From: Kurt Stephens <ks@...>
Date: 2011-09-22T13:40:07+09:00
Subject: [ruby-core:39668] Re: [Ruby 1.9 - Bug #4576] Range#step miss the last value, if end-exclusive and has float number

A basis that sums to unity reduces error, due to convex hull property:

int n; double min, max;
for ( int i = 0; i <= n; ++ i ) {
   double u    = (double) i / (double) n;
   double _u_1 = 1.0 - u;
   double v    = min * u1 + max * u;
}

-- KAS

On 9/21/11 9:16 AM, Michal Suchanek wrote:
> 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
>