Re: Issue in man page clog.3

[Alejandro Colomar <alx@kernel.org>]

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Hi Branden,

On Wed, Nov 01, 2023 at 11:56:32AM -0500, G. Branden Robinson wrote:
> Hi Alex,
> 
> At 2023-11-01T16:00:16+0100, Alejandro Colomar wrote:
> > On Wed, Nov 01, 2023 at 02:02:10PM +0000, Helge Kreutzmann wrote:
> > > Without further ado, the following was found:
> > > 
> > > Issue:    [-pi,pi] means both -pi and pi are included, this does not
> > > make sense, either one must be out of the interval?
> > > 
> > > "The logarithm B<clog>()  is the inverse function of the exponential
> > > " "B<cexp>(3).  Thus, if I<y\\ =\\ clog(z)>, then I<z\\ =\\
> > > cexp(y)>.  The " "imaginary part of I<y> is chosen in the interval
> > > [-pi,pi]."
> > 
> > I don't know this function.  Please suggest a fix, and CC glibc so
> > that they can review the change.
> 
> The complex logarithm is the inverse function of the complex
> exponential, with which you may be familiar if you've taken a course in
> ordinary differential equations.

Yup, I have.  This reminds me of some old TODO I had: take a course on
multivariable calculus from a professor I like:
<https://www2.math.upenn.edu/~ghrist/BLUE.html>

The course on single-variable calculus from him was quite good.
<https://www2.math.upenn.edu/~ghrist/calculus.html>

> 
> Euler's formula famously relates the trigonometric and exponential
> functions.
> 
> exp(i*x) = cos x + i sin x
> 
> While the exponential function is aperiodic, that is for y=exp(x), no
> value of y ever occurs more than once for any real x, this is not true
> of a complex x or (equivalently[1]), the complex exponential, or real x
> multiplied by the imaginary unit i as seen in Euler's formula.
> 
> (Recall that for any x, sin(x) and cos(x) take on values in the interval
> [-1, 1], and i is a constant imaginary unit that we can interpret as a
> y axis.  Thus the complex exponential maps any real x to a point on the
> unit circle.)

e^(ix) is something I can visualize thanks to Euler, and e^(ix + y),
well, not so much, but by combination of e^(ix) and e^y I can
understand, but

> 
> In other words, for us to have an inverse function for the complex
> exponential, we must impose a restriction on its range, lest it give us
> an infinite vector of solutions.  Geometrically, the complex logarithm
> asks, "given a point on the unit circle, which value of x to the complex
> exponential corresponds to it?"  But there is so single answer to that
> question.  It is still a useful one to ask, so we can apply a constraint
> on the range of the solution which will make the complex logarithm
> one-to-one.

I can't yet visualize a complex or simple imaginary logarithm.  I'm not
so imaginative at the moment.  :|

> 
> We do a similar thing for the arc sine function.  Given a value c in
> [-1, 1], what angle theta has c as its sine?  There is an unbounded
> number of answers.  If you plot y=sin(x), you will see that y takes on
> every real value from -1 to 1,[2] repeatedly and unendingly.
> 
> At any rate, I think this statement:
> 
> > > Issue:    [-pi,pi] means both -pi and pi are included, this does not
> > >           make sense, either one must be out of the interval?
> 
> Needs more support.  clog(1) = pi and clog(-1) = -pi.  The limits of
> machine representation are applicable here, so you'll only ever get
> values "close to" ±pi anyway.
> 
> Also, POSIX and the ISO C committee didn't seem troubled by this;
> the same closed interval is issued in POSIX Issue 8 Draft 3, which says
> (in so many words) that the function's definition comes from ISO C99.
> 
> Oh, now that I'm about ready to send this, I see Jakub Wilk made the
> same point far more concisely.  Who's surprised?
> 
> Regards,
> Branden
> 
> [1] ...by applying of properties of powers such as x^(ab) = x^a * x^b.
> 
> [2] I, uh, don't actually have a proof of the claim "every value".  And
>     in fact I am unlikely to ever have one.  Per Niven's theorem, the
>     elementary trigonometric functions never take any rational values
>     _except_ 0, ±1, and ±1/2.[3]  This may be a startling result to
>     insightful students of elementary calculus, as, if one is paying
>     attention, one should then wonder why we can say with certainty that
>     any of these functions are differentiable, since there are jump
>     discontinuities in them--in fact a countably infinite number of
>     such discontinuities.[4]

You mean discontinuities in the representation, right?  Not in the
function, I presume.

Students should have in mind that a representation cannot be perfect in
some cases.  You can ask the same student to measure the rope that
circles a circumference of radius 1; it'll have a hard time with the
ruler.  :P

Cheers,
Alex

> 
>     Since our domain of discourse is computer arithmetic with finite
>     precision, it's an academic question, as with sufficient precision
>     you can have a sine function that will produce every representable
>     value within the function's range[5].
> 
> [3] https://en.wikipedia.org/wiki/Niven%27s_theorem
> 
> [4] I think it's a set of measure zero but I am now exceeding the limits
>     of my training.  I need more topology and theory of functions of a
>     real variable.  And to learn Lebesgue integration.
> 
> [5] I don't actually have a proof of that, either.  It seems not
>     impossible to me that the way IEEE floating point is defined means
>     that functions with the properties that trig functions have might
>     skip some representable values due to the nonlinearity of that
>     representation (the magnitude of error in floating-point math is not
>     constant, which is a reason some applications prefer fixed-point).
>     An "obvious" possibility is the set of rational values that have a
>     precise floating point representation in base 2, so 1/4th, -1/8th,
>     and so.  I need more numerical analysis, too.[6]  Anyway, one
>     counterexample disproves my claim, so let's settle for "almost
>     every", where we are using "almost" in a hand-wavy manner, not the
>     way someone studying the Dirichlet indicator function might.
> 
> [6] Obligatory citation of mandatory reading (Goldberg 1991):
> 
>     https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/02Numerics/Double/paper.pdf
> 
>     Press et al.'s books on numerical methods ("numerical recipes in
>     $LANG") are surprisingly controversial.



-- 
<https://www.alejandro-colomar.es/>

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