Legendre Elliptic Integral First Kind

  phi sin(alpha)      actual           ellf(phi,ak)
  0.09   0.03       8.726660e-02       8.726659e-02
  0.09   0.50       8.729413e-02       8.729413e-02
  0.09   1.00       8.737730e-02       8.737730e-02
  0.52   0.03       5.236264e-01       5.236264e-01
  0.52   0.50       5.294286e-01       5.294286e-01
  0.52   1.00       5.492704e-01       5.492704e-01
  1.57   0.03       1.571275e+00       1.571275e+00
  1.57   0.50       1.685750e+00       1.685750e+00
  1.57   1.00       4.742717e+00       4.742727e+00