The Floppy Disk of Floating Point
evanmiller.org
evanmiller.org
No. It computes pow(2, x) - 1.
Suppose x is close to 0, so that pow(2, x) is close to 1, like 1.00000000000000002345678. Because of that leading 1 (to the left of the radix point), all those zeros also have to be stored, meaning some of the low-end digits will have to be dropped, so you end up rounding to 1.00000000000000002. But if you can subtract 1 from the exact result before rounding, then you can store more of the low-end digits: 2.345678e-18. Hence the F2XM1 instruction.
I do a reasonable amount of computational programming but to my recollection I've never used either of those functions.
But you can thank (or blame!) me for extending its life - back when Microsoft was developing their 64 bit Windows, they were all set to not save the x87 state during a context switch, effectively killing the x87. I talked them out of it.
Microsoft's VC still doesn't support the x87, their "long double" is 64 bits. I suppose that's for compatibility with other processors it supports.
I'm proud of the Digital Mars C/C++ compiler which still sports full 80 bit long doubles, as well as the D programming language which still (despite opposition) delivers 80 bit precision for type real, even for 64 bit Windows.
An experienced eye could easily see the difference between the tiny x87-related indeterminacy and other kinds of changes, but we were uncomfortable automating this comparison, and it took a while for someone without strong domain knowledge (such as myself or any other software engineer being hired) to become comfortable eyeballing it. With deterministic output, we could use automated tests to verify that, for example, the work we did to add a new computational feature did not change the output when that feature was not enabled, or that small changes intended as performance optimizations did not inject tiny numerical errors.
Our customers were also a lot more comfortable when they could use "diff" to validate that our latest X-times-faster release was really doing the same computation as the last one :-)
EDIT: We also got a noticeable speed-up by enabling the SSE2 instructions. The bulk of the numeric work was done in hardware, so it wasn't dramatic, but it was measurable.
As for speed, Intel has neglected to keep up the x87.
As C# is jit-compiled, you could never be sure what code would actually run at the end user machine, and where the truncation from 80 to 64 bits would occur.
In the end the best cure is to ensure you never ever use x87, which happened automatically when dropping 32 bit support.
Determinism is too important to give up for 16 extra bits.
Nowadays maybe that sort of diversity is less of an issue? Expecting determinism in the sense you mean it just seems weird to me.
The x87 can load/store 80-bit floats from/to memory, so it can definitely save intermediate results in full precision if asked to; I'd call that more of a compiler flaw than anything else.
Do you mean '64-bit Windows'? (When 32-bit Windows was being developed, killing x87 would have meant killing all hardware floating point on x86.)
80-bit, 128-bit or sum of 64-bit floats, doesn't matter. Summing an arbitrarily large list of floating point numbers that maximizes precision is actually a hard problem, because the order in which you add the numbers really matters due to incremental errors. Even when this problem is recognized, it turns out the most precise algorithm escaped us for decades--it's not as simple as adding them up low to high. It's only been within the past 10 years that this optimal algorithm has been discovered.
80-bit is a non-portable abomination. It's caused us decades of us "average" programmers headaches and generally has not even given experts better numerical precision.
Good riddance IMHO!
I'm curious - link please?
https://en.wikipedia.org/wiki/Kahan_summation_algorithm
This paper from 1993 analyzes five algorithms:
https://www.semanticscholar.org/paper/The-Accuracy-of-Floati...
Sorry I don't a reference for the "optimal" algorithm--I had it explained to me in person by someone working in the field and they didn't give me a specific place to go track it down. The gist of it is that you need to "bin" the numbers by their approximate order of magnitude and then even in each bin you need to keep a separate running estimate of the error within each bin. It's a kind of divide-and-conquer approach. AFAIR it is "optimal" with respect to minimum error, but not optimally efficient.
It's apparently a hard problem!
http://code.activestate.com/recipes/393090/
but according to the comments on it it isn't entirely accurate, either. It is indeed a hard problem, and just having more bits is easy :-)
I am actually pretty OK with just having an 80-bit float type in the language, as long as it is always 80 bits, and no 64-bit floats get silently promoted. But C didn't have that because it's not what x87 does the fastest. C just has whatever crap is the fastest. I hate that! I imagine that you did a better job of it in D.
In designing WebAssembly we were very careful to go for as much determinism as possible and stick closely to IEEE 754. There were loud voices calling for nondeterminism (even pushing for the default to be nondeterminism as to whether FTZ mode was enabled!) but in the end the portability considerations won out. I was very heavily on the determinism side.
Interesting, do you have a link for us lazy but curious folk?
Truth is that successive rounding errors accumulate complex calculations can lose a lot of bits of precision. That’s why you sometimes need to do calculations in a lot more bits than needed in the answer.
Also, the x87 had 80 bits floats, but not all instructions computed all of them correctly (https://www.infania.net/misc/coproc.html)
I don’t know how current FPUs fare there, but would hope they have become optimal.
