References: CLtL p.208, 212
Related issues: IEEE-ATAN-BRANCH-CUT
Edit history: Version 1, 13-Dec-88, Steele
The formula that defines ATAN results in a branch cut that is at
variance with the recommendations of Prof. W. Kahan and with the
implementations of that function in many computing systems and
Replace the formula
arctan z = - i log ((1+iz) sqrt (1/(1+z^2)))
with the formula
arctan z = (log (1+iz) - log (1-iz)) / (2i)
This leaves the branch cuts pretty much in place; the only change is
that the upper branch cut (on the positive imaginary axis above i)
is continuous with quadrant I, where the old formula has it continuous
with quadrant II.
(atan #c(0 2)) => #c(-1.57... 0.549...) ;Current
(atan #c(0 2)) => #c(1.57... -0.549...) ;Proposed
Note: 1.57... = pi/2, and 0.549... = (log 3)/2.
Compatibility with what seems to be becoming standard practice.
(atan #c(0 2)) => #c(-1.57... 0.549...) ;Symbolics CL
(atan #c(0 2)) => #c(-1.57... 0.549...) ;Allegro CL 1.1 (Macintosh)
(atan #c(0 2)) => #c(-1.57... 0.549...) ;Sun-4 CL 2.1.3 of 10-Nov-88
(atan #c(0 2)) => #c(1.57... -0.549...) ;Sun CL 2.0.3 of 30-Jun-87
(atan #c(0 2)) => #c(1.57... 0.549...) ;KCL of 3-Jun-87
Note that in KCL the upper branch cut is thus continuous with
quadrant I, but its lower branch cut is continuous with quadrant III!
Cost to Implementors:
ATAN must be rewritten. It is not a very difficult fix.
Cost to Users:
It is barely conceivable that some user code could depend on this.
Note that the proposed change invalidates the identities
arctan i z = i arctanh z
and arctanh i z = i arctan z
on the upper branch cut.
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:
Incompatibility with HP calculators.
Numerical analystsmay find the new definition easier to use.
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.
Paul Penfield of MIT, after whose article the Common Lisp branch
cuts were originally patterned, endorses this change.