حسابات دقيقة وسريعة لأرقام الفاصلة العائمة باستخدام مثال دالة الجيب. مقدمة وجزء 1

#include <iostream>
#include <iomanip>
#include <cmath>
#include <array>
#include <bitset>
#include <quadmath.h>
//      clang
//#include "/usr/lib/gcc/x86_64-linux-gnu/10/include/quadmath.h"
#include <numeric>
#include <limits>
#include <vector>

#include <boost/timer/timer.hpp>
#include <boost/math/special_functions/factorials.hpp>

namespace bm = boost::math;

using namespace std;

typedef union { uint32_t i[2]; double x; } mynumber;

array<double, 25> fc;

double sin_e1(double x) {
  double result = 0;
  int sign = 1;
  for(int i = 1; i < 25; i += 2) {
    result += sign * pow(x, i) / bm::unchecked_factorial<double>(i);
    sign = -sign;
  }
  return result;
}

double sin_e2(double x) {
  double result = 0;
  int sign = 1;
  double xx = x * x;
  double pw = x;
  double fti = 1.0;
  for(int i = 1; i < 25; i += 2) {
    fti /= i;
    result += sign * pw * fti;
    fti /= ( i + 1 );
    sign = -sign;
    pw  *= xx;
  }
  return result;
}

double sin_e3(double x) {
  double result = 0;
  for(int i = 25; i >= 1; i -= 2) {
    result += (((i - 1) % 4 == 0) ? 1 : -1 ) * pow(x, i) / bm::unchecked_factorial<double>(i);
  }
  return result;
}

double sin_e4(double x) {
  double xx = x * x;
  double res = fc[25];
  for(int i = 23; i >= 1; i -= 2) {
    res = fc[i] + xx * res;
  }
  return x * res;
}

double sin_e5(double x) {
  double xx = x * x;
  double res = fc[25];
  for(int i = 23; i >= 3; i -= 2) {
    res = fc[i] + xx * res;
  }
  return x + x * xx * res;
}

inline
double fsin(double x) {
  double result;
  asm ("fsin" :"=t" (result) : "0" (x));
  return result;
}

#define SIN(a) fsin(a)
//#define SIN(a) sin(a)
//#define SIN(a) sin_e5(a)
// ^^     . ^^

int main() {

  __uint128_t ft = 1;
  fc[1] = 1.0; //3 * 5;
  for(int i = 2; i < fc.size(); i++) {
    ft *= i;
    // factorial with sign for Taylor series
    fc[i] = (1.0 / ft) * (( (i - 2) % 4 < 2) ? -1 : 1);
  }
  vector<double> xv;
  xv.resize(8 * 2000000);
  //      0  M_PI/2
  for (int i = 0; i < xv.size(); i++) {
    //      .
    xv[i] = (M_PI / 2) * i / double(xv.size());
  }

  double res = 0;
  {
    boost::timer::auto_cpu_timer at;
    for(int i = 0; i < xv.size(); i++) {
      res += SIN(xv[i]);
    }
  }

  int co = 0, cn = 0;
  //      .
  __float128 avg = 0.0, div = 0.0;
  for(int i = 0; i < xv.size(); i++) {
    mynumber dold, dnew;
    dold.x = sin(xv[i]);
    dnew.x = SIN(xv[i]);
    __float128 q = sinq(xv[i]); // <= sinq  .
    __float128 dd = __float128(dnew.x) - q;
    //     .
    div += dd * dd;
    avg += dd;
    //  ,         .
    //   ,         .
    if( dold.i[0] != dnew.i[0] || dold.i[1] != dnew.i[1] ) {
      if( fabsq(q - dold.x) <= fabsq(q - dnew.x) )
        co++;
      else
        cn++;
    }
  }
  avg /= xv.size();
  div /= xv.size();

  cout << res << endl;

  //  ,           .
  cout << "Better libm: " <<  co << " / " << xv.size() << "(" << 100.0 * co / xv.size() << "%)" << endl;

  //  ,  ""         .
  cout << "Better new: " <<  cn << " / " << xv.size() << "(" << 100.0 * cn / xv.size() << "%)" << endl;

  //         .
  cout << "  Avg / std new: " << double(avg) << " / " << double(sqrtq( div - avg * avg )) << endl;
  return 0;
}

double sin_e2(double x) {
  double result = 0;
  int sign = 1;
  double xx = x * x;
  double pw = x;
  double fti = 1.0;
  for(int i = 1; i < 25; i += 2) {
    fti /= i;
    result += sign * pw * fti;
    fti /= ( i + 1 );
    sign = -sign;
    pw  *= xx;
  }
  return result;
}