Actually the trend is in the opposite direction -- fewer of the math functions are implemented in hardware than used to be. There are many reasons (optimized out-of-order CPUs and old/slow transcendental implementations) but one significant reason is that the new glibc math functions are generally correctly rounded -- exactly correct. Whereas the hardware versions are often not -- as I discussed in this recent blog post:
High-precision math is an excellent time to use assembly language. Assembly languages generally have a way to express ideas like a 32x32->64-bit multiply (and 64x64->128-bit multiply), and add-with-carry. High-level languages generally support neither of those options directly. To tell the compiler that you want a 32x32->64-bit multiply you generally have to have two 32-bit inputs, then cast one of them to 64-bit, and hope that the compiler doesn't actually generate a 64x64 multiply.
For 64x64->128-bit multiplies the problem is more difficult because many languages don't have a 128-bit type, and yet these multiplies are crucial for getting maximal multi-precision performance on x64.
Without access to the carry flag a programmer in a high-level language has to do things like:
a0 += b0;
if (a0 https://randomascii.wordpress....
As the article mentions, inflation adjusted gas taxes have been dropping for 21 years. That doesn't make sense. At the very least they should be returned to their 1993 levels and indexed for inflation. Roads are crowded and a gas tax would relieve that by encouraging alternatives. It would also reduce pollution, reduce carbon emissions, reduce oil imports that kill the balance of trade and finance people who use the money to try to kill us.
Gas taxes really are good.
As to the complaints that the gas taxes are being used to fund other things, such as bicycle paths and mass transit -- I'm not sure how true that is, but you would be foolish to fight it. Alternatives to driving are crucial. Paving huge amounts of land makes walking and biking very difficult so drivers *owe* the non-drivers a bit of help. And drivers benefit *greatly* from mass transit. With no mass transit the traffic congestion would be even worse.
And, given that drivers park free almost everywhere it is truly rich to hear drivers complaining about having to subsidize transit. The implicit subsidy that cars get through free parking is orders of magnitude greater (read "The High Cost of Free Parking" for all the details).
It is of course well known that, for double precision, sin(x) == x if x 1.49e-8. They teach that in kindergarten these days.
However the article is about sin(double(pi)), and pi is actually greater than 1.49e-8. Therefore range reduction needs to be done, and that is where things go awry.
Yes, the caller could do the range reduction, but this is not trivial to do correctly and it really should be done by the sin() function. With glibc 2.19 it is.
But the parent you refer to (this comment's great grandparent?) says "Any serious calculation requires an error calculation to go with it." Sure. I can agree with that.
And that's the whole point of the article. If somebody does an error calculation based on Intel's documentation then they will have an incorrect error calculation -- in some cases grossly incorrect. So the claim that an error calculation is needed actually *supports* the article (mine, BTW), while arguing against it.
I think ledow is violently agreeing with my article, perhaps without realizing it.
As the other reply said, the function contract and an explanation of the implementation are different beasts. The Intel manual said that fsin is accurate across a huge range of values. Elsewhere the Manual hinted at Intel's range reduction algorithm such that a numerical analyst could suspect that there was a problem. So, to a numerical analyst the documentation was contradictory. To anybody else it was clear -- guaranteed one ulps precision. The documentation was therefore at best misleading, but for most people it was just wrong. Linus Torvalds was fooled, for example.
> If you let x = pi, then people would ordinarily expect that sin (x) = 0.
Many people would expect that, although the article (my article) most certainly did not expect that.
The calculation being done in the article takes into account the fact that double(pi) is not the same as the mathematical constant pi and it uses that and the properties of sine to measure the error in double(pi). This is an unusual but perfectly reasonable calculation to make. It failed because of the limitations of fsin. Those limitations are contradictory to the documentation. Hence the article.
> A more precise approximation according to Wikipedia would have been...
You don't need to go to Wikipedia, you just have to RTA. It lists a 192-bit hexadecimal approximation of pi.
> The reality is that the correct result would have been zero
No. Zero is the correct answer if doing symbolic or infinite precision math, but I did not make that assumption because I was doing double-precision math.
Can you give an example of when the fsin instruction's accuracy would be insufficient? A few people have asked and a clear answer would be quite helpful.
There's an alternate test that works on the Windows 7 calculator to show that it is not implemented on the FPU. Calculate square root of four, and then subtract two. You should get zero but you don't.
That is an error that no FPU would make. The IEEE standard requires a correctly rounded result for square root, and the Windows 7 calculator fails to deliver that on even very simple inputs like four. The Windows 7 calculator does its calculations to more digits of precision than double precision, but it doesn't do its calculations accurately. Oops.
How would you like that example number to be given? In hexadecimal for maximum readability? Giving it in decimal is necessary for communicating the issue.
But the reality is that that number can show up during calculations. Intel promises to calculate it's tangent to a specific precision, but Intel's documentation is incorrect. That is the problem -- hugely misleading documentation.
The issue of not being able to fully specify a particular real number is actual crucial to the article though, but in a different way. The example is sin(double(pi)), what it should be, what that should allow, and how that fails. You should read the article. I think it's excellent.
> Any serious calculation requires an error calculation to go with it.
Sure. That sounds good. And in order to make that error calculation you need to consult the documentation to know how accurate the instructions you are using are. Let me see -- Intel says that fsin is accurate to one ULP. That's sufficient. Error calculation done.
That's why the inaccuracies in their documentation matters.
> I'll tell you now that I wouldn't rely on a FPU instruction to be anywhere near accurate.
Really. Well that seems foolish. As required by the IEEE standard the x87 FPU supplies correctly rounded results for add, subtract, multiply, divide, and square root. At double and long-double precision you can't do better. Those can be composed into higher-level functions with well defined accuracy if you know what you are doing.
It's funny that there is one group of people asking when this tiny error could even matter (http://randomascii.wordpress.com/2014/10/09/intel-underestimates-error-bounds-by-1-3-quintillion/#comment-13638) and another group saying that even without this error the precision is insufficient. You guys should talk.
Visual C++ has this handy
A performance benchmark which doesn't turn on the go-fast option is not going to produce meaningful results.
The author also doesn't discuss debug symbols. VC++ generates debug symbols by default, whereas the other compilers do not. Generating builds without symbols is not a reasonable scenario for most builds, so this makes the file size comparisons rather meaningless.
"Engineering without management is art." -- Jeff Johnson
