The ELI5 explanation of floating point: they approximately give you the same accuracy (in terms of bits) independently of the scale. Whether your number if much below 1, around 1, or much above 1, you can expect to have as much precision in the leading bits.
This is the key property, but internalizing it is difficult.
So between 1/2 and 1 there are the same number of numbers as between 1024 and 2048. If you have 1024 numbers between each power of 2, then each interval is 1/2048 in the first case and 1 in the second case.
I reality there are usually:
bfloat16: 128 numbers between each power of 2
float16: 1024 numbers between each power of 2
float32: 2*23 numbers (~8 million) between each power of 2
float64: 2*52 numbers (~4.5 quadrillion) between each power of 2
(keep in mind there is a subnormal range where there's implicit 0. at the beginning of the mantissa instead)
To reiterate, increasing the exponent by 1 doubles the window size, so the exponent describes how many times the window size was doubled while the number of bits of mantissa describes how many times you can do the reverse and "half" it, hence the exponent to mantissa bits relation.
For example, if you use single precision floats, then you need up to 9 digits of decimal precision to uniquely identify a float. So you would need to use a printf pattern like %.9g to print it. But then 0.1 would be output as 0.100000001, which is ugly. So a common approach is to round to 6 decimal digits: If you use %.6g, you are guaranteed that any decimal input up to 6 significant digits will be printed just like you stored it.
But you would no longer be round-trip safe when the number is the result of a calculation. This is important when you do exact comparisons between floats (eg. to check if data has changed).
So one idea I had was to try printing the float with 6 digits, then scanning it and seeing if it resulted in the same binary representation. If not, try using 7 digits, and so on, up to 9 digits. Then I would have the shortest decimal representation of a float.
This is my algorithm:
int out_length;
char buffer[32];
for (int prec = 6; prec<=9; prec++) {
out_length = sprintf(buffer, "%.*g", prec, floatValue);
if (prec == 9) {
break;
}
float checked_number;
sscanf(buffer, "%g", &checked_number);
if (checked_number == floatValue) {
break;
}
}
I am surprised how complex the issue seems to be. I assumed there might be an elegant solution, but the problem seems to be a lot harder than I thought.
[1] Some (e.g. Windows CRT) do use the shortest representation as a basis, in which case you can actually extract it with large enough precision (where all subsequent digits will be zeros). But many libcs print the exact representation instead (e.g. 3.140000000000000124344978758017532527446746826171875 for `printf("%.51f", 3.14)`), so they are useless for our purpose.
printf("%f\n", 3.14); // 3.140000
printf("%g\n", 3.14); // 3.14> g, G: A double argument representing a floating-point number is converted in style f or e (or in style F or E in the case of a G conversion specifier), depending on the value converted and the precision. Let P equal the precision if nonzero, 6 if the precision is omitted, or 1 if the precision is zero. Then, if a conversion with style E would have an exponent of X: if P > X ≥ −4, the conversion is with style f (or F) and precision P − (X + 1). otherwise, the conversion is with style e (or E) and precision P − 1.
Note that it doesn't say anything about, say, the inherent precision of number. It is a simple remapping to %f or %e depending on the precision value.
I once wanted to find a vector for which Euler rotation (5°, 5°, 0) will result with the same vector, so I just ran a loop of million iterations or so which would take a starting vector, translate it randomly slightly (add a small random vector) and see if after rotation it's closer to original than the previous vector would be after rotation, if not, discard the change otherwise keep it. The script ran for a couple seconds on Python and with decreasing translation vector based on iteration number, I got perfect result (based on limited float precision of the vector). 2s would be terribly slow in a library but completely satisfying for my needs :D
[1]: https://en.wikipedia.org/wiki/Axis%E2%80%93angle_representat...
[2]: https://en.wikipedia.org/wiki/Axis%E2%80%93angle_representat...
I just tested it:
from bpy import context as C
from mathutils import Vector, Euler
from math import radians as rad
r = Euler((rad(5), rad(5), 0))
ob = C.object
ob.rotation_euler = r
ob.rotation_mode = 'AXIS_ANGLE'
a, x, y, z = ob.rotation_axis_angle
v = Vector((x, y, z))
print(v)
v.rotate(r)
print(v)
print("--")
from mathutils import Vector, Euler
from math import radians as rad
r = Euler((rad(5), rad(5), 0))
v = Vector(r.to_quaternion().axis)
print(v)
v.rotate(r)
print(v)
print("--")Yes, just `printf("%f", ...);` will get you that.
