step=0.2;

coordinates=Select[Flatten[Table[{x,y,z},{x,-0.99,0.99,step},{y,-0.99,0.99,step},{z,-0.99,0.99,step}],2],#.#<=1&];

L=Length[coordinates]

{time1,A}=AbsoluteTiming[Flatten[Table[NonNegative[ 2*(4*Norm[coordinates[[i]]-coordinates[[k]]]-2*(1-Sqrt[1-Norm[coordinates[[i]]]^2])-2*(1-Sqrt[1-Norm[coordinates[[k]]]^2]))^(1/2)*(4*Norm[coordinates[[k]]-coordinates[[j]]]-2*(1-Sqrt[1-Norm[coordinates[[k]]]^2])-2*(1-Sqrt[1-Norm[coordinates[[j]]]^2]))^(1/2)-
(6-2coordinates[[i]].coordinates[[j]]-4Norm[coordinates[[i]]-coordinates[[k]]]-4Norm[coordinates[[k]]-coordinates[[j]]]+4*(1-Sqrt[1-Norm[coordinates[[k]]]^2]))],{i,1,Length[coordinates]},{j,1,Length[coordinates]},{k,1,Length[coordinates]}],2]];

r1=Count[A,True]/Length[A];

N[r1]

time1/60

Length[A]

pts1=RandomPoint[Ball[],L];

pts2=RandomPoint[Ball[],L];

pts3=RandomPoint[Ball[],L];

{time2,B}=
AbsoluteTiming[
Flatten[
Table[
NonNegative[
2*(4*Norm[pts1[[i]]-pts3[[k]]]-2*(1-Sqrt[1-Norm[pts1[[i]]]^2])-2*(1-Sqrt[1-Norm[pts3[[k]]]^2]))^(1/2)*(4*Norm[pts3[[k]]-pts2[[j]]]-2*(1-Sqrt[1-Norm[pts3[[k]]]^2])-2*(1-Sqrt[1-Norm[pts2[[j]]]^2]))^(1/2)-
(6-2pts1[[i]].pts2[[j]]-4Norm[pts1[[i]]-pts3[[k]]]-4Norm[pts3[[k]]-pts2[[j]]]+4*(1-Sqrt[1-Norm[pts3[[k]]]^2]))],{i,1,L},{j,1,L},{k,1,L}],2]];

r2=Count[B,True]/Length[B];

N[r2]

time2/60

Length[B]

N[r1-r2]

N[r1]

N[r2]