Two numbers that should be equal are not, fractional part showing different values after 17th fractional digit

Hello everyone,

While working on a project I came accross a problem that I don't understand, so I would very much appreciate your help.

I've created a script to demonstrate the issue. The script, by itself, doesn't do anything meaningful, it's just taken out of context to reproduce the problem.

font_size = 72;
d = 0.01;

[h,w,hm,wm,cd,sp,ld] = genTLparams(font_size);

hf = sp;
vf = sp + h;

fs = 's';

fs_found = [];

fs_sp = containers.Map;
fs_sp(fs) = cell(0);
fs_ep = containers.Map;
fs_ep(fs) = cell(0);

hf = hf + w + cd;

fs_found = [fs_found fs];
fs_sp(fs) = [fs_sp(fs) {{hf vf}}];

hf = hf + sp;

hfsp = fs_sp(fs);
hfsp = hfsp{1,end}{1,1};
vfsp = fs_sp(fs);
vfsp = vfsp{1,end}{1,2};

hfsp == hf-sp

function [h,w,hm,wm,cd,sp,ld] = genTLparams(font_size)
p2mm = 0.3527777778; % point to milimeter
h = font_size*p2mm/1000;

w  = 6/14*h;
hm = 10/14*h;
wm = 6/14*h;
cd = 2/14*h;
sp = 8/14*h;
ld = 22/14*h;
end

And the output is:

ans =
  logical
   0

I'm expecting the hfsp and (hf-sp) to have the same value, but they end up being slightly different. Their values up to the 55th fraction are:

>> sprintf('%.55f%',hfsp)
ans =
    '0.0290285714304000033725294827036123024299740791320800781'

>> sprintf('%.55f%',hf-sp)
ans =
    '0.0290285714304000068419764346572264912538230419158935547'

Why does this happen? What could have caused such imprecision during calculation?

[James Tursa]

See this link:

https://www.mathworks.com/matlabcentral/answers/703427-why-is-0-3-0-1-0-2-0-1-0-2-0-3?s_tid=ta_ans_results

Answer 1

[Stephen23]

"Why does this happen? What could have caused such imprecision during calculation?"

Operations on binary floating point numbers accumulate floating point error. It is the task of the programmer to understand this and to write their code such that they assume all numeric operations will accumulate floating point error. Commonly this means not testing for exact equality, instead comparing the absolute difference against a tolerance:

abs(A-B)<tol

Read more:

https://www.mathworks.com/matlabcentral/answers/57444-faq-why-is-0-3-0-2-0-1-not-equal-to-zero

https://www.mathworks.com/matlabcentral/answers/316889-why-two-equal-numbers-are-not-equal

http://www.mathworks.com/matlabcentral/answers/102419-how-do-i-determine-if-the-error-in-my-answer-is-the-result-of-round-off-error-or-a-bug

http://www.mathworks.com/matlabcentral/answers/140656-about-numerical-difficulty-of-equality-comparison-operation-between-two-double-real-numbers-in-matlab

This is worth reading as well:

https://docs.oracle.com/cd/E19957-01/806-3568/ncg_goldberg.html

http://faculty.tarleton.edu/agapie/documents/cs_343_arch/papers/1991_Goldberg_FloatingPoint.pdf

[Aleksandar Ritan]

Thank you.