The actual algorithms to do the float->string conversion are quite complicated. Here is a recent pretty good one: https://github.com/ulfjack/ryu
I think there's been an even more recent one that is even more efficient than Ryu but I don't remember the name.
https://en.cppreference.com/w/cpp/types/numeric_limits/max_d...
This is totally pointless when serialization to and from an unsigned integer that's binary equivalent would be perfectly reversible and therefore wouldn't lose any information.
double f = 0.0/0.0; // might need some compiler flags to make this a soft error.
double g;
char s[9];
assert(sizeof double == sizeof uint64_t);
snprintf(s, 9, "%0" PRIu64, *(uint64_t *)(&f));
snscanf(s, 9, "%0" SCNu64, (uint64_t *)(&g));
union {
float f;
int i;
} foo;
foo.f = 3.14;
printf(“%x”, foo.i);
In C++, you have to use memmove (compilers can and often do recognize that idiom)
The implication is that the next biggest float is (almost) always what you get when you reinterpret its bits as an integer, and add one. For example, start with the zero float: all bits zero. Add one using integer arithmetic. In int-speak it's just one; in float-speak it's a tiny-mantissa denormal. But that's the next float; and `nextafter` is implemented using integer arithmetic.
Learning that floats are ordered according to integer comparisons makes it feel way more natural. But of course there's the usual asterisks: this fails with NaNs, infinities, and negative zero. We get a few nice things, but only a few.
A more correct version of the statement would be that comparison is the same as on sign-magnitude integers. Of course, this still has the caveats you already mentioned.
let mut left = self.to_bits() as i32;
let mut right = other.to_bits() as i32;
// In case of negatives, flip all the bits except the sign
// to achieve a similar layout as two's complement integers
left ^= (((left >> 31) as u32) >> 1) as i32;
right ^= (((right >> 31) as u32) >> 1) as i32;
left.cmp(&right)
It's like the paranormal trope of an expedition encountering things being disconcertingly "off" at first, and then eventually the laws of nature start breaking down as well. All because of float precision.
For example, the Assassin's Creed series.
From the wiki, far lands ("spongy walls of terrain") aren't caused by precision loss but integer overflow in the terrain generation.
Donald Knuth has all of that covered in one of his "The Art of Computer Programming" books, with estimations about the error introduced, some basic facts like a + (b + c) != (a + b) + c with floats and similar things.
And believe it or not, there have been real world issues coming out of that corner. I remember Patriot missile systems requiring a restart because they did time accounting with floats and one part of the software didn't handle the corrections for it properly, resulting in the missiles going more and more off-target the longer the system was running. So they had to restart them every 24 hours or so to keep that within certain limits until the issue got fixed (and the systems updated). There have been massive constructions breaking apart due to float issues (like material thicknesses calculated too thin), etc.
Really though, games are theater tech, not science. Double-Precision will be more than enough for anything but the most exotic use-case.
The most important thing is just to remember not to add very-big and very-small numbers together.
imagine if integer arithmetic gave wrong answers in certain conditions lol why did we choose the current compromise?
It is odd to me that every major CPU instruction set has ALU codes to indicate when these conditions have occurred, and yet many programming languages ignore them entirely or make it hard to access them. Rust at least has the quartet of saturating, wrapping, checked, and unchecked arithmetic operations.
Signed Integer Overflow OTOH is Undefined Behavior, so it's worse.
https://www.h-schmidt.net/FloatConverter/IEEE754.html
This one has the extra feature of showing the conversion error, but it doesn't support double precision.
However the one in OP has an amazing graph intuitively explaining the numeric space partitioning - the vertical axis is logarithmic, and the horizontal is linear for each row on its own, but rows are normalized to fit the range between the logarithmic values on the vertical axis. I guess it's obvious once you're comfortably understanding floats and could do with some explanations for those still learning it.
These types of visualizations are super useful.
[1] - https://cidr.xyz
It's sometimes fun to have these kinds of edge cases up your sleeve when testing things.
For other curious readers, these are one beyond the largest integer values that can be represented accurately. In other words, the next representable value away from zero after ±16,777,216.0 is ±16,777,218.0 in 32 bits -- the value ±16,777,217.0 cannot be represented, and will be rounded somewhat hand-wavingly (usually towards zero).
Precision rounding is one of those things that people often overlook.
