vectorize pi

In this exercises you will vectorize the computation of pi using the arc integral

1. start from the code used in the OpenMP course or from http://goo.gl/zI3WqS or use the code below

    #include <limits>
    #include <algorithm>
    #include <type_traits>
    #include <cstring>
    #include <x86intrin.h>
       
    template<typename Float>
    float pi(int num_steps) {
      if(num_steps <=0) __builtin_unreachable();
      Float step =  Float(1.0)/(Float) num_steps;
      Float sum = 0;
      // num_steps = 4*(num_steps/4);  // make sure is a multiple of 4
      for (int i=0;i< num_steps; i++){
        auto x = (Float(i)+Float(0.5))*step;
        sum += Float(4.0)/(Float(1.0)+x*x);
      }
      return step * sum;
    }
       
    #include<iostream>
    #include <chrono>
       
    template<typename T>
    void go(int num_steps) {
      auto start = std::chrono::high_resolution_clock::now();
      auto res = pi<T>(num_steps);
      auto total_time = std::chrono::high_resolution_clock::now() -start;
      std::cout << "pi = " << res << " in " << total_time.count() << std::endl;
    }
       
    int main ()
    {
   
      auto total_time = std::chrono::high_resolution_clock::duration{};
     
      constexpr int num_steps = 1000000;
         
      go<float>(num_steps);
      go<double>(num_steps);
       
      return 0;
    }
   

2. compile with c++ -O2 pi.cpp -fopt-info-vec -march=native
3. compile with -Ofast (try also -funroll-loops) (why O3 will not vectorize?)

4. try to vectorize it by yourself using native-vectors

    typedef float __attribute__( ( vector_size( 16 ) ) ) float32x4_t;
    constexpr float32x4_t zero4= {0.f,0.f,0.f,0.f};
   
    template<>
    float pi<float32x4_t>(int num_steps) {
       // fill in
    }