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<TITLE>CLHS: Issue IEEE-ATAN-BRANCH-CUT Writeup</TITLE>
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<H1><A REV=MADE HREF="http://www.lispworks.com/"><IMG WIDTH=80 HEIGHT=65 ALT="[LISPWORKS]" SRC="../Graphics/LWSmall.gif" ALIGN=Bottom></A><A REL=TOP HREF="../Front/index.htm"><IMG WIDTH=237 HEIGHT=65 ALT="[Common Lisp HyperSpec (TM)]" SRC="../Graphics/CLHS_Sm.gif" ALIGN=Bottom></A> <A REL=PREV HREF="../Issues/iss191_w.htm"><IMG WIDTH=40 HEIGHT=40 ALT="[Previous]" SRC="../Graphics/Prev.gif" ALIGN=Bottom></A><A REL=UP HREF="../Issues/iss192.htm"><IMG WIDTH=40 HEIGHT=40 ALT="[Up]" SRC="../Graphics/Up.gif" ALIGN=Bottom></A><A REL=NEXT HREF="../Issues/iss193_w.htm"><IMG WIDTH=40 HEIGHT=40 ALT="[Next]" SRC="../Graphics/Next.gif" ALIGN=Bottom></A></H1>
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<H2>Issue IEEE-ATAN-BRANCH-CUT Writeup</H2>
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<PRE><B>Status:</B> Passed, Jan 89 X3J13<P>
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<B>Forum:</B> Cleanup<P>
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<B>Issue:</B> <A HREF="iss192.htm">IEEE-ATAN-BRANCH-CUT</A><P>
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<B>References:</B> CLtL p.203-214<P>
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Related issues: <A HREF="iss069.htm">COMPLEX-ATAN-BRANCH-CUT</A><P>
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<B>Category:</B> CHANGE<P>
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<P>
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<B>Edit history:</B> Version 1, 13-Dec-88, Steele<P>
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Version 2, 11-Jan-89, Masinter <P>
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<P>
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<B>Problem description:<P>
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</B><P>
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If an implementation provides a floating-point minus zero as well as a<P>
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floating-point plus zero, most notably as in IEEE 754 floating-point<P>
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arithmetic, then slight adjustments in the branch cuts of transcendental<P>
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functions are appropriate.<P>
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<P>
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If there is a minus zero and a plus zero, then *no* complex number<P>
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is actually "on the axis" whether real or imaginary. Instead,<P>
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numbers of the form x+0i and x-0i straddle the real axis, and those<P>
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of the form 0+xi and -0+xi straddle the imginary axis. Branch cuts<P>
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that lie on the axes therefore run between such numbers, and directions<P>
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of continuity are not an issue.<P>
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<P>
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<B>Proposal (IEEE-ATAN-BRANCH-CUT:SPLIT):<P>
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</B><P>
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Redefine the branch cut for two-argument <A REL=DEFINITION HREF="../Body/f_asin_.htm#atan"><B>ATAN</B></A>, covering<P>
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the cases where there is or is not a minus zero, and then redefine *all*<P>
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other functions that have branch cuts in terms of two-argument <A REL=DEFINITION HREF="../Body/f_asin_.htm#atan"><B>ATAN</B></A>.<P>
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Specifically, we first define <A REL=DEFINITION HREF="../Body/f_phase.htm#phase"><B>PHASE</B></A> in terms of two-argument <A REL=DEFINITION HREF="../Body/f_asin_.htm#atan"><B>ATAN</B></A>,<P>
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and complex <A REL=DEFINITION HREF="../Body/f_abs.htm#abs"><B>ABS</B></A> in terms of real <A REL=DEFINITION HREF="../Body/f_sqrt_.htm#sqrt"><B>SQRT</B></A> (which has no branch cut problems);<P>
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then define complex <A REL=DEFINITION HREF="../Body/f_log.htm#log"><B>LOG</B></A> in terms of <A REL=DEFINITION HREF="../Body/f_phase.htm#phase"><B>PHASE</B></A>, <A REL=DEFINITION HREF="../Body/f_abs.htm#abs"><B>ABS</B></A>, and real LOG; then define<P>
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complex <A REL=DEFINITION HREF="../Body/f_sqrt_.htm#sqrt"><B>SQRT</B></A> in terms of LOG; and then define all others in terms of these.<P>
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<P>
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In this propoal Lisp expressions should be taken as mathematical<P>
