From: masa <masa16.tanaka@...>
Date: 2011-09-21T21:25:45+09:00
Subject: [ruby-core:39656] Re: [Ruby 1.9 - Bug #4576] Range#step miss the last value, if end-exclusive and has float number

I haven't explained the reason of the error estimation in
Range#step for Float;

      double n = (end - beg)/unit;
      double err = (fabs(beg) + fabs(end) + fabs(end-beg)) / fabs(unit) * epsilon;

The reason is as follows. (including unicode characters)
This is based on the theory of the error propagation;
    http://en.wikipedia.org/wiki/Propagation_of_uncertainty

If f(x,y,z) is given as a function of x, y, z,
��f (the error of f) can be estimated as:

    ��f^2 = |���f/���x|^2*��x^2 + |���f/���y|^2*��y^2 + |���f/���z|^2*��z^2

This is a kind of `statistical' error.  Instead, `maximum' error
can be expressed as:

    ��f = |���f/���x|*��x + |���f/���y|*��y + |���f/���z|*��z

I considered the latter is enough for this case.
Now, the target function here is:

    n = f(e,b,u) = (e-b)/u

The partial differentiations of f are:

    ���f/���e = 1/u
    ���f/���b = -1/u
    ���f/���u = -(e-b)/u^2

The errors of floating point values are estimated as:

    ��e = |e|*��
    ��b = |b|*��
    ��u = |u|*��

Finally, the error is derived as:

  ��n = |���n/���e|*��e + |���n/���b|*��b + |���n/���u|*��u
     = |1/u|*|e|*�� + |1/u|*|b|*�� + |(e-b)/u^2|*|u|*��
     = (|e| + |b| + |e-b|)/|u|*��

Masahiro Tanaka