[Question]
  [
I have this code which I have written in Python/NumPy
```
from __future__ import division
import numpy as np
import itertools
n = 6
iters = 1000
firstzero = 0
bothzero = 0
""" The next line iterates over arrays of length n+1 which contain only -1s and 1s """
for S in itertools.product([-1, 1], repeat=n+1):
    """For i from 0 to iters -1 """
    for i in xrange(iters):
        """ Choose a random array of length n.
            Prob 1/4 of being -1, prob 1/4 of being 1 and prob 1/2 of being 0. """
        F = np.random.choice(np.array([-1, 0, 0, 1], dtype=np.int8), size=n)
        """The next loop just makes sure that F is not all zeros."""
        while np.all(F == 0):
            F = np.random.choice(np.array([-1, 0, 0, 1], dtype=np.int8), size=n)
        """np.convolve(F, S, 'valid') computes two inner products between
        F and the two successive windows of S of length n."""
        FS = np.convolve(F, S, 'valid')
        if FS[0] == 0:
            firstzero += 1
        if np.all(FS == 0):
            bothzero += 1
print("firstzero: %i" % firstzero)
print("bothzero: %i" % bothzero)
```
It is counting the number of times the convolution of two random arrays, one which is one longer than the other, with a particular probability distribution, has a 0 at the first position or a 0 in both positions.
I had a bet with a friend who says Python is a terrible language to write code in that needs to be fast. It takes 9s on my computer. He says it could be made 100 times faster if written in a "proper language".
The challenge is to see if this code can indeed by made 100 times faster in any language of your choice. I will test your code and the fastest one week from now will win. If anyone gets below 0.09s then they automatically win and I lose.
**Status** 
* **Python**. 30 times speed up by Alistair Buxon! Although not the fastest solution it is in fact my favourite.
* **Octave**. 100 times speed up by @Thethos.
* **Rust**. 500 times speed up by @dbaupp.
* **C++**. 570 times speed up by Guy Sirton.
* **C**. 727 times speed up by @ace.
* **C++**. Unbelievably fast by @Stefan.
The fastest solutions are now too fast to sensibly time. I have therefore increased n to 10 and set iters = 100000 to compare the best ones. Under this measure the fastest are.
* **C**. 7.5s by @ace.
* **C++**. 1s by @Stefan.
**My Machine** The timings will be run on my machine. This is a standard ubuntu install on an AMD FX-8350 Eight-Core Processor. This also means I need to be able to run your code.
**Follow up posted** As this competition was rather too easy to get a x100 speedup, I have posted a followup for those who want to exercise their speed guru expertise. See [How Slow Is Python Really (Part II)?](https://codegolf.stackexchange.com/questions/26371/how-slow-is-python-really-part-ii)
  
[Answer]
# Python2.7 + Numpy 1.8.1: 10.242 s
# Fortran 90+: ~~0.029 s~~ ~~0.003 s~~ ~~0.022 s~~ 0.010 s
Damn straight you lost your bet! Not a drop of parallelization here too, just straight Fortran 90+.
**EDIT** I've taken Guy Sirton's algorithm for permuting the array `S` (good find :D). I apparently also had the `-g -traceback` compiler flags active which were slowing this code down to about 0.017s. Currently, I am compiling this as 
```
ifort -fast -o convolve convolve_random_arrays.f90
```
For those who don't have `ifort`, you can use
```
gfortran -O3 -ffast-math -o convolve convolve_random_arrays.f90
```
**EDIT 2**: The decrease in run-time is because I was doing something wrong previously and got an incorrect answer. Doing it the right way is apparently slower. I still can't believe that C++ is faster than mine, so I'm probably going to spend some time this week trying to tweak the crap out of this to speed it up.
**EDIT 3**: By simply changing the RNG section using one [based on BSD's RNG](http://www.gnu.org/software/gsl/manual/html_node/Unix-random-number-generators.html) (as suggested by Sampo Smolander) and eliminating the constant divide by `m1`, I cut the run-time to the same as the [C++ answer by Guy Sirton](https://codegolf.stackexchange.com/a/26336/11376). Using static arrays (as suggested by Sharpie) drops the run-time to under the C++ run-time! Yay Fortran! :D
**EDIT 4** Apparently this doesn't compile (with gfortran) and run correctly (incorrect values) because the integers are overstepping their limits. I've made corrections to ensure it works, but this requires one to have either ifort 11+ or gfortran 4.7+ (or another compiler that allows `iso_fortran_env` and the F2008 `int64` kind).
Here's the code:
```
program convolve_random_arrays
   use iso_fortran_env
   implicit none
   integer(int64), parameter :: a1 = 1103515245
   integer(int64), parameter :: c1 = 12345
   integer(int64), parameter :: m1 = 2147483648
   real, parameter ::    mi = 4.656612873e-10 ! 1/m1
   integer, parameter :: n = 6
   integer :: p, pmax, iters, i, nil(0:1), seed
   !integer, allocatable ::  F(:), S(:), FS(:)
   integer :: F(n), S(n+1), FS(2)
   !n = 6
   !allocate(F(n), S(n+1), FS(2))
   iters = 1000
   nil = 0
   !call init_random_seed()
   S = -1
   pmax = 2**(n+1)
   do p=1,pmax
      do i=1,iters
         F = rand_int_array(n)
         if(all(F==0)) then
            do while(all(F==0))
               F = rand_int_array(n)
            enddo
         endif
         FS = convolve(F,S)
         if(FS(1) == 0) then
            nil(0) = nil(0) + 1
            if(FS(2) == 0) nil(1) = nil(1) + 1
         endif
      enddo
      call permute(S)
   enddo
   print *,"first zero:",nil(0)
   print *," both zero:",nil(1)
 contains
   pure function convolve(x, h) result(y)
!x is the signal array
!h is the noise/impulse array
      integer, dimension(:), intent(in) :: x, h
      integer, dimension(abs(size(x)-size(h))+1) :: y
      integer:: i, j, r
      y(1) = dot_product(x,h(1:n-1))
      y(2) = dot_product(x,h(2:n  ))
   end function convolve
   pure subroutine permute(x)
      integer, intent(inout) :: x(:)
      integer :: i
      do i=1,size(x)
         if(x(i)==-1) then
            x(i) = 1
            return
         endif
         x(i) = -1
      enddo
   end subroutine permute
   function rand_int_array(i) result(x)
     integer, intent(in) :: i
     integer :: x(i), j
     real :: y
     do j=1,i
        y = bsd_rng()
        if(y <= 0.25) then
           x(j) = -1
        else if (y >= 0.75) then
           x(j) = +1
        else
           x(j) = 0
        endif
     enddo
   end function rand_int_array
   function bsd_rng() result(x)
      real :: x
      integer(int64) :: b=3141592653
      b = mod(a1*b + c1, m1)
      x = real(b)*mi
   end function bsd_rng
end program convolve_random_arrays
```
I suppose the question now is will you stop using slow-as-molasses Python and use fast-as-electrons-can-move Fortran ;).
[Answer]
# C++ bit magic
## 0.84ms with simple RNG, 1.67ms with c++11 std::knuth
0.16ms with slight algorithmic modification (see edit below)
The python implementation runs in 7.97 seconds on my rig. So this is 9488 to 4772 times faster depending on what RNG you choose.
