Why Computers Screw up Floating Point Math 

always use AT LEAST decimals.

@Anon Some languages give you a built in fixed-point decimal type, if you don't have one then you'd probably just go for a long or an actual object

Either doubles or decimals

@Anon double is short for “double-precision floating point” By decimal they mean fixed precision or integers

@Anon Sorry. i stroked out a little. I meant that its preferable to use double in finance software. decimals would be too computing intensive and needlessly sloww down the programm. integers can only hold full numbers so thats not too accurate.:D

Split proton in a perfect half

"split one of the atoms" that'd cost you more than the piece of gold you already have xd

@@groszak1 problem: proton is made of 3 quarks, and those are indivisible

@@no-better-name Again, REEE. The universe is broken. We must fix it.

@@censoredterminalautism4073 god is yandere dev confirmed, who else would mess up this badly? xd

Welp, not for positions in bedrock tho

@@charautreal bedrock is shit

@@DoggyBingBong Yeah, a lot of things were crap in 2018

@@charautreal +now

@@charautreal they are shit now too

1 decimal place != 1 bit

in this context it should be binary places

Should've just used digits

@@hossamdash digits are for decimal, bits are the same thing but in binary - decimal places would be binary places when talking about binary numbers

@@sodiboo lol no a digit is the general name for any base number system. BIT is literally short for Binary digIT.

It depends if you need that level of precision or not. Machine learning requires multiply add operations with billions of floats but since it doesn't need the precision, you can get away with it. If you're doing some high precision scientific math and need something like float64, then yes, you need to be extremely careful about your math because of this rounding error.

or if you ad 1 to ten million, you get ten million not ten million 1, if you try to add 1 in a loop 1 million times, you'd still have 10 million

PAIN

Found that out trying to build a Rendering-Tool for OpenStreetMap data. I didn't want to draw nodes that arent even in the field of view, so I created a function to return the distance from one coordinate to another. During testing on the same Map-Data I got stumped on why during execution I would get different amounts of Rendered Nodes until I figured out that I use 4byte floats for coordinates...

Thanks, you're a life saver

NEVER use equals with floats or doubles that had any operation done to it, most languages have some variation of a "Close Enough?" function for that purpose. but even then you need to be careful to some extent. if you did a lot of operations with that value the error propagation can be so out of whack that even that might fail

@@khhnator example of those? I see people say be careful but none explaining what's careful

@@ko-Daegu is not something you need to worry about often, only be aware of. but if you doing lots of math or have a program that runs for a very long time you might have a issue. the quick version of it is, don't accumulate results of operations indefinitely. myfloat = something * myfloat; because you just introduced a approximation error there. and every time you repeat it, the difference between the number you got and the real result will only increase if that will ever be a problem depends on how many operations you doing, the range of the numbers you using and what you doing with those numbers easy way to not have a problem, is to never reuse results of operations. rewrite your math to use know error less values and recalculate it from the start when you need the result. instead of modifying the value of a previous operation. myfloat = something * initial value * number of operations; of course, that's not always doable, and then you need to figure out if a error margin will break your code or not.

@@ko-Daegu Usually you do an operation like Math.Abs(a - b) < PRECISION when you need to check equality with a given precision. Using the cash register example you would use a 0.01 precision because if the difference between the two numbers is smaller than 0.01 you don't care and you can treat them as equal.

yup, perfectly alignes with how the mantissa and the exponent works in floats

The PS1 does not do floating point at all, which is why the polygons flicker as the polygons snaps into place and gives PS1 games that recognizable look.

That's called fixed point computation, not floating point

It's not floating point math if you don't use floating point. But yeah that's why the polygons and textures look so wonky on the PS1. It's because the pixel position snaps to the nearest integer coordinate as they can't do fractions.

​@61 Cygni You don't had to just look at PS1 polygons(triangles) how mess they are, I read once that game-cube had way better geometry calculation than PS1 but weaker texture filling power, I think the games changes in gaming industry started with Voodoo

@@letusplay2296 Yeah I mean N64, I always forgot that console name, since it was never released in my county(we were under kinda sanctions back then)

He's using FFMPEG or whatever that cringe program was called that's terminal based instead of using OBS which isn't such a CPU hog.

