For anyone turned off by this document and its proofs, I recommend Numerical Methods for Scientists and Engineers (Hamming). Still a math text, but more approachable.
The five key ideas from that book, enumerated by the author:
(1) the purpose of computing is insight, not numbers
(2) study families and relationships of methods, not individual algorithms
Shared this because I was having fun thinking through floating point numbers the other day.
I worked through what fp6 (e3m2) would look like, doing manual additions and multiplications, showing cases where the operations are non-associative, etc. and then I wanted something more rigorous to read.
For anyone interested in floating point numbers, I highly recommend working through fp6 as an activity! Felt like I truly came away with a much deeper understanding of floats. Anything less than fp6 felt too simple/constrained, and anything more than fp6 felt like too much to write out by hand. For fp6 you can enumerate all 64 possible values on a small sheet of paper.
For anyone not (yet) interested in floating point numbers, I’d still recommend giving it a shot.
Well, there are many legitimate cases for using the equality operator. Insisting someone is doing something wrong is downright wrong and you shouldn't be teaching floating-point numbers.
A few use cases are: Floating-points differing from default or initial values and carrying meaning, e.g. 0 or 1 translates to omitting entire operations. Then there is also the case for measuring the tinyest possible variation when using relative tolerances are not what you want. Not exhaustive.
If you use == with fp, it only means you should've thought about it thoroughly.
Absolutely nobody will think this is 'clearer', this is a leaky abstraction and personally I think that the OP is right and == in combination with floating point constants should be limited to '0' and that's it.
We all know that 1/3 + 1/3 + 1/3 = 1, but 0.33 + 0.33 + 0.33 = 0.99. We're sufficiently used to decimal to know that 1/3 doesn't have a finite decimal representation. Decimal 1/10 doesn't have a finite binary representation, for the exact same reason that 1/3 doesn't have one in decimal — 3 is co-prime with 10, and 5 is co-prime with 2.
The only leaky abstraction here is our bias towards decimal. (Fun fact: "base 10" is meaningless, because every base calls itself base 10)
There’s plenty of cases where ‘==‘ is correct. If you understand how floating point numbers work at the same depth you understand integers, then you may know the result of each side and know there’s zero error.
Anything to do “approximately close” is much slower, prone to even more subtle bugs (often trading less immediate bugs for much harder to find and fix bugs).
For example, I routinely make unit tests with inputs designed so answers are perfectly representable, so tests do bit exact compares, to ensure algorithms work as designed.
I’d rather teach students there’s subtlety here with some tradeoffs.
Sure, division might be a tad more surprising though since most don't do that on an every-day basis. The specific case we had was when a colleague had rewritten
(a / b) * (c / d) * (e / f)
to
(a * c * e) / (b * d * f)
as a performance optimization. The result of each division in the original was all roughly one due to how the variables were computed, but the latter was sometimes unstable because the products could produce denomalized numbers.
If they were taught what was representable and why they’d learn it quickly. And those that forget details later know to chase it down again if they need it. Making it voodoo hides that it’s learnable, deterministic, and useful to understand.
Tell them that they can only store integer powers of 2 and their sums exactly. 2^0 == 1. 2^-2 == .25. Then say it's the same with base 10. 10^-1 == 0.1. 1/9 isn't a power of 10, you you can't have an exact representation.
One thing that really did it for me was programming something where you would normally use floats (audio/DSP) on a platform where floats were abysmally slow. This forced me to explore Fixed-Point options which in turn forced me to explore what the differences to floats are.
Fixed point gave rise to the old programmers meme 'if you need floating point you don't understand your problem'. It's of course partially in jest but there is a grain of truth in it as well.
What Every Computer Scientist Should Know About Floating-Point Arithmetic (1991) - https://news.ycombinator.com/item?id=23665529 - June 2020 (85 comments)
What Every Computer Scientist Should Know About Floating-Point Arithmetic - https://news.ycombinator.com/item?id=3808168 - April 2012 (3 comments)
What Every Computer Scientist Should Know About Floating-Point Arithmetic - https://news.ycombinator.com/item?id=1982332 - Dec 2010 (14 comments)
What Every Computer Scientist Should Know About Floating-Point Arithmetic - https://news.ycombinator.com/item?id=1746797 - Oct 2010 (2 comments)
Weekend project: What Every Programmer Should Know About FP Arithmetic - https://news.ycombinator.com/item?id=1257610 - April 2010 (9 comments)
What every computer scientist should know about floating-point arithmetic - https://news.ycombinator.com/item?id=687604 - July 2009 (2 comments)
For anyone turned off by this document and its proofs, I recommend Numerical Methods for Scientists and Engineers (Hamming). Still a math text, but more approachable.
The five key ideas from that book, enumerated by the author:
(1) the purpose of computing is insight, not numbers
(2) study families and relationships of methods, not individual algorithms
(3) roundoff error
(4) truncation error
(5) instability
(1991). This article is also available in HTML: https://docs.oracle.com/cd/E19957-01/806-3568/ncg_goldberg.h...
Shared this because I was having fun thinking through floating point numbers the other day.
I worked through what fp6 (e3m2) would look like, doing manual additions and multiplications, showing cases where the operations are non-associative, etc. and then I wanted something more rigorous to read.
For anyone interested in floating point numbers, I highly recommend working through fp6 as an activity! Felt like I truly came away with a much deeper understanding of floats. Anything less than fp6 felt too simple/constrained, and anything more than fp6 felt like too much to write out by hand. For fp6 you can enumerate all 64 possible values on a small sheet of paper.
For anyone not (yet) interested in floating point numbers, I’d still recommend giving it a shot.
Well, there are many legitimate cases for using the equality operator. Insisting someone is doing something wrong is downright wrong and you shouldn't be teaching floating-point numbers. A few use cases are: Floating-points differing from default or initial values and carrying meaning, e.g. 0 or 1 translates to omitting entire operations. Then there is also the case for measuring the tinyest possible variation when using relative tolerances are not what you want. Not exhaustive. If you use == with fp, it only means you should've thought about it thoroughly.
That has more to do with decimal <-> binary conversion than arithmetic/comparison. Using hex literals makes it clearer
Absolutely nobody will think this is 'clearer', this is a leaky abstraction and personally I think that the OP is right and == in combination with floating point constants should be limited to '0' and that's it.
We all know that 1/3 + 1/3 + 1/3 = 1, but 0.33 + 0.33 + 0.33 = 0.99. We're sufficiently used to decimal to know that 1/3 doesn't have a finite decimal representation. Decimal 1/10 doesn't have a finite binary representation, for the exact same reason that 1/3 doesn't have one in decimal — 3 is co-prime with 10, and 5 is co-prime with 2.
The only leaky abstraction here is our bias towards decimal. (Fun fact: "base 10" is meaningless, because every base calls itself base 10)
> Fun fact: "base 10" is meaningless, because every base calls itself base 10
Maybe we should name the bases by the largest digit they have, so that we are using base 9 most of the time.
Repeating the exercise with something that is exactly representable in floating point like 1/8 instead of 1/10 highlights the difference.
There’s plenty of cases where ‘==‘ is correct. If you understand how floating point numbers work at the same depth you understand integers, then you may know the result of each side and know there’s zero error.
Anything to do “approximately close” is much slower, prone to even more subtle bugs (often trading less immediate bugs for much harder to find and fix bugs).
For example, I routinely make unit tests with inputs designed so answers are perfectly representable, so tests do bit exact compares, to ensure algorithms work as designed.
I’d rather teach students there’s subtlety here with some tradeoffs.
I also like how a / b can result in infinity even if both a and b are strictly non-zero[1]. So be careful rewriting floating-point expressions.
[1]: https://www.cs.uaf.edu/2011/fall/cs301/lecture/11_09_weird_f... (division result matrix)
Anything that overflows the max float turns into infinity. You can multiply very large numbers, or divide large numbers into small ones.
Sure, division might be a tad more surprising though since most don't do that on an every-day basis. The specific case we had was when a colleague had rewritten
to as a performance optimization. The result of each division in the original was all roughly one due to how the variables were computed, but the latter was sometimes unstable because the products could produce denomalized numbers.I have a relaxed rule for myself: if I’m using the == operator on floats, I must write a comment explaining why. I use == for maybe once a year.
.125 + .375 == .5
You should be using == for floats when they're actually equal. 0.1 just isn't an actual number.
Are you saying that my students should memorize which numbers are actual floats and which are not?
If they were taught what was representable and why they’d learn it quickly. And those that forget details later know to chase it down again if they need it. Making it voodoo hides that it’s learnable, deterministic, and useful to understand.
Your students should be able to figure out if a computation is exact or not, because they should understand binary representation of numbers.
Tell them that they can only store integer powers of 2 and their sums exactly. 2^0 == 1. 2^-2 == .25. Then say it's the same with base 10. 10^-1 == 0.1. 1/9 isn't a power of 10, you you can't have an exact representation.
They shouldn’t “memorize” this per se, but it should take them only a few seconds to work out in their head.
> 0.1 just isn't an actual number.
A finitist computer scientists only accepts those numbers as real that can be expressed exactly in finite base-two floating point?
Yes. A computer scientist should know how numbers are represented and not expect non-representable numbers in that format to be representable.
0.1 is just as non-representable in floating point as is pi as is 100^100 in a 32 bit integer.
Terminating dyadic rationals (up to limits based on float size) are the representable values.
That's essentially what you already do for integer arithmetic.
I would argue that
can avoid having to carry around, set and check some extra m_HasValueBeenSet booleans.Of course, it might not be something you want to overload beginner programmers with.
One thing that really did it for me was programming something where you would normally use floats (audio/DSP) on a platform where floats were abysmally slow. This forced me to explore Fixed-Point options which in turn forced me to explore what the differences to floats are.
Fixed point gave rise to the old programmers meme 'if you need floating point you don't understand your problem'. It's of course partially in jest but there is a grain of truth in it as well.
Also heavily used in FPGA based DSP.