Oops, correction to the car analogy: The rubber sheet is 6000 miles long when the car starts driving, is 46.6 billion miles long when the car stops driving, and the car itself thinks it has driven 13.7 billion miles relative to the sheet. (If space behind you expands, you don't have to redrive that extra distance.) Replace miles with light-years and we have a description of the photon traveling from this star to the Milky Way galaxy: they started 6000 light-years apart, they ended 46.6 billion light-years apart, and the photon thinks it spent 13.7 billion years in flight because it traveled 13.7 billion light-years with respect to the background of expanding space.

The phrasing is a bit misleading. The star was 6000 light-years away when it first emitted the light, but the empty space between stars (and galaxies, now) has been expanding continuously (but at a nonconstant rate) since then. Imagine a car driving along a rubber sheet that's stretching. The sheet *starts* 6000 miles long and the car drives at one mile per hour, but since the sheet is growing as the car drives along it's 13.7 billion years long by the time the car reaches the other end.

So we're seeing the star as it looked 13.7 billion years, ago, not as it looked 6000 years ago, the current physical distance between us and the star is actually a whopping 46.6 billion light-years, and the 6000 light-year number corresponds to the distance between us and the star when the light was first emitted. (The universe was much smaller then.)

Note that this goofy universe-expansion correction factor doesn't apply to such short distances nowadays. The center of the Milky Way galaxy is 30000 light-years away, but since space itself isn't expanding so rapidly today, we see the center of the galaxy the way it looked about 30000 light-years ago, as expected. The most ridiculous rates of expansion took place shortly after the Big Bang.

"If I asked you to divide 500 by 21 and I asked you whether the answer is greater than one, you would say yes right away," Raghunathan said. "You are doing division but not to the full accuracy. If I asked you whether it is greater than 30, you would probably take a little longer, but if I ask you if it's greater than 23, you might have to think even harder. The application context dictates different levels of effort, and humans are capable of this scalable approach, but computer software and hardware are not like that. They often compute to the same level of accuracy all the time."

To determine if a/b is greater than 1, it is sufficient to check if a > b.

To determine if a/b is greater than c, it is sufficient to check if a > bc.

Multiplication already consumes less time and energy than division on modern computers. I do not see why they needed to modify their instruction set to realize such gains.

Not quite. IEEE754 mandates correct rounding for addition, subtraction, mutiplication, division, and square roots, but it only requires "faithful rounding" for the transcendental functions like sin, cos, and exp. That means that, for example, that even the floating point number nearest arcsin(1) is above it (i.e. correct rounding in this case requires that you round up), a math library that rounds arcsin(1) *down* is still compliant. The only requirement is that it round ot one of the two nearest floating point numbers.

While that's true in many cases, there are some situations in which we need . Read Shewchuk's excellent paper on the subject.

The easiest way to think about it is with a made-up problem about sorting. Let's say that you have a list of mathematical expressions like sin(pi*e^2), sqrt(14*pi*ln(8)), tan(10/13), etc and you want to sort them, but some numbers in the list are so close to each other that they might compare differently on different computers that round differently, (e.g. one computer says that sin(-10) is greater than ln(100)-ln(58) and the other says it's less).

Imagine now that this list has billions of elements and you're trying to sort the items using some sort of distributed algorithm. For the sorting to work properly, you *need* to be sure that a < b implies that b > a. There are situations (often in computational geometry) where it's OK if you get the wrong answer for borderline cases (e.g. it doesn't matter whether you can tell whether sin(-10) is bigger than ln(100)-ln(58) because they're close enough for graphics purposes) as long as you get the wrong answer consistently, so the next algorithm out (sorting in my example, or triangulation in Shewchuk's) doesn't get stuck in infinite loops.

Mathematica in particular uses adaptive precision; if you ask it to compute some quantity to fifty decimal places, it will do so.

In general, if you want bit-for-bit reproducible calculations to arbitrary precision, the MPFR library may be right for you. It computes correctly-rounded special functions to arbitrary accuracy. If you write a program that calls MPFR routines, then even if your own approximations are not correctly-rounded, they will at least be reproducible.

