Euler's constant 0.5772...: Difference between revisions
C++ entry
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=={{header|C++}}==
<syntaxhighlight lang="cpp">#include <array>
#include <cmath>
#include <iomanip>
#include <iostream>
double ByVaccaSeries(int numTerms)
{
// this method is simple but converges slowly
// calculate gamma by:
// 1 * (1/2 - 1/3) +
// 2 * (1/4 - 1/5 + 1/6 - 1/7) +
// 3 * (1/8 - 1/9 + 1/10 - 1/11 + 1/12 - 1/13 + 1/14 - 1/15) +
// 4 * ( . . . ) +
// . . .
double gamma = 0;
size_t next = 4;
for(double numerator = 1; numerator < numTerms; ++numerator)
{
double delta = 0;
for(size_t denominator = next/2; denominator < next; denominator+=2)
{
// calculate terms two at a time
delta += 1.0/denominator - 1.0/(denominator + 1);
}
gamma += numerator * delta;
next *= 2;
}
return gamma;
}
// based on the C entry
double ByEulersMethod()
{
//Bernoulli numbers with even indices
const std::array<double, 8> B2 {1.0, 1.0/6, -1.0/30, 1.0/42, -1.0/30,
5.0/66, -691.0/2730, 7.0/6};
const int n = 10;
//n-th harmonic number
const double h = [] // immediately invoked lambda
{
double sum = 1;
for (int k = 2; k <= n; k++) { sum += 1.0 / k; }
return sum - log(n);
}();
//expansion C = -digamma(1)
double a = -1.0 / (2*n);
double r = 1;
for (int k = 1; k < ssize(B2); k++)
{
r *= n * n;
a += B2[k] / (2*k * r);
}
return h + a;
}
int main()
{
std::cout << std::setprecision(16) << "Vacca series: " << ByVaccaSeries(32);
std::cout << std::setprecision(16) << "\nEulers method: " << ByEulersMethod();
}
</syntaxhighlight>
{{out}}
<pre>
Vacca series: 0.5772156610598274
Eulers method: 0.5772156649015325
</pre>
=={{header|FreeBASIC}}==
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