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<TITLE>CLHS: Issue COMPLEX-RATIONAL-RESULT Writeup</TITLE>
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<H2>Issue COMPLEX-RATIONAL-RESULT Writeup</H2>
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<PRE><B>Issue:</B> <A HREF="iss071.htm">COMPLEX-RATIONAL-RESULT</A><P>
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<P>
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<B>References:</B> CLtL p.203<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, 20-Mar-89, by Moon<P>
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Version 2, 08-Apr-89, by Steele<P>
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Version 3, 28-Apr-89, by Steele<P>
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<P>
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<B>Problem description:<P>
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</B> <P>
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Referring to irrational and transcendental functions, CLtL says:<P>
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<P>
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When the arguments to a function in this section are all rational and<P>
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the true mathematical result is also (mathematically) rational, then<P>
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unless otherwise noted an implementation is free to return either an<P>
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accurate result of type rational or a single-precision floating-point<P>
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approximation. If the arguments are all rational but the result cannot<P>
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be expressed as a rational number, then a single-precision<P>
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floating-point approximation is always returned.<P>
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<P>
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Referring to <A REL=DEFINITION HREF="../Body/f_exp_e.htm#expt"><B>EXPT</B></A>, CLtL says:<P>
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<P>
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If the base-number is of type <A REL=DEFINITION HREF="../Body/a_ration.htm#rational"><B>RATIONAL</B></A> and the power-number is an<P>
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<A REL=DEFINITION HREF="../Body/t_intege.htm#integer"><B>INTEGER</B></A>, the calculation will be exact and the result will be of<P>
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type RATIONAL; otherwise a floating-point approximation may result.<P>
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<P>
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What about arguments of type (complex rational)?<P>
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<P>
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<B>Proposal (COMPLEX-RATIONAL-RESULT:EXTEND):<P>
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</B><P>
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Extend the paragraph quoted above as follows to cover the components<P>
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of complex numbers. If the arguments to a function are all of type<P>
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(<A REL=DEFINITION HREF="../Body/a_or.htm#or"><B>OR</B></A> <A REL=DEFINITION HREF="../Body/a_ration.htm#rational"><B>RATIONAL</B></A> (<A REL=DEFINITION HREF="../Body/a_comple.htm#complex"><B>COMPLEX</B></A> <A REL=DEFINITION HREF="../Body/a_ration.htm#rational"><B>RATIONAL</B></A>)) and the true mathematical result is<P>
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(mathematically) a complex number with rational real and imaginary<P>
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parts, then unless otherwise noted an implementation is free to return<P>
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either an accurate result of type (<A REL=DEFINITION HREF="../Body/a_or.htm#or"><B>OR</B></A> <A REL=DEFINITION HREF="../Body/a_ration.htm#rational"><B>RATIONAL</B></A> (<A REL=DEFINITION HREF="../Body/a_comple.htm#complex"><B>COMPLEX</B></A> <A REL=DEFINITION HREF="../Body/a_ration.htm#rational"><B>RATIONAL</B></A>)) or<P>
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a single-precision floating-point approximation of type <A REL=DEFINITION HREF="../Body/t_short_.htm#single-float"><B>SINGLE-FLOAT</B></A><P>
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(permissible only <A REL=DEFINITION HREF="../Body/s_if.htm#if"><B>if</B></A> <A REL=DEFINITION HREF="../Body/s_the.htm#the"><B>the</B></A> imaginary part of <A REL=DEFINITION HREF="../Body/s_the.htm#the"><B>the</B></A> true mathematical<P>
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result is zero) or (<A REL=DEFINITION HREF="../Body/a_comple.htm#complex"><B>COMPLEX</B></A> <A REL=DEFINITION HREF="../Body/t_short_.htm#single-float"><B>SINGLE-FLOAT</B></A>). If the arguments are<P>
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all of type (<A REL=DEFINITION HREF="../Body/a_or.htm#or"><B>OR</B></A> <A REL=DEFINITION HREF="../Body/a_ration.htm#rational"><B>RATIONAL</B></A> (<A REL=DEFINITION HREF="../Body/a_comple.htm#complex"><B>COMPLEX</B></A> <A REL=DEFINITION HREF="../Body/a_ration.htm#rational"><B>RATIONAL</B></A>)) but the result cannot be<P>
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expressed as a rational or complex rational number, then the returned<P>
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value will be of type <A REL=DEFINITION HREF="../Body/t_short_.htm#single-float"><B>SINGLE-FLOAT</B></A> (permissible only if the imaginary<P>
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part of the true mathematical result is zero) or (<A REL=DEFINITION HREF="../Body/a_comple.htm#complex"><B>COMPLEX</B></A> <A REL=DEFINITION HREF="../Body/t_short_.htm#single-float"><B>SINGLE-FLOAT</B></A>).<P>
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<P>
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For <A REL=DEFINITION HREF="../Body/f_exp_e.htm#expt"><B>EXPT</B></A> of a (<A REL=DEFINITION HREF="../Body/a_comple.htm#complex"><B>COMPLEX</B></A> <A REL=DEFINITION HREF="../Body/a_ration.htm#rational"><B>RATIONAL</B></A>) to an integer power, the<P>
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calculation must be exact and the result will be of type<P>
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(<A REL=DEFINITION HREF="../Body/a_or.htm#or"><B>OR</B></A> <A REL=DEFINITION HREF="../Body/a_ration.htm#rational"><B>RATIONAL</B></A> (<A REL=DEFINITION HREF="../Body/a_comple.htm#complex"><B>COMPLEX</B></A> <A REL=DEFINITION HREF="../Body/a_ration.htm#rational"><B>RATIONAL</B></A>)).<P>
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<P>
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<B>Examples:<P>
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</B><P>
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[a] (log #c(-16 16) #c(2 2)) => 3 or approximately #c(3.0 0.0)<P>
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or approximately 3.0 (unlikely)<P>
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[b] (abs #c(3/5 4/5)) => 1 or approximately 1.0<P>
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[c] (<A REL=DEFINITION HREF="../Body/f_exp_e.htm#expt"><B>expt</B></A> #c(2 2) 3) => #c(-16 16)<P>
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[d] (<A REL=DEFINITION HREF="../Body/f_exp_e.htm#expt"><B>expt</B></A> #c(2 2) 4) => -64 <P>
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<P>
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<B>Rationale:<P>
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</B> <P>
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This seems most consistent with the treatment of real numbers.<P>
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<P>
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<B>Current practice:<P>
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</B> <P>
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[a] [b] [c] [d]<P>
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Symbolics Genera 7.4 #c(3.00... 1.40...e-7) 1 #c(-16 16) -64<P>
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Sun Common Lisp 3.0.1 #c(3.0 2.61...e-16) 1.0 #c(-16 16) -64<P>
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KCL of 9/16/86 on VAX #c(3.0s0 -1.40...s-7) 1.0s0 #c(-16 16) -64<P>
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Allegro CL (Mac II) #c(3.0 2.05...e-16) 1.0 #c(-16 16) -64<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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Only <A REL=DEFINITION HREF="../Body/f_exp_e.htm#expt"><B>EXPT</B></A> would have to change, since the type of the other results<P>
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is at the discretion of the implementation.<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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Probably none, but it is hard to predict.<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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Slightly less self-consistent language.<P>
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<P>
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<B>Performance impact:<P>
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</B><P>
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None of any significance.<P>
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<P>
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<B>Benefits/Esthetics:<P>
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</B><P>
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More self-consistent language.<P>
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<P>
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<B>Discussion:<P>
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</B><P>
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None.<P>
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</PRE>
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