So you're arguing that the scalar is irrelevant? That `2*inf == inf`?

I disagree because: ```2*inf > inf```

And:

```# Given that: inf / inf = 1

# When we solve for symbol x: 2*inf*x = inf 2*x = 1 x = 1/2

# If we discard the scalar instead: 2*inf*x = inf inf*x = inf x = 1

# I think it's specious to argue that there are infinity solutions; that axioms of symbolic mathematics do not apply because infinity ```

This is relevant to the (now-forked) main thread if the plan is to return inf/-inf/+inf instead of raising ZeroDivisionError; so I'm replying to the main thread.

On Sun, Oct 11, 2020, 4:10 PM Chris Angelico rosuav@gmail.com wrote: On Mon, Oct 12, 2020 at 5:06 AM Wes Turner wes.turner@gmail.com wrote:

SymPy ComplexInfinity, 1/0 < 2/0, *tests* for symbolic results

FWIW, SymPy (a CAS: Computer Algebra System) has Infinity,

NegativeInfinity, ComplexInfinity.

Regarding a symbolic result for 1/0:

If 1/0 is infinity (because 0 goes into 1 infinity times), is 2/0 2*inifnity (because 0 goes into 2 2 times more than into 1)

If you try to treat "infinity" as an actual number, you're inevitably going to run into paradoxes. Consider instead: 1/x tends towards +∞ as x tends towards 0 (if x starts out positive), therefore we consider that 1/0 is +∞. By that logic, the limit of 2/0 is the exact same thing. It's still not a perfect system, and division by zero is always going to cause problems, but it's far less paradoxical if you don't try to treat 2/0 as different from 1/0 :)

BTW, you're technically correct, in that 2/0 would be the same as 2 * (whatever 1/0 is), but that's because 2*x tends towards +∞ as x tends towards +∞, meaning that 2*∞ is also ∞.

ChrisA

On Sun, Oct 11, 2020 at 2:03 PM Wes Turner wes.turner@gmail.com wrote:

SymPy ComplexInfinity, 1/0 < 2/0, *tests* for symbolic results

FWIW, SymPy (a CAS: Computer Algebra System) has Infinity, NegativeInfinity, ComplexInfinity.

Regarding a symbolic result for 1/0:

If 1/0 is infinity (because 0 goes into 1 infinity times), is 2/0 2*inifnity (because 0 goes into 2 2 times more than into 1)

A proper CAS really is advisable. FWIU, different CAS have different outputs for the above problem (most just disregard the scalar because it's infinity so who care if that cancels out later).

Where are the existing test cases for arithemetic calculations with (scalar times) IEEE-754 int, +inf, or -inf as the output?

On Tue, Sep 15, 2020 at 1:54 AM David Mertz mertz@gnosis.cx wrote:

Thanks so much Ben for documenting all these examples. I've been frustrated by the inconsistencies, but hasn't realized all of those you note.

It would be a breaking change, but I'd really vastly prefer if almost all of those OverflowErrors and others were simply infinities. That's much closer to the spirit of IEEE-754.

The tricky case is 1./0. Division is such an ordinary operation, and it's so easy to get zero in a variable accidentally. That one still feels like an exception, but yes 1/1e-323 vs. 1/1e-324 would them remain a sore spot.

Likewise, a bunch of operations really should be NaN that are exceptions now.

On Mon, Sep 14, 2020, 5:26 PM Ben Rudiak-Gould benrudiak@gmail.com wrote:

On Mon, Sep 14, 2020 at 9:36 AM Stephen J. Turnbull < turnbull.stephen.fw@u.tsukuba.ac.jp> wrote:

Christopher Barker writes:

IEEE 754 is a very practical standard -- it was well designed, and is widely used and successful. It is not perfect, and in certain use

cases, it

may not be the best choice. But it's a really good idea to keep to

that

standard by default.

I feel the same way; I really wish Python was better about following IEEE 754.

I agree, but Python doesn't. It raises on some infs (generally

speaking, true infinities), and returns inf on others (generally speaking, overflows).

It seems to be very inconsistent. From testing just now:

math.lgamma(0) raises "ValueError: math domain error"

math.exp(1000) raises "OverflowError: math range error"

math.e ** 1000 raises "OverflowError: (34, 'Result too large')"

(math.e ** 500) * (math.e ** 500) returns inf

sum([1e308, 1e308]) returns inf

math.fsum([1e308, 1e308]) raises "OverflowError: intermediate overflow

in fsum"

math.fsum([1e308, inf, 1e308]) returns inf

math.fsum([inf, 1e308, 1e308]) raises "OverflowError: intermediate

overflow in fsum"

float('1e999') returns inf

float.fromhex('1p1024') raises "OverflowError: hexadecimal value too

large to represent as a float"

I get the impression that little planning has gone into this. There's no consistency in the OverflowError messages. 1./0. raises ZeroDivisionError which isn't a subclass of OverflowError. lgamma(0) raises a ValueError, which isn't even a subclass of ArithmeticError. The function has a pole at 0 with a well-defined two-sided limit of +inf. If it isn't going to return +inf then it ought to raise ZeroDivisionError, which should obviously be a subclass of OverflowError.

Because of the inconsistent handling of overflow, many functions aren't even monotonic. exp(2*x) returns a float for x <= 709.782712893384, raises OverflowError for 709.782712893384 < x <= 8.98846567431158e+307, and returns a float for x > 8.98846567431158e+307.

1./0. is not a true infinity. It's the reciprocal of a number that may have underflowed to zero. It's totally inconsistent to return inf for 1/1e-323 and raise an exception for 1/1e-324, as Python does.

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