```
#include <iostream>
#include <bitset>
#include <random>
#include <chrono>
#include <stdint.h>
#include <cassert>
#include <tuple>
#if 0
// C++11 random
std::random_device rd;
std::knuth_b gen(rd());
uint32_t genRandom()
{
    return gen();
}
#else
// bad, fast, random.
uint32_t genRandom()
{
    static uint32_t seed = std::random_device()();
    auto oldSeed = seed;
    seed = seed*1664525UL + 1013904223UL; // numerical recipes, 32 bit
    return oldSeed;
}
#endif
#ifdef _MSC_VER
uint32_t popcnt( uint32_t x ){ return _mm_popcnt_u32(x); }
#else
uint32_t popcnt( uint32_t x ){ return __builtin_popcount(x); }
#endif
std::pair<unsigned, unsigned> convolve()
{
    const uint32_t n = 6;
    const uint32_t iters = 1000;
    unsigned firstZero = 0;
    unsigned bothZero = 0;
    uint32_t S = (1 << (n+1));
    // generate all possible N+1 bit strings
    // 1 = +1
    // 0 = -1
    while ( S-- )
    {
        uint32_t s1 = S % ( 1 << n );
        uint32_t s2 = (S >> 1) % ( 1 << n );
        uint32_t fmask = (1 << n) -1; fmask |= fmask << 16;
        static_assert( n < 16, "packing of F fails when n > 16.");
        for( unsigned i = 0; i < iters; i++ )
        {
            // generate random bit mess
            uint32_t F;
            do {
                F = genRandom() & fmask;
            } while ( 0 == ((F % (1 << n)) ^ (F >> 16 )) );
            // Assume F is an array with interleaved elements such that F[0] || F[16] is one element
            // here MSB(F) & ~LSB(F) returns 1 for all elements that are positive
            // and  ~MSB(F) & LSB(F) returns 1 for all elements that are negative
            // this results in the distribution ( -1, 0, 0, 1 )
            // to ease calculations we generate r = LSB(F) and l = MSB(F)
            uint32_t r = F % ( 1 << n );
            // modulo is required because the behaviour of the leftmost bit is implementation defined
            uint32_t l = ( F >> 16 ) % ( 1 << n );
            uint32_t posBits = l & ~r;
            uint32_t negBits = ~l & r;
            assert( (posBits & negBits) == 0 );
            // calculate which bits in the expression S * F evaluate to +1
            unsigned firstPosBits = ((s1 & posBits) | (~s1 & negBits));
            // idem for -1
            unsigned firstNegBits = ((~s1 & posBits) | (s1 & negBits));
            if ( popcnt( firstPosBits ) == popcnt( firstNegBits ) )
            {
                firstZero++;
                unsigned secondPosBits = ((s2 & posBits) | (~s2 & negBits));
                unsigned secondNegBits = ((~s2 & posBits) | (s2 & negBits));
                if ( popcnt( secondPosBits ) == popcnt( secondNegBits ) )
                {
                    bothZero++;
                }
            }
        }
    }
    return std::make_pair(firstZero, bothZero);
}
int main()
{
    typedef std::chrono::high_resolution_clock clock;
    int rounds = 1000;
    std::vector< std::pair<unsigned, unsigned> > out(rounds);
    // do 100 rounds to get the cpu up to speed..
    for( int i = 0; i < 10000; i++ )
    {
        convolve();
    }
    auto start = clock::now();
    for( int i = 0; i < rounds; i++ )
    {
        out[i] = convolve();
    }
    auto end = clock::now();
    double seconds = std::chrono::duration_cast< std::chrono::microseconds >( end - start ).count() / 1000000.0;
#if 0
    for( auto pair : out )
        std::cout << pair.first << ", " << pair.second << std::endl;
#endif
    std::cout << seconds/rounds*1000 << " msec/round" << std::endl;
    return 0;
}
```
Compile in 64-bit for extra registers. When using the simple random generator the loops in convolve() run without any memory access, all variables are stored in the registers.
How it works: rather than storing `S` and `F` as in-memory arrays, it is stored as bits in an uint32\_t.  
For `S`, the `n` least significant bits are used where an set bit denotes an +1 and an unset bit denotes an -1.  
`F` requires at least 2 bits to create an distribution of [-1, 0, 0, 1]. This is done by generating random bits and examining the 16 least significant (called `r`) and 16 most significant bits (called `l`). If `l & ~r` we assume that F is +1, if `~l & r` we assume that `F` is -1. Otherwise `F` is 0. This generates the distribution we're looking for.
Now we have `S`, `posBits` with an set bit on every location where F == 1 and `negBits` with an set bit on every location where F == -1.
We can prove that `F * S` (where \* denotes multiplication) evaluates to +1 under the condition `(S & posBits) | (~S & negBits)`. We can also generate similar logic for all cases where `F * S` evaluates to -1. And finally, we know that `sum(F * S)` evaluates to 0 if and only if there is an equal amount of -1's and +1's in the result. This is very easy to calculate by simply comparing the number of +1 bits and -1 bits.
This implementation uses 32 bit ints, and the maximum `n` accepted is 16. It is possible to scale the implementation to 31 bits by modifying the random generate code, and to 63 bits by using uint64\_t instead of uint32\_t. 
## edit
The folowing convolve function:
```
std::pair<unsigned, unsigned> convolve()
{
    const uint32_t n = 6;
    const uint32_t iters = 1000;
    unsigned firstZero = 0;
    unsigned bothZero = 0;
    uint32_t fmask = (1 << n) -1; fmask |= fmask << 16;
    static_assert( n < 16, "packing of F fails when n > 16.");
    for( unsigned i = 0; i < iters; i++ )
    {
        // generate random bit mess
        uint32_t F;
        do {
            F = genRandom() & fmask;
        } while ( 0 == ((F % (1 << n)) ^ (F >> 16 )) );
        // Assume F is an array with interleaved elements such that F[0] || F[16] is one element
        // here MSB(F) & ~LSB(F) returns 1 for all elements that are positive
        // and  ~MSB(F) & LSB(F) returns 1 for all elements that are negative
        // this results in the distribution ( -1, 0, 0, 1 )
        // to ease calculations we generate r = LSB(F) and l = MSB(F)
        uint32_t r = F % ( 1 << n );
        // modulo is required because the behaviour of the leftmost bit is implementation defined
        uint32_t l = ( F >> 16 ) % ( 1 << n );
        uint32_t posBits = l & ~r;
        uint32_t negBits = ~l & r;
        assert( (posBits & negBits) == 0 );
        uint32_t mask = posBits | negBits;
        uint32_t totalBits = popcnt( mask );
        // if the amount of -1 and +1's is uneven, sum(S*F) cannot possibly evaluate to 0
        if ( totalBits & 1 )
            continue;
        uint32_t adjF = posBits & ~negBits;
        uint32_t desiredBits = totalBits / 2;
        uint32_t S = (1 << (n+1));
        // generate all possible N+1 bit strings
        // 1 = +1
        // 0 = -1
        while ( S-- )
        {
            // calculate which bits in the expression S * F evaluate to +1
            auto firstBits = (S & mask) ^ adjF;
            auto secondBits = (S & ( mask << 1 ) ) ^ ( adjF << 1 );
            bool a = desiredBits == popcnt( firstBits );
            bool b = desiredBits == popcnt( secondBits );
            firstZero += a;
            bothZero += a & b;
        }
    }
    return std::make_pair(firstZero, bothZero);
}
```
cuts the runtime to 0.160-0.161ms. Manual loop unroll (not pictured above) makes that 0.150. The less trivial n=10, iter=100000 case runs under 250ms. I'm sure i can get it under 50ms by leveraging additional cores but that's too easy.
This is done by making the inner loop branch free and swapping the F and S loop.  
If `bothZero` is not required i can cut down the run time to 0.02ms by sparsely looping over all possible S arrays.
