Log(1+x)
========

Author: Momchil Ivanov <momchil@xaxo.eu>
Date  : 2014.11.05

Introduction
------------

  This article contains notes on the numerical stability of
  log(1+x). There are claims that directly computing log(1+x) is
  inaccurate for small x when 1+x=1 in machine arithmetic.

Method 1
--------

  Trivial method to compute:

    z=1.0+x
    y=log(z)

Method 2
--------

  Claims to cancel errors with IEEE arithmetic [2]:

    z=1.0+x
    y=log(z)-((z-1.0)-x)/z

Method 3
--------

  Uses Taylor expansion for small x [1]:

  if (abs(x) > 1e-4)
    y=log(1.0+x)
  else
    y=(1.0-0.5*x)*x

References
----------

  1: http://www.johndcook.com/cpp_log_one_plus_x.html
  2: http://www.davidgsimpson.com/software/invhyp_f90.txt