# How does the mod function compensate for floating point round off?

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Joel Handy on 19 Sep 2022
Commented: Bobby Cheng on 20 Sep 2022
The documentation for the mod function states, "mod attempts to compensate for floating-point round-off effects to produce exact integer results when possible." What exactly does this mean?

Joel Handy on 20 Sep 2022
Answering my own question. I wrapped the mod function and generated c code. if the quotient is within eps * quotient of bieing an integer value, the mod function returns zero.
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John D'Errico on 20 Sep 2022
x = 1 + eps;
mod(x,1)
ans = 2.2204e-16
Does that suggest your answer may have a subtle flaw? There is no number repersentable in MATLAB between 1 and (1 + eps).
Bobby Cheng on 20 Sep 2022
Two things:
1) The compensation only happens when the second input is not of integer values. That is why mod(1+eps,1) returns eps. Otherwise it is what Joel said.
2) I can see there is room for improvement for MATLAB documentation to better clarify this.

### More Answers (3)

Matt J on 19 Sep 2022
Edited: Matt J on 19 Sep 2022
I suspect it is just meant to tell you that it doesn't do a naive computation like in modNaive below.
m=1000*pi; n=pi;
mod(m,n)
ans = 0
modNaive(m,n)
ans = 3.1416
It isn't something you should try to rely on, though. There are definitely cases where the result won't be an exact integer, even when it's clearly what would be ideal, e.g.,
mod(117*sqrt(1001)+1,sqrt(1001))-1
ans = -3.5527e-14
function out=modNaive(m,n)
out=m-floor(m/n)*n;
end
Matt J on 20 Sep 2022
Edited: Matt J on 20 Sep 2022
@Joel Handy It is additional logic, and we don't know authoritatively what it is. However, the only thing it could rationally be doing is assessing a tolerance on the error,
err=abs(m/n-round(m/n))
If err<tolerance, it is assumed that m/n is intended to be an integer and the output of mod defaults to zero. Otherwise, mod is computed as in modNaive.
Matt J on 20 Sep 2022
I 'm not sure I can agree, mod must be identical sequence of instructions based on IEEE 754 division and remainder. Any CPU architecture should give identical result.
That may apply to the operations executed within mod itself. However, the preceding computations that generated the inputs to mod may have different floating point errors affecting them. That would be enough to generate different results in the neighborhood of mod's discontinuities.

Bruno Luong on 20 Sep 2022
Edited: Bruno Luong on 20 Sep 2022
I try to replicate MATLAB mod ith this function, it seems working well for 2 examples, no warranty beside that.
x=(rand(1,1e6)-0.5)*(10*pi);
all(rmod(x,pi) == mod(x,pi))
ans = logical
1
x = (-1000:1000)*pi;
all(rmod(x,pi) == mod(x,pi))
ans = logical
1
function r = rmod(x, a)
k = round(x/a);
r = x - a*k;
r(abs(r) < eps(x)) = 0; % EDIT test probably non effective
r(r < 0) = r(r < 0) + a;
end

Bruno Luong on 20 Sep 2022
Second version, correction under stricter condition than in the first version
function r = rmod2(x, a)
k = round(x/a);
r = x - a*k;
r(abs(r) < eps(a)) = 0;
r(r < 0) = r(r < 0) + a;
end
Bruno Luong on 20 Sep 2022
I think my test with espsilon is actually useless, this seem to match mod, unless someone can come with a counter example
function r = rmod(x, a)
k = round(x/a);
r = x - a*k;
r(r < 0) = r(r < 0) + a;
end