"""
GROWING CONTINUED FRACTIONS

A continued fraction with remainder looks
like [a,b,c,r], where r is a nonnegative
number less than one.  We can "grow"
this continued fraction by replacing
the remainder r by the pair

   int(r), 1/frac(r)

Here is an example, done by hand

  [1.4142135623730951]
  [1, 2.4142135623730945]
  [1, 2, 2.4142135623730985]
  [1, 2, 2, 2.4142135623730749]

Here is another example, done using
the code in "growcf.py" listed below.

  >>> from growcf import *
  >>> from math import *
  >>> [sqrt(13)]
  [3.6055512754639891]
  >>> growcf(_)
  [3, 1.6513878188659978]
  >>> growcf(_)
  [3, 1, 1.5351837584879953]
  >>> growcf(_)
  [3, 1, 1, 1.8685170918213339]

NOTE: shortencf below is the inverse of growcf:

  >>> shortencf([3, 1, 1, 1.8685170918213339])
  [3, 1, 1.5351837584879953]

"""

frac = lambda x: x - int(x)

growcf = lambda x: x[:-1] + [int(x[-1]), 1/frac(x[-1]) ]

shortencf = lambda x: x[:-2] + [(1 + x[-2]*x[-1])/float(x[-1])]

"""
NOTES:

  1.  If x = [a,b] and y = [c,d,e] are lists, then
      x + y = [a,b,c,d,e] is the list obtained by
      putting x and y together.

  2.  x[-1] is the last element of a list.  x[-2] is the
      next to the last element.

  3.  x[:-1] is the list consisting of everything but the
      last element.

"""