Issue COMPLEX-ATAN-BRANCH-CUT Writeup

Forum:         Cleanup

Issue:         COMPLEX-ATAN-BRANCH-CUT

References:    CLtL p.208, 212

Related issues: IEEE-ATAN-BRANCH-CUT

Category:      CHANGE


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


Problem description:


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

calculators.


Proposal (COMPLEX-ATAN-BRANCH-CUT:TWEAK):

  

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.


Examples:


(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.


Rationale:


Compatibility with what seems to be becoming standard practice.


  

Current 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.


Benefits:


Numerical analystsmay find the new definition easier to use.


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.


Paul Penfield of MIT, after whose article the Common Lisp branch

cuts were originally patterned, endorses this change.