If you want to do your calculations to machine precision, you can probably rely on C to behave reproducibly if you do two things: use a compiler flag like -mpc64 on GCC to force the elementary floating point operations (addition, subtraction, multiplication, division, and square root) to behave predictably, and use a correctly-rounded floating point library like crlibm (Sun also released a version of this at one point) to make the transcendental functions behave predictably.

Whoa, now. While it's true that the NSA has a history of disregarding the law, it's bad to fall into the trap of believing that there's no point to creating such laws at all.

What do you want Congressman Holt do? Rip off his shirt and physically attack James Clapper? That's not going to help curtail the powers of the NSA and you know it. Congress creates laws. That's what they're supposed to do. If you think the law is a good idea, then proposing the law isn't "pointless posturing," it's Congress' job.

It's easy to get so lost in cynicism that you stop believing that forward progress is possible. But it's an ugly fact that many of the NSA's recent activities have had explicit Congressional approval. Revoking that approval is an essential step to fixing the situation, and Congressman Holt should be applauded for attempting to do so.

I hate to troll what could be a pleasant discussion about a new release of a popular web browser, but Chrome sends lots of extra data to Google and Google sends lots of extra data to the NSA. I therefore do not trust Chrome.

I understand that this is irrelevant to the current topic. If I don't want to use Chrome, I should just use another web browser, right? Like how I didn't want to use GMail, so I got a Lavabit account in 2009. (Seriously!) But Lavabit is gone now. And when I updated my friends with my new email address, I sullenly noted that well over half of them use GMail because the convenience outweighs the loss of privacy.

So... I'm tired and afraid. Google is integrating more features into the browser that are based on the "recency of the websites visited"? That information will almost certainly be radioed back home to Google. And when the NSA is finally forced to answer for their sins before Congress, they'll point to how many people downloaded Google Chrome 29 and they'll say, "People don't mind being tracked! They sign up for this stuff voluntarily!" Regardless of whether that's a good reason for the NSA to be able to read my email, it's going to come up repeatedly in the inquiry, and that one fact might be enough to allow the program continue.

I have no choice but to see the existence of Google Chrome 29 as a threat to my freedom. I wish I could be more reasonable about it, or at least keep my stupid opinions to myself, but being reasonable about Snowden wasn't enough to save my email account.

*sigh*

I want my email back. Show me a plan that restores my Lavabit access and I'll take this effort seriously. This isn't a game.

I've been modded up, which is fantastic, but to be honest I was hoping to provoke discussion.

What I was thinking when I wrote the above post (and was more sober) was that this issue is affecting regular people. I'm a real person! I live in an apartment in midtown Atlanta! I have a trilobite collection and I like to take long walks. I'm preparing dinner for some friends tomorrow evening. I'm a savvy Internet user, like the rest of you, who reflexively discounts conspiracy theories. But my email provider was just taken down because it provided too much privacy.

Godwin's Law prevents me from typing many of my thoughts right now. I know from experience that everyone's weary from constant political combat. I was even in Washington D.C. (well, Richmond; the DC subways weren't working that day) for the Stewart/Colbert Rally to Restore Sanity, which emphasized the importance of getting along with your neighbors, even if they disagreed with your politics. And besides, I didn't do anything about the Snowden revelation, even though I consciously understood it was a Bad Thing.

But here I am today with no email because (we assume) the Federal government presented Ladar Levison with an ultimatum: either break his own security and tell nobody, or stop providing the service altogether. (Fortunately Levison did what I paid him to do: he stood up for my privacy and let Lavabit go down.) But we can't be sure that this is what transpired because of Levison has been under a gag order for six weeks. This is a terrifying concept.

Has anyone on Slashdot watched Babylon 5? It had a long story arc in which the Earth government gradually became more and more repressive. There was an episode in the middle of Season 3 called "Severed Dreams" in which the Interstellar News Network (ISN) was forcibly brought offline by the military. Right before they went off the air, an anchor came onscreen, apologized for interrupting, reported that some colonies had declared independence and that the president didn't want that information let out, and that many things had been going on for a year that ISN was not allowed to report. Some explosions damaged the building, ISN went offline, and a week later, it came back with a new (completely unfamiliar) news anchor who calmly explained that terrorists had faked the previous broadcast.

