Unexpected behaviour of math.floor, round and int functions (rounding)

[Chris Angelico]

On Sat, Nov 20, 2021 at 3:41 PM Ben Bacarisse <ben.usenet at bsb.me.uk> wrote:
>
> Chris Angelico <rosuav at gmail.com> writes:
>
> > On Sat, Nov 20, 2021 at 12:43 PM Ben Bacarisse <ben.usenet at bsb.me.uk> wrote:
> >>
> >> Chris Angelico <rosuav at gmail.com> writes:
> >>
> >> > On Sat, Nov 20, 2021 at 9:07 AM Ben Bacarisse <ben.usenet at bsb.me.uk> wrote:
> >> >>
> >> >> Chris Angelico <rosuav at gmail.com> writes:
> >> >>
> >> >> > On Sat, Nov 20, 2021 at 5:08 AM ast <ast at invalid> wrote:
> >> >>
> >> >> >>  >>> 0.3 + 0.3 + 0.3 == 0.9
> >> >> >> False
> >> >> >
> >> >> > That's because 0.3 is not 3/10. It's not because floats are
> >> >> > "unreliable" or "inaccurate". It's because the ones you're entering
> >> >> > are not what you think they are.
> >> >> >
> >> >> > When will people understand this?
> >> >> >
> >> >> > (Probably never. Sigh.)
> >> >>
> >> >> Most people understand what's going on when it's explained to them.  And
> >> >> I think that being initially baffled is not unreasonable.  After all,
> >> >> almost everyone comes to computers after learning that 3/10 can be
> >> >> written as 0.3.  And Python "plays along" with the fiction to some
> >> >> extent.  0.3 prints as 0.3, 3/10 prints as 0.3 and 0.3 == 3/10 is True.
> >> >
> >> > In grade school, we learn that not everything can be written that way,
> >> > and 1/3 isn't actually equal to 0.3333333333.
> >>
> >> Yes.  We learn early on that 0.3333333333 means 3333333333/10000000000.
> >> We don't learn that 0.3333333333 is a special notation for machines that
> >> have something called "binary floating point hardware" that does not
> >> mean 3333333333/10000000000.  That has to be learned later.  And every
> >> generation has to learn it afresh.
> >
> > But you learn that it isn't the same as 1/3. That's my point. You
> > already understand that it is *impossible* to write out 1/3 in
> > decimal. Is it such a stretch to discover that you cannot write 3/10
> > in binary?
> >
> > Every generation has to learn about repeating fractions, but most of
> > us learn them in grade school. Every generation learns that computers
> > talk in binary. Yet, putting those two concepts together seems beyond
> > many people, to the point that they feel that floating point can't be
> > trusted.
>
> Binary is a bit of a red herring here.  It's the floating point format
> that needs to be understood.  Three tenths can be represented in many
> binary formats, and even decimal floating point will have some surprises
> for the novice.

Not completely a red herring; binary floating-point as used in Python
(IEEE double-precision) is defined as a binary mantissa and a scale,
just as "blackboard arithmetic" is generally defined as a decimal
mantissa and a scale. (At least, I don't think I've ever seen anyone
doing arithmetic on a blackboard in hex or octal.)

> >> Yes, agreed, but I was not commenting on the odd (and incorrect) view
> >> that floating point operations are not reliable and well-defined, but on
> >> the reasonable assumption that a clever programming language might take
> >> 0.3 to mean what I was taught it meant in grade school.
> >
> > It does mean exactly what it meant in grade school, just as 1/3 means
> > exactly what it meant in grade school. Now try to represent 1/3 on a
> > blackboard, as a decimal fraction. If that's impossible, does it mean
> > that 1/3 doesn't mean 1/3, or that 1/3 can't be represented?
>
> As you know, it is possible, but let's say we outlaw any finite notation
> for repeated digits...  Why should I convert 1/3 to this particular
> apparently unsuitable representation?  I will write 1/3 and manipulate
> that number using factional notation.

If you want that, the fractions module is there for you. And again,
grade school, we learned about ratios as well as decimals (or vulgar
fractions and decimal fractions). They have different tradeoffs. For
instance, I learned pi as both 22/7 and 3.14, because sometimes it'd
be convenient to use the rational form and other times the decimal.

> The novice programmer might similarly expect that when they write 0.3,
> the program will manipulate that number as the faction it clearly is.
> They may well be surprised by the fact that it must get put into a
> format that can't represent what those three characters mean, just as I
> would be surprised if you insisted I write 1/3 as a finite decimal (with
> no repeat notation).

Except that 0.3 isn't written as a fraction, it's written as a decimal.

> I'm not saying your analogy would not help someone understand, but you
> first have to explain why 0.3 is not treated as three tenths -- why I
> (to use your analogy) must not keep 1/3 as a proper fraction, but I must
> instead write it using a finite number of decimal digits.  Neither is,
> in my view, obvious to the beginner.

Try adding 1/3 + e; either you have to convert 1/3 to a decimal, or
find a rational approximation for e (there aren't any really good ones
but 193/71 seems promising - that's 2.7183, close enough) and then go
to the work of rational addition. No, more likely you'll go for a
finite number of decimal digits.

> >> > But lack of knowledge is never a problem. (Or rather, it's a solvable
> >> > problem, and I'm always happy to explain things to people.) The
> >> > problem is when, following that lack of understanding, people assume
> >> > that floats are "unreliable" or "inaccurate", and that you should
> >> > never ever compare two floats for equality, because they're never
> >> > "really equal". That's what leads to horrible coding practices and
> >> > badly-defined "approximately equal" checks that cause far more harm
> >> > than a simple misunderstanding ever could on its own.
> >>
> >> Agreed.  Often, the "explanations" just make things worse.
> >
> > When they're based on a fear of floats, yes. Explanations like "never
> > use == with floats because 0.1+0.2!=0.3" are worse than useless,
> > because they create that fear in a way that creates awful cargo-cult
> > programming practices.
> >
> > If someone does something in Python, gets a weird result, and comes to
> > the list saying "I don't understand this", that's not a problem. We
> > get it all the time with mutables. Recent question about frozensets
> > appearing to be mutable, same thing. I have no problem with someone
> > humbly asking "what's happening?", based on an internal assumption
> > that there's a reason things are the way they are. For some reason,
> > floats don't get that same respect from many people.
>
> On all this, I agree.  As a former numerical analyst, I want maximal
> respect for all the various floating-point representations!
>

Yeah - or if not the full meaning of the representation, at least the
fact that it involves no more than 53 bits of "number", and all the
consequences of that.

Oh, and at least once, everyone should play with a stringified number
system, where you actually truly store the string representation and
do arithmetic on it. It has very interesting consequences that really
make you appreciate IEEE binary floating-point.

ChrisA