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<TITLE>CLHS: Issue EXPT-RATIO 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/iss152_w.htm"><IMG WIDTH=40 HEIGHT=40 ALT="[Previous]" SRC="../Graphics/Prev.gif" ALIGN=Bottom></A><A REL=UP HREF="../Issues/iss153.htm"><IMG WIDTH=40 HEIGHT=40 ALT="[Up]" SRC="../Graphics/Up.gif" ALIGN=Bottom></A><A REL=NEXT HREF="../Issues/iss154_w.htm"><IMG WIDTH=40 HEIGHT=40 ALT="[Next]" SRC="../Graphics/Next.gif" ALIGN=Bottom></A></H1>
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<H2>Issue EXPT-RATIO Writeup</H2>
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<PRE><B>Forum:</B> Cleanup<P>
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<B>Issue:</B> <A HREF="iss153.htm">EXPT-RATIO</A><P>
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<P>
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<B>References:</B> CLtL pages 204 and 211<P>
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<P>
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<B>Category:</B> CLARIFICATION<P>
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<P>
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<B>Edit history:</B> Version 1, 4-Oct-88, by Aspinall and Moon<P>
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Version 2, 6-Oct-88, Masinter (very minor discussion)<P>
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Version 3, 31-Oct-88, Masinter (fix typo)<P>
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<P>
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<B>Problem description:<P>
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</B><P>
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The comment (page 204, 2nd para) that "... an implementation [of expt]<P>
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might choose to compute (<A REL=DEFINITION HREF="../Body/f_exp_e.htm#expt"><B>expt</B></A> x 3/2) as if it had been written<P>
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(<A REL=DEFINITION HREF="../Body/f_sqrt_.htm#sqrt"><B>sqrt</B></A> (<A REL=DEFINITION HREF="../Body/f_exp_e.htm#expt"><B>expt</B></A> x 3))" disagrees with the principal value definition on<P>
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page 211. See the example below for a case where the two disagree. We<P>
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believe the principal value definitions are consistent and reasonable,<P>
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therefore the implementation comment is wrong.<P>
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<P>
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<B>Proposal (EXPT-RATIO:P.211):<P>
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</B><P>
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Clarify that (<A REL=DEFINITION HREF="../Body/f_sqrt_.htm#sqrt"><B>sqrt</B></A> (<A REL=DEFINITION HREF="../Body/f_exp_e.htm#expt"><B>expt</B></A> x 3)) is not equivalent to (<A REL=DEFINITION HREF="../Body/f_exp_e.htm#expt"><B>expt</B></A> x 3/2)<P>
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and that page 211 rules.<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/m_defpar.htm#defvar"><B>defvar</B></A> x (<A REL=DEFINITION HREF="../Body/f_exp_e.htm#exp"><B>exp</B></A> (/ (* 2 <A REL=DEFINITION HREF="../Body/v_pi.htm#pi"><B>pi</B></A> #c(0 1)) 3))) ;exp(2.<A REL=DEFINITION HREF="../Body/v_pi.htm#pi"><B>pi</B></A>.i/3)<P>
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(<A REL=DEFINITION HREF="../Body/f_exp_e.htm#expt"><B>expt</B></A> x 3) => 1 (except for round-off error)<P>
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(<A REL=DEFINITION HREF="../Body/f_sqrt_.htm#sqrt"><B>sqrt</B></A> (<A REL=DEFINITION HREF="../Body/f_exp_e.htm#expt"><B>expt</B></A> x 3)) => 1 (except for round-off error)<P>
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(<A REL=DEFINITION HREF="../Body/f_exp_e.htm#expt"><B>expt</B></A> x 3/2) => -1 (except for round-off error)<P>
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<P>
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There can be no question that <P>
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(<A REL=DEFINITION HREF="../Body/f_exp_e.htm#expt"><B>expt</B></A> x 3) ==> 1<P>
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because <A REL=DEFINITION HREF="../Body/f_exp_e.htm#expt"><B>expt</B></A> is single-valued with an integer second argument, and<P>
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(<A REL=DEFINITION HREF="../Body/f_sqrt_.htm#sqrt"><B>sqrt</B></A> 1) ==> 1<P>
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definitely follows the principal branch of the square root function.<P>
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<P>
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But (<A REL=DEFINITION HREF="../Body/f_exp_e.htm#expt"><B>expt</B></A> x 3/2) is defined as (exp (* (log x) 3/2)) (page 211).<P>
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(log x) ==> 2.pi.i/3<P>
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according to the definition of the logarithm's branch cuts on page 211<P>
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(which really comes down to the branch cuts of phase - page 210), so<P>
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(* (log x) 3/2) ==> pi.i<P>
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and<P>
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exp(pi.i) is -1.<P>
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<P>
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<B>Rationale:<P>
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</B><P>
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We believe the principal value definitions are consistent and<P>
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reasonable, therefore the implementation comment is wrong.<P>
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<P>
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<B>Current practice:<P>
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</B><P>
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Symbolics Genera 7.3 currently returns the wrong answer, following page<P>
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204 rather than page 211. Lucid Common Lisp, and <P>
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Envos Medley implement the proposal.<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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The obvious code changes in complex <A REL=DEFINITION HREF="../Body/f_exp_e.htm#expt"><B>expt</B></A>.<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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None.<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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Self-contradictory language specification.<P>
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<P>
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<B>Benefits:<P>
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</B><P>
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Users can better predict the branch cuts in <A REL=DEFINITION HREF="../Body/f_exp_e.htm#expt"><B>expt</B></A>.<P>
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<P>
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<B>Discussion:<P>
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</B><P>
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Mathematical Explanation: When the <A REL=DEFINITION HREF="../Body/f_exp_e.htm#expt"><B>expt</B></A> function returns a complex result<P>
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in CL (Cartesian) form, the phase of the complex number is effectively<P>
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canonicalized. Information is lost, and that information is necessary to<P>
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specify upon which branch of the <A REL=DEFINITION HREF="../Body/f_sqrt_.htm#sqrt"><B>sqrt</B></A> function the final result should lie.<P>
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<P>
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Another way to put it would be that although<P>
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<A REL=DEFINITION HREF="../Body/f_sqrt_.htm#sqrt"><B>sqrt</B></A>(<A REL=DEFINITION HREF="../Body/f_exp_e.htm#expt"><B>expt</B></A>(x,3)) = <A REL=DEFINITION HREF="../Body/f_exp_e.htm#expt"><B>expt</B></A>(x,3/2)<P>
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where <A REL=DEFINITION HREF="../Body/f_exp_e.htm#expt"><B>expt</B></A> and <A REL=DEFINITION HREF="../Body/f_sqrt_.htm#sqrt"><B>sqrt</B></A> are the mathematical multi-valued functions, it is not<P>
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true that:<P>
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pvsqrt(pvexpt(x,3)) = pvexpt(x,3/2)<P>
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where pvexpt and pvsqrt denote the principal value versions of those functions.<P>
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<P>
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</PRE>
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<A REL=COPYRIGHT HREF="../Front/Help.htm#Legal"><I>Copyright 1996-2005, LispWorks Ltd. All rights reserved.</I></A><P>
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