@@sharoyveduchi OBS uses ffmpeg

@@notuxnobux and? Pulse audio uses Alsa but nobody actually uses Alsa by itself.

@@sharoyveduchi Stupid comparison. How do you expect OBS to be better than ffmpeg when it uses ffmpeg directly? you can use hardware acceleration with ffmpeg if you provide the correct options. Its very easy. Its just one command.

@@notuxnobux no it's not a stupid comparison. OBS knows how to configure FFMPEG better than some dumb config file some amateur wrote and posted on his blog. If it was "very easy" then Mental Outlaw's CPU usage wouldn't be at 99% unless he's recording at some absurdly high bitrate or he's recording on a one core toaster. There is literally no excuse for this, even fraps and bandicam run better than this. What do you mean "how do you expect OBS to better"? Of course I know it's better! I use OBS and I didn't have that happen even on hardware from 10 years ago! I speak from experience. I'd rather use FREE SOFTWARE that "just works" than free software that requires more effort and hinders my performance even more than non-free solutions unless I know exactly what I'm doing.

why are the character speeds set to 250 u/s?

@@tankfire20 to avoid floats, is my best guess. I mean, I could be wrong, but I'd bet my left nut on it. Basically, 1 meter was about 100 units. These days, modern game engines expect 1 meter = 1 unit.

Who ever says that? People who only work with double precision?

@@skaruts Makes sense

180 units tall... Sounds like centimeters ;)

Without capital C specifically? Ok, chad

This sounds hilariously obvious out of context.

ok random guy who cant code

@@AlizerLeHaxor how do you know he can't code? lmao

@@eter9nitysslut131 just people assuming and acting suprerior.

Where are the over-complicated ones?

hi

@Nicefisher bruh

@@test-cp8uz bruh moment

@@ClifffSVK bruh

German moustache man? You meant Chaplin, right? EDIT: I know , I know you meant the other guy, but the others probably didn't.

oh yeah, i remember implementing a fraction class of two long ints and using GCD to reduce them every time one is constructed. expensive but worth it

Was it in python? That way you get something that runs at the speed of snail^2

I'm pretty sure both C# and Python have linear algebra libraries fyi. Once made Numpy try to invert a Matrix with a zero determinant. Unfortunately, it crashed, so use the libraries with caution

I did that too, it was pretty satisfying to see how my matrix inversion spit out the exact answer while regular floating point tools had huge errors.

Tbh that was some of the most fun I had in highschool (or the equivalent to it). Implementing the stuff from math class into code was both a great way to better understand the mathematical concept and the programming concept.

@pfp is a distro logo in my photos I did work with fixed point numbers in the past but didn't know that Python has a module for this.Thank you very much.

@linlinö önilnil The classic is 0.1+0.2 == 0.3 and it returns false, but yeah, something like that.

@@WandererOfWorlds0 In those cases you just have to be more cautious, like: If(Abs(0.1+0.2 - 0.3)

Terrible suggestion. Learn floating point.

Well, no, you can. You just usually shouldn’t check for strict equality, instead test for being in a certain range of values. Some languages provide built in functions for this, but you could always just chain together some comparisons. Floating point shouldn’t be used in situations where absolute precision is necessary, but for something like a physics engine in a game or normalized positions in graphics that don’t need to be perfect, floats are great, because they are much faster to compute and use less memory than representing non-integers as fractions.

I'd compare Computational Math to atomic physics, and "Math" as the general field of study to the whole of Physics, including quantum physics; it has the elements of atomic physics present, but when we get into anything more complex, like theories on sub-subatomic particles and whether forces are a Particle or not it alot like the more existential portions of math that are abstract and far beyond any basic understanding. You wouldn't expect an 11th grader learning what friction is to understand nuanced theories on whether gravity is a particle, and you shouldn't expect the computer to understand what the hell the number 0.25 means.

What exactly are they used to? I'm curious. I always turn everything into int.

