New issue: IEEE-ATAN-BRANCH-CUT

[Guy Steele <gls@Think.COM>]

Forum:         Cleanup
Issue:         IEEE-ATAN-BRANCH-CUT
References:    CLtL p.203-214
Related issues: COMPLEX-ATAN-BRANCH-CUT
Category:      CHANGE

Edit history:  Version 1, 13-Dec-88, Steele

Problem description:

If an implementation provides a floating-point minus zero as well as a
floating-point plus zero, most notably as in IEEE 754 floating-point
arithmetic, then slight adjustments in the branch cuts of transcendental
functions are appropriate.

If there is a minus zero and a plus zero, then *no* complex number
is actually "on the axis" whether real or imaginary.  Instead,
numbers of the form x+0i and x-0i straddle the real axis, and those
of the form 0+xi and -0+xi straddle the imginary axis.  Branch cuts
that lie on the axes therefore run between such numbers, and directions
of continuity are not an issue.

Proposal (IEEE-ATAN-BRANCH-CUT:SPLIT):

We propose to redefine the branch cut for two-argument ATAN, covering
the cases where there is or is not a minus zero, and then redefine *all*
other functions that have branch cuts in terms of two-argument ATAN.
Specifically, we first define PHASE in terms of two-argument ATAN,
and complex ABS in terms of real SQRT (which has no branch cut problems);
then define complex LOG in terms of PHASE, ABS, and real LOG; then define
complex SQRT in terms of LOG; and then define all others in terms of these.

When I write Lisp expressions, I mean them to be taken as mathematical
formulas that could be implemented in other ways for improved accuracy.

(1) If there is no minus zero, two-argument ATAN behaves as in CLtL.
    If there is a minus zero, then some cases are modified:
	Condition			result
	y=+0  x>0			   +0
	y=-0  x>0			   -0
	y=+0  x<0			   +pi
	y=-0  x<0			   -pi
	y=+0  x=+0			   +0
	y=-0  x=+0			   -0
	y=+0  x=-0			   +pi
	y=-0  x=-0			   -pi
    The range of two-argument atan therefore includes -pi in this case.

    Note that the case y=0 x=0 is an error in the absence of minus zero,
    but is defined in the presence of minus zero.

(2) (PHASE X) = (ATAN (IMAGPART X) (REALPART X)), as before, but now
    the range of PHASE may include -PI if there is a minus zero.

(3) (ABS X) = (SQRT (+ (* (REALPART X) (REALPART X))
		       (* (IMAGPART X) (IMAGPART X)))), as before

(4) (LOG X) = (COMPLEX (LOG (ABS X)) (PHASE X))

(5) (SQRT X) = (EXP (/ (LOG X) 2))

(6) For EXPT, ASIN, ACOS, ATAN, ASINH, ACOSH, ATANH use the formulas
    in CLtL pp. 211-213, but use the formulas (1-5) above as the
    definitions of LOG and SQRT in order to determine the branch cuts
    properly.

Examples:

(LOG #c(-1.0 +0.0)) => #c(0.0 3.14159...)	;Current
(LOG #c(-1.0 -0.0)) => #c(0.0 3.14159...)	;Current

(LOG #c(-1.0 +0.0)) => #c(0.0 3.14159...)	;Proposed (= current)
(LOG #c(-1.0 -0.0)) => #c(0.0 -3.14159...)	;Proposed (conjugate)

Rationale:

The current specification ignores some natural consequences of IEEE
floating-point arithmetic.  The proposed specification produces results
more natural to that arithmetic.

  
Current practice:

Of implementations that support a minus zero that I have checked,
such as Sun-4 CL 2.1.3 of 10-Nov-88 and Symbolics CL, all follow
the current CLtL specification.

The IEEE trig library atan2 routine written by K.C. Ng (with the advice
or supervision of W. Kahan, I believe), and distributed with BSD UNIX
(it is file /usr/src/usr.lib/libm/IEEE/atan2.c on my machine) follows
this proposal for treatment of minus zero.


Cost to Implementors:

Some of the trig routines will require some rewriting.  Probably certain
tests that are now written using ZEROP need to be rewritten to use
FLOAT-SIGN instead.


Cost to Users:

It is barely conceivable that some user code could depend on this.
Probably there is no cost.

The compatibility note on p. 210 of CLtL gave users fair warning that
a change of this kind might be adopted.

Cost of non-adoption:

Unnatural treatment of some functions on machines supporting IEEE
floating-point arithmetic.

Benefits:

Natural treatment, etc.

Esthetics:

A toss-up, except to those who care.

Discussion:

Steele has sent a letter to W. Kahan at Berkeley to get any last
comments he may have on the matter.