## Half precision floating point question — smallest non-zero number

There’s a floating point question that popped up and I’m confused about the solution. It states that

IEEE 754-2008 introduces half precision, which is a binary
floating-point representation that uses 16 bits: 1 sign bit, 5
exponent bits (with a bias of 15) and 10 significand bits. This format
uses the same rules for special numbers that IEEE754 uses. Considering
this half-precision floating point format, answer the following
questions: ….

What is the smallest positive non-zero number it can represent?

bias = 15
Binary representation is: $$0 , 00000 , 0000000001 = 2^{-14} * 2^{-10}=2^{-24}$$

I’ve understood the binary representation part, but how does it get to those exponents of 2??

## Padding Inside a Floating div

I am in the middle of creating a basic, plain jane, old school div layout. But for the life of me, I can’t figure out how to get padding in the content area and side menu without the template distorting with extra padding. I’ve been searching online for hours — and I’m finding ‘apparent’ solutions, but when I try to implement them, they don’t work. I now have a migraine. If anyone has a solution, I’d be rapt to know what it is.

Note: The template isn’t complete, per se. I’m just trying to get the basic layout/functionality right.

``````html {
height: 100%;
font-size: 1.8vw;
}
body {
height: 100%;
background-color: #EEEEEE;
font-family: arial;
}
.wrap {
width: 80%;
height: 100%;
background-color: #FFFFFF;
max-width: 1200px;
margin: 0px auto 0px auto;
border-left: 1px solid #999999;
border-right: 1px solid #999999;
}
margin-top: 10px;
}
.logo {
background-color: #FFFFFF;
float: left;
width: 30%;
height: 100px;
line-height: 100px;
text-align: center;
}
background-color: #FFFFFF;
float: right;
width: 70%;
text-align: center;
height: 100px;
line-height: 100px;
}
clear: both;
background-color: #FFFFFF;
}
.content-wrap {
background-color: yellow;
}
.content {
background-color: #CCCCCC;
float: left;
width: 70%;
}
.content-box {

}
.content-2 {
clear: both;
border-top: 1px solid #999999;
background-color: #FFFFFF;
}
.content-2-box {
}
background-color: #999999;
float: right;
width: 30%;
}

}
.foot {
clear: both;
background-color: #FFFFFF;
margin-top: 10px;
}
ul {
list-style-type: none;
margin: 0;
overflow: hidden;
background-color: green;
text-align: center;
}

li {
display: inline-block;
}

li a {
display: block;
color: white;
text-align: center;
text-decoration: none;
}

li a:hover {
background-color: #111111;
}
}
/* Responsive */
@media screen and (max-width: 600px) {
html {
font-size: 2.5vw;
}
.logo {
float: none;
width: 100%;
text-align: center;
}
float: none;
width: 100%;
}
.content {
float: none;
width: 100%;
}
float: none;
width: 100%;
}
}``````

Code (CSS):

