/*
     error estimator for repeated randomized experiments.
     every scientists should know how to do this...
     requires c++17 or above.

     Copyright (c) 2021,2022 zhaoyongwei<zhaoyongwei@ict.ac.cn>
     IPRC, ICT CAS. Visit https://yongwei.site/error-estimator

     Permission is hereby granted, free of charge, to any person obtaining a copy
     of this software and associated documentation files (the "Software"), to deal
     in the Software without restriction, including without limitation the rights
     to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
     copies of the Software, and to permit persons to whom the Software is
     furnished to do so, subject to the following conditions:
     
     The above copyright notice and this permission notice shall be included in all
     copies or substantial portions of the Software.
     
     THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
     IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
     FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
     AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
     LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
     OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
     SOFTWARE.
*/

// ====================================================================
//           contents before the DEMO rule is a header.
// ====================================================================
// #pragma once // <-- don't forget opening this in a header.
                //     compilers don't like this in the main file.

#define __STDCPP_WANT_MATH_SPEC_FUNCS__ 1 // for std::beta
#include <cmath>
#include <iterator>
#include <tuple>

namespace errest {

// gaussian distribution. works as an approximation in most cases
// due to CLT, but is over-optimistic, especially when samples are
// few.
template<class RealType = double>
struct gaussian {
  static RealType critical_score(RealType cl, size_t df) {

    // due to CLT, mean of samples approaches a normal distribution.
    // `erf` is used to estimate confidence level.
    // now we set a confidence level and try obtain corresponding
    // confidence interval, thus we need the inverse of `erf`,
    // i.e. `erfinv`.
    using std::log; using std::sqrt; using std::erf; using std::exp;

    // solve erfinv by newton-raphson since c++stl lacks it.
    auto erfinv = [](RealType x) {
      constexpr RealType k = 0.88622692545275801; // 0.5 * sqrt(pi)
      RealType step, y = [](RealType x) { // a rough first approx.
           RealType sign = (x < 0) ? -1.0 : 1.0;
           x = (1 - x)*(1 + x);
           RealType lnx = log(x);
           RealType t1 = RealType(4.330746750799873)+lnx/2;
           RealType t2 = 1 / RealType(0.147) * lnx;
           return sign * sqrt(-t1 + sqrt(t1 * t1 - t2));
      }(x);
      do {
        step = k * (erf(y) - x) / exp(-y * y);
        y -= step;
      } while (step); // converge until zero step.
                      // this maybe too accurate for our usecase?
                      // add a threshold or cache if too slow.
      return y;
    };
    // we hope that the "truth" E[mean] lands in the interval
    // [mean-x*sigma, mean+x*sigma] with a probability of
    //                conf_level = erf( x/sqrt(2) )
    // conversely,
    //                x = sqrt(2) * erfinv(conf_level)
    return sqrt(RealType(2)) * erfinv(cl);
    // to get interval, mult this multiplier on the standard error of mean.
  }
};

// student's t-distribution. yields an accurate estimation even for few
// samples, but is more computational expensive.
template<class RealType = double>
struct student {
  static RealType critical_score(RealType cl, size_t df) {
    
    // similar to gaussian, we need to evaluate ppf for critical score.
    // ppf is the inverse function of cdf, and pdf is the derivative
    // function of cdf. we find the inverse of cdf by newton-raphson.
    using std::beta; using std::sqrt; using std::pow; using std::lgamma;
    using std::exp; using std::log; using std::nextafter;

    auto pdf = [](RealType x, RealType df) {
      return pow(df / (df + x * x), (df + 1)/2) /
        (sqrt(df) * beta(df/2, RealType(0.5)));
    };

    auto cdf = [](RealType x, RealType df) {
      // ibeta, derived from https://codeplea.com/incomplete-beta-function-c
      // original code is under zlib License,
      // Copyright (c) 2016, 2017 Lewis Van Winkle
      // i believe this snippet should be relicensed due to thorough rework.
      auto ibeta = [](RealType a, RealType b, RealType x){
        const RealType lbeta_ab = lgamma(a)+lgamma(b)-lgamma(a+b);
        const RealType first = exp(log(x)*a+log(1-x)*b-lbeta_ab) / a;
        RealType f = 2, c = 2, d = 1, step; int m = 0;
        do {
            RealType o = -((a+m)*(a+b+m)*x)/((a+2*m)*(a+2*m+1));
            m++; RealType e = (m*(b-m)*x)/((a+2*m-1)*(a+2*m));
            RealType dt = 1 + o/d; d = 1 + e/dt;
            RealType ct = 1 + o/c; c = 1 + e/ct;
            f *= (step = (ct+e)/(dt+e));
        } while(step - 1.);
        return first * (f - 1);
      };
      RealType root = sqrt(x*x+df);
      return ibeta(df/2, df/2, (x + root)/(2*root));
    };

