[math-fun] π puzzle

How does this work? In[787]:= Nest[#/(Sqrt[1 - #] + Sqrt[1 + #]) &, 1`33, 98] 2^99 Out[787]= 3.141592653589793 (I confess not understanding where half the significance goes. Forcing Mathematica to retain 33 digits throughout only rescues three or four.) --rwg