[Answer]
**Python 2.7 - ~~0.882s~~ 0.283s**
**(OP's original: 6.404s)**
**Edit:** Steven Rumbalski's optimization by precomputing F values. With this optimization cpython beats pypy's 0.365s.
```
import itertools
import operator
import random
n=6
iters = 1000
firstzero = 0
bothzero = 0
choicesF = filter(any, itertools.product([-1, 0, 0, 1], repeat=n))
for S in itertools.product([-1,1], repeat = n+1):
    for i in xrange(iters):
        F = random.choice(choicesF)
        if not sum(map(operator.mul, F, S[:-1])):
            firstzero += 1
            if not sum(map(operator.mul, F, S[1:])):
                bothzero += 1
print "firstzero", firstzero
print "bothzero", bothzero
```
OP's original code uses such tiny arrays there is no benefit to using Numpy, as this pure python implementation demonstrates. But see also [this numpy implementation](https://codegolf.stackexchange.com/a/26451) which is three times faster again than my code.
I also optimize by skipping the rest of the convolution if the first result isn't zero.
[Answer]
# Rust: 0.011s
### Original Python: 8.3
A straight translation of the original Python.
```
extern crate rand;
use rand::Rng;
static N: uint = 6;
static ITERS: uint = 1000;
fn convolve<T: Num>(into: &mut [T], a: &[T], b: &[T]) {
    // we want `a` to be the longest array
    if a.len() < b.len() {
        convolve(into, b, a);
        return
    }
    assert_eq!(into.len(), a.len() - b.len() + 1);
    for (n,place) in into.mut_iter().enumerate() {
        for (x, y) in a.slice_from(n).iter().zip(b.iter()) {
            *place = *place + *x * *y
        }
    }
}
fn main() {
    let mut first_zero = 0;
    let mut both_zero = 0;
    let mut rng = rand::XorShiftRng::new().unwrap();
    for s in PlusMinus::new() {
        for _ in range(0, ITERS) {
            let mut f = [0, .. N];
            while f.iter().all(|x| *x == 0) {
                for p in f.mut_iter() {
                    match rng.gen::<u32>() % 4 {
                        0 => *p = -1,
                        1 | 2 => *p = 0,
                        _ => *p = 1
                    }
                }
            }
            let mut fs = [0, .. 2];
            convolve(fs, s, f);
            if fs[0] == 0 { first_zero += 1 }
            if fs.iter().all(|&x| x == 0) { both_zero += 1 }
        }
    }
    println!("{}\n{}", first_zero, both_zero);
}
/// An iterator over [+-]1 arrays of the appropriate length
struct PlusMinus {
    done: bool,
    current: [i32, .. N + 1]
}
impl PlusMinus {
    fn new() -> PlusMinus {
        PlusMinus { done: false, current: [-1, .. N + 1] }
    }
}
impl Iterator<[i32, .. N + 1]> for PlusMinus {
    fn next(&mut self) -> Option<[i32, .. N+1]> {
        if self.done {
            return None
        }
        let ret = self.current;
        // a binary "adder", that just adds one to a bit vector (where
        // -1 is the zero, and 1 is the one).
        for (i, place) in self.current.mut_iter().enumerate() {
            *place = -*place;
            if *place == 1 {
                break
            } else if i == N {
                // we've wrapped, so we want to stop after this one
                self.done = true
            }
        }
        Some(ret)
    }
}
```
* Compiled with `--opt-level=3`
* My rust compiler is a recent [nightly](http://www.rust-lang.org/install.html): (`rustc 0.11-pre-nightly (eea4909 2014-04-24 23:41:15 -0700)` to be precise)
[Answer]
**C++ (VS 2012) - ~~0.026s~~ 0.015s**
**Python 2.7.6/Numpy 1.8.1 - 12s**
Speedup ~x800.
The gap would be a lot smaller if the convolved arrays were very large...
```
#include <vector>
#include <iostream>
#include <ctime>
using namespace std;
static unsigned int seed = 35;
int my_random()
{
   seed = seed*1664525UL + 1013904223UL; // numerical recipes, 32 bit
   switch((seed>>30) & 3)
   {
   case 0: return 0;
   case 1: return -1;
   case 2: return 1;
   case 3: return 0;
   }
   return 0;
}
bool allzero(const vector<int>& T)
{
   for(auto x : T)
   {
      if(x!=0)
      {
         return false;
      }
   }
   return true;
}
void convolve(vector<int>& out, const vector<int>& v1, const vector<int>& v2)
{
   for(size_t i = 0; i<out.size(); ++i)
   {
      int result = 0;
      for(size_t j = 0; j<v2.size(); ++j)
      {
         result += v1[i+j]*v2[j];
      }
      out[i] = result;
   }
}
void advance(vector<int>& v)
{
   for(auto &x : v)
   {
      if(x==-1)
      {
         x = 1;
         return;
      }
      x = -1;
   }
}
void convolve_random_arrays(void)
{
   const size_t n = 6;
   const int two_to_n_plus_one = 128;
   const int iters = 1000;
   int bothzero = 0;
   int firstzero = 0;
   vector<int> S(n+1);
   vector<int> F(n);
   vector<int> FS(2);
   time_t current_time;
   time(&current_time);
   seed = current_time;
   for(auto &x : S)
   {
      x = -1;
   }
   for(int i=0; i<two_to_n_plus_one; ++i)
   {
      for(int j=0; j<iters; ++j)
      {
         do
         {
            for(auto &x : F)
            {
               x = my_random();
            }
         } while(allzero(F));
         convolve(FS, S, F);
         if(FS[0] == 0)
         {
            firstzero++;
            if(FS[1] == 0)
            {
               bothzero++;
            }
         }
      }
      advance(S);
   }
   cout << firstzero << endl; // This output can slow things down
   cout << bothzero << endl; // comment out for timing the algorithm
}
```
A few notes:
* The random function is being called in the loop so I went for a very light weight linear congruential generator (but generously looked at the MSBs).
* This is really just the starting point for an optimized solution.
* Didn't take that long to write...
* I iterate through all the values of S taking `S[0]` to be the "least significant" digit.
Add this main function for a self contained example:
```
int main(int argc, char** argv)
{
  for(int i=0; i<1000; ++i) // run 1000 times for stop-watch
  {
      convolve_random_arrays();
  }
}
```
[Answer]
## C
Takes 0.015s on my machine, with OP's original code taking ~ 7.7s. Tried to optimize by generating the random array and convolving in the same loop, but it doesn't seem to make a lot of difference.
The first array is generated by taking an integer, write it out in binary, and change all 1 to -1 and all 0 to 1. The rest should be very straightforward.
**Edit:** instead of having `n` as an `int`, now we have `n` as a macro-defined constant, so we can use `int arr[n];` instead of `malloc`.
**Edit2:** Instead of built-in `rand()` function, this now implements an xorshift PRNG. Also, a lot of conditional statements are removed when generating the random array.
