There is an old epigram which assigns the empire of the sea to the English, of the land to the French, and of the clouds to the Germans. Surely it was from the clouds that the Germans fetched + and −; the ideas which these symbols have generated are much too important for the welfare of humanity to have come from the sea or from the land.
It is a safe rule to apply that, when a mathematical or philosophical author writes with a misty profundity, he is talking nonsense.Intel (and AMD) CPUs include math processors that don't just do simple arithmetic but also transcendental functions, e.g. sin, cos, exponential, ... . As they calculate sine and cosine with polynomial approximations, it's nice to know how accurate the methods are. For, like, forever now, Intel has been claiming their trig functions are really accurate. From a 1996 manual: where ulps means units in the last place, i.e. the least significant bit of the mantissa.
You know what's coming next, but wanna guess how bad it is? Hint: an Intel processor's idea of sin(x) can be off by quite a bit.
This is reminiscent of the infamous FDIV bug. It hasn't gotten as much publicity but compiler developers are wise to it and know to calculate sin by other means. Intel doesn't deem this a processor bug, in the sense that they don't fix it. But it kinda sucks that their manuals have been making a wildly false claim of accuracy for two decades now.
How far off does Intel's sin(x) get? For values of x close to π, the error is about one quintillion times the claimed worst case.
But let it not be said that Intel doesn't care. Just this past week, Intel said they'll fix their documentation.
A few missed words in the earlier editions of the "Intel® 64 and IA-32 Architectures Software Developer's Manual" that propagated to this day did not help.I love that phrasing: words that weren't there did not help. The US government restricts what communications providers can say about orders they've received for aid in surveillance. Twitter sued the US Department of Justice et al. today, saying that such restrictions are unconstitutional prior restraint of speech.
Google, Microsoft, Facebook, Yahoo!, and LinkedIn had previously filed court motions requesting permission to publish detailed reports. But they all struck a deal where they accepted only modest freedoms to publish (in ranges, e.g. "we received 0-999 orders"). Such rules implicitly forbid the publication of so-called warrant canaries: statements that exactly zero orders have been received, statements whose disappearance would be telling.
Good for Twitter that they didn't settle for that crumb the government threw out. Reading Twitter's complaint spurred me to put up my own warrant canary page, trivial and symbolic though it may be. A metaphorical explanation of a probability question:
The most probable suit distribution in bridge hands is 4-4-3-2, as compared to 4-3-3-3, which is the most evenly distributed. This is because the world likes to have unequal numbers: a thermodynamic effect saying things will not be in the state of lowest energy, but in the state of lowest disordered energy.1Where "suit distribution" is "hand pattern" as described in (where else?) Wikipedia.
The prevalence of 4-4-3-2 over 4-3-3-3 reflects that there are twelve permutations of 4-4-3-2 and only four permutations of 4-3-3-3. If the parallel to thermodynamics isn't evident, I don't know if I can explain. I think I get what the above quote was trying to say but I'm having a hard time putting it into words. I could say variety is the spice of life but I don't know if that helps much.
The most likely suit distribution in ten-card2 hands is 4-3-2-1, which probably would've thrilled the Pythagoreans to no end.
Computer Go programs have improved considerably in the past few years. They still don't beat the best human players but they're getting closer. The progress has come from algorithms quite unlike those used in computer chess.
Whereas chess programs consider how the game stands after a handful of carefully-chosen moves by each player, the best computer Go programs play out lots of hypothetical continuations to the bitter end but with most of the moves chosen at random. If you're surprised that this works, you're not alone. As one of the designers of Deep Blue (the chess program that beat Garri Kasparov in 1997) put it, "I still find it kind of magical that it [random play] works as well as it does."
A match between the top Go program and a strong human player started today. The program won the first game by a slim margin. The programmer and the human contender are interviewed here.
The best Hex program at the moment also relies on random play. I described to a friend this morning what Nixie tubes are: neon-filled tubes used in the mid 20th century for numeric displays. Each tube has ten digit-shaped wire cathodes stacked one behind the other. My friend (whose attention to detail in all things I find endearing) asked what sequence the digits are stacked in. I said I didn't know but I had a few Nixie tubes on hand and I would check. He thought 1 would be first, but maybe 0 or 9. "If it's 3, we're in trouble," he said.
Happy nineteenth, everyone. A ton is 2000 pounds in the USA and 2240 in the UK, because a British hundredweight isn't 100 pounds but rather 112 which is (of course) 8 stone.
Stone is a great name for a unit, but you're hearing this from someone who once chose to live in a town named Boulder.
Climbing routes are graded by difficulty, with different systems in use around the world. The systems are largely numeric—except in England, where climbers also use adjectival grades: moderate, difficult, very difficult, severe, hard severe, very severe, hard very severe, extremely severe one, extremely severe two, ... . British routes have both a numeric and an adjectival grade because the two systems take different aspects of a route into account.
A few climbing grade systems date from when climbing standards were not so high (and before specialized climbing shoes were common). The US system was intended to be decimal, with ratings from 5.0 through 5.9—but has since been extended to 5.10, 5.11, and so on. British "severe" is easy by today's standards.
The USA ditched its system of named and colored threat levels in 2010, but the UK still rates the likelihood of terrorist attack as low, moderate, substantial, severe, or critical. I wonder if someday, maybe when everyone and his brother has armed drones, the UK will see fit to add threat levels beyond critical.
Around 20 years ago, while sharing a ledge several hundred feet off the ground with another climber, I had a conversation that started like this:
When I do this, I synthesize a bird's-eye view of the scene rather than recalling what it might have looked like from my own eye level at the time. Visual recollection constructed from an elevated point of view outside one's body is typical in such cases, so I hear.
There are potentially intreresting implications of this quirk of human memory, but I leave them to the reader to consider.
I wonder whether birds do it too. When a bird remembers something that happened while airborne, does he/she picture it from higher still? I installed new motherboard/CPU/memory/OS in my computer today, the same day my snake shed his skin. Not every shed goes well (sometimes the skin doesn't all come off) and I won't list all the stuff that can go wrong with building a computer. But snake and I both had smooth experiences today. The new machine's first boot greeted me with 19:19 at the top of the screen, and no I didn't wait until then on purpose.
I got ten years out of the old machine, more than I expected to when I put it together. The hardware was still working but recent versions of Linux don't seem to like it. Time marches on.
I got AMD CPUs ten years ago because I like supporting underdogs and their speed was competitive. I got Intel this time because they have since trounced AMD in CPU performance. The Intel processor came with a sticker subject to the same dopey license I wrote about a while back. And there's this notice: ... which took me a moment to understand. It doesn't mean that if the pages fall out of the manual, that's not covered. While napping this afternoon, I had a dream that included a sequence where I was being chased. At a point where there was no dialogue but where action got tense, foreboding minor‑key background music started playing. I think this means I watch too many movies.