m3, s, s2, t1, t2, 
       s1] == 0, s1] // Simplify;
ConditionRegion[ma_, mb_, m1_, m2_, m3_, s_, s2_, t1_, 
   t2_] = (s1sol[ma, mb, m1, m2, m3, s, s2, t1, t2][[2]] - 
       s1sol[ma, mb, m1, m2, m3, s, s2, t1, t2][[1]] // Simplify)^2 //
     Simplify // Numerator;
Region3Particle[ma_, mb_, m1_, m2_, m3_, s_] = 
  ImplicitRegion[
   ConditionRegion[ma, mb, m1, m2, m3, s, s2, t1, t2] >= 0, {s2, 
    t1, t2}];


rp[ma_, mb_, m1_, m2_, m3_, s_] := 
 RandomPoint[Region3Particle[ma, mb, m1, m2, m3, s]]
maval=0;
mbval=0.938;
m1val=RandomReal[{0.,13}];
m2val=0.105;
m3val=0.938;
sval=RandomReal[{(m1val+m2val+m3val)^2,1000*(m1val+m2val+m3val)^2}];

However, the evaluation turns out to be extremely slow:

rp[maval, mbval, m1val, m2val, m3val, sval]//AbsoluteTiming

{4.957888,{0.383624,0.578836,0.790515}}

Could you please tell me how to improve the performance of this evaluation?

My attempt is to compile the overall calculation:

ConditionRegionCompiled = 
  Compile[{{ma, _Real}, {mb, _Real}, {m1, _Real}, {m2, _Real}, {m3, \
_Real}, {s, _Real}, {s2, _Real}, {t1, _Real}, {t2, _Real}}, 
   ConditionRegion[ma, mb, m1, m2, m3, s, s2, t1, t2]];
ConditionRegionCompiledN[ma_?NumericQ, mb_?NumericQ, m1_?NumericQ, 
  m2_?NumericQ, m3_?NumericQ, s_?NumericQ, s2_?NumericQ, t1_?NumericQ,
   t2_?NumericQ] := 
 ConditionRegionCompiled[ma, mb, m1, m2, m3, s, s2, t1, t2]
Region3ParticleCompiled[ma_, mb_, m1_, m2_, m3_, s_] := 
  ImplicitRegion[
   ConditionRegion[ma, mb, m1, m2, m3, s, s2, t1, t2] >= 0, {s2, t1, 
    t2}];
rpCompiled[ma_, mb_, m1_, m2_, m3_, s_] := 
 RandomPoint[Region3ParticleCompiled[ma, mb, m1, m2, m3, s]]

But rpCompiled does not show any improvement over rp...

Answer 1

Edit

When we limit the range of BoundaryDiscretizeRegion, the RandomPoint work.

Clear[reg, pts];
reg = ImplicitRegion[{ConditionRegion[maval, mbval, m1val, m2val, 
      m3val, sval, s2, t1, t2] >= 0}, {s2, t1, t2}];
dreg = BoundaryDiscretizeRegion[
   reg, {{-1000, 1000}, {-1000, 1000}, {-1000, 1000}}];
pts = RandomPoint[dreg, 
  1000]; 
Graphics3D[{FaceForm[{Opacity[.5], Cyan}], EdgeForm[], dreg, 
  Red, Point[pts]}, Boxed -> False]

Original

It seems that it is not easy to use DiscretizeRegion to such region to speedup the RandomPoint. We try to use FindInstance instead of RandomPoint.

Clear[reg, sols, pts, bd];
reg = ImplicitRegion[{ConditionRegion[maval, mbval, m1val, m2val, 
      m3val, sval, s2, t1, t2] >= 0}, {s2, t1, t2}];
sols = FindInstance[{s2, t1, t2} ∈ reg, {s2, t1, t2}, 1000];
pts = {s2, t1, t2} /. sols;
bd = MinMax[pts] // N;
Show[Region[Style[reg, Directive[Yellow, Opacity[.5]]], 
  PlotRange -> bd], ListPointPlot3D[pts, PlotStyle -> Red]]