$f(x) = \sqrt{x^{2}+1}-1$ (Loss of Significance)

Let us say that I want to compute $ f(x) = \sqrt{x^{2}+1}-1$ for small values of $ x$ in a Marc-32 architecture. I can avoid loss of significance by rewriting the function

$ $ f(x)=\left(\sqrt{x^{2}+1}-1\right)\left(\frac{\sqrt{x^{2}+1}+1}{\sqrt{x^{2}+1}+1}\right)=\frac{x^{2}}{\sqrt{x^{2}+1}+1}$ $

Even though I can solve the problem, I do not know/understand why the solution has avoid the loss of significance?

What is the significance of monster subtypes?

A number of subtype tags are added after some monsters’ general cassification. With the exception of Shapechanger and a few other specific exceptions these tags are linked to a particular monster type

For example Fiends as classification sometimes have the subtypes Devil, Demon, and Yugoloth in parenthesis after their monster type in their stat blocks.

You can consider an imp, a quasit, a hellhound, and a mezzoloth. The imp is listed as Tiny fiend (devil). The quasit is listed as Tiny fiend (demon). The hellhound has no listed subtype and the mezzoloth has the Yugoloth subtype. There is also the unusual Gnoll Fang of Yeenoghu that is listed as Medium fiend (Gnoll).

This distinction is not afforded to every common language "subtype" of monster. Hags, for instance, have their own section in the Monster Manual but are only listed as Medium Fey or, in the case of Night Hags, Medium Fiend without any listed subtype.

The following tags are the subtypes I could find listed in monsters’ stat blocks

General: Shapechanger, Titan
Aberration: None
Beast: None
Celestial: None, (Titan)
Construct: None
Dragon: None
Elemental: None
Fey: None
Fiend: Demon, Devil, Yugoloth, (Gnoll)
Giant: None
Humanoid: Goblinoid, Elf, Gith, Gnoll, Gnome, Grimlock, Human, Kobold, Kua-toa, Lizardfolk, Merfolk, Orc, Quaggoth, Sahuagin, Thri-kreen, Troglodyte, Yuan-ti, Aarakockra, Dwarf, etc…
Monstrosity: None, (Titan, Yuan-ti)
Ooze: None
Plant: None
Undead: None

With the exception of "shapechanger" that is affected by a number of spells what is the significance of these tags? Does a particular subtype having a tag (such as devils and most humanoids) have any particular meaning that is different from a variety of monster not having a tag (such as hags and beholders)?

Significance of model in arithmetic coding

I am trying to understand the concept of arithmetic coding, i understand how the range is subdivided after each character is read from the string. But i am unable to understand why using a more accurate model compresses the string. Whether the model is static/adaptive either way the number of characters in the compressed string would be the same. What am i missing?

What’s the significance of different Laser Types in Starfinder

As the title says, What is up with all the different types of lasers, there’s Zenith, Parallax, Corona and others. But I can’t seem to find the significance of these subtypes of laser? I am assuming there are maybe certain armors or something that resist say a zenith laser, but not a corona laser?

using crtl+f doesn’t seem to be picking up any relevant info in my PDF. The index is of equal uselessness.

Any help on determining what all these types mean and what i should be looking for in arming myself with a laser weapon would be highly appreciated.

What is the significance of ML in Information Security?

This is not a technical question rather

  • I am seeking an advice about how ML and AI is going to take over the our information security tasks that are currently done manually.
  • How should I prepare myself future ready against such digital labour that is going to be cheap and more efficient.
  • Which other areas should I start exploring apart from just IS which would be beneficial in securing job.

How to deal with loss of significance in the case $f(x) = \sqrt{x+2} -\sqrt{x}$?

I would like to evaluate the expression $ f(x) = \sqrt{x+2} -\sqrt{x}$ with cases when $ x = 3.2 \times 10^{30}$ and $ x= 3.2 \times 10^{16}$ . I tried using N[Sqrt[x+2] - Sqrt[x], 100] and

ScientificForm[Sqrt[x+2] - Sqrt[x], 100], both yielding 0 as an output. How can I obtain the desired output?

I also tried the method in Making a calculation with high precision by applying .`100 as a suffix but my mathematica 12 doesn’t seem to be recognizing it.

what time complexity is more significance

my question: a certian algorithm executes n operations of 3 types, inesert delete and find. we know that n/10 of the operations are insert and the rest are delete and find. you are given 2 options of implementation to the algorithm. The first implementation implements insert at the worst case of theta(n) time complexity and the cost of every operation (including insert) is theta(1) amortized cost. The second implementation implemets insert at the worst case of theta(logn) worst case and every operation(including insert) at theta(logn) amortized cost. which implementation whould you recommend? why? thanks:)

Significance of integrally closed in an affinoid algebra

A Tate affinoid k-algebra is defined as a pair $ (R,R^+)$ where $ R$ is a Tate algebra and $ R^+$ is an open and integrally closed subring of $ R$ contained in the ring of powerbounded elements.

See for instance Defn 2.6 here.

What is the (geometric) significance of requiring $ R^+ $ to be integrally closed?

On a related note, what is the difference (geometrically) between total integral closure and integrally closed. It is surprisingly hard to find anything about total integral closures…