Compile instructions:
```
gcc -O3 -march=native -fwhole-program -fstrict-aliasing -ftree-vectorize -Wall ./test.c -o ./test
```
Code:
```
#include <stdio.h>
#include <time.h>
#define n (6)
#define iters (1000)
unsigned int x,y=34353,z=57768,w=1564; //PRNG seeds
/* xorshift PRNG
 * Taken from https://en.wikipedia.org/wiki/Xorshift#Example_implementation
 * Used under CC-By-SA */
int myRand() {
    unsigned int t;
    t = x ^ (x << 11);
    x = y; y = z; z = w;
    return w = w ^ (w >> 19) ^ t ^ (t >> 8);
}
int main() {
    int firstzero=0, bothzero=0;
    int arr[n+1];
    unsigned int i, j;
    x=(int)time(NULL);
    for(i=0; i< 1<<(n+1) ; i++) {
        unsigned int tmp=i;
        for(j=0; j<n+1; j++) {
            arr[j]=(tmp&1)*(-2)+1;
            tmp>>=1;
        }
        for(j=0; j<iters; j++) {
            int randArr[n];
            unsigned int k, flag=0;
            int first=0, second=0;
            do {
                for(k=0; k<n; k++) {
                    randArr[k]=(1-(myRand()&3))%2;
                    flag+=(randArr[k]&1);
                    first+=arr[k]*randArr[k];
                    second+=arr[k+1]*randArr[k];
                }
            } while(!flag);
            firstzero+=(!first);
            bothzero+=(!first&&!second);
        }
    }
    printf("firstzero %d\nbothzero %d\n", firstzero, bothzero);
    return 0;
}
```
[Answer]
# J
I don't expect to beat out any compiled languages, and something tells me it'd take a miraculous machine to get less than 0.09 s with this, but I'd like to submit this J anyway, because it's pretty slick.
```
NB. constants
num =: 6
iters =: 1000
NB. convolve
NB. take the multiplication table                */
NB. then sum along the NE-SW diagonals           +//.
NB. and keep the longest ones                    #~ [: (= >./) #/.
NB. operate on rows of higher dimensional lists  " 1
conv =: (+//. #~ [: (= >./) #/.) @: (*/) " 1
NB. main program
S  =: > , { (num+1) # < _1 1                NB. all {-1,1}^(num+1)
F  =: (3&= - 0&=) (iters , num) ?@$ 4       NB. iters random arrays of length num
FS =: ,/ S conv/ F                          NB. make a convolution table
FB =: +/ ({. , *./)"1 ] 0 = FS              NB. first and both zero
('first zero ',:'both zero ') ,. ":"0 FB    NB. output results
```
This takes about 0.5 s on a laptop from the previous decade, only about 20x as fast as the Python in the answer. Most of the time is spent in `conv` because we write it lazily (we compute the entire convolution) and in full generality.
Since we know things about `S` and `F`, we can speed things up by making specific optimizations for this program. The best I've been able to come up with is `conv =: ((num, num+1) { +//.)@:(*/)"1`—select specifically the two numbers that correspond from the diagonal sums to the longest elements of the convolution—which approximately halves the time.
[Answer]
**Perl - 9.3X faster...830% improvement**
On my ancient netbook, the OP's code takes 53 seconds to run; Alistair Buxton's version takes about 6.5 seconds, and the following Perl version takes about 5.7 seconds.
```
use v5.10;
use strict;
use warnings;
use Algorithm::Combinatorics qw( variations_with_repetition );
use List::Util qw( any sum );
use List::MoreUtils qw( pairwise );
my $n         = 6;
my $iters     = 1000;
my $firstzero = 0;
my $bothzero  = 0;
my $variations = variations_with_repetition([-1, 1], $n+1);
while (my $S = $variations->next)
{
  for my $i (1 .. $iters)
  {
    my @F;
    until (@F and any { $_ } @F)
    {
      @F = map +((-1,0,0,1)[rand 4]), 1..$n;
    }
    # The pairwise function doesn't accept array slices,
    # so need to copy into a temp array @S0
    my @S0 = @$S[0..$n-1];
    unless (sum pairwise { $a * $b } @F, @S0)
    {
      $firstzero++;
      my @S1 = @$S[1..$n];  # copy again :-(
      $bothzero++ unless sum pairwise { $a * $b } @F, @S1;
    }
  }
}
say "firstzero ", $firstzero;
say "bothzero ", $bothzero;
```
[Answer]
Python 2.7 - numpy 1.8.1 with mkl bindings - 0.086s
(OP's original: 6.404s)
(Buxton's pure python: 0.270s)
```
import numpy as np
import itertools
n=6
iters = 1000
#Pack all of the Ses into a single array
S = np.array( list(itertools.product([-1,1], repeat=n+1)) )
# Create a whole array of test arrays, oversample a bit to ensure we 
# have at least (iters) of them
F = np.random.rand(int(iters*1.1),n)
F = ( F < 0.25 )*-1 + ( F > 0.75 )*1
goodrows = (np.abs(F).sum(1)!=0)
assert goodrows.sum() > iters, "Got very unlucky"
# get 1000 cases that aren't all zero
F = F[goodrows][:iters]
# Do the convolution explicitly for the two 
# slots, but on all of the Ses and Fes at the 
# same time
firstzeros = (F[:,None,:]*S[None,:,:-1]).sum(-1)==0
secondzeros = (F[:,None,:]*S[None,:,1:]).sum(-1)==0
firstzero_count = firstzeros.sum()
bothzero_count = (firstzeros * secondzeros).sum()
print "firstzero", firstzero_count
print "bothzero", bothzero_count
```
As Buxton points out, OP's original code uses such tiny arrays there is no benefit to using Numpy. This implementation leverages numpy by doing all
of the F and S cases at once in an array oriented way. This combined
with mkl bindings for python leads to a very fast implementation.
Note also that just loading the libraries and starting the interpreter takes 0.076s so the actual computation is taking ~ 0.01 seconds, similar to the C++ solution.
[Answer]
# MATLAB 0.024s
Computer 1
* Original Code: ~ 3.3 s
* Alistar Buxton's Code: ~ 0.51 s
* Alistar Buxton's new Code: ~0.25 s
* Matlab Code: ~ 0.024 s (Matlab already running)
Computer 2
* Original Code: ~ 6.66 s
* Alistar Buxton's Code: ~ 0.64 s
* Alistar Buxton's new Code: ?
* Matlab: ~ 0.07 s (Matlab already running)
* Octave: ~ 0.07 s
I decided to give the oh so slow Matlab a try. If you know how, you can get rid of most of the loops (in Matlab), which makes it quite fast. However, the memory requirements are higher than for looped solutions but this will not be an issue if you don't have very large arrays...
```
function call_convolve_random_arrays
tic
convolve_random_arrays
toc
end
function convolve_random_arrays
n = 6;
iters = 1000;
firstzero = 0;
bothzero = 0;
rnd = [-1, 0, 0, 1];
S = -1 *ones(1, n + 1);
IDX1 = 1:n;
IDX2 = IDX1 + 1;
for i = 1:2^(n + 1)
    F = rnd(randi(4, [iters, n]));
    sel = ~any(F,2);
    while any(sel)
        F(sel, :) = rnd(randi(4, [sum(sel), n]));
        sel = ~any(F,2);
    end
    sum1 = F * S(IDX1)';
    sel = sum1 == 0;
    firstzero = firstzero + sum(sel);
    sum2 = F(sel, :) * S(IDX2)';
    sel = sum2 == 0;
    bothzero = bothzero + sum(sel);
    
    S = permute(S); 
end
fprintf('firstzero %i \nbothzero %i \n', firstzero, bothzero);
end
function x = permute(x)
for i=1:length(x)
    if(x(i)==-1)
        x(i) = 1;
            return
    end
        x(i) = -1;
end
end
```
Here is what I do:
* use Kyle Kanos function to permute through S
* calculate all n \* iters random numbers at once
* map 1 to 4 to [-1 0 0 1]
* use Matrix multiplication (elementwise sum(F \* S(1:5)) is equal to matrix multiplication of F \* S(1:5)'
* for bothzero: only calculate members that fullfill the first condition
I assume you don't have matlab, which is too bad as I really would have liked to see how it compares...
(The function can be slower the first time you run it.)
