Legendre Elliptic Integral Third Kind

   phi    -en sin(alpha)     actual           ellpi(phi,ak)
  0.26  -0.40   0.50       2.649530e-01       2.649535e-01
  1.05  -0.60   1.00       1.719510e+00       1.719512e+00
  1.57   0.00   0.50       1.685750e+00       1.685750e+00
  1.57  -0.90   0.97       1.246407e+01       1.246409e+01
  0.26  -1.00   0.26       2.681560e-01       2.681556e-01
  0.79  -0.06   0.50       8.138450e-01       8.138455e-01
  0.79  -0.62   0.50       9.211300e-01       9.211296e-01
  0.79  -1.25   0.50       1.132136e+00       1.132136e+00
  0.79   0.25   0.50       7.698720e-01       7.698724e-01
  1.57  -0.06   0.50       1.743055e+00       1.743055e+00
  1.57  -0.62   0.50       2.800989e+00       2.800989e+00
  1.57   0.25   0.50       1.501762e+00       1.501762e+00