    cl = 1 - (1 - cl) / 2;
    // solve the root of cdf(x) - cl = 0.
    // first use newton-raphson, generally starting at 0 since cdf is
    // monotonically increasing. disallow overshoot. when overshoot
    // is detected, an inclusive interval is also determined, then
    // switch to bisection for the accurate root.
    RealType l = 0, r = 0, step;
    while((step = (cdf(r, df) - cl) / pdf(r, df)) < 0) {
      l = r; r -= step;
    }
    while (r > nextafter(l, r)) {
      RealType m = (l + r) / 2;
      (cdf(m, df) - cl < 0 ? l : r) = m;
    }
    return nextafter(l, r); // let cdf(return val) >= 0 always holds.
  }
};

template<template<class> class Distribution = gaussian, class RealType = double>
struct estimator {

  // std::tie(mean, error) = estimator::test(first, last, cl)
  //
  // perform error estimation over the range [`first`, `last`). 
  // the ground truth should be included in the interval
  // [`mean-error`, `mean+error`] with a probability of `cl`.
  template<class InputIt>
  static std::tuple<RealType,RealType> test
          (InputIt first, InputIt last, RealType cl = 0.95) {

    // sampling has 1 less degrees of freedom.
    auto n = distance(first, last);
    auto df = n - 1;

    // pairwise sum to reduce rounding error.
    using std::distance; using std::next; using std::sqrt;
    auto sum = [](InputIt first, InputIt last,
                         auto&& trans, auto&& sum)->RealType {
      if (distance(first, last) == 1) {
        return trans(*first);
      } else {
        auto mid = first + distance(first, last) / 2;
        return sum(first, mid, trans, sum) + sum(mid, last, trans, sum);
      }
    };

    auto trans_mean = [n](RealType x)->RealType {
      return x/n;
    };
    auto mean = sum(first, last, trans_mean, sum);
    auto trans_var = [mean, df](RealType x)->RealType {
      return (x - mean) * (x - mean) / df;
    };
    auto var = sum(first, last, trans_var, sum);

    // assuming samples are i.i.d., we estimate `mean` as
    //
    //       samples[i] ~ N(mean, var)
    //              sum = Sum(samples)
    //                  ~ N(n*mean, n*var)
    //             mean = sum/n
    //                  ~ N(mean, var/n)
    // the standard error of mean, namely `sigma`,
    //            sigma = sqrt(var/n).
    //
    // due to Levy-Lindeberg CLT, above estimation to `mean`
    // approximately holds for any sequences of random variables when
    // `m_iterations`->inf, as long as:
    //
    //   * with finite variance
    //   * independently distributed
    
    auto sigma = sqrt(var/n);

    // finally, confidence interval is (+-) `critical_score*sigma`.
    // here critical_score is dependent on the assumed distribution.
    // we provide `gaussian` and `student` here.
    //
    //  - `gaussian` works well for many (>30) samples due to CLT,
    //    but are over-optimistic when samples are few;
    //  - `student` is more conservative, depends on number of
    //    samples, but is much more computational expensive.
    
    return { mean, Distribution<RealType>::critical_score(cl, df) * sigma };
  }
};

using gaussian_estimator = estimator<gaussian>;
using student_estimator = estimator<student>;

};

// ====================================================================
//      DEMO: experiment with RNG, try to obtain its expectation.
// ====================================================================
#include <random>
#include <vector>
#include <iostream>
#include <iomanip>

int main(int argc, char** argv) {

  // the "truth" hides behind a normal distribution N(5,2).
  // after this demo converged, we expect the mean very close to 5.
  // we use student's t-distribution here, which works better for
  // small sets.
  // if you change the distribution to gaussian, you will likely
  // see an increased chance to fail the challenge. 
  std::random_device rd;
  std::mt19937 gen(rd());
  std::normal_distribution<double> dist{5,2}; // <-- the "truth"

  // demo configuration.
  constexpr double confidence_level = 0.95; // 95% confidence to pass the challenge.
  constexpr double error_tolerance = 0.10;  // stop below +-10% error, ~[4.5, 5.5].