> .EXPOSED will be utilized by registrants seeking new avenues for expression on the Internet. There is a deep history of progressivity and societal advancement resulting from the online free expressions of criticism. Individuals and groups will register names in .EXPOSED when interested in editorializing, providing input, revealing new facts or views, interacting with other communities, and publishing commentary.
Like what? A risque lingerie shop at balls.exposed or something? And new TLDs don't in any way facilitate "better search", you know, nor "information sharing".
> Along with the other TLDs in the Donuts family
Sorry, the what family?
> online identities and expression that do not currently exist.
What does this phrase even mean?
> the TLD will introduce significant consumer choice and competition to the Internet namespace – the very purpose of ICANN’s new TLD program.
"Considered harmful" etc.
> Individuals and groups will register names in .EXPOSED when interested in editorializing, providing input, revealing new facts or views, interacting with other communities, and publishing commentary.
Still not sure how "provision of legitimate goods" fits into this. Or the floating point formats, for that matter.
On the other hand linux.rocks and windows.rocks are taken (no website), vi.rocks is 200 USD/year and emacs.rocks is just 14 USD/year.
microsoft.sucks redirects to microsoft.com, but microsoft.rocks is just taken :thinking:
On that note, I've been trying to see if GoDaddy will buy a domain and resell for higher price by searching for some plausibly nice domain names on their site. They haven't took the "bait" yet.
https://raku.org defaults to Rationals (Rats) and provides FatRat for arbitrary precision
otherwise even relatively normal calculations (eg what’s the mass of electron in quectogram) fail
- 8-bit floating point types: (S1)E4M3 (finite only, has no-negative-zero variant), (S1)E5M2 (has no-negative-zero variant), E8M0
- 6-bit floating point types: (S1)E2M3, (S1)E3M2
- 4-bit floating point types: (S1)E2M1, NF4
- Block floating point types: MXFP8 (E5M2/E4M3*32+E8M0), MXFP6 (E3M2/E2M3*32+E8M0), MXFP4 (E2M1*32+E8M0), NXFP [1]
[1] https://arxiv.org/pdf/2412.19821
Do you have anything to add on?
The real shocking fact about floating point is that they are even used at all.
It's throwing out of the window the most basic property operations on number should have : "associativity" and all that for a gain in dynamic range which is not necessary most of the time.
The associativity we expect to hold is (a+b)+c == a+(b+c) and (ab)c == a(bc) and these don't hold for floats even though most math formulas and compiler optimizations rely on these to hold. It's a sad miracle that everything somehow still works out OK most of the time.
You lose determinism most of the time with respect to compiler optimizations, and platform reproducibility if processor don't exactly respect IEE-754 (or is it IEE-854).
The real problem comes when you want to use parallelism. With things like atomic operations and multiple processor doing things out of order, you lose determinism and reproducibility, or add a need for synchronisation or casting operations everywhere.
Even more problematic, is that because number operations are used so often, they are set in "stone", and are implemented at the hardware level. And they use much more transistor because they are more complex than integer arithmetic.
Real programmers don't use floating points, only sloppy lazy ones do.
Real programmers use fixed point representation and make sure the bounds don't overflow/underflow unexpectedly.
Let's ban all hardware floating-point implementation : Just imagine future alien archeologists having a laugh at us when they look at our chips and think "no wonder they were doomed they can't even do a+b right : its foundations were built on sand".
Amazing confidence on display here.
Most compilers have an option like -fassociative-math that explicitly allows optimization under the assumption of associativity and distributivity.
> Real programmers use fixed point representation and make sure the bounds don't overflow/underflow unexpectedly.
So you complain that floating-point is bad because it's not associative but then suggest that we use fixed-point instead (which is also nonassociative), but it's okay, because it's fine as long as you do thing that programmers rarely do.
> Let's ban all hardware floating-point implementation : Just imagine future alien archeologists having a laugh at us when they look at our chips and think "no wonder they were doomed they can't even do a+b right : its foundations were built on sand".
Ah, you're the kind of person who sees that 0.1 + 0.2 != 0.3 and decides to go on a crusade against floating-point because of it. By the way, fixed point has that exact same bug: it's a fault that is caused by the base being different more than the other principles of the floating-point type.
Floating-point has trade-offs; a good programmer understands the trade-offs of floating-point and will not ask more of it than it can provide.