[Answer]
## Haskell: ~2000x speedup per core
Compile with 'ghc -O3 -funbox-strict-fields -threaded -fllvm', and run with '+RTS -Nk' where k is the number of cores on your machine.
```
import Control.Parallel.Strategies
import Data.Bits
import Data.List
import Data.Word
import System.Random
n = 6 :: Int
iters = 1000 :: Int
data G = G !Word !Word !Word !Word deriving (Eq, Show)
gen :: G -> (Word, G)
gen (G x y z w) = let t  = x `xor` (x `shiftL` 11)
                      w' = w `xor` (w `shiftR` 19) `xor` t `xor` (t `shiftR` 8)
                  in (w', G y z w w')  
mask :: Word -> Word
mask = (.&.) $ (2 ^ n) - 1
gen_nonzero :: G -> (Word, G)
gen_nonzero g = let (x, g') = gen g 
                    a = mask x
                in if a == 0 then gen_nonzero g' else (a, g')
data F = F {zeros  :: !Word, 
            posneg :: !Word} deriving (Eq, Show)
gen_f :: G -> (F, G)       
gen_f g = let (a, g')  = gen_nonzero g
              (b, g'') = gen g'
          in  (F a $ mask b, g'')
inner :: Word -> F -> Int
inner s (F zs pn) = let s' = complement $ s `xor` pn
                        ones = s' .&. zs
                        negs = (complement s') .&. zs
                    in popCount ones - popCount negs
specialised_convolve :: Word -> F -> (Int, Int)
specialised_convolve s f@(F zs pn) = (inner s f', inner s f) 
    where f' = F (zs `shiftL` 1) (pn `shiftL` 1)
ss :: [Word]
ss = [0..2 ^ (n + 1) - 1]
main_loop :: [G] -> (Int, Int)
main_loop gs = foldl1' (\(fz, bz) (fz', bz') -> (fz + fz', bz + bz')) . parMap rdeepseq helper $ zip ss gs
    where helper (s, g) = go 0 (0, 0) g
                where go k u@(fz, bz) g = if k == iters 
                                              then u 
                                              else let (f, g') = gen_f g
                                                       v = case specialised_convolve s f
                                                               of (0, 0) -> (fz + 1, bz + 1)
                                                                  (0, _) -> (fz + 1, bz)
                                                                  _      -> (fz, bz)
                                                   in go (k + 1) v g'
seed :: IO G                                        
seed = do std_g <- newStdGen
          let [x, y, z, w] = map fromIntegral $ take 4 (randoms std_g :: [Int])
          return $ G x y z w
main :: IO ()
main = (sequence $ map (const seed) ss) >>= print . main_loop
```
[Answer]
# Julia: 0.30 s
# Op's Python: 21.36 s (Core2 duo)
**71x speedup**
```
function countconv()                                                                                                                                                           
    n = 6                                                                                                                                                                      
    iters = 1000                                                                                                                                                               
    firstzero = 0                                                                                                                                                              
    bothzero = 0                                                                                                                                                               
    cprod= Iterators.product(fill([-1,1], n+1)...)                                                                                                                             
    F=Array(Float64,n);                                                                                                                                                        
    P=[-1. 0. 0. 1.]                                                                                                                                                                                                                                                                                                             
                                                                                                                                                                           
    for S in cprod                                                                                                                                                             
        Sm=[S...]                                                                                                                                                              
        for i = 1:iters                                                                                                                                                        
            F=P[rand(1:4,n)]                                                                                                                                                  
            while all(F==0)                                                                                                                                                   
                F=P[rand(1:4,n)]                                                                                                                                              
            end                                                                                                                                                               
            if  dot(reverse!(F),Sm[1:end-1]) == 0                                                                                                                           
                firstzero += 1                                                                                                                                                 
                if dot(F,Sm[2:end]) == 0                                                                                                                              
                    bothzero += 1                                                                                                                                              
                end                                                                                                                                                            
            end                                                                                                                                                                
        end                                                                                                                                                                    
    end
    return firstzero,bothzero
end
```
I did some modifications of Arman's Julia answer: First of all, I wrapped it in a function, as global variables make it hard for Julia's type inference and JIT: A global variable can change its type at any time, and must be checked every operation. Then, I got rid of the anonymous functions and array comprehensions. They aren't really necessary, and are still pretty slow. Julia is faster with lower-level abstractions right now.
There's lots more ways to make it faster, but this does a decent job.
[Answer]
Ok I am posting this just because I feel Java needs to be represented here. I am terrible with other languages and I confess to not understand the problem exactly, so I will need some help to fix this code. I stole most of the code ace's C example, and then borrowed some snippets from others. I hope that isn't a faux pas...
One thing I would like to point out is that languages that optimize at run time need to be run several/many times to get up to full speed. I think it is justified to take the fully optimized speed (or at least the average speed) because most things you are concerned with running fast will be run a bunch of times.
The code still needs to be fixed, but I ran it anyways to see what times I would get.
Here are the results on an Intel(R) Xeon(R) CPU E3-1270 V2 @ 3.50GHz on Ubuntu running it 1000 times:
server:/tmp# time java8 -cp . Tester
firstzero 40000
bothzero 20000
first run time: 41 ms
last run time: 4 ms
real 0m5.014s
user 0m4.664s
sys 0m0.268s
Here is my crappy code:
```
public class Tester 
{
    public static void main( String[] args )
    {
        long firstRunTime = 0;
        long lastRunTime = 0;
        String testResults = null;
        for( int i=0 ; i<1000 ; i++ )
        {
            long timer = System.currentTimeMillis();
            testResults = new Tester().runtest();
            lastRunTime = System.currentTimeMillis() - timer;
            if( i ==0 )
            {
                firstRunTime = lastRunTime;
            }
        }
        System.err.println( testResults );
        System.err.println( "first run time: " + firstRunTime + " ms" );
        System.err.println( "last run time: " + lastRunTime + " ms" );
    }
    private int x,y=34353,z=57768,w=1564; 
    public String runtest()
    {
        int n = 6;
        int iters = 1000;
        //#define iters (1000)
        //PRNG seeds
        /* xorshift PRNG
         * Taken from https://en.wikipedia.org/wiki/Xorshift#Example_implementation
         * Used under CC-By-SA */
            int firstzero=0, bothzero=0;
            int[] arr = new int[n+1];
            int i=0, j=0;
            x=(int)(System.currentTimeMillis()/1000l);
            for(i=0; i< 1<<(n+1) ; i++) {
                int tmp=i;
                for(j=0; j<n+1; j++) {
                    arr[j]=(tmp&1)*(-2)+1;
                    tmp>>=1;
