cmath module documentation is misleading on branch cuts

[mdickinson]

BPO	41245
Nosy	@rhettinger, @mdickinson

^{Note: these values reflect the state of the issue at the time it was migrated and might not reflect the current state.}

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assignee = None
closed_at = None
created_at = <Date 2020-07-08.17:19:22.882>
labels = ['3.7', '3.8', '3.9', '3.10', 'docs']
title = 'cmath module documentation is misleading on branch cuts'
updated_at = <Date 2020-07-08.18:52:27.472>
user = 'https://github.com/mdickinson'

bugs.python.org fields:

activity = <Date 2020-07-08.18:52:27.472>
actor = 'mark.dickinson'
assignee = 'docs@python'
closed = False
closed_date = None
closer = None
components = ['Documentation']
creation = <Date 2020-07-08.17:19:22.882>
creator = 'mark.dickinson'
dependencies = []
files = []
hgrepos = []
issue_num = 41245
keywords = []
message_count = 5.0
messages = ['373323', '373325', '373328', '373329', '373330']
nosy_count = 3.0
nosy_names = ['rhettinger', 'mark.dickinson', 'docs@python']
pr_nums = []
priority = 'normal'
resolution = None
stage = None
status = 'open'
superseder = None
type = None
url = 'https://bugs.python.org/issue41245'
versions = ['Python 3.7', 'Python 3.8', 'Python 3.9', 'Python 3.10']

Linked PRs

• gh-85417: Clarify behaviour on branch cuts in cmath module #102046
• gh-85417: remove misleaning remarks about branch cuts for platforms without signed zero #102047
• [3.10] gh-85417: Clarify behaviour on branch cuts in cmath module (GH-102046) #102275
• [3.11] gh-85417: Clarify behaviour on branch cuts in cmath module (GH-102046) #102276

[mdickinson]

The documentation for the cmath module is misleading on the behaviour near branch cuts. For example, the documentation for cmath.acos says:

Return the arc cosine of x. There are two branch cuts: One
extends right from 1 along the real axis to ∞, continuous
from below. The other extends left from -1 along the real
axis to -∞, continuous from above.

That "continuous from below" and "continuous from above" language is misleading; in fact what happens on the vast majority of systems (those for which the floating-point format used is IEEE 754 binary64), if the imaginary part of x is zero, the sign of x is used to determine which side of the branch cut x lies.

[mdickinson]

the sign of x is used [...]

Correction: That should say "the sign of the imaginary part of x is used [...]"

[rhettinger]

+1 for changing the language to match the actual mechanics.

"the sign of the imaginary part of x is used [...]"

I'm trying to see where this happens. Is this part of cmath_sqrt?

    if (z.real >= 0.) {
        r.real = s;
        r.imag = copysign(d, z.imag); 
    } else {
        r.real = d;
        r.imag = copysign(s, z.imag);
    }

"continuous from below" and "continuous from above"
language is misleading;

I'm curious, is that language incorrect? My mental image of a branch cut is a helical graph with the edge cases being continuous from above and below. Likewise, my mental model for branch cut logic is it resolves multiple possible output values in a way preserves continuity from one side or the other.

In other words, I think about branch cuts in terms of continuity rather than sign preservation. Is that incorrect?

[mdickinson]

Yes, that looks like the right part of the sqrt code.

For the acos docstring, "continuous from below" implies that for any complex number z that lies exactly _on_ the branch cut, acos(z) is close to acos(w) for a nearby value w just _below_ the branch cut. But that's demonstrably not true: see the change of sign in the real part below:

>>> acos(complex(2.3, -1e-10))  # value just "below" the branch cut
(4.828045495852677e-11+1.475044781241425j)
>>> acos(complex(2.3, 0.0))  # nearby value exactly _on_ the branch cut
-1.475044781241425j

In effect, for a branch cut along the real axis, the sign of the zero in the imaginary part of the argument allows us to be continuous from both sides at once.

[mdickinson]

[...] see the change of sign in the real part below [...]

Grr. Stupid fingers. That should say "imaginary part", not "real part"

[skirpichev]

in fact what happens on the vast majority of systems (those for which the floating-point format used is IEEE 754 binary64), if the imaginary part of x is zero, the sign of x is used to determine which side of the branch cut x lies.

@mdickinson, it seems that some MP libraries use different convention, e.g.:

>>> from mpmath import acos
>>> acos(2 + 1e-10j)
mpc(real='5.773502691896258e-11', imag='-1.3169578969248168')
>>> acos(complex(2))
mpc(real='0.0', imag='1.3169578969248168')
>>> acos(2 - 1e-10j)
mpc(real='5.773502691896258e-11', imag='1.3169578969248168')
>>> acos(-2 + 1e-10j)
mpc(real='3.1415926535320584', imag='-1.3169578969248168')
>>> acos(complex(-2))
mpc(real='3.1415926535897931', imag='-1.3169578969248168')
>>> acos(-2 - 1e-10j)
mpc(real='3.1415926535320584', imag='1.3169578969248168')

Ditto Mathematica. That's exactly "continuous from below ... and ... continuous from above." from the cmath docs.

Shouldn't we adjust the implementation instead?

[mdickinson]

Shouldn't we adjust the implementation instead?

Definitely not! The implementation was very deliberately written this way. It's based on Kahan's "Branch Cuts for Complex Elementary Functions" paper, and it agrees with C99's Annex G and many other languages.

[skirpichev]

Ok, I see. Looks a bit surprising for complex analysis, but I see why it does make sense with signed zero floating-point arithmetics.

But it seems, that the current docs contains (since 2008) a note ("On platforms with hardware and system-level support for signed zeros, functions involving branch cuts are continuous on both sides of the branch cut: the sign of the zero distinguishes one side of the branch cut from the other. On platforms that do not support signed zeros the continuity is as specified below."). With this remark in mind - the acos docs seems correct, isn't?

Edit: So far, after 5ea052b - IEEE 754 support is a strict requirement. IIUIC, we can just leave the above remark (or like C11's Annex G "The functions are continuous onto both sides of their branch cuts, taking into account the sign of
zero. For example, csqrt (−2±i0) = ±i sqrt(2).") and skip all "continuous from" notes for all functions. Does it make sense as pr?