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<TITLE>CLHS: Issue COMPLEX-ATAN-BRANCH-CUT Writeup</TITLE>
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<H2>Issue COMPLEX-ATAN-BRANCH-CUT Writeup</H2>
<PRE><B>Forum:</B> Cleanup<P>
<B>Issue:</B> <A HREF="iss069.htm">COMPLEX-ATAN-BRANCH-CUT</A><P>
<B>References:</B> CLtL p.208, 212<P>
Related issues: <A HREF="iss192.htm">IEEE-ATAN-BRANCH-CUT</A><P>
<B>Category:</B> CHANGE<P>
<P>
<B>Edit history:</B> Version 1, 13-Dec-88, Steele<P>
<P>
<B>Problem description:<P>
</B><P>
The formula that defines <A REL=DEFINITION HREF="../Body/f_asin_.htm#atan"><B>ATAN</B></A> results in a branch cut that is at<P>
variance with the recommendations of Prof. W. Kahan and with the<P>
implementations of that function in many computing systems and<P>
calculators.<P>
<P>
<B>Proposal (COMPLEX-ATAN-BRANCH-CUT:TWEAK):<P>
</B> <P>
Replace the formula<P>
<P>
arctan z = - i log ((1+iz) <A REL=DEFINITION HREF="../Body/f_sqrt_.htm#sqrt"><B>sqrt</B></A> (1/(1+z^2)))<P>
<P>
with the formula<P>
<P>
arctan z = (log (1+iz) - log (1-iz)) / (2i)<P>
<P>
This leaves the branch cuts pretty much in place; the only change is<P>
that the upper branch cut (on the positive imaginary axis above i)<P>
is continuous with quadrant I, where the old formula has it continuous<P>
with quadrant II.<P>
<P>
<B>Examples:<P>
</B><P>
(<A REL=DEFINITION HREF="../Body/f_asin_.htm#atan"><B>atan</B></A> #c(0 2)) =&gt; #c(-1.57... 0.549...) ;Current<P>
(<A REL=DEFINITION HREF="../Body/f_asin_.htm#atan"><B>atan</B></A> #c(0 2)) =&gt; #c(1.57... -0.549...) ;Proposed<P>
<P>
Note: 1.57... = pi/2, and 0.549... = (log 3)/2.<P>
<P>
<B>Rationale:<P>
</B><P>
Compatibility with what seems to be becoming <A REL=DEFINITION HREF="../Body/07_ffb.htm#standard"><B>standard</B></A> practice.<P>
<P>
<P>
<B>Current practice:<P>
</B><P>
(<A REL=DEFINITION HREF="../Body/f_asin_.htm#atan"><B>atan</B></A> #c(0 2)) =&gt; #c(-1.57... 0.549...) ;Symbolics CL<P>
(<A REL=DEFINITION HREF="../Body/f_asin_.htm#atan"><B>atan</B></A> #c(0 2)) =&gt; #c(-1.57... 0.549...) ;Allegro CL 1.1 (Macintosh)<P>
(<A REL=DEFINITION HREF="../Body/f_asin_.htm#atan"><B>atan</B></A> #c(0 2)) =&gt; #c(-1.57... 0.549...) ;Sun-4 CL 2.1.3 of 10-Nov-88<P>
(<A REL=DEFINITION HREF="../Body/f_asin_.htm#atan"><B>atan</B></A> #c(0 2)) =&gt; #c(1.57... -0.549...) ;Sun CL 2.0.3 of 30-Jun-87<P>
(<A REL=DEFINITION HREF="../Body/f_asin_.htm#atan"><B>atan</B></A> #c(0 2)) =&gt; #c(1.57... 0.549...) ;KCL of 3-Jun-87<P>
<P>
Note that in KCL the upper branch cut is thus continuous with<P>
quadrant I, but its lower branch cut is continuous with quadrant III!<P>
<P>
<B>Cost to Implementors:<P>
</B><P>
<A REL=DEFINITION HREF="../Body/f_asin_.htm#atan"><B>ATAN</B></A> must be rewritten. It is not a very difficult fix.<P>
<P>
<B>Cost to Users:<P>
</B><P>
It is barely conceivable that some user code could depend on this.<P>
Note that the proposed change invalidates the identities<P>
arctan i z = i arctanh z<P>
and arctanh i z = i arctan z<P>
on the upper branch cut.<P>
<P>
The compatibility note on p. 210 of CLtL gave users fair warning that<P>
a change of this kind might be adopted.<P>
<P>
<B>Cost of non-adoption:<P>
</B><P>
Incompatibility with HP calculators.<P>
<P>
<B>Benefits:<P>
</B><P>
Numerical analystsmay find the new definition easier to use.<P>
<P>
<B>Esthetics:<P>
</B><P>
A toss-up, except to those who care.<P>
<P>
<B>Discussion:<P>
</B><P>
Steele has sent a letter to W. Kahan at Berkeley to get any last<P>
comments he may have on the matter.<P>
<P>
Paul Penfield of MIT, after whose article the Common Lisp branch<P>
cuts were originally patterned, endorses this change.<P>
<P>
</PRE>
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