                }
                for(j=0; j<iters; j++) {
                    int[] randArr = new int[n];
                    int k=0;
                    long flag = 0;
                    int first=0, second=0;
                    do {
                        for(k=0; k<n; k++) {
                            randArr[k]=(1-(myRand()&3))%2;
                            flag+=(randArr[k]&1);
                            first+=arr[k]*randArr[k];
                            second+=arr[k+1]*randArr[k];
                        }
                    } while(allzero(randArr));
                    if( first == 0 )
                    {
                        firstzero+=1;
                        if( second == 0 )
                        {
                            bothzero++;
                        }
                    }
                }
            }
         return ( "firstzero " + firstzero + "\nbothzero " + bothzero + "\n" );
    }
    private boolean allzero(int[] arr)
    {
       for(int x : arr)
       {
          if(x!=0)
          {
             return false;
          }
       }
       return true;
    }
    public int myRand() 
    {
        long t;
        t = x ^ (x << 11);
        x = y; y = z; z = w;
        return (int)( w ^ (w >> 19) ^ t ^ (t >> 8));
    }
}
```
And I tried running the python code after upgrading python and installing python-numpy but I get this:
```
server:/tmp# python tester.py
Traceback (most recent call last):
  File "peepee.py", line 15, in <module>
    F = np.random.choice(np.array([-1,0,0,1], dtype=np.int8), size = n)
AttributeError: 'module' object has no attribute 'choice'
```
[Answer]
Golang version 45X of python on my machine on below Golang codes:
```
package main
import (
"fmt"
"time"
)
const (
n     = 6
iters = 1000
)
var (
x, y, z, w = 34353, 34353, 57768, 1564 //PRNG seeds
)
/* xorshift PRNG
 * Taken from https://en.wikipedia.org/wiki/Xorshift#Example_implementation
 * Used under CC-By-SA */
func myRand() int {
var t uint
t = uint(x ^ (x << 11))
x, y, z = y, z, w
w = int(uint(w^w>>19) ^ t ^ (t >> 8))
return w
}
func main() {
var firstzero, bothzero int
var arr [n + 1]int
var i, j int
x = int(time.Now().Unix())
for i = 0; i < 1<<(n+1); i = i + 1 {
    tmp := i
    for j = 0; j < n+1; j = j + 1 {
        arr[j] = (tmp&1)*(-2) + 1
        tmp >>= 1
    }
    for j = 0; j < iters; j = j + 1 {
        var randArr [n]int
        var flag uint
        var k, first, second int
        for {
            for k = 0; k < n; k = k + 1 {
                randArr[k] = (1 - (myRand() & 3)) % 2
                flag += uint(randArr[k] & 1)
                first += arr[k] * randArr[k]
                second += arr[k+1] * randArr[k]
            }
            if flag != 0 {
                break
            }
        }
        if first == 0 {
            firstzero += 1
            if second == 0 {
                bothzero += 1
            }
        }
    }
}
println("firstzero", firstzero, "bothzero", bothzero)
}
```
and the below python codes copyed from above:
```
import itertools
import operator
import random
n=6
iters = 1000
firstzero = 0
bothzero = 0
choicesF = filter(any, itertools.product([-1, 0, 0, 1], repeat=n))
for S in itertools.product([-1,1], repeat = n+1):
    for i in xrange(iters):
        F = random.choice(choicesF)
        if not sum(map(operator.mul, F, S[:-1])):
            firstzero += 1
            if not sum(map(operator.mul, F, S[1:])):
                bothzero += 1
print "firstzero", firstzero
print "bothzero", bothzero
```
and the time below:
```
$time python test.py
firstzero 27349
bothzero 12125
real    0m0.477s
user    0m0.461s
sys 0m0.014s
$time ./hf
firstzero 27253 bothzero 12142
real    0m0.011s
user    0m0.008s
sys 0m0.002s
```
[Answer]
# C# 0.135s
C# based on Alistair Buxton's [plain python](https://codegolf.stackexchange.com/a/26337/4163): 0.278s  
Parallelised C#: 0.135s  
Python from the question: 5.907s  
Alistair's plain python: 0.853s
I'm not actually certain this implementation is correct - its output is different, if you look at the results down at the bottom.
There's certainly more optimal algorithms. I just decided to use a very similar algorithm to the Python one.
### Single-threaded C
```
using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
namespace ConvolvingArrays
{
    static class Program
    {
        static void Main(string[] args)
        {
            int n=6;
            int iters = 1000;
            int firstzero = 0;
            int bothzero = 0;
            int[] arraySeed = new int[] {-1, 1};
            int[] randomSource = new int[] {-1, 0, 0, 1};
            Random rand = new Random();
            foreach (var S in Enumerable.Repeat(arraySeed, n+1).CartesianProduct())
            {
                for (int i = 0; i < iters; i++)
                {
                    var F = Enumerable.Range(0, n).Select(_ => randomSource[rand.Next(randomSource.Length)]);
                    while (!F.Any(f => f != 0))
                    {
                        F = Enumerable.Range(0, n).Select(_ => randomSource[rand.Next(randomSource.Length)]);
                    }
                    if (Enumerable.Zip(F, S.Take(n), (f, s) => f * s).Sum() == 0)
                    {
                        firstzero++;
                        if (Enumerable.Zip(F, S.Skip(1), (f, s) => f * s).Sum() == 0)
                        {
                            bothzero++;
                        }
                    }
                }
            }
            Console.WriteLine("firstzero {0}", firstzero);
            Console.WriteLine("bothzero {0}", bothzero);
        }
        // itertools.product?
        // http://ericlippert.com/2010/06/28/computing-a-cartesian-product-with-linq/
        static IEnumerable<IEnumerable<T>> CartesianProduct<T>
            (this IEnumerable<IEnumerable<T>> sequences)
        {
            IEnumerable<IEnumerable<T>> emptyProduct =
              new[] { Enumerable.Empty<T>() };
            return sequences.Aggregate(
              emptyProduct,
              (accumulator, sequence) =>
                from accseq in accumulator
                from item in sequence
                select accseq.Concat(new[] { item }));
        }
    }
}
```
### Parallel C#:
```
using System;
using System.Collections.Concurrent;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading;
using System.Threading.Tasks;
namespace ConvolvingArrays
{
    static class Program
    {
        static void Main(string[] args)
        {
            int n=6;
            int iters = 1000;
            int firstzero = 0;
            int bothzero = 0;
            int[] arraySeed = new int[] {-1, 1};
            int[] randomSource = new int[] {-1, 0, 0, 1};
            ConcurrentBag<int[]> results = new ConcurrentBag<int[]>();
            // The next line iterates over arrays of length n+1 which contain only -1s and 1s
            Parallel.ForEach(Enumerable.Repeat(arraySeed, n + 1).CartesianProduct(), (S) =>
            {
                int fz = 0;
                int bz = 0;
                ThreadSafeRandom rand = new ThreadSafeRandom();
                for (int i = 0; i < iters; i++)
                {
                    var F = Enumerable.Range(0, n).Select(_ => randomSource[rand.Next(randomSource.Length)]);
                    while (!F.Any(f => f != 0))
                    {
                        F = Enumerable.Range(0, n).Select(_ => randomSource[rand.Next(randomSource.Length)]);
                    }
                    if (Enumerable.Zip(F, S.Take(n), (f, s) => f * s).Sum() == 0)
                    {
                        fz++;
                        if (Enumerable.Zip(F, S.Skip(1), (f, s) => f * s).Sum() == 0)
                        {
                            bz++;
                        }
                    }
                }
                results.Add(new int[] { fz, bz });
            });
            foreach (int[] res in results)
            {
                firstzero += res[0];
                bothzero += res[1];
            }
            Console.WriteLine("firstzero {0}", firstzero);
            Console.WriteLine("bothzero {0}", bothzero);
        }
        // itertools.product?
        // http://ericlippert.com/2010/06/28/computing-a-cartesian-product-with-linq/
        static IEnumerable<IEnumerable<T>> CartesianProduct<T>
            (this IEnumerable<IEnumerable<T>> sequences)
        {
            IEnumerable<IEnumerable<T>> emptyProduct =
              new[] { Enumerable.Empty<T>() };
            return sequences.Aggregate(
              emptyProduct,
              (accumulator, sequence) =>
                from accseq in accumulator
                from item in sequence
                select accseq.Concat(new[] { item }));
        }
    }
    // http://stackoverflow.com/a/11109361/1030702
    public class ThreadSafeRandom
    {
        private static readonly Random _global = new Random();
        [ThreadStatic]
        private static Random _local;
        public ThreadSafeRandom()
        {
            if (_local == null)
            {
                int seed;
                lock (_global)
                {
                    seed = _global.Next();
                }
                _local = new Random(seed);
            }
        }
        public int Next()
        {
            return _local.Next();
        }
        public int Next(int maxValue)
        {
            return _local.Next(maxValue);
        }
    }
}
```
## Test output:
### Windows (.NET)
The C# is much faster on Windows. Probably because .NET is faster than mono.
