Even faster asin() was staring right at me
16bpp.net
16bpp.net
ArcCos(x)= Π √((1-x)/(4+x)).
This tradition came to its own as Madhava school of mathematics from Kerala. https://en.wikipedia.org/wiki/Kerala_school_of_astronomy_and...
Note the approximation is for 0 < x < 1. For the range [-1, 0] Bhaskara used symmetry.
If I remember correctly, Aryabhatta had derived a rational approximation about a hundred years before this.
Lot of interesting mathematics was done by Indians, Persians, Arabs, Mayans.
Indian mathematics has an additional layer of obscurity. Very little was written down and when it was it was written down in picturesque and poetic verses (as a mnemonic device) that used a lot of symbolism and imagery. For the number one they will mention the Sun, for two the moon and so on, these mappings would also change from work to work, chapter to chapter. So one needs a lot of context to understand what a document is saying.
For example the source of the approximation above is described as follows (literal translation) [1]
The degree of the arc, subtracted from the total degrees of half a circle, multiplied by the remainder from that [subtraction], are put down twice. [In one place] they are subtracted from sky-cloud-arrow-sky-ocean [40500]; [in] the second place, [divided] by one-fourth of [that] remainder [and] multiplied by the final result [i.e., the trigonometric radius].
[1] Kim Plofker, Mathematics in India.
For reference, the coefficients given are [1.5707288, -0.2121144, 0.0742610, -0.0187293]: if we optimize P(x) = (π/2 - arcsin(x))/sqrt(1-x) ourselves, we can extend them to double precision as [1.5707288189560218, -0.21211524058527342, 0.0742623449400704, -0.018729868776598532]. Increasing the precision reduces the max error, at x = 0, by 0.028%.
Adjusting our error function to optimize the absolute error of arcsin(x) = π/2 - P(x)*sqrt(1-x) on [0,1], we get the coefficients [1.5707583404833712, -0.2128751841625164, 0.07689738736091772, -0.02089203710669022]. The max error is reduced by 44%, from 6.75e-5 to 3.80e-5. If we plot the error function [0], we see that the new max error is achieved at five points, x = 0, 0.105, 0.386, 0.730, 0.967.
(Alternatively, adjusting our error function to optimize the relative error of arcsin(x), we get the coefficients [1.5707963267948966, -0.21441792645252514, 0.08365774237116316, -0.02732304481232744]. The max absolute error is 2.24e-4, but the max relative error is now 0.0181%, even in the vicinity of the root at x = 0. Though we'd almost certainly want to use a different formula to avoid catastrophic cancellation.)
So it goes to show, we can nearly double our accuracy, without modifying the code, just by optimizing for the right error metric.
Adjusting the "1" upward in sqrt(1-x) does not seem to help at all.
Let α represent a roll rotation, and β a pitch rotation.
Let R(α) be:
( cos α sin α 0)
(-sin α cos α 0)
( 0 0 1)
(1 0 0 )
(0 cos β -sin β)
(0 sin β cos β)
R(β).R(α) = ( cos α sin α 0 )
((-sin α*cos β) (cos α*cos β) -sin β)
((-sin α*sin β) (cos α*sin β) cos β)
( 1 α 0)
(-α 1 -β)
( 0 β 1)
x' = x + αy
y' = y - αx - βz
z' = z + βy
This is a great technique for cheaply doing 3D starfields etc on 8-bit machines.
Look ma, no sine table!
Large angular deflections while being linear transforms, do not in general commute.
It will spoil the linear relation in your elegant expression, but a slightly better approximation for cos for small θ is
1 - 0.5θ²This precisions should be measured in avg and worst ULP, not in "charts". A good approximation should also give exact results in critical points (-1/0/-1 in this case).
The "faster" version gives this:
asin(0) = 6.75268e-05 (double precision)
constexpr double a0 = 1.5707288;Maybe it's folded into expressions, propagated through constant expressions, or used it in contexts that require compile-time constants (template parameters, array sizes, static_assert, other constexpr expressions).
