# Function not evaluating to zero - numerical errors

I have a complex but analytical expression that should always evaluate to zero for certain values, but I'm getting strange numerical noise I think I can avoid. The problem seems to occur in a subsection involving logs. Here's the offending section in Mathematica code:

  rn = 250*10^-6; e = 0.75;
f = e*rn;

test1[r_, t_] := Log[(r^2 - (f^2) (Cos[t]^2) )/(rn^2  - f^2)];
t1out = Plot[test1[rn*Sqrt[1 - (e*Sin[t])^2], t], {t, 0, 2*Pi}]


Now, I know when $r = r_{n}\sqrt{1 - (e\sin(t))^2}$, then for any value of $t$,

$\frac{r^2 - f^2\cos^2 (t)}{r_{n}^2 - f^2} = \frac{r_{n}^2 - f^2(\sin^2(t) + \cos^2 (t))}{r_{n}^2 - f^2} = 1$

and so the log should always be zero. And for some values of $t$this holds - but for others, I get weird oscillating errors...

These are very small (order $10^{-16}$) but I think they're avoidable, especially as uncorrected they might mess up some other outputs I need. Is there a way to eradicate this numerical noise, given there's an analytic answer?

• Since you are working with machine numbers, this numerical noise is unavoidable. But you can wrap test1 in Chop for suppressing the noise.. – Fred Simons May 17 '18 at 11:56
• Though this question is a "simple mistake" as per the close reasons, machine numbers are lately one of my favorite topics. +1 – LLlAMnYP May 17 '18 at 19:19
• Thanks, very interesting! – DRG May 21 '18 at 13:57

It is a precision issue.

rn = 250*10^-6;
e = 3/4; (* use exact numbers for parameters *)
f = e*rn;

test1[r_, t_] := Log[(r^2 - (f^2) (Cos[t]^2))/(rn^2 - f^2)];


Use arbitrary precision rather than machine precision.

t1out = Plot[test1[rn*Sqrt[1 - (e*Sin[t])^2], t], {t, 0, 2*Pi},
WorkingPrecision -> 10]


EDIT: Alternatively, use Simplify and Evaluate

t1out = Plot[
test1[rn*Sqrt[1 - (e*Sin[t])^2], t] // Simplify // Evaluate, {t, 0, 2*Pi}]

• Cheers Bob - makes sense now! I figured it'd evaluate first but doing this forces it to do so! – DRG May 21 '18 at 13:57
• @DRG - If you evaluate Attributes[Plot] you will see that Plot has the attribute HoldAll so evaluation does not occur initially unless forced by use of Evaluate. – Bob Hanlon May 21 '18 at 14:25