User and sys timing doesn't seem to work (used `git bash` for timing).
```
$ time /c/Python27/python.exe numpypython.py
firstzero 27413
bothzero 12073
real    0m5.907s
user    0m0.000s
sys     0m0.000s
$ time /c/Python27/python.exe plainpython.py
firstzero 26983
bothzero 12033
real    0m0.853s
user    0m0.000s
sys     0m0.000s
$ time ConvolvingArrays.exe
firstzero 28526
bothzero 6453
real    0m0.278s
user    0m0.000s
sys     0m0.000s
$ time ConvolvingArraysParallel.exe
firstzero 28857
bothzero 6485
real    0m0.135s
user    0m0.000s
sys     0m0.000s
```
### Linux (mono)
```
bob@phoebe:~/convolvingarrays$ time python program.py
firstzero 27059
bothzero 12131
real    0m11.932s
user    0m11.912s
sys     0m0.012s
bob@phoebe:~/convolvingarrays$ mcs -optimize+ -debug- program.cs
bob@phoebe:~/convolvingarrays$ time mono program.exe
firstzero 28982
bothzero 6512
real    0m1.360s
user    0m1.532s
sys     0m0.872s
bob@phoebe:~/convolvingarrays$ mcs -optimize+ -debug- parallelprogram.cs
bob@phoebe:~/convolvingarrays$ time mono parallelprogram.exe
firstzero 28857
bothzero 6496
real    0m0.851s
user    0m2.708s
sys     0m3.028s
```
[Answer]
# F# solution
Runtime is 0.030s when compiled to x86 on the CLR Core i7 4 (8) @ 3.4 Ghz
I have no idea if the code is correct.
* Functional optimization (inline fold) -> 0.026s
* Building via Console Project -> 0.022s
* Added a better algorithm for generation of the permutation arrays -> **0.018s**
* Mono for Windows -> 0.089s
* Running Alistair's Python script -> 0.259s
```
let inline ffoldi n f state =
    let mutable state = state
    for i = 0 to n - 1 do
        state <- f state i
    state
let product values n =
    let p = Array.length values
    Array.init (pown p n) (fun i ->
        (Array.zeroCreate n, i)
        |> ffoldi n (fun (result, i') j ->
            result.[j] <- values.[i' % p]
            result, i' / p
        )
        |> fst
    )
let convolute signals filter =
    let m = Array.length signals
    let n = Array.length filter
    let len = max m n - min m n + 1
    Array.init len (fun offset ->
        ffoldi n (fun acc i ->
            acc + filter.[i] * signals.[m - 1 - offset - i]
        ) 0
    )
let n = 6
let iters = 1000
let next =
    let arrays =
        product [|-1; 0; 0; 1|] n
        |> Array.filter (Array.forall ((=) 0) >> not)
    let rnd = System.Random()
    fun () -> arrays.[rnd.Next arrays.Length]
let signals = product [|-1; 1|] (n + 1)
let firstzero, bothzero =
    ffoldi signals.Length (fun (firstzero, bothzero) i ->
        let s = signals.[i]
        ffoldi iters (fun (first, both) _ ->
            let f = next()
            match convolute s f with
            | [|0; 0|] -> first + 1, both + 1
            | [|0; _|] -> first + 1, both
            | _ -> first, both
        ) (firstzero, bothzero)
    ) (0, 0)
printfn "firstzero %i" firstzero
printfn "bothzero %i" bothzero
```
[Answer]
## Ruby
**Ruby (2.1.0) 0.277s**  
**Ruby (2.1.1) 0.281s**  
**Python (Alistair Buxton) 0.330s**  
**Python (alemi) 0.097s**
```
n = 6
iters = 1000
first_zero = 0
both_zero = 0
choices = [-1, 0, 0, 1].repeated_permutation(n).select{|v| [0] != v.uniq}
def convolve(v1, v2)
  [0, 1].map do |i|
    r = 0
    6.times do |j|
      r += v1[i+j] * v2[j]
    end
    r
  end
end
[-1, 1].repeated_permutation(n+1) do |s|
  iters.times do
    f = choices.sample
    fs = convolve s, f
    if 0 == fs[0]
      first_zero += 1
      if 0 == fs[1]
        both_zero += 1
      end
    end
  end
end
puts 'firstzero %i' % first_zero
puts 'bothzero %i' % both_zero
```
[Answer]
# *thread wouldnt be complete without* PHP
## 6.6x faster
### PHP v5.5.9 - ~~1.223~~ 0.646 sec;
### vs
### Python v2.7.6 - 8.072 sec
```
<?php
$n = 6;
$iters = 1000;
$firstzero = 0;
$bothzero = 0;
$x=time();
$y=34353;
$z=57768;
$w=1564; //PRNG seeds
function myRand() {
    global $x;
    global $y;
    global $z;
    global $w;
    $t = $x ^ ($x << 11);
    $x = $y; $y = $z; $z = $w;
    return $w = $w ^ ($w >> 19) ^ $t ^ ($t >> 8);
}
function array_cartesian() {
    $_ = func_get_args();
    if (count($_) == 0)
        return array();
    $a = array_shift($_);
    if (count($_) == 0)
        $c = array(array());
    else
        $c = call_user_func_array(__FUNCTION__, $_);
    $r = array();
    foreach($a as $v)
        foreach($c as $p)
            $r[] = array_merge(array($v), $p);
    return $r;
}
function rand_array($a, $n)
{
    $r = array();
    for($i = 0; $i < $n; $i++)
        $r[] = $a[myRand()%count($a)];
    return $r;
}
function convolve($a, $b)
{
    // slows down
    /*if(count($a) < count($b))
        return convolve($b,$a);*/
    $result = array();
    $w = count($a) - count($b) + 1;
    for($i = 0; $i < $w; $i++){
        $r = 0;
        for($k = 0; $k < count($b); $k++)
            $r += $b[$k] * $a[$i + $k];
        $result[] = $r;
    }
    return $result;
}
$cross = call_user_func_array('array_cartesian',array_fill(0,$n+1,array(-1,1)));
foreach($cross as $S)
    for($i = 0; $i < $iters; $i++){
        while(true)
        {
            $F = rand_array(array(-1,0,0,1), $n);
            if(in_array(-1, $F) || in_array(1, $F))
                break;
        }
        $FS = convolve($S, $F);
        if(0==$FS[0]) $firstzero += 1;
        if(0==$FS[0] && 0==$FS[1]) $bothzero += 1;
    }
echo "firstzero $firstzero\n";
echo "bothzero $bothzero\n";
```
* Used a custom random generator (stolen from C answer), PHP one sucks and numbers dont match
* `convolve` function simplified a bit to be more fast
* Checking for array-with-zeros-only is very optimized too (see `$F` and `$FS` checkings).
