code: """Compute a Pade approximation for the principle branch of the
  Lambert W function around 0 and compare it to various other
  approximations.
  
  """
  import numpy as np
  
  try:
      import mpmath  # type: ignore[import]
      import matplotlib.pyplot as plt
  except ImportError:
      pass
  
  
  def lambertw_pade():
      derivs = [mpmath.diff(mpmath.lambertw, 0, n=n) for n in range(6)]
      p, q = mpmath.pade(derivs, 3, 2)
      return p, q
  
  
  def main():
      print(__doc__)
      with mpmath.workdps(50):
          p, q = lambertw_pade()
          p, q = p[::-1], q[::-1]
          print("p = {}".format(p))
          print("q = {}".format(q))
  
      x, y = np.linspace(-1.5, 1.5, 75), np.linspace(-1.5, 1.5, 75)
      x, y = np.meshgrid(x, y)
      z = x + 1j*y
      lambertw_std = []
      for z0 in z.flatten():
          lambertw_std.append(complex(mpmath.lambertw(z0)))
      lambertw_std = np.array(lambertw_std).reshape(x.shape)
  
      fig, axes = plt.subplots(nrows=3, ncols=1)
      # Compare Pade approximation to true result
      p = np.array([float(p0) for p0 in p])
      q = np.array([float(q0) for q0 in q])
      pade_approx = np.polyval(p, z)/np.polyval(q, z)
      pade_err = abs(pade_approx - lambertw_std)
      axes[0].pcolormesh(x, y, pade_err)
      # Compare two terms of asymptotic series to true result
      asy_approx = np.log(z) - np.log(np.log(z))
      asy_err = abs(asy_approx - lambertw_std)
      axes[1].pcolormesh(x, y, asy_err)
      # Compare two terms of the series around the branch point to the
      # true result
      p = np.sqrt(2*(np.exp(1)*z + 1))
      series_approx = -1 + p - p**2/3
      series_err = abs(series_approx - lambertw_std)
      im = axes[2].pcolormesh(x, y, series_err)
  
      fig.colorbar(im, ax=axes.ravel().tolist())
      plt.show()
  
      fig, ax = plt.subplots(nrows=1, ncols=1)
      pade_better = pade_err < asy_err
      im = ax.pcolormesh(x, y, pade_better)
      t = np.linspace(-0.3, 0.3)
      ax.plot(-2.5*abs(t) - 0.2, t, 'r')
      fig.colorbar(im, ax=ax)
      plt.show()
  
  
  if __name__ == '__main__':
      main()
  
apis: numpy.meshgrid matplotlib.pyplot.subplots numpy.exp matplotlib.pyplot.show numpy.log numpy.array numpy.polyval numpy.linspace