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formulas that actually are implemented in other ways for improved accuracy.<P>
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<P>
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(1) If there is no minus zero, two-argument <A REL=DEFINITION HREF="../Body/f_asin_.htm#atan"><B>ATAN</B></A> behaves as in CLtL.<P>
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If there is a minus zero, then some cases are modified:<P>
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Condition result<P>
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y=+0 x>0 +0<P>
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y=-0 x>0 -0<P>
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y=+0 x<0 +pi<P>
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y=-0 x<0 -pi<P>
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y=+0 x=+0 +0<P>
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y=-0 x=+0 -0<P>
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y=+0 x=-0 +pi<P>
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y=-0 x=-0 -pi<P>
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The range of two-argument <A REL=DEFINITION HREF="../Body/f_asin_.htm#atan"><B>atan</B></A> therefore includes -pi in this case.<P>
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<P>
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Note that the case y=0 x=0 is an error in the absence of minus zero,<P>
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but is defined in the presence of minus zero.<P>
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<P>
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(2) (<A REL=DEFINITION HREF="../Body/f_phase.htm#phase"><B>PHASE</B></A> X) = (<A REL=DEFINITION HREF="../Body/f_asin_.htm#atan"><B>ATAN</B></A> (<A REL=DEFINITION HREF="../Body/f_realpa.htm#imagpart"><B>IMAGPART</B></A> X) (<A REL=DEFINITION HREF="../Body/f_realpa.htm#realpart"><B>REALPART</B></A> X)), as before, but now<P>
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the range of <A REL=DEFINITION HREF="../Body/f_phase.htm#phase"><B>PHASE</B></A> may include -PI if there is a minus zero.<P>
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<P>
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(3) (<A REL=DEFINITION HREF="../Body/f_abs.htm#abs"><B>ABS</B></A> X) = (<A REL=DEFINITION HREF="../Body/f_sqrt_.htm#sqrt"><B>SQRT</B></A> (+ (* (<A REL=DEFINITION HREF="../Body/f_realpa.htm#realpart"><B>REALPART</B></A> X) (<A REL=DEFINITION HREF="../Body/f_realpa.htm#realpart"><B>REALPART</B></A> X))<P>
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(* (<A REL=DEFINITION HREF="../Body/f_realpa.htm#imagpart"><B>IMAGPART</B></A> X) (<A REL=DEFINITION HREF="../Body/f_realpa.htm#imagpart"><B>IMAGPART</B></A> X)))), as before<P>
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<P>
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(4) (<A REL=DEFINITION HREF="../Body/f_log.htm#log"><B>LOG</B></A> X) = (<A REL=DEFINITION HREF="../Body/a_comple.htm#complex"><B>COMPLEX</B></A> (<A REL=DEFINITION HREF="../Body/f_log.htm#log"><B>LOG</B></A> (<A REL=DEFINITION HREF="../Body/f_abs.htm#abs"><B>ABS</B></A> X)) (<A REL=DEFINITION HREF="../Body/f_phase.htm#phase"><B>PHASE</B></A> X))<P>
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<P>
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(5) (<A REL=DEFINITION HREF="../Body/f_sqrt_.htm#sqrt"><B>SQRT</B></A> X) = (<A REL=DEFINITION HREF="../Body/f_exp_e.htm#exp"><B>EXP</B></A> (/ (<A REL=DEFINITION HREF="../Body/f_log.htm#log"><B>LOG</B></A> X) 2))<P>
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<P>
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(6) For <A REL=DEFINITION HREF="../Body/f_exp_e.htm#expt"><B>EXPT</B></A>, <A REL=DEFINITION HREF="../Body/f_asin_.htm#asin"><B>ASIN</B></A>, <A REL=DEFINITION HREF="../Body/f_asin_.htm#acos"><B>ACOS</B></A>, <A REL=DEFINITION HREF="../Body/f_asin_.htm#atan"><B>ATAN</B></A>, <A REL=DEFINITION HREF="../Body/f_sinh_.htm#asinh"><B>ASINH</B></A>, <A REL=DEFINITION HREF="../Body/f_sinh_.htm#acosh"><B>ACOSH</B></A>, <A REL=DEFINITION HREF="../Body/f_sinh_.htm#atanh"><B>ATANH</B></A> use the formulas<P>
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in CLtL pp. 211-213, but use the formulas (1-5) above as the<P>
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definitions of <A REL=DEFINITION HREF="../Body/f_log.htm#log"><B>LOG</B></A> and <A REL=DEFINITION HREF="../Body/f_sqrt_.htm#sqrt"><B>SQRT</B></A> in order to determine the branch cuts<P>
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properly.<P>
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<P>
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<B>Examples:<P>
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</B><P>
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(<A REL=DEFINITION HREF="../Body/f_log.htm#log"><B>LOG</B></A> #c(-1.0 +0.0)) => #c(0.0 3.14159...) ;Current<P>
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(<A REL=DEFINITION HREF="../Body/f_log.htm#log"><B>LOG</B></A> #c(-1.0 -0.0)) => #c(0.0 3.14159...) ;Current<P>