I failed to find which instructions weren't computed correctly in the text linked, and I'd like to know about that, that is, if it's something I wasn't aware of. Can you specify a bit more?
it says “Intel claims that maximum error in the 8087, 80287, and 80387 for all transcendental functions does not exceed two bits in the mantissa of the double extended format”. Typically, those reposts get rounded to double format, but that doesn’t guarantee that such errors get dropped.
Edit: http://www.math.utah.edu/~beebe/software/ieee/ has lots of links and code for testing floats.
http://li.mit.edu/Archive/Activities/Archive/CourseWork/Ju_L... gives quite a few corner cases, so it would be helpful when writing a test suite.
https://slixe.de/stash/projects/MAS/repos/libfirm/browse/ir/... has source code for a/the compliance checker
https://core.ac.uk/download/pdf/151193633.pdf describes what seems to be a Better version of that test framework (it adds “++” to the name). Unfortunately I couldn’t find its source code.
But that's still only for the "convenient" input range for the implemented algorithm.
In practice, the programmers sometimes want to pass the values from the less convenient input range, and then it got much worse when the library functions in high level languages have expected too much from the implementation in the processor:
https://randomascii.wordpress.com/2014/10/09/intel-underesti...
The resulting problem in the high level implementation described described was blind belief in the wrong (in not correctly stating the input range) documentation by some that then implemented the C library functions for some environments, only a few years ago:
The Intel's documentation was "buggy. By claiming near-perfect accuracy across a huge domain and then only providing it in a tiny domain Intel has misled developers and caused them to make poor decisions."
And it was fixed in 2014:
https://software.intel.com/blogs/2014/10/09/fsin-documentati...
Read about Unum computing, pioneered by John Gustafson, and implemented on many modern software and hardware platforms.
Maybe 12 years. The PowerPC had fused multiply-add in 1992. For whatever reason, it didn't really appear in x86 until 2012.
This sounds nice, but in practice was quite problematic, as results depended on whether or not intermediate results spilled into memory.
It is very easy to build small invertible matrices that cannot be inverted with 32-bit floats, or even 64-bit floats, thus Kahan's insistence on very high precision floating points.
Edit: small invertible matrices
Presumably these all have eigenvalues near zero and away from zero (i.e. the condition number is large)?
"The Hilbert matrices are canonical examples of ill-conditioned matrices, being notoriously difficult to use in numerical computation."
The man lives still, not to be numbered among the dead:
It's totally possible that scientists in the future will laugh at our extremely wide 16-bit neural networks. Computation at the biological synapse level is considered to have an accuracy of 1 to 3 bits only - opinions differ.
As it turns out, though, most people don't need more than 3 or 4 decimal digits of precision, so while a float may easily accumulate enough error to corrupt that last needed digit, a double tends to be more than roomy enough for almost everybody.
[1] http://web.mit.edu/tabbott/Public/quaddouble-debian/qd-2.3.4...
(i mean this is a legitimate reason, it's just amusing the extremes you'd have to go to to break 64bits).
Microseconds gets you to the year 2255; milliseconds to heat death.
1. https://www.wolframalpha.com/input/?i=January+1%2C+1970+%2B+...
It is easy to run out of bits when trying to use a single number as both an absolute date/time, and to compute relative durations between timestamps with reasonable precision. I've lost track of the number of bugs I've fixed where someone assumed a double would be big enough without doing the math.
The NTP nanokernel uses a 64.64 integer represention https://papers.freebsd.org/2000/phk-nanokernel/ because 32 bits isn't enough above the binary point because it runs out in 2038, and below the binary point you need 64 bits to accurately represent the nanosecond-scale frequency of the CPU's cycle counter to translate it to wall-clock time.
Algorithms stability. As precision decreases when magnitude rises, many tricks are used in sci. comp. to scale everything into ranges where computations won't diverge due to a lack of precision.
===Update===
Couldn't find the one I was looking for, but I did dig up Eckert, Punched Card Methods in Scientific Computation (1940).
From Chapter VII "The Multiplication of Series":
p.68 "By a continuation of this process we obtain about 100 groups of cards each with its rate card. Since the capacity of the multiplier is eight digits, those groups which have 9 and 10 digit multipliers and eight digit multiplicand are done in two groups. The few terms with large multiplier and multiplicand are done by hand. In a series with reasonable convergence there may be about 20,000 cards altogether. The reproducer is of course used to check the punching."
10 decimal digits is what, 33 bits?
p.74 "The machine time for 100 series is as follows:
Original punching and verifying . . . . 2 days
Listing and summary punching checksums. 1/4 day
Sorting and gang punching duplicates. . 1 day
Multiplying . . . . . . . . . . . . . . 1 day
Sorting, tabulating, and summary punch. 1 day
I guess in principle a punched card could have contained 80 decimal places, but it seemed like the ALU equivalents were much narrower.
(this was published by the Thomas J. Watson Astronomical Computing Bureau)
https://www.jpl.nasa.gov/edu/news/2016/3/16/how-many-decimal...
* JPL's highest accuracy calculations, which are for interplanetary navigation, we use 3.141592653589793.