C# has a Decimal type, with 128 bit precision. Works pretty well, but it's orders of magnitude slower than float/double.

Base-12 and the imperial system

That also comes with a very rare error, if you change the country which uses comma, it will give you errors

But very few cpu's support the decimal type natively.

@@alexvitkov No, you add those 5 layers of abstraction at the API with the DB. Computing is all about abstraction anyway

@@alexvitkov there is a standard SQL type specifically for fixed decimal precision (ie. for representing decimal fractions). It's called NUMERIC (there's also BIGNUMERIC with less precision but a higher scale). The overwhelming majority of SQL implementations will have many many more than that with configurable precision for specific applications (DECIMAL, SMALLMONEY, MONEY). You will learn that FLOAT is *always just an approximation* by reading just one sentence in your engine-of-choice's documentation.

@@alexvitkov you are perfectly capable of implementing any of the data types your engine has in any language. People who wrote the engine did that before you after all. Even if you don't have the necessary information to recreate the exact datatype or "reinvent" it for your use case, the library you're using to operate on the database will have to provide those data types to you. Since the dawn of SQL, you can even just construct queries to make the database engine do all the math for you; due to this interoperability and "nativeness" is irrelevant. Beyond that, many languages natively support fixed-point arithmetic (that includes COBOL) and even C has had extensions for that for almost 20 years (ISO/IEC DTR 18037). If you're working on x86, even assembly has support for binary coded decimals you can use to easily implement fixed-point arithmetic (as native as it gets). I don't know what you mean in that last part, but fractions of a cent are needed so that money doesn't just "disappear" in calculations due to rounding.

Doesn't even has to be any of that. If it was C it could be a quirk from your compiler of choice, similarly if it was C++... jesus almight deliver me from this evil.

Bruh

Easy, use Rust

Because at the transistor and logic gate level (i.e., the flow of electrons along nanometer-sized "wires"), everything is effectively "on" or "off" with no meaningful in between.

Probably because they can't be bothered. idk.

I think the first computers did use base-10, then they switched to base-2 because it is faster.

Intel has FMA

@@tolkienfan1972 keyword of the day is "SOME"

@@SebiBubble that is true. You can have both implementations and use cpuid to choose which to use. Tis a common technique.

Language that magically changes computer architecture when?

Python isnt doing the math, the cpu is. Youd get the same result be it in C or in visual basic.

@@laharl2k I'll create a GUI user interface in visual basic, see if I can track the floating point calculations.

@@laharl2k lmao i use C++, no problem with error rates with such simple calculations. I depends on what and how you use it. Python is like a scratch from programming. You dont write big amd complex stuff on python for a reason.

@@corejake You know Tensorflow exists?

Binary representation of 1/3: 1/11 If you're allowed to use Σ, i'm allowed to use /

C# decimal type. It's a floating point number type, rather than floating binary type. Safe for finance. So, technically, you can store money using floating point, just not float

Inevitable on a tech channel.

what about my boi the Unary numeral system?

There is a range of floating points which work perfectly as integers. JS just uses that range. It does do floating point math on your integers. It's just that integers happen to be represented with 100% accuracy within a certain range. So it's the other way around, it's floats disguised as integers.

Second one is fine. He gets 45. If it were wrong he would get 60. The "fp magic" is what he is showcasing here. For new programmers.

@@JuddMan03 and i kind of watched it with my speakers off. oops.

No 0.89999999999999999999... does not equal 0.9. The repeating decimal approaches 0.9 as a limit, but if you subtract the repeating decimal from 0.9 you will always get 0.00000000000000000000...1. I think some chips might round numbers at an arbitrary precision or test for fuzzy equality, but mathematically they differ, so a test for equality should fail.