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<P>
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(<A REL=DEFINITION HREF="../Body/f_log.htm#log"><B>LOG</B></A> #c(-1.0 +0.0)) => #c(0.0 3.14159...) ;Proposed (= current)<P>
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(<A REL=DEFINITION HREF="../Body/f_log.htm#log"><B>LOG</B></A> #c(-1.0 -0.0)) => #c(0.0 -3.14159...) ;Proposed (conjugate)<P>
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<P>
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<B>Rationale:<P>
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</B><P>
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The current specification ignores some natural consequences of IEEE<P>
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floating-point arithmetic. The proposed specification produces results<P>
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more natural to that arithmetic.<P>
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<P>
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<P>
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<B>Current practice:<P>
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</B><P>
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Of implementations that support a minus zero that I have checked,<P>
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such as Sun-4 CL 2.1.3 of 10-Nov-88 and Symbolics CL, all follow<P>
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the current CLtL specification.<P>
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<P>
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The IEEE trig library atan2 routine written by K.C. Ng (with the advice<P>
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or supervision of W. Kahan, I believe), and distributed with BSD UNIX<P>
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(it is file /usr/src/usr.lib/libm/IEEE/atan2.c on my machine) follows<P>
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this proposal for treatment of minus zero.<P>
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<P>
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<P>
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<B>Cost to Implementors:<P>
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</B><P>
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Some of the trig routines will <A REL=DEFINITION HREF="../Body/f_provid.htm#require"><B>require</B></A> some rewriting. Probably certain<P>
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tests that are now written using <A REL=DEFINITION HREF="../Body/f_zerop.htm#zerop"><B>ZEROP</B></A> need to be rewritten to use<P>
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<A REL=DEFINITION HREF="../Body/f_dec_fl.htm#float-sign"><B>FLOAT-SIGN</B></A> instead.<P>
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<P>
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<P>
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<B>Cost to Users:<P>
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</B><P>
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It is barely conceivable that some user code could depend on this.<P>
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Probably there is no cost.<P>
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<P>
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The compatibility note on p. 210 of CLtL gave users fair warning that<P>
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a change of this kind might be adopted.<P>
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<P>
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<B>Cost of non-adoption:<P>
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</B><P>
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Unnatural treatment of some functions on machines supporting IEEE<P>
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floating-point arithmetic.<P>
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<P>
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<B>Benefits:<P>
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</B><P>
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Natural treatment, etc.<P>
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<P>
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<B>Esthetics:<P>
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</B><P>
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A toss-up, except to those who care.<P>
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<P>
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<B>Discussion:<P>
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</B><P>
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Steele has sent a letter to W. Kahan at Berkeley to get any last<P>
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comments he may have on the matter.<P>
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</PRE>
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