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57 lines
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<TITLE>CLHS: Section 12.1.3.3</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="12_acb.htm"><IMG WIDTH=40 HEIGHT=40 ALT="[Previous]" SRC="../Graphics/Prev.gif" ALIGN=Bottom></A><A REL=UP HREF="12_ac.htm"><IMG WIDTH=40 HEIGHT=40 ALT="[Up]" SRC="../Graphics/Up.gif" ALIGN=Bottom></A><A REL=NEXT HREF="12_ad.htm"><IMG WIDTH=40 HEIGHT=40 ALT="[Next]" SRC="../Graphics/Next.gif" ALIGN=Bottom></A></H1>
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<H2>
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12.1.3.3 Rule of Float Substitutability</H2> <P>
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When the arguments to an irrational mathematical <A REL=DEFINITION HREF="26_glo_f.htm#function"><I>function</I></A> are all <A REL=DEFINITION HREF="26_glo_r.htm#rational"><I>rational</I></A> and the true mathematical result is also (mathematically) rational, then unless otherwise noted an implementation is free to return either an accurate <A REL=DEFINITION HREF="26_glo_r.htm#rational"><I>rational</I></A> result or a <A REL=DEFINITION HREF="26_glo_s.htm#single_float"><I>single float</I></A> approximation. If the arguments are all <A REL=DEFINITION HREF="26_glo_r.htm#rational"><I>rational</I></A> but the result cannot be expressed as a <A REL=DEFINITION HREF="26_glo_r.htm#rational"><I>rational</I></A> number, then a <A REL=DEFINITION HREF="26_glo_s.htm#single_float"><I>single float</I></A> approximation is always returned. <P>
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If the arguments to an irrational mathematical <A REL=DEFINITION HREF="26_glo_f.htm#function"><I>function</I></A> are all of type <TT>(or rational (complex rational))</TT> and the true mathematical result is (mathematically) a complex number with rational real and imaginary parts, then unless otherwise noted an implementation is free to return either an accurate result of type <TT>(or rational (complex rational))</TT> or a <A REL=DEFINITION HREF="26_glo_s.htm#single_float"><I>single float</I></A> (permissible only if the imaginary part of the true mathematical result is zero) or <TT>(complex single-float)</TT>. If the arguments are all of type <TT>(or rational (complex rational))</TT> but the result cannot be expressed as a <A REL=DEFINITION HREF="26_glo_r.htm#rational"><I>rational</I></A> or <A REL=DEFINITION HREF="26_glo_c.htm#complex_rational"><I>complex rational</I></A>, then the returned value will be of <A REL=DEFINITION HREF="26_glo_t.htm#type"><I>type</I></A> <A REL=DEFINITION HREF="t_short_.htm#single-float"><B>single-float</B></A> (permissible only if the imaginary part of the true mathematical result is zero) or <TT>(complex single-float)</TT>. <P>
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Float substitutability applies neither to the rational <A REL=DEFINITION HREF="26_glo_f.htm#function"><I>functions</I></A> <A REL=DEFINITION HREF="f_pl.htm#PL"><B>+</B></A>, <A REL=DEFINITION HREF="f__.htm#-"><B>-</B></A>, <A REL=DEFINITION HREF="f_st.htm#ST"><B>*</B></A>, and <A REL=DEFINITION HREF="f_sl.htm#SL"><B>/</B></A> nor to the related <A REL=DEFINITION HREF="26_glo_o.htm#operator"><I>operators</I></A> <A REL=DEFINITION HREF="f_1pl_1_.htm#1PL"><B>1+</B></A>, <A REL=DEFINITION HREF="f_1pl_1_.htm#1-"><B>1-</B></A>, <A REL=DEFINITION HREF="m_incf_.htm#incf"><B>incf</B></A>, <A REL=DEFINITION HREF="m_incf_.htm#decf"><B>decf</B></A>, and <A REL=DEFINITION HREF="f_conjug.htm#conjugate"><B>conjugate</B></A>. For rational <A REL=DEFINITION HREF="26_glo_f.htm#function"><I>functions</I></A>, if all arguments are <A REL=DEFINITION HREF="26_glo_r.htm#rational"><I>rational</I></A>, then the result is <A REL=DEFINITION HREF="26_glo_r.htm#rational"><I>rational</I></A>; if all arguments are of type <TT>(or rational (complex rational))</TT>, then the result is of type <TT>(or rational (complex rational))</TT>. <P>
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<PRE>
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Function Sample Results
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<A REL=DEFINITION HREF="f_abs.htm#abs">abs</A> (abs #c(3 4)) => 5 or 5.0
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<A REL=DEFINITION HREF="f_asin_.htm#acos">acos</A> (acos 1) => 0 or 0.0
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<A REL=DEFINITION HREF="f_sinh_.htm#acosh">acosh</A> (acosh 1) => 0 or 0.0
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<A REL=DEFINITION HREF="f_asin_.htm#asin">asin</A> (asin 0) => 0 or 0.0
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<A REL=DEFINITION HREF="f_sinh_.htm#asinh">asinh</A> (asinh 0) => 0 or 0.0
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<A REL=DEFINITION HREF="f_asin_.htm#atan">atan</A> (atan 0) => 0 or 0.0
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<A REL=DEFINITION HREF="f_sinh_.htm#atanh">atanh</A> (atanh 0) => 0 or 0.0
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<A REL=DEFINITION HREF="f_cis.htm#cis">cis</A> (cis 0) => 1 or #c(1.0 0.0)
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<A REL=DEFINITION HREF="f_sin_c.htm#cos">cos</A> (cos 0) => 1 or 1.0
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<A REL=DEFINITION HREF="f_sinh_.htm#cosh">cosh</A> (cosh 0) => 1 or 1.0
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<A REL=DEFINITION HREF="f_exp_e.htm#exp">exp</A> (exp 0) => 1 or 1.0
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<A REL=DEFINITION HREF="f_exp_e.htm#expt">expt</A> (expt 8 1/3) => 2 or 2.0
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<A REL=DEFINITION HREF="f_log.htm#log">log</A> (log 1) => 0 or 0.0
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(log 8 2) => 3 or 3.0
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<A REL=DEFINITION HREF="f_phase.htm#phase">phase</A> (phase 7) => 0 or 0.0
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<A REL=DEFINITION HREF="f_signum.htm#signum">signum</A> (signum #c(3 4)) => #c(3/5 4/5) or #c(0.6 0.8)
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<A REL=DEFINITION HREF="f_sin_c.htm#sin">sin</A> (sin 0) => 0 or 0.0
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<A REL=DEFINITION HREF="f_sinh_.htm#sinh">sinh</A> (sinh 0) => 0 or 0.0
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<A REL=DEFINITION HREF="f_sqrt_.htm#sqrt">sqrt</A> (sqrt 4) => 2 or 2.0
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(sqrt 9/16) => 3/4 or 0.75
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<A REL=DEFINITION HREF="f_sin_c.htm#tan">tan</A> (tan 0) => 0 or 0.0
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<A REL=DEFINITION HREF="f_sinh_.htm#tanh">tanh</A> (tanh 0) => 0 or 0.0
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</PRE>
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<P><B>Figure 12-8. Functions Affected by Rule of Float Substitutability</B> <P>
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<P><HR>The following <A REL=META HREF="../Front/X3J13Iss.htm">X3J13 cleanup issue</A>, <I>not part of the specification</I>, applies to this section:<P><UL><LI> <A REL=CHILD HREF="../Issues/iss071.htm">COMPLEX-RATIONAL-RESULT:EXTEND</A><P></UL><HR>
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