Question: How to fasten this simulation?

restart:

with(Statistics):

# This is the naive coding of the problem
#
# It has the advantage to explain clearly (at least I hope so) what is to be done.

NaiveProc := proc(REP, a, b, N)
  local roll, rep, p, K, N0, N1:
  roll := rand(a .. b);
  N0   := 0:
  N1   := 0:
  for rep from 1 to REP do
    p := roll();
    K := Sample(Binomial(N, p), 1)[1]:
    if K=0 then
      N0 := N0+1:
    elif K=1 then
      N1 := N1+1:
    end if:
  end do:
  N0, N1:
end proc:
    
CodeTools:-Usage(NaiveProc(10^3, 0, 5e-4, 25000));

memory used=78.92MiB, alloc change=68.00MiB, cpu time=1.21s, real time=6.30s, gc time=24.80ms

# As the previous code is very slow I implemented several variants based on the fact
# that I care only on values of K equal to 0 and 1 and that it is then useless to sample
# a Binomial random variable to do this.
#
# Each of them is much faster than the naive coding... but can we even faster?
# For comparison, the same simulation realized in R takes less than 15s to run with REP=10^8

pi := (n, p, k) -> binomial(n, k) * (1-p)^(n-k) * p^k;


f1 := proc(REP, a, b, N)
  local Q, P0, P1, P01, U, Got01, Got0, N01, N0, N1:
  Q     := Sample(Uniform(a, b), REP)^+:
  P0    := pi~(N, Q, 0):
  P1    := pi~(N, Q, 1):
  P01   := P0 + P1:
  U     := Sample(Uniform(0, 1), REP)^+:
  Got01 := Select(t -> is(t <= 0), U-P01):
  Got0  := Select(t -> is(t <= 0), U-P0 );
  N01   := numelems(Got01):
  N0    := numelems(Got0):
  N1    := N01-N0:
  N0, N1;
end proc:

CodeTools:-Usage(f1(10^5, 0, 5e-4, 25000));

print():

f2 := proc(REP, a, b, N)
  local Q, P0, P1, P01, U, Got01, Got0, N01, N0, N1:
  Q     := Sample(Uniform(a, b), REP)^+:
  P0    := pi~(N, Q, 0):
  P1    := pi~(N, Q, 1):
  P01   := P0 + P1:
  U     := Sample(Uniform(0, 1), REP)^+:
  Got01 := select[flatten](`<=`, U-P01, 0):
  Got0  := select[flatten](`<=`, U-P0 , 0):
  N01   := numelems(Got01):
  N0    := numelems(Got0):
  N1    := N01-N0:
  N0, N1;
end proc:

CodeTools:-Usage(f2(10^5, 0, 5e-4, 25000));

print():

f3 := proc(REP, a, b, N)
  local Q, P0, P1, P01, U, Got01, Got0, N01, N0, N1:
  Q     := Sample(Uniform(a, b), REP)^+:
  P0    := pi~(N, Q, 0):
  P1    := pi~(N, Q, 1):
  P01   := P0 + P1:
  U     := Sample(Uniform(0, 1), REP)^+:
  Got01 := Vector(REP, i -> `if`(U[i] <= P01[i], 1, 0)):
  Got0  := Vector(REP, i -> `if`(U[i] <= P0 [i], 1, 0)):
  N01   := add(Got01):
  N0    := add(Got0):
  N1    := N01-N0:
  N0, N1;
end proc:

CodeTools:-Usage(f3(10^5, 0, 5e-4, 25000));

memory used=376.83MiB, alloc change=296.77MiB, cpu time=4.12s, real time=4.03s, gc time=430.38ms

memory used=173.26MiB, alloc change=0 bytes, cpu time=2.30s, real time=1.98s, gc time=461.22ms

memory used=154.19MiB, alloc change=0 bytes, cpu time=2.28s, real time=2.01s, gc time=432.40ms