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