I'm looking for ways to do that in my home-made compiler back-end. If you've got an example (on compiler explorer, a paper, blog or whatever), I'd be interested in reading about it.
I'd agree that fixed point would have sufficed in many cases where people use floating point. But floating point can be more convenient, with the increased range providing some protection against overflow.
If the order of the operations is important because you don't have associativity then you can't legally do it.
You can have special flags (-fassociative-math) for floats which allows to treat them as if they are associative but these mean your program result will depend on which optimization the compiler picked.
And it turns out that these loop reordering optimizations are really useful when you need to do some backward automatic differentiation. Because all the loops are basically iterated in reverse for the automatically generated code of the backward pass.
But the memory access pattern for the backward pass are not contiguous if you don't interchange the loop order, which the compiler can't do legally because of floats. Nor can he then merge loops together. Which is really useful because if you can merge the forward pass with the backward pass then you don't have to keep values inside a "tape".
So basically you can't rely on compiler optimizations, so your auto-differentiator can't benefit from existing compiler progress. (You can have look either at Julia Zygote, or enzyme which rely on compiler optimizations chaining well). Or you write backward passes manually.
The limitation is the minimal quantization level. But for a 3d engine let's say your base increment is nanometers. Then you set your maximum dimension let's say 1000km. You only have to be able to represent number up 10^20 so 64-bit fixed point number is good enough.
Do everything in 128-bit fixed point numbers, and float are no more problem for anything scientific.
nevertheless, us Weitek guys made 32-bit FPUs to do 3D graphics (pipeline, 1 instruction per clock) P754, IBM, DEC standards to power SGI, Sun etc
this is still the best format to get graphics throughout per transistor (although the architectures have got a bit more parallel)
then 64-bit became popular for CAD (32-bit means the wallpaper in your aircraft carrier might sometimes be under the surface of your wall)
It does of course work with base 2 and exponents as well so you could still be using floating-point format, only with additional meta-data indicating the repeating range. When a result degenerates into a number that can't fit within the number of digits, you would be left with a regular floating-point number.
I'd want to write a simple calculator that uses this numerical format but I have only been able to find algorithms for addition and subtraction. Every description I've found of the format has converted to the regular numerator/denominator form before multiplication and division.
For games and most simulation, the "soft failure" of gradual precision loss is much more desirable than the wildly wrong effects you would get from fixed-point overflow.
And the solution to this problem is to adjust your coordinate space, e.g. make every nanometer represented as `1` but have the containing object matrix have scale fields set to 1e-9.
So this is not a theoretical problem, just a practical one: the z-fighting you get with floats, would happen much more often with integers - you absolutely can avoid it in both cases, but practically 3D engines are designed with performance in mind, and so some assumptions lead to limitations and you would get more of them with integers.
It's kind of a chicken and egg problem where people use floats because there are FPUs available. All the engineering effort which went into dealing with floats and the problem that comes with them, would have been better invested in making integers faster.
We went onto the wrong path, and inertia keep us going on the wrong path. And now the wrong path is even more tempting because all efforts have made it more practical and almost as good. We hide the precision complexity to the programmer but it's still lurking around instead of being tamed.
The absolute GPU cluster-fuck with as many floating types as you can write on a napkin while drunk at the bar, mean that at the end of the day your neural network is non-deterministic, and you can't replicate any result from your program from 6 month ago, or last library version. Your simulations results therefore are perishable.
Inability to replicate results mean that you can't verify weight modifications to your neural networks haven't been tampered by an adversary. So you just lose all fighting chance to build a secure system.
You also can't share work in a distributed fashion because since verification is not possible you can't trust any computation that you haven't done yourself.
Regarding 64 bit double vs 64 bit fixed width, I don't think there is a really good reason to bother with fixed width, it adds more instructions, and will require a custom debug visualizer to inspect the values.
Bit shifts, at least in SSE/AVX2 etc, are only able to run on a single port, so they actually aren't such a great idea(not sure about scalar, I don't bother to optimize scalar code in this way).
Edit:
For those who want more context:
https://vbn.aau.dk/ws/portalfiles/portal/494103077/WPMC22_Ko...
Here is a big fat warning:
>There is one important downside though, which relates to the fact that designing fixed-point algorithms is a significantly more complicated task as compared to similar floating-point based algorithms. This fact essentially has two major consequences: >1) The designer must have an extended mathematical knowledge about the numerical characteristics of the algorithms, and >2) The development time is in some cases longer than for equivalent floating-point systems.