Contradiction occurred trying to calculate the value of $\sqrt{2}+Re\Bigl(^{10}\hspace{-1mm}\left(-\sqrt{2}\right)\Bigr)$

Question

Today I was trying to calculate the value of $\sqrt{2}+Re\Bigl(^{10}\hspace{-1mm}\left(-\sqrt{2}\right)\Bigr)$ (i.e., $\sqrt{2}$ plus the real part of the tenth tetration of the base $-\sqrt{2}$) up to the $100$th significant figure.
Surprisingly, WolframAlpha returned the exact value $4.2758814860817447418784921638871352 \cdot 10^{-45}$, without the More digits option available. Then, I asked wether $\sqrt{2}+Re\Bigl(^{10}\hspace{-1mm}\left(-\sqrt{2}\right)\Bigr)=4.2758814860 \cdot 10^{-45}$ and also $\sqrt{2}+Re\Bigl(^{10}\hspace{-1mm}\left(-\sqrt{2}\right)\Bigr)=4.2758814861 \cdot 10^{-45}$, and the answer was "true" for both the entries, whereas it says "false" for $\sqrt{2}+Re\Bigl(^{10}\hspace{-1mm}\left(-\sqrt{2}\right)\Bigr)=4.27588148 \cdot 10^{-45}$ and also for $\sqrt{2}+Re\Bigl(^{10}\hspace{-1mm}\left(-\sqrt{2}\right)\Bigr)=4.275881481 \cdot 10^{-45}$.
Finally, I tried to ask it to calculate the value of $\sqrt{2}+Re\Bigl(^{10}\hspace{-1mm}\left(-\sqrt{2}\right)\Bigr)-4.275881486081744741878492163887 \cdot 10^{-45}$, and this time the answer was "indeterminate".

Now, I think this outcome is pretty uncommon, so I am just curious to know why such a contradiction occurs.

Answer 1

To allow to close the case I took the well received comments together.

a) I agree to the previous comment (= question has a non-mathematical aspect).

b) The only idea I have is that W/A has some automatic routine to decide for decimal precision in interactive dialogue for "good user experience" ;-) and switches to this or that precision depending on user input.

Using Pari/GP with internal precision 200 dec digits I get this:
4.2758814860817447418784921638871352 E10−45, \\ this is your value by W|A 4.2758814860817447418784921638871352 408515814618335003649498522824092577 28313290846812136221703432305 E-45 (100 digits displayed using Pari/GP)

---

Looking further and leaving away the addition of $\sqrt2$ we find that the $10$'th iterate under discussion is (already) very near a 9-periodic-point.
Iterating to heights $(10,19,28,37,...)$ converges quickly and does so towards the complex value $$ z_\infty = \small -1.414213562373095048801688724209698 07856967187110106658709493 \\ \small 4996112240314574971798000347580082307702 \\\small -2.29302767777696818611693848679464474833732772671198199282030 \\ \small 0591647340434697133855458987589449941309 E-44 \cdot î$$ (100 dec digits displayed)
The set of 9-periodic points containing this value is attracting under the iterated exponentiating and converges fairly quickly.