Q (Number Format)
en.wikipedia.org
en.wikipedia.org
The FMA thing is the thing that made me realize that expecting identical output from FP algorithms is a lost cause. Knuth's choice was the only reasonable choice if you're trying to define a deterministic platform that makes your results reproducible on future hardware. We can expect with great confidence that 50 or 100 years from now, much less 1000 or 5000 years from now, there will be other optimizations like FMA that change the results of our programs, rendering their executions irreproducible. But TeX will still produce the same page layout when run on the same input files.
Or you can just use infinite precision libraries at the cost of CPU.
Do you have specific knowledge that it is actually difficult due to some nuance specific to layout because you’ve spent time on the problem or is that an institution you have from first principles?
Dr. Knuth could have plausibly gotten results similar to yours by writing a 'classic' floating point algorithm: but IEEE 734 [sic: 754] wasn't even published until 1985, so relying on hardware, let alone stable results from hardware, was completely impossible.
This would also have been very, very, very. Very slow. Adding a constant factor of 20-50X to TeX wouldn't have been fun at all.
TeX is also a famous program, the algorithms involved are well-known to anyone with an interest. The statement that it uses an expensive global strategy for layout is accurate.
When a problem is linear, a small pertubation in your input will result in a similarly small difference in the results, bounded by some constant factor. When a problem is non-linear, there is no such constant upper bound to the output error.
There are differences in the amount of nonlinearity however. It seems that your algorithm was nonlinear, e.g. using log and exp functions, but otherwise pretty well behaved. So while the factor between input and output error might not be constant, but rather dependent on input values, it is still the case that in the limit of the input error to zero, the output error will also vanish. (obviously in the real domain, not considering floating point).
Contrast this with a problem that has discontinuities in it. In that case it might happen that however small you make your input error, any nonzero pertubation will cause a significant change in the solution. The TeX layout problem is an example of this, but also happens often enough in physical simulations.
If a single tiny rounding difference makes a letter even a millionth of its width wider, that may mean line breaks move to different locations, that the number of lines in the text changes, or even that the number of pages in the text changes.
• Suppose the width of each line is L, and the inter-word space is "S plus P minus M" in TeX notation (i.e. default S with stretchability of P and shrinkability of M), and that after N words with total width C, we encounter the next word of width W, which would have to either go on the next line, or be squeezed onto that line.
.....................|
This is an example line
• If we break the line after that word, we'd have to shrink the default space of (C + W + NS) to L, so a shrink factor of (C + W + NS - L)/M.
So we'd pick one choice or the other depending on which of these two fractions is smaller.
(This is not exactly TeX's algorithm — the above is a greedy/local "best-fit" rather than TeX's "optimum fit" that considers the paragraph as a whole; see the paper at http://www.eprg.org/G53DOC/pdfs/knuth-plass-breaking.pdf — but it's hopefully illustrative.)
Even in this simple case that just needs to compare two ratios (in practice we have to at least compare two sums of such fractions), if you compute these two fractions as real numbers, and you don't have any consistent floating-point arithmetic available across different machines and compilers (or across time), how would you ensure that the system would be deterministic / numerically stable as you said? (Would always make the same decision between the two choices?)
(And would that method have worked in 1980? The initial version of TeX, now called TeX78, was written in 1977–1978 in the SAIL language and only had to run on one computer/compiler (the ones at SAIL in Stanford), and it did use the "real" type. By 1980, already (programs based on) TeX had proliferated to various places—people were porting/rewriting TeX into their own OS/languages—and Knuth started the rewrite of TeX into the current TeX82 in WEB (based on Pascal), and that time both hardware and Pascal compilers varied widely in their support for and implementation of floating-point arithmetic, so Knuth pretty much had to use fixed-point (i.e. only integer arithmetic staying within 2^31) to ensure that the same output would be produced everywhere. But I'm curious how we could get numerical stability in the sense you mentioned even today, as the problem seems to inherently have the property that small perturbations can lead to different output.)
Of course the whole point of things like Kalman filters is that they are insensitive to small quantitative errors, and as you say you worked hard to ensure your programs as a whole were also insensitive; experience with things like that can lead your intuition astray.
You are correct that requiring both perfect reproducibility and optimal accuracy requires sacrificing performance, but actually it is worse than that; this change makes some ordinary terminating algorithms fail to terminate.
It is a shame x87 was so poorly thought out. If it had a better defined programming model, it would be incredibly useful.
For high-precision numbers, e.g. for 64 bits, the standard double-precision floating-point numbers are the best.
Any number format using a given number of bits has the same number of values that can be represented. The only difference between various formats, e.g. integers, fixed-point numbers, floating-point numbers or posits, is how the same number of points is distributed over the line of the real numbers, being more dense in some parts and more rare in other parts.
