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184 lines
9.3 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>12.5.1. Exponential and Logarithmic Functions</TITLE>
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<BODY>
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<meta name="description" value=" Exponential and Logarithmic 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=tex2html3117 HREF="node128.html"><IMG ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif"></A> <A NAME=tex2html3115 HREF="node126.html"><IMG ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif"></A> <A NAME=tex2html3109 HREF="node126.html"><IMG ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif"></A> <A NAME=tex2html3119 HREF="node1.html"><IMG ALIGN=BOTTOM ALT="contents" SRC="icons/contents_motif.gif"></A> <A NAME=tex2html3120 HREF="index.html"><IMG ALIGN=BOTTOM ALT="index" SRC="icons/index_motif.gif"></A> <BR>
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<B> Next:</B> <A NAME=tex2html3118 HREF="node128.html"> Trigonometric and Related </A>
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<B>Up:</B> <A NAME=tex2html3116 HREF="node126.html"> Irrational and Transcendental </A>
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<B> Previous:</B> <A NAME=tex2html3110 HREF="node126.html"> Irrational and Transcendental </A>
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<HR> <P>
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<H2><A NAME=SECTION001651000000000000000>12.5.1. Exponential and Logarithmic Functions</A></H2>
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<P>