@@mikechappell4156 1/inf converges to 0. So 1/inf = 0. A nice way to illustrate 1/inf=0 is a line and points on that line. How much Volume does a square have? None - a square is a two dimensioanl plane, Volume is a property, only avaliable in three or higher dimensional space. How much area, does a line have? None, since area is a property of two or higher dimensional space. You can guess how long a point is then, right? It obviously has not volume, area or length. Thats what makes it a point. A point can be inside any dimensional space, but does not extend in it in any way - not even length. If you now ask yourself how many points you could fit on a line with a length of 1, you would come to the conclusion that you can fit any amount of points on that line. And its the same for a line, with half the length. This must mean that inf*0 is 1, right? But it could also be 2. So does 1 = 2? No, of course it doesnt. In reality 0*inf is undefined. You gotta be very careful, when messing with 0 and infinity. But keep in mind that you can fit any amount of points on any line, any area, any room, any hyper room and so on. Which leads us back to 1/inf = 0.

@@disco.jellyfish Limits convey approximation, possibly infinitely precise, not equality. Good enough for government work does not work for logical comparisons.

@@mikechappell4156 Incorrect, it has been proven that 0.89999... (continuing) = 0.9. Its not approximately but exactly 0.9. Mathematically, those two expressions are the same number! Your example with the difference does not hold up, because you are treating 0.89999... as a finite number. Which it is not. Computers simply can't deal with infinity - this is where the float error occurs.

Good catch! 0.011011... is actually 3/7, not 1/3

ok but 99 would only be 99 in 2 digit unsigned integers

also 1111 1111 signed char is -1 isn't it. -128 is 1000 0000 -128 signed int would be 11111111 11111111 11111111 10000000

it's interesting, a ton of AAA games right now clearly use floats in their damage calculations when they could have just ran it as an integer and converted it to decimal for human reading later. The one's I know personally are Monster Hunter World and Smash Bros Ultimate (there was one screen where someone showed a move in training registering different in total and in individual hit values. When I saw the point difference, I knew it had to be caused by floats in some way.) It's not a bad practice for physics processes as they're already really complex anyways, but damage calculations probably waste by using anything but integers.

No. In every base, if you multiply by the base you just shift the number. Base 10 is not special. Multiplying by 16 in base 16 moves the number to the left.

From practical point of view base 12 or something would be more useful, since you get one more denominator than in base 10 without going too much into inventing new "digits" for this system. And actually dividing things in half (1/2) or thirds (1/3) or fourths (1/4) is more common in real life, than dividing by 1/5.

Wrong. This is true for every base, the same thing happens if you multiply by 2 in binary for example.

except you have no process that will get to infinite decimal places, so it will never get there after all

@@shiinondogewalker2809 and then we get to the definition of convergence, that combats that

@@emmata98 nah not really, I don't deny adding 3s forever converges to 1/3, I meant that because infinite time is needed it will never happen

US Customary*

It will always be a thing. There is no way around it, as you are trying to represent an uncountably infinite set of numbers into a digital machine. The only thing you can do is be very careful and control error. There are some arbitrary-precision arithmetic libraries, but there is no free lunch.

This is the real problem with floats, but everyone just focus on the simple rounding errors. Duh! Of course rounding errors happen. Anyone who's tried using a third in decimal should know that. Floating point being non-associative, and equality being non-reflexive? _Those_ can make one want to tear their hair out. Heck, multiplying by and adding 0 doesn't compile to a no-op because of -0.

@@angeldude101 right? And some compiler options *assume associativity* and other invalid assumptions... meaning the code can be correct but be turned into incorrect assembly!

@@tolkienfan1972 I can understand when floats are non-associative _because of intermediate rounding, but I have to seriously question any developer who actually wants this kind of behavior. Personally, I have a project with a bunch of specialized types that all defer to a single 32 x 32 monster product with unused fields say to 0. Unfortunately, -0 (and NaN) means that every single term of that Monster product needs to be evaluated unless I use fast-math. Then the problem becomes the fact that a NaN or infinity sneaking into the code causes undefined behavior, which could just crash, or it could put my program into an infinite loop.

@@angeldude101 are you saying you have values known at compile time to be zero, but the compiler won't optimize them away because of negative zero?

@@tolkienfan1972 Yes.

class Float def sorta_eq?(another_float, delta=0.001) (self - another_float).abs < delta end end (0.1+0.2).sorta_eq?(0.3) #=> true