``````
<!DOCTYPE html>
<html xmlns="http://www.w3.org/1999/xhtml">
<meta charset="utf-8" />
<title></title>
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
<body>
<div class="wrap">
<div class="logo">
LOGO
</div>
</div>
</div>
<ul>
<li>
<a href="#">Home</a>
</li>
<li>
</li>
<li>
<a href="#">Contact</a>
</li>
<li>
<a href="#">Blog</a>
</li>
</ul>
</div>
<div class="content-wrap">
<div class="content content-box">
<p>This is some content.</p>
</div>
</div>
<div class="content-2 content-2-box">
Content 2
</div>
</div>
<div class="foot">
<div class="content content-box">
<p>Foot Left</p>
</div>
Foot Right
</div>
</div>
</div>
</body>
</html>

``````

HTML:

## usability – Is the Floating Side Panel / Icons affect the UX?

I have been asked to include a floating side Menu Icons. Here is an example:

I am not convinced about this idea and I think it is intrusive & distracting, so I want to have your opinion about that. Would implementing such thing would affect the UX in a negative way?

Thank you guys!

## bitcoind: why sendmany allows only double-precision floating point numbers?

I am switching from electrum to bitcoind. On `electrum` is function `paytomany` which accept usual bitcoin amounts like `0.0001`. But why method sendtomany accept only double-precision floating point numbers? Maybe there is another option for that purpose and I just missed it?

## dnd 3.5e – How does a Chameleon’s “floating” Bonus Feat interact with the various sacred vows from the Book of Exalted Deeds?

At 2nd level, the Chameleon prestige class gets a bonus feat. The description of the chameleon’s Bonus Feat ability states (Races of Destiny, p. 113):

Bonus Feat (Ex): At 2nd level, you gain a bonus feat. (…) At the start of each day, you can choose to change your bonus feat to any other feat for which you meet the prerequisites.

Assuming I have taken the Sacred Vow feat (Book of Exalted Deeds, p. 45) earlier in my career from another feat source, my character would then meet the prerequisites for other feats such as the “Vow of Poverty” or “Vow of Chastity” using this rotating feat slot.

These “Exalted” feats all have this nice little blurb on it (BoED, p. 47-48):

If you break your vow, you immediately and irrevocably lose the benefit of this feat. You may not take another feat to replace it.

Mechanically, if I were to swap out my exalted vows using the Bonus Feat ability from the Chameleon prestige class, I lost the vow without breaking the criteria of the oath. Or have I broken the vow by removing the feat voluntarily?

Is there an appropriate or articulate RAW interpretation of this ability interaction?
Or is this a “DM’s prerogative” type ruling?

I wonder if anyone knows why there is no panel for floating ip address in the Horizon dashboard?

here is my Horizon dashboard version:
3:18.3.2-0ubuntu0.20.04.4
I can create and assign using the openstack cli.

## computer architecture – Negative Numbers in 32 bit Floating Point IEEE Numbers

Contrary to two’s complement representation of negative integers, the negative numbers in IEEE floating-point are represented with only a sign bit change, as shown in there.

For example, $$0cdot01111100cdot01000000000000000000000$$ is the representation of $$0.15625$$, and $$1cdot01111100cdot01000000000000000000000$$ is the representation of $$-0.15625$$.

In particular, there are two representations of zero (the positive one and the negative one).

## Floating point bitwise comparator. If f1 and f2 are floating point numbers with the following properties can we always say f1 > f2?

Recall floating-point representation:

Suppose $$f$$ is a floating-point number then we can express f as,
If $$f$$ is normal:
$$(-1)^{s}cdot2^{e-127}(1 + sumlimits_{k=1}^{23} b_{23-k}cdot 2^{-k})$$
If $$f$$ is denormal/subnormal: $$(e = 0)$$
$$(-1)^{s}cdot2^{-126}(0 + sumlimits_{k=1}^{23} b_{23-k}cdot 2^{-k})$$

where

• $$s$$ is the sign of $$f$$.
• $$e$$ is the stored exponent of $$f$$. This means $$e-127$$ is the effective exponent of $$f$$.
• $$b_k$$ is the $$k$$-th bit of $$f$$ where $$b_0$$ is the LSB and $$b_{22}$$ is the MSB.

Let $$f_1$$ be a floating-point number with constants $$s_1, e_1,$$ and $$b_{1k}$$.

Let $$f_2$$ be a floating-point number with constants $$s_2, e_2,$$ and $$b_{2k}$$.

I’m trying to find out if the following statement is true: $$e_1 > e_2 > 0$$ and $$s_1 = s_1$$ then $$f_1 > f_2$$.

My initial strategy was to subtract $$f_2$$ from $$f_1$$ and show that it must be strictly greater than $$0$$.

What I have so far.

begin{align*} f_1-f_2 &= 2^{e_1-127}(1 + sumlimits_{k=1}^{23} b_{1,23-k}cdot 2^{-k}) – 2^{e_2-127}(1 + sumlimits_{k=1}^{23} b_{2,23-k}cdot 2^{-k})\ &= 2^{-127}(2^{e_1}(1 + sumlimits_{k=1}^{23} b_{1,23-k}cdot 2^{-k}) – 2^{e_2}(1 + sumlimits_{k=1}^{23} b_{2,23-k}cdot 2^{-k})) \ &= 2^{-127}(2^{e_1} + sumlimits_{k=1}^{23} b_{1,23-k}cdot 2^{e_1-k} – 2^{e_2} – sumlimits_{k=1}^{23} b_{2,23-k}cdot 2^{e_2-k})\ &=2^{-127}(2^{e_1} – 2^{e_2} + sumlimits_{k=1}^{23} b_{1,23-k}cdot 2^{e_1-k} – sumlimits_{k=1}^{23} b_{2,23-k}cdot 2^{e_2-k})\ &=2^{-127}(2^{e_1} – 2^{e_2} + sumlimits_{k=1}^{23}(b_{1,23-k}cdot 2^{e_1-k}-b_{2,23-k}cdot 2^{e_2-k})) end{align*}

Now since $$e_1 > e_2 iff e_1-e_2 > 0 iff 2^{e_1}-2^{e_2} > 1$$ we have that:

begin{align*} f_1-f_2 & > 2^{-127}(1 + sumlimits_{k=1}^{23}(b_{1,23-k}cdot 2^{e_1-k}-b_{2,23-k}cdot 2^{e_2-k})) end{align*}

In order for the above to always be greater than zero I require that $$sumlimits_{k=1}^{23}(b_{1,23-k}cdot 2^{e_1-k}-b_{2,23-k}cdot 2^{e_2-k}) > -1$$.

However, I don’t know how to formally show this. Any help/hints are much appreciated as inequalities are not my strong suit.