That's how I feel about Lavabit right now. I've been watching the Snowden news for months. Then my email went down... And then suddenly I'm hit with this speech that for the last six weeks, Lavabit's founder has been fighting to protect my privacy while under a gag order, and twice has tried to get that order lifted. But he failed and now I have to go change my email address everywhere it's used. Wow! I never imagined that the drama on the news, where the United States tried to promise Russia that we wouldn't torture one of our own citizens were he extradited, would have a direct impact on my insignificant life! But it did. And apparently Lavabit has been fighting for the last six weeks, while I've been going to work, trying to talk pretty girls into dinner, and going jogging around the neighborhood. This is real! These issues aren't going away. I ignored them and I lost my email account. What will I lose next?

Please, Slashdot. Please, please, please take this seriously. This isn't just another petty Internet squabble. This is serious business. I got caught in the crossfire early because I cared enough about my privacy to use Lavabit. Other people got caught in it earlier---Manning and Snowden because they had both moral courage and access to incriminating information, and probably many other people of Pakistani descent because that's just how things go. I got caught in it today. When will the rest of you get caught? GMail users are safe from shutdowns because even in 2009 we knew that Google didn't care about your privacy, but I wouldn't be surprised that the stakes will continue to increase as time wears on.

Maybe I deserve to be alone in this mess because I left Manning, Snowden, and probably untold others in the lurch when they needed support. Yes, I probably do...

My name is Anthony Coulter. I signed up for Lavabit on October 5, 2009 with the address anthonycoulter(at)lavabit.com. I chose Lavabit very consciously. My university email address was about to expire and I had concerns about Google's privacy policies. Lavabit was created specifically for privacy-conscious people. They offered server-side encryption to paying customers; when I became a paying customer a year or two later I decided to check that box because, hell, why not?

[Note that I never did ask how server-side encryption worked. They said that things were rigged up so that they could not decrypt my on-server email even if they were coerced into it. My guess was that they used a hash of your login password to decrypt your email. I didn't know whether it was true or not, but I didn't think it really mattered. Apparently it did matter.]

I use my Lavabit account for everything. My bank statements are mailed to it. Most of my internet login IDs created since 2009 depend on it. All of my friends use it. And now it's gone.

I last checked my email around 9pm on Tuesday, August 6. When I woke up the next morning my connection attempts to the Lavabit server timed out. That was inconvenient; I had to send some information to my parents about an upcoming family reunion, so I sent them a text message promising to email it to them when the service was restored Wednesday night. It wasn't; I finally sent the email from an old family account I used back in the late 1990s. When I woke up *this* morning and Lavabit was still down, I did a couple of Google searches to see if anyone else noticed that an email provider had been gone for twenty-four straight hours. I found this discussion, which I quote for the benefit of people who will read this post long after the forum has ceased to exist:

RobertPaulsen
Junior Member
Join Date: Aug 2013
has anybody considered that if edward snowden did use lavabit then the Gouvernment is maybe interested in his mails which he wrote and sended to Glenn Greenwald. Maybe they seized the server and waved with a national security letter. just a thought !

This was posted at 10:55pm last night; when I saw it this morning I instantly dismissed the poster as a childish Internet revolutionary. The idea that the Federal government would clog up Lavabit for an entire day and a half just to get at Snowden is silly! They can't disrupt business like that!

Then I ran another Google search for "lavabit down" before getting off work today, and... here we are. Emails sent to my lavabit account still don't get bounce warnings, so noone who's emailed me since 9pm on Tuesday will know that I didn't get their email, or that I never will. I also have to go through the long and tedious process of reassociating all of my Internet accounts with a new email address. But which provider will I choose? I still don't trust Google. I don't know what I'll do yet; it was only two hours ago when discovered that my four-year-old email address had been taken down by the Federal government.

I just donated two thousand dollars to Lavabit's legal defense fund. (The confirmation email from Paypal just arrived in my old Cox account.) I cannot prove this to the Internet, and it's debatably silly for someone so privacy-conscious to want to do so. But at some point we will have to take this issue seriously. I watched the Snowden news from a distance; I didn't say or do anything about it because it wasn't really my problem. Now I lost my email, and if I had used IMAP this would have been a tragedy of enormous proportions.

--Anthony Coulter, a.k.a. Red Jesus

--Salvor Hardin, paraphrased