I mean, not in this case of pi/2, where it's more about announcing semantics, but in general those are the purposes and uses.
If you just want the optimizer to be able to constant-fold a value, then yes, either of those will work.
If you want to be able to use the value in the other contexts the parent mentioned that require constant expressions as a language rule, then you generally need constexpr. As an exception, non-constexpr variable values can be used if they’re const (not ‘happens to not vary’) and have integer or enum type (no floats, structs, pointers, etc.). This exception exists for legacy reasons and there’s no particular reason to rely on it unless you’re aiming for compatibility with older versions of C++ or C.
Even if you don’t need to use a variable in those contexts, constexpr evaluation is different from optimizer constant evaluation, and generally better if you can use it. In particular, the optimizer will give up if an expression is too hard to evaluate (depending on implementation-specific heuristics), whereas constexpr will either succeed or give an error (depending only on language rules). It’s also a completely separate code path in the compiler. There are some cases where optimizer constant evaluation can do things constexpr can’t, but most of those have been removed or ameliorated in recent C++ standards.
So it’s often an improvement to tag anything you want to be evaluated at compile time as constexpr, and rarely worse. However, if an expression is so trivial that it’s obvious the optimizer will be able to evaluate it, and you don’t need it in contexts that require a constant expression, then there’s no concrete benefit either way and it becomes a matter of taste. Personally, I wouldn’t tag this particular pi/2 variable constexpr or const, because it does satisfy those criteria and I personally prefer brevity. But I understand why some people prefer a rule of “always constexpr if possible”, either because they like the explicitness or because it’s a simpler rule.
Something like a struct that I can say "this struct is global to the whole program and everyone can see it, but once this function exits those values are locked in". Maybe something like that one function is allowed to unlock and update it, but nowhere else.
Think in terms of storing a bunch of precomputed coefficients that are based on the samplerate of a system, where you really only need to set it up once on startup and it is unlikely to change during the application's running lifetime.
I feel like there probably is a way to do this, and if I was good at high level languages like C I'd know what it is. If you know, tell me what I'm not understanding ;-)
An optimizing compiler might see through a non-constexpr declaration like 'double a0 = ...' or it might not. Constexpr is somewhat more explicit, especially with more complicated initializer expressions.
That might explain why the original Cg (a GPU programming language) code did not use Estrin's, since at least the code in the post does add 1 extra op (squaring `abs_x`).
(AMD GPUs used to use VLIW (very long instruction word) which is "static" ILP).
That’s quite subjective. I happen to find trigonometry to be elegant and true.
I also agree that trigonometric functions lack efficiency in software.
Where did that come from in the article?
The only thing I remember at this point is that I copied and pasted that sentence (I didn't type it.) Even search doesn't find the sentence anywhere but HN.
Sin/cos must be borders of sqrt(x²+y²). It is also cached indeed.
It looks like there is a reasonable explanation when it is written math form but there is no.
We can compute these things using iteration or polynomial approximations (sufficient for 64 bit).
Many algorithms require iteration.
This doesn't make a ton of sense.
I think it is stored like sintable[deg]. The degree is index.
In some way vaguely like this: https://github.com/jeremybarnes/cephes/blob/master/cmath/sin...
> I think it is stored like sintable[deg]. The degree is index.
I can think of a few reasons why this is a bad idea.
1. Why would you use degrees? Pretty much everybody uses and wants radians.
2. What are you going to do about fractional degrees? Some sort of interpretation, right?
3. There's only so much cache available, are you willing to spend multiple kilobytes of it every time you want to calculate a sine? If you're imagining doing this in hardware, there are only so many transistors available, are you willing to spend that many thousands of them?
4. If you're keeping a sine table, why not keep one half the size, and then add a cosine table of equal size. That way you can use double and sum angle formulae to get the original range back and pick up cosine along the way. Reflection formulae let you cut it down even further.