Outputs:
```
$ time python num.py 
firstzero 27050
bothzero 11990
real    0m8.072s
user    0m8.037s
sys 0m0.024s
$ time php num.php
firstzero 27407
bothzero 12216
real    0m1.223s
user    0m1.210s
sys 0m0.012s
```
Edit. Second version of script works for for just `0.646 sec`:
```
<?php
$n = 6;
$iters = 1000;
$firstzero = 0;
$bothzero = 0;
$x=time();
$y=34353;
$z=57768;
$w=1564; //PRNG seeds
function myRand() {
    global $x;
    global $y;
    global $z;
    global $w;
    $t = $x ^ ($x << 11);
    $x = $y; $y = $z; $z = $w;
    return $w = $w ^ ($w >> 19) ^ $t ^ ($t >> 8);
}
function array_cartesian() {
    $_ = func_get_args();
    if (count($_) == 0)
        return array();
    $a = array_shift($_);
    if (count($_) == 0)
        $c = array(array());
    else
        $c = call_user_func_array(__FUNCTION__, $_);
    $r = array();
    foreach($a as $v)
        foreach($c as $p)
            $r[] = array_merge(array($v), $p);
    return $r;
}
function convolve($a, $b)
{
    // slows down
    /*if(count($a) < count($b))
        return convolve($b,$a);*/
    $result = array();
    $w = count($a) - count($b) + 1;
    for($i = 0; $i < $w; $i++){
        $r = 0;
        for($k = 0; $k < count($b); $k++)
            $r += $b[$k] * $a[$i + $k];
        $result[] = $r;
    }
    return $result;
}
$cross = call_user_func_array('array_cartesian',array_fill(0,$n+1,array(-1,1)));
$choices = call_user_func_array('array_cartesian',array_fill(0,$n,array(-1,0,0,1)));
foreach($cross as $S)
    for($i = 0; $i < $iters; $i++){
        while(true)
        {
            $F = $choices[myRand()%count($choices)];
            if(in_array(-1, $F) || in_array(1, $F))
                break;
        }
        $FS = convolve($S, $F);
        if(0==$FS[0]){
            $firstzero += 1;
            if(0==$FS[1])
                $bothzero += 1;
        }
    }
echo "firstzero $firstzero\n";
echo "bothzero $bothzero\n";
```
[Answer]
# Q, 0.296 seg
```
n:6; iter:1000  /parametrization (constants)
c:n#0           /auxiliar constant (sequence 0 0.. 0 (n))
A:B:();         /A and B accumulates results of inner product (firstresult, secondresult)
/S=sequence with all arrays of length n+1 with values -1 and 1
S:+(2**m)#/:{,/x#/:-1 1}'m:|n(2*)\1 
f:{do[iter; F:c; while[F~c; F:n?-1 0 0 1]; A,:+/F*-1_x; B,:+/F*1_x];} /hard work
f'S               /map(S,f)
N:~A; +/'(N;N&~B) / ~A is not A (or A=0) ->bitmap.  +/ is sum (population over a bitmap)
                  / +/'(N;N&~B) = count firstResult=0, count firstResult=0 and secondResult=0
```
Q is a collection oriented language (kx.com)
Code rewrited to explote idiomatic Q, but no other clever optimizations
Scripting languages optimize programmer time, not execution time
* Q is not the best tool for this problem
First coding attempt = not a winner, but reasonable time (approx. 30x speedup)
* quite competitive among interpreters
* stop and choose another problem
NOTES.- 
* program uses default seed (repeteable execs) To choose another
seed for random generator use `\S seed`
* Result is given as a squence of two ints, so there is a final i-suffix at second value 
27421 12133i -> read as (27241, 12133)
* Time not counting interpreter startup. `\t sentence` mesures time
consumed by that sentence
[Answer]
# Rust, 6.6 ms, 1950x speedup
Pretty much a direct translation of [Alistair Buxton's code](https://codegolf.stackexchange.com/a/26337/3103) to Rust. I considered making use of [multiple cores with rayon](https://crates.io/crates/rayon) (fearless concurrency!), but this didn't improve the performance, probably because it's very fast already.
```
extern crate itertools;
extern crate rand;
extern crate time;
use itertools::Itertools;
use rand::{prelude::*, prng::XorShiftRng};
use std::iter;
use time::precise_time_ns;
fn main() {
    let start = precise_time_ns();
    let n = 6;
    let iters = 1000;
    let mut first_zero = 0;
    let mut both_zero = 0;
    let choices_f: Vec<Vec<i8>> = iter::repeat([-1, 0, 0, 1].iter().cloned())
        .take(n)
        .multi_cartesian_product()
        .filter(|i| i.iter().any(|&x| x != 0))
        .collect();
    // xorshift RNG is faster than default algorithm designed for security
    // rather than performance.
    let mut rng = XorShiftRng::from_entropy(); 
    for s in iter::repeat(&[-1, 1]).take(n + 1).multi_cartesian_product() {
        for _ in 0..iters {
            let f = rng.choose(&choices_f).unwrap();
            if f.iter()
                .zip(&s[..s.len() - 1])
                .map(|(a, &b)| a * b)
                .sum::<i8>() == 0
            {
                first_zero += 1;
                if f.iter().zip(&s[1..]).map(|(a, &b)| a * b).sum::<i8>() == 0 {
                    both_zero += 1;
                }
            }
        }
    }
    println!("first_zero = {}\nboth_zero = {}", first_zero, both_zero);
    println!("runtime {} ns", precise_time_ns() - start);
}
```
And Cargo.toml, as I use external dependencies:
```
[package]
name = "how_slow_is_python"
version = "0.1.0"
[dependencies]
itertools = "0.7.8"
rand = "0.5.3"
time = "0.1.40"
```
Speed comparison:
```
$ time python2 py.py
firstzero: 27478
bothzero: 12246
12.80user 0.02system 0:12.90elapsed 99%CPU (0avgtext+0avgdata 23328maxresident)k
0inputs+0outputs (0major+3544minor)pagefaults 0swaps
$ time target/release/how_slow_is_python
first_zero = 27359
both_zero = 12162
runtime 6625608 ns
0.00user 0.00system 0:00.00elapsed 100%CPU (0avgtext+0avgdata 2784maxresident)k
0inputs+0outputs (0major+189minor)pagefaults 0swaps
```
6625608 ns is about 6.6 ms. This means 1950 times speedup. There are many optimizations possible here, but I was going for readability rather than performance. One possible optimization would be use arrays instead of vectors for storing choices, as they will always have `n` elements. It's also possible to use RNG other than XorShift, as while Xorshift is faster than the default HC-128 CSPRNG, it's slowest than naivest of PRNG algorithms.
[Answer]
# Julia: ~~12.149~~ 6.929 s
Despite their [claims to speed](http://julialang.org/benchmarks/), the [initial JIT compilation time](https://stackoverflow.com/a/10711762/1642693) holds us back!
Note that the following Julia code is effectively a direct translation of the original Python code (no optimisations made) as a demonstration that you can easily transfer your programming experience to a faster language ;)
```
require("Iterators")
n = 6
iters = 1000
firstzero = 0
bothzero = 0
for S in Iterators.product(fill([-1,1], n+1)...)
    for i = 1:iters
        F = [[-1 0 0 1][rand(1:4)] for _ = 1:n]
        while all((x) -> round(x,8) == 0, F)
            F = [[-1 0 0 1][rand(1:4)] for _ = 1:n]
        end
        FS = conv(F, [S...])
        if round(FS[1],8) == 0
            firstzero += 1
        end
        if all((x) -> round(x,8) == 0, FS)
            bothzero += 1
        end
    end
end
println("firstzero ", firstzero)
println("bothzero ", bothzero)
```
# Edit
Running with `n = 8` takes 32.935 s. Considering that the complexity of this algorithm is `O(2^n)`, then `4 * (12.149 - C) = (32.935 - C)`, where `C` is a constant representing the JIT compilation time. Solving for `C` we find that `C = 5.2203`, suggesting that actual execution time for `n = 6` is 6.929 s.
]