[Cfp-interest 3336] Re: special cases of cpow

[Jim Thomas]

Paul, 

Here’s a try at deducing an answer to your question using mathematical functions extended to match C and IEEE 754 special cases. As you note, Annex G defines cpow(z, w) to have a branch cut for z along the negative real axis. This fits with clog(z) which is defined to have a branch cut along the negative real axis, and the range of the strip [-pi i, +pi i] along the imaginary axis. Evaluating z ^ w = (-4 + 0 i) ^ (1/2 + 0 i) as e ^ (w ln(z)) ...

ln(-4 + 0i) = ln(4) + pi i

(1/2 + 0 i) (ln(4) + pi i) = ln(4)/2 + pi/2 i

Using the Annex F (and IEEE 754) specification that cospi(n + 1/2) returns +0 for integers n ...

e^(ln(4)/2 + pi/2 i) = e^(ln(4)/2) cis(pi/2) = 2 (+0 + i) = +0 + 2 i


- Jim Thomas



> On Nov 21, 2024, at 6:42 AM, Paul Zimmermann <Paul.Zimmermann at inria.fr> wrote:
> 
>       Hi,
> 
> with Andreas Enge (in cc), co-author of GNU MPC, we wonder about the special cases of
> the complex power, in particular when one of the arguments has a zero part (+0 or -0).
> 
> For the real power, IEEE 754 completely specifies the special cases of pow.
> 
> Is there a similar document for the complex power? N3301 only says "The cpow functions compute
> the complex power function xy, with a branch cut for the first parameter along the negative real
> axis." and "This allows cpow(z, c) to be implemented as cexp(c*clog(z)) without precluding
> implementations that treat special cases more carefully".
> 
> For example, what should (-4,+0) ^ (0.5, +0) return? (+0, 2) or (-0,2)?
> If we replace each +0 by a tiny value, we get different signs for the real part of the
> result depending on the ratio of these tiny values:
> 
> sage: e=1e-6
> sage: (-4+e*i)^(0.5+e^2*i)
> 2.49997227485645e-7 + 1.99999999999373*I
> sage: (-4+e^2*i)^(0.5+e*i)
> -2.77257976169088e-6 + 1.99999371682264*I
> 
> Paul
> 
> 
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