Friday question: negative zero

Hello everyone

The IEEE floating point standard allows for negative zero, but it's  
hard
to know that you have one in R.  One reliable test is to take the
reciprocal.  For example,

y <- 0
1/y

[1] Inf

y <- -y
1/y

[1] -Inf

The other day I came across one in complex numbers, and it took me a
while to figure out that negative zero was what was happening:

x <- complex(real = -1)
x

[1] -1+0i

1/x

[1] -1+0i

x^(1/3)

[1] 0.5+0.8660254i

(1/x)^(1/3)

[1] 0.5-0.8660254i

(The imaginary part of 1/x is negative zero.)

As a Friday question:  are there other ways to create and detect
negative zero in R?

And another somewhat more serious question:  is the behaviour of
negative zero consistent across platforms?  (The calculations above  
were
done in Windows in R-devel.)

I have been pondering branch cuts and branch points
for some functions which I am implementing.

In this area, it is very important to know whether one has
+0 or -0.

Take the log() function, where it is sometimes
very important to know whether one is just above the
imaginary axis or just below it:



(i).  Small y

 > y <- 1e-100
 > log(-1 +  1i*y)
[1] 0+3.141593i
 > y <- -y
 > log(-1 +  1i*y)
[1] 0-3.141593i


(ii)  Zero y.


 > y <- 0
 > log(-1 +  1i*y)
[1] 0+3.141593i
 > y <- -y
 > log(-1 +  1i*y)
[1] 0+3.141593i
 >


[ie small imaginary jumps have  a discontinuity, infinitesimal jumps  
don't].

This behaviour is undesirable (IMO): one would like log (-1+0i) to be  
different from log(-1-0i).

Tony Plate's example shows that
even though  y<- 0 ;  identical(y, -y) is TRUE,  one has identical(1/ 
y, 1/(-y)) is FALSE,
  so the sign is not discarded.

My complex function does have a branch cut that follows a portion of  
the negative real axis
but the other cuts follow absurdly complicated implicit equations.

At this point
one needs the IEEE requirement that x=x == +0  [ie not -0] for any  
real x; one then finds that
(s-t) and -(t-s) are numerically equal but not necessarily  
indistinguishable.

One of my earlier questions involved branch cuts for the inverse trig  
functions but
(IIRC) the patch I supplied only tested for the imaginary part being  
 >0; would it be
possible to include information about signed zero in these or other  
functions?

Duncan Murdoch

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Robin Hankin
Uncertainty Analyst and Neutral Theorist,
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