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#include <ciso646>
#include <cmath>
#include <ctime>
#include <iomanip>
#include <iostream>
#include <limits>
#include <sstream>
#include <string>
#include <vector>
using namespace std;
//
// The recursive way
long double an( long double x, unsigned n )
{
long double a0 = 1;
long double g0 = sqrt( 1 + x*x );
while (n)
{
a0 = 0.5 * (a0 + g0);
g0 = sqrt( a0*g0 );
}
return a0;
}
long double d( long double x, unsigned k, unsigned n )
{
if (k == 0) return an( x, n );
long double d0 = d( x, k1, n );
long double d1 = d( x, k1, n1 );
long double p = 1/pow( 2.0, 2 * k );
return (d0  (p * d1)) / (1  p);
}
long double arctan_r( long double x, unsigned n = 10 )
{
return x / d( x, n, n );
}
//
// Now to tweak it up fine
long double arctan( long double x, unsigned N = 10 )
{
if (N < 2) N = 1;
long double a0, g0; // ongoing calculation of a(n)
vector <long double> d( N+1 ); // results of d(k,n), where d[k] == d(k,n) for largest n
g0 = sqrt( 1 + (x * x) );
a0 = 0.5 * (1 + g0);
g0 = sqrt( a0 * g0 );
d[ 0 ] = a0; // d[0] == d(0,1) == a(1)
d[ 1 ] = (a0  0.25) / 0.75; // d[1] == d(1,1)
// for n = 2 to N
for (unsigned n = 1; n++ < N; )
{
// calc a(n)
a0 = 0.5 * (a0 + g0);
g0 = sqrt( a0 * g0 );
long double d0 = a0;
// for k = 1 to n
for (unsigned k = 0; k++ < n; )
{
// d0 = d(k1,n)
// d[k1] = d(k1,n1)
// p = 1/(2**(2*k))
// d(k,n) = (d0  (p * d[k1])) / (1  p)
long double p = 1 / pow( 2.0, 2*k );
long double d1 = (d0  (p * d[ k1 ])) / (1  p);
d[ k1 ] = d0;
d0 = d1;
}
d[ n ] = d0;
}
return x / d[ N ];
}
//
int main( int argc, char** argv )
{
double x;
int n = 8;
if (argc == 3)
{
string s( argv[ 2 ] );
istringstream ss( s );
ss >> n;
argc = 2;
}
if (argc == 2)
{
string s( argv[ 1 ] );
istringstream ss( s );
ss >> x;
if (!ss) argc = 1;
}
if (argc != 2)
{
cout << "calculate atan(x); input x> ";
cin >> x;
if (!cin) { cout << "fooey!\n"; return 1; }
cin >> n;
}
cout << setprecision( numeric_limits <long double> ::digits10 + 1 );
cout << "recursive = " << arctan_r( x, n ) << "\n";
cout << "iterative = " << arctan( x, n ) << "\n";
cout << "standard = " << atan( x ) << "\n";
cout << "test iterations? ";
unsigned long iterations;
{
string s;
getline( cin, s );
istringstream ss( s );
ss >> iterations;
if (ss and iterations)
{
clock_t time;
time = clock();
for (unsigned long n = 0; n < iterations; n++)
arctan( x, n );
time = clock()  time;
cout << "iterative * " << iterations << " = "
<< ((long double)time/CLOCKS_PER_SEC) << " seconds\n";
time = clock();
for (unsigned long n = 0; n < iterations; n++)
arctan_r( x, n );
time = clock()  time;
cout << "recursive * " << iterations << " = "
<< ((long double)time/CLOCKS_PER_SEC) << " seconds\n";
}
}
return 0;
}