For scientific computation, the optimal distribution of the representable numbers is when the relative error is constant. The ideal number format would be logarithmic, but it is too difficult to add and subtract logarithmic numbers, so the second best format is used, i.e. floating-point numbers, which allow fast implementations for all important arithmetic operations.
Posits have a non-uniform relative error, low close too 1 and high for large and small numbers, so they are unsuitable for complex physics modelling, which needs more computations with large numbers and small numbers than with numbers close to 1.
On the other hand, for the applications where low-precision numbers are suitable, e.g. machine learning, graphics or some kinds of DSP, posits could be better than floating-point numbers.
Logarithmic distribution is optimal for complex physics models where there are relationships between many different kinds of physical quantities, so it is not possible or convenient to choose a scale factor that would make all quantities have values close to 1. In such applications many numbers are very large or very small and any other distribution except logarithmic leads to increased errors in the results. Such models also typically need a 64-bit precision. Moreover, having a much greater exponent range than provided by the 32-bit floating-point format, in order to avoid overflows and underflows, is frequently even more important for them than having a greater precision.
If for an application the range of the 32-bit floating-point numbers is good enough, then there are indeed chances that posits might be better for it.
The real disadvantage appears to be that a fast implementation is likely to be more expensive, i.e. it requires more complex circuits, thus a greater chip area. Most designers of recent CPUs or GPUs appear to be much more concerned with obtaining a greater speed and a minimum chip area than with obtaining adequate numeric accuracy, because most buyers are influenced by speed benchmark results and know little or nothing about the accuracy required by their computations.
Moreover, the subnormals are an optional feature of the floating-point standard, the alternative to using subnormals is to enable the underflow exception. However, an overwhelming majority of the programmers are too lazy to enable and handle overflow exceptions, much less to enable and handle underflow exceptions.
Unlike using either underflow exceptions or subnormals, the use of flush-to-zero and denormals-are-zero is acceptable only in the programs where computation errors do not matter at all, such as games.
That is not true. FTZ and DAZ are perfectly reasonable in a lot of scientific computing scenario's, where computation errors in general are closely scrutinized.
FTZ and DAZ and any other methods that are certain to introduce errors can be accepted only inside a computation for which there is an independent way to verify the results and that way is always used, without exceptions.
For example, it is indeed OK to use FTZ and DAZ when solving a system of equations, if, and only if, after obtaining a solution it is inserted in the original equations and a residual error is computed, this time without using FTZ and DAZ, and an excessive residual error triggers an appropriate means of notifying the user about the failure.
It differs from vendor to vendor. I know at least one major GPU vendor that has full denormal support, at least for normal arithmetic (not sure about transcendentals etc). In Vulkan, it's possible to ask the GPU to either always flush denormals to zero, or (if the HW supports it) to never do so, but I think nobody uses this, and there may be a (huge) performance penalty from requesting either of these!
Er why not
For the latter likely because you have to do layout. You could try defining a fundamental page unit but some layout decisions will likely still put you into fractional space where you need to preserve the fractions to get a correct result.
Thought it was "integers not allowed" which would be strange.
Actually it is "integers not sufficient"
Basically it is a way to push primitive values of any type and pointers into a 64-bit double, to reduce memory/cache footprint in dynamic languages.
[1] https://en.m.wikipedia.org/wiki/Unum_(number_format)#Unum_II...
The "saturation" thing in the article is a joke; no way you can get correct results at the end if you cap the result of computations on overflow. That's slightly better than letting the overflow happen, but the right thing to do is to throw an error, not silently min/max the result.
It's not designed to not have errors, it's a way to detect overflows without floating point capability. You can't always just throw an error in some hard real time systems (though you obviously need to take care of saturated results "carefully", which depends on what you're up to).
(careful defending the sign bit unless you believe zero to be a positive number)
one place where "the sign bit interpretation" shows up, is still not a slam dunk for sign bits. Using C as the example, when casting a signed char to a signed word, you do have to take the "sign bit" and extend it: but you don't just move it to the new sign bit, you extended it through all those other "data/value" bits.
and one's complement is a complement, but two's complement isn't really. It does turn out that if you take the complement and add one you get the right value, but a better way to think of it (on byte scale e.g.) is "take a modulo-256 dial, now change the labeling counter clockwise from the top to negative numbers because you note that 255+1 is 0 just like -1+1 is 0". I just think this explanation serves beginning C or ASM programmers better because it's more mentally agile.
I learned it what I would call "the dumb way", but since I was amazed that it worked I studied it to convince myself it was true and then I realized what was actually going on.