From: Michal Suchanek <hramrach@...>
Date: 2011-09-21T23:16:13+09:00
Subject: [ruby-core:39657] Re: [Ruby 1.9 - Bug #4576] Range#step miss the last value, if end-exclusive and has float number

On 21 September 2011 14:25, masa <masa16.tanaka@gmail.com> wrote:
> 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|*��
>

Well, if you can calculate the maximum error and minimum error then
you can get the range over which you need to check *every* value if it
exceeds the end of the range. The estimated (~expected ~average) error
is not useful in this case. Or you can iterate over the intersection
of the original range and error range again with smaller error.

Thanks

Michal