  double mean, error;
  size_t iteration = 5; // experiments per test.
  std::vector<double> samples;

  while (1) {
    // draw 5 more samples then test.
    std::cout << "Single Experiment Results: ";
    for (size_t i = 0; i < iteration; i++) {
      double sample = dist(gen); // <-- perform a single experiment.
      samples.push_back(sample);
      if (i < 10)
        std::cout << sample << " ";
      else if (i < 11)
        std::cout << "...";
    }
    std::cout << std::endl;

    // test the confidence interval with our estimator.
    std::tie(mean, error) = errest::student_estimator::test(samples.begin(), samples.end(), confidence_level);
    // or with structured binding declaration [c++17]:
    // auto [mean, error] = errest::student_estimator::test(samples.begin(), samples.end(), confidence_level);

    std::cout << " * Error Estimator Report: "
      "collected " << samples.size() << " results, "
      "mean: " << mean << ", error: " << error << std::endl
      << std::endl;

    // stop if error is below 10%,
    if (error < mean * error_tolerance) break;

  }

  std::cout << "demo converged with " << (confidence_level * 100) << " percent confidence "
    "within " << (error_tolerance * 100) << " percent error range. "
    "is the \"truth\" 5 within the interval "
    "[" << mean-error << ", " << mean+error << "]?" << std::endl;
  return 0;

}

#include <iostream>
#include <fstream>
#include <cstring>
#include <random>
#include <map>
#include <Magick++.h>
#include <fcgio.h>

std::ifstream t10k_images("./t10k-images-idx3-ubyte", std::ios_base::binary | std::ios_base::in);
std::ifstream t10k_labels("./t10k-labels-idx1-ubyte", std::ios_base::binary | std::ios_base::in);
size_t count = 0;

enum {
  IMAGE_IN = 16,
  IMAGE_NUM = 5000,
  IMAGE_SIZE = 28,
  IMAGE_BYTE = IMAGE_SIZE * IMAGE_SIZE,
  LABEL_IN = 8,
};

std::multimap<int, size_t> categories;
std::random_device rd;
std::mt19937 mt(rd());

void init() {
  t10k_labels.seekg(LABEL_IN);
  for (size_t i = 0; i < IMAGE_NUM; i++) {
    unsigned char c;
    t10k_labels.read(reinterpret_cast<char*>(&c), 1);
    categories.insert({c, i * IMAGE_BYTE + IMAGE_IN});
  }
  std::ifstream fcounter("./fcounter.db", std::ios_base::binary | std::ios_base::in);
  fcounter.read(reinterpret_cast<char*>(&count), sizeof(count));
  fcounter.close();
}

void select(std::array<unsigned char, IMAGE_BYTE>& img, unsigned char c) {
  auto range = categories.equal_range(c);
  auto first = range.first; auto last = range.second;
  auto n = std::distance(first, last);
  std::uniform_int_distribution<> dist(0, n - 1);
  auto sk = std::next(first, dist(mt))->second;
  t10k_images.seekg(sk);
  t10k_images.read(reinterpret_cast<char*>(img.data()), IMAGE_BYTE);
}

void hit(std::ostream& os) {
  count++;
  std::ofstream fcounter("./fcounter.db", std::ios_base::binary | std::ios_base::out);
  fcounter.write(reinterpret_cast<char*>(&count), sizeof(count));
  fcounter.close();
  std::string str = std::to_string(count);
  if (str.length() < 6)
    str = std::string(6 - str.length(), '0') + str;
  size_t w = IMAGE_SIZE * str.length(), h = IMAGE_SIZE;
  std::vector<unsigned char> canvas(w*h, 0);
  size_t i = 0;
  for (auto&& c : str) {
    std::array<unsigned char, IMAGE_BYTE> img;
    select(img, c - '0');
    for (int y = 0; y < IMAGE_SIZE; y++) {
      std::memcpy(&canvas[y * w + i * IMAGE_SIZE], &img[y * IMAGE_SIZE], IMAGE_SIZE);
    }
    i++;
  }
  Magick::Image image(IMAGE_SIZE*str.length(), IMAGE_SIZE, "I", Magick::CharPixel, canvas.data());
  Magick::Blob blob;
  image.type(Magick::GrayscaleType);
  image.magick("PNG");
  image.write(&blob);
  os << "Content-Type: image/png\r\n";
  os << "Content-length: " << blob.length() << "\r\n\r\n";
  os.write(reinterpret_cast<const char*>(blob.data()), blob.length()) << std::flush;
}

int main() {
  FCGX_Request request;
  init();
  FCGX_Init();
  FCGX_InitRequest(&request, 0, 0);
  while (FCGX_Accept_r(&request) == 0) {
    fcgi_streambuf osbuf(request.out);
    std::ostream os(&osbuf);
    hit(os);
  }
  return 0;
}