Status: Passed, Jan 89 X3J13
References: CLtL p.203-214
Related issues: COMPLEX-ATAN-BRANCH-CUT
Edit history: Version 1, 13-Dec-88, Steele
Version 2, 11-Jan-89, Masinter
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.
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.
In this propoal Lisp expressions should be taken as mathematical
formulas that actually are 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:
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
(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)
The current specification ignores some natural consequences of IEEE
floating-point arithmetic. The proposed specification produces results
more natural to that arithmetic.
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
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
Natural treatment, etc.
A toss-up, except to those who care.
Steele has sent a letter to W. Kahan at Berkeley to get any last
comments he may have on the matter.