* How many digits of pi would we need to calculate the circumference of a circle with a radius of 46 billion light years to an accuracy equal to the diameter of a hydrogen atom (the simplest atom)? The answer is that you would need 39 or 40 decimal places.
By 1596 Ludolph van Ceulen had 20 digits, and even more by the time he designed his tombstone: https://upload.wikimedia.org/wikipedia/commons/e/e8/Monument...
39 or 40 decimal places would have to wait until 1699.
https://en.wikipedia.org/wiki/Chronology_of_computation_of_π
If you are using a 64-bit float, then you have 53 bits of precision in 1D, 53 bits of precision in 2D, and 53 bits of precision in 3D. The precision doesn’t change as you add or remove dimensions.
Addition has problems where x is close to negative y, sure.
So of course you want to avoid error states but a double already has an exponent range well beyond beyond astronomical. The width of the universe is around 10^27, a planck length is around 10^-35, and the limit of the format is ±10^308.
This implementation of mod2pi in Julia, for example, uses double doubles ("TwicePrecision") here and there:
https://github.com/pkofod/RemPiO2.jl/blob/master/src/RemPiO2...
And, pertinent to the article, here's a bug in mod2pi that was probably triggered by 80 bit extended floats:
If 64 bit IEEE-754 floating point numbers do not support your needs, adding more bits will almost certainly not fix the problem. The problem isn't lack of bits, it's going to be something fundamental about the nature of IEEE-754. You'll probably need something like a rational data type, or a symbolic algebra system that can abstractly represent everything that shows up in your domain without losing any precision. Generally you'll be required to do this in software.
32 bit is the sweet spot between precision and performance. There are only a handful of things that require 64 bits, (notably latitude and longitude) but these things are common enough that hardware support is (IMO) valuable.
Basically, they're summing up a bunch of intermediate calculations. The intermediate calculations are 64 bit floats, and are precise to 17 decimal digits or whatever. They sum all the intermediate components, and get a result that's only precise to 13 digits or whatever. They use double doubles to perform the summation, and they get a result that's back to being precise to 17 decimal digits. Great.
Now imagine doing this with 128 bit floats, which are precise to 33 digits. So you sum the intermediate results, now you're precise to 29 digits. So you've lost 4 digits of precision again. So you add a 192 bit floating point type...
IEEE-754 floating points are always going to have that problem. If you add two numbers of differing magnitude, you're going to lose precision. Finding the sum of a large sequence is generally going to result in the addition of a very large running total with small individual members.
(I perused the other examples, but they seem to be a variation on the same theme)
> This permitted benchmark results to be accurately reproduced for a significantly longer time, with virtually no change in total run time
They clearly understand that bigger floats do not magically change the fundamental behavior, but the additional headroom makes a significant difference
> A larger example of this sort arose in an atmospheric model (a component of large climate model). While such computations are by their fundamental nature “chaotic,” so that computations will eventually depart from any benchmark standard case, nonetheless it is essential to distinguish avoidable numerical error from fundamental chaos.
> Researchers working with this atmospheric model were perplexed by the difficulty of reproducing benchmark results. Even when their code was ported from one system to another, or when the number of processors used was changed, the computed data diverged from a benchmark run after just a few days of simulated time. As a result, they could never be sure that in the process of porting their code or changing the number of processors that they did not introduce a bug into their code.
> After an in-depth analysis of this code, He and Ding found that merely by employing double-double arithmetic in two critical global summations, almost all of this numerical variability was eliminated. This permitted benchmark results to be accurately reproduced for a significantly longer time, with virtually no change in total run time [3].
The keyword is "numerical variability." Here's the referenced article: https://link.springer.com/article/10.1023/A:1008153532043 Choice quote from the article:
> In climate model simulations, for example, the initial conditions and boundary forcings can seldomly be measured more accurately than a few percent. Thus in most situations, we only require 2 decimal digits accuracy in final results. But this does not imply that 2 decimal digits accuracy arithmetic (or 6-7 bits mantissa plus exponents) can be employed during the internal intermediate calculations. In fact, double precision arithmetic is usually required.
The problem isn't lack of precision. The problem is numerical instability when adding up a bunch of numbers with high absolute values but since they were roughly evenly positive/negative, their sum was approximately 1. IEEE-754 floats, as useful as they are, are just bad at this, and adding more bits isn't a solution, it's a punt. They used Kahan summation or Bailey summation and the problem went away. No 128 bit hardware floats required. (Kahan summation is very well known, Bailey summation is new to me)
Here's my point: if double precision floating point doesn't satisfy your needs, you should dig into the problem and understand why. Understand first, write code second. 999/1000 the solution isn't "we need 128 bit floats", and for that .1%, we're waaaay better off telling those people "Sorry, do it in software and take the performance hit."
Interestingly, you don't need multiprecision floating point for every pixel: you can do precise calculations for a few tentpole pixels and calculate most of the rest at adouble precision based on the differences between their location and a tentpole. But you also need some significant cleverness to accurately fill in the gap betwen "most" and "all":
http://www.fractalforums.com/announcements-and-news/pertubat...
Incorrect. Scroll to the bottom of this page to find works dated 2016: