[Cuis-dev] Documentation for the different kinds of divisions

[Andres Valloud]

Hi,

> In Squeak trunk we still have cos defined as ^ (self + Halfpi) sin, a 
> proof that we value simplicity more than correctness.

I'd like to believe there was a good reason for such an approach, I just 
can't imagine what that might be.  Right now, all I can see is that it's 
slow, it gets the wrong results, and it also makes porting serious 
numerical analysis algorithms impossible.  Also, it's clever :/.

> Couldn't we ask the same question about div:/mod: ?

The thing about div: and mod: is that, since the remainder is never 
negative, it facilitates doing many commonplace things.  For example, 
back in 2015, there used to be a sender of rem: (I believe) that would 
essentially give you the HSV hue given a number of radians.  Except of 
course when the receiver was negative, you'd get a negative hue, and 
that would break stuff because HSV hues are not negative.  It's much 
better to use mod:, rather than... you know... rem:, < 0 ifTrue: [2 pi - 
rem]... blah... this was an actual bug that was fixed with mod:, see 
Color>>hue:.

I think it could have been fixed by \\ too, but now you have to remember 
that \\ will give you the right sign of the remainder when the 
receiver's sign... blah blah... and all that intellectual overhead goes 
away using mod:, the one with non-negative remainders no matter what. 
Also, in that method, the code really means mod:, not \\ or rem:.

In number theory, the mod: operation with its never-negative remainders 
is the most natural for many applications (but not all, of course, 
finite calculus is much better with // and \\ because they lead to 
stable behavior of floor and ceiling).  Now it doesn't matter which 
integer you divide by: you always get positive remainders, which feed 
naturally into equivalence classes with non-negative representatives, as 
well as histograms that do not have to account for signs.

> The integer division makes sense for specialized applications.
> I used it 30 years back for polynomial factorization, if I remember, 
> because it leads to smaller LargeIntegers in p-adic methods.

Do you know much about Dixon's factorization method?

> Does it make sense in some Core/Kernel image?

I think it makes at least some sense there, seeing how it's used helps 
developing an opinion.  Apparently Luciano is already using the idea, 
why have it twice.  However...

> The float remainder might have less applications, I haven't one myself, 
> apart for some numerical analysis...
... so, let's take a closer look then.  Maybe we can find better names, 
kind of along the lines of what Luciano says?

> I even asked the question here: 
> https://cs.stackexchange.com/questions/24362/why-does-floating-point-modulus-exactness-matters
> Extending IEC 60559 compliancy could be a goal per se for a generalist 
> language, but that's a mixture of engineering/marketing decisions.
> The kind of little decisions which might make a difference in the end, 
> if you read recent post from Gilad, 
> https://gbracha.blogspot.com/2020/05/bits-of-history-words-of-advice.htm
What you don't learn by comprehension, you will learn by suffering.

Andres.