There's a certain train of thought that leads from (2).
a. I'm going to be interpreting values anyway
b. How few support points can I get away with?
c. Are there better choices than evenly spaced points?
d. Wait, do I want to limit myself to polynomials?
Following it you get answers "b: just a handful" and "c: oh yeah!" and "d: you can if you want but you don't have to". Then if you do a bunch of thinking you end up with something very much like what everybody else in these two threads have been talking about.
I try to understand how Math.sin works. There is Math.cos. It is sin +90 degrees. So not all of them is something that completes a big puzzle.
for x in range(0, 90): for y in range(0, 90): if xx + yy < 90*90: # it is in circle
So for the each x, the one that has the greatest y will be the sin.
Something like floatrange(0,1,0.001) may work too.
You can think whatever you want, that's no substitute for being correct.
[1] https://www.researchgate.net/profile/Nikki-Mirghafori/public...
TLDR: it depends on the usage and we actually should have multiple functions that are specialized based on the properties of the caller’s needs where the caller can try a cache or compute approach.
I make things faster all the time by leveraging various CPU caches, sometimes even disk or networked disks. As a general principle though, memory lookups are substantially slower than CPU (and that has indeed changed over time; a decade or three ago they were close to equal), and even cache lookups are fairly comparatively slow, especially when you consider whole-program optimization.
That isn't to say that you can't speed things up with caches, but that you have to be replacing a lot of computations for even very small caches to be practically better (and even very small caches aren't helpful if the whole-program workload is such that you'll have to pull those caches from main RAM each time you use them).
To your paper in particular, their technique still assumes reasonably small caches which you constantly access (so that you never have to reach out to main RAM), even when it was written, and part of what makes it faster is that it's nowhere near as accurate as 1ULP.
Logarithms are interesting because especially across their entire domain they can take 40-120 cycles to compute, more if you're not very careful with the implementation. Modern computers have fairly fast floating-point division and fused multiply-add, so something I often do nowadays is represent them as a ratio of two quadratics (usually rescaling the other math around the problem to avoid the leading coefficient on one of those quadratics) to achieve bounded error in my domain of interest. It's much faster than a LUT (especially when embedded in a larger computation and not easily otherwise parallelizable) and much faster than full-precision solutions. It's also pretty trivially vectorizable in case your problem is amenable to small batches. Other characteristics of your problem might cause you to favor other solutions.
An example of an exp and a log respectively from my personal library of bit hacks:
bit_cast<float>((int32_t)(fma(12102203.2f, x, 0x3f800000)));
bit_cast<float>((uint32_t)(-0x3f800000 - 36707.375f*x)) + 7;L1D caches have also gotten bigger -- as big as 128KB. A Deflate/zlib implementation, for instance, can use a brute force full 32K entry LUT for the 15-bit Huffman decoding on some chips, no longer needing the fast small table.
If curious, On a N64 the graphics chip is also the memory controller so every thing the cpu can do to stay off the memory bus has an additive effect allowing the graphics to do more graphics. This is also why the n64 has weird 9-bit ram, it is so they could use a 18-bit pixel format, only taking two bytes per pixel, for cpu requests the memory controller ignored the 9th bit, presenting a normal 8 bit byte.
They were hoping that by having high speed memory, 250 mHz, the cpu ran at 90mHz, it could provide for everyone and it did ok, there are some very impressive games on the n64. but on most of them the cpu is running fairly light, gotta stay off that memory bus.
https://www.youtube.com/watch?v=xFKFoGiGlXQ (Kaze Emanuar: Finding the BEST sine function for Nintendo 64)
The Ensoniq EPS sampler (the first version) used 13-bit RAM for sample memory. Why 13 and not 12? Who knows? Possibly because they wanted it "one louder", possibly because the Big Rival in the E-Mu Emulator series used μ-law codecs which have the same effective dynamic range as 13-bit linear.
Anyway you read a normal 16-bit word using the 68000's normal 16-bit instructions but only the upper 13 were actually valid data for the RAM, the rest were tied low. Haha, no code space for you!