90 lines
5.4 KiB
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90 lines
5.4 KiB
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<!DOCTYPE HTML PUBLIC "-//W3O//DTD W3 HTML 2.0//EN">
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<!Converted with LaTeX2HTML 0.6.5 (Tue Nov 15 1994) by Nikos Drakos (nikos@cbl.leeds.ac.uk), CBLU, University of Leeds >
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<HEAD>
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<TITLE>12.5. Irrational and Transcendental Functions</TITLE>
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</HEAD>
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<BODY>
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<meta name="description" value=" Irrational and Transcendental Functions">
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<meta name="keywords" value="clm">
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<meta name="resource-type" value="document">
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<meta name="distribution" value="global">
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<P>
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<b>Common Lisp the Language, 2nd Edition</b>
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<BR> <HR><A NAME=tex2html3102 HREF="node127.html"><IMG ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif"></A> <A NAME=tex2html3100 HREF="node121.html"><IMG ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif"></A> <A NAME=tex2html3094 HREF="node125.html"><IMG ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif"></A> <A NAME=tex2html3104 HREF="node1.html"><IMG ALIGN=BOTTOM ALT="contents" SRC="icons/contents_motif.gif"></A> <A NAME=tex2html3105 HREF="index.html"><IMG ALIGN=BOTTOM ALT="index" SRC="icons/index_motif.gif"></A> <BR>
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<B> Next:</B> <A NAME=tex2html3103 HREF="node127.html"> Exponential and Logarithmic </A>
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<B>Up:</B> <A NAME=tex2html3101 HREF="node121.html"> Numbers</A>
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<B> Previous:</B> <A NAME=tex2html3095 HREF="node125.html"> Arithmetic Operations</A>
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<HR> <P>
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<H1><A NAME=SECTION001650000000000000000>12.5. Irrational and Transcendental Functions</A></H1>
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<P>
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<A NAME=TRANSCENDENTALSECTION>Common</A>
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Lisp provides no data type that can accurately represent irrational
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numerical values.
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The functions in this section are described as if the results
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were mathematically accurate, but actually they all produce floating-point
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approximations to the true mathematical result in the general case.
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In some places
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mathematical identities are set forth that are intended to elucidate the
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meanings of the functions; however, two mathematically identical
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expressions may be computationally different because of errors
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inherent in the floating-point approximation process.
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<P>
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When the arguments to
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a function in this section are all rational and the true mathematical result
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is also (mathematically) rational, then unless otherwise noted
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an implementation is free to return either an accurate result of
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type <tt>rational</tt> or a single-precision floating-point approximation.
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If the arguments are all rational but the result cannot be expressed
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as a rational number, then a single-precision floating-point
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approximation is always returned.
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<P>
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<img align=bottom alt="change_begin" src="gif/change_begin.gif"><br>
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X3J13 voted in March 1989
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(COMPLEX-RATIONAL-RESULT) <A NAME=11759> </A>
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to clarify that the provisions of the previous paragraph apply to complex
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numbers. If the arguments to a function are all of type
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<tt>(or rational (complex rational))</tt> and the true mathematical result
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is (mathematically) a complex number with rational real
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and imaginary parts, then unless otherwise noted
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an implementation is free to return either an accurate result of
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type <tt>(or rational (complex rational))</tt>
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or a single-precision floating-point approximation
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of type <tt>single-float</tt> (permissible only if the imaginary part
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of the true mathematical result is zero) or <tt>(complex single-float)</tt>.
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If the arguments are all of type
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<tt>(or rational (complex rational))</tt> but the result cannot be expressed
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as a rational or complex rational number, then the returned value
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will be of type <tt>single-float</tt> (permissible only if the imaginary part
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of the true mathematical result is zero) or <tt>(complex single-float)</tt>.
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<br><img align=bottom alt="change_end" src="gif/change_end.gif">
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<P>
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The rules of floating-point contagion and complex contagion are
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effectively obeyed by all the functions in this section except <tt>expt</tt>,
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which treats some cases of rational exponents specially.
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When, possibly after contagious conversion, all of the arguments are of
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the same floating-point or complex floating-point type,
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then the result will be of that same type unless otherwise noted.
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<P>
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<hr>
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<b>Implementation note:</b> There is a ``floating-point cookbook'' by
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Cody and Waite [<A HREF="node368.html#CODYANDWAITE">14</A>] that may be a useful aid
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in implementing the functions defined in this section.
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<hr>
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<P>
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<HR>
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<UL>
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<LI> <A NAME=tex2html3106 HREF="node127.html#SECTION001651000000000000000"> Exponential and Logarithmic Functions</A>
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<LI> <A NAME=tex2html3107 HREF="node128.html#SECTION001652000000000000000"> Trigonometric and Related Functions</A>
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<LI> <A NAME=tex2html3108 HREF="node129.html#SECTION001653000000000000000"> Branch Cuts, Principal Values, and Boundary Conditions in the Complex Plane</A>
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</UL>
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<BR> <HR><A NAME=tex2html3102 HREF="node127.html"><IMG ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif"></A> <A NAME=tex2html3100 HREF="node121.html"><IMG ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif"></A> <A NAME=tex2html3094 HREF="node125.html"><IMG ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif"></A> <A NAME=tex2html3104 HREF="node1.html"><IMG ALIGN=BOTTOM ALT="contents" SRC="icons/contents_motif.gif"></A> <A NAME=tex2html3105 HREF="index.html"><IMG ALIGN=BOTTOM ALT="index" SRC="icons/index_motif.gif"></A> <BR>
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<B> Next:</B> <A NAME=tex2html3103 HREF="node127.html"> Exponential and Logarithmic </A>
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<B>Up:</B> <A NAME=tex2html3101 HREF="node121.html"> Numbers</A>
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<B> Previous:</B> <A NAME=tex2html3095 HREF="node125.html"> Arithmetic Operations</A>
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<HR> <P>
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<HR>
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<P><ADDRESS>
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AI.Repository@cs.cmu.edu
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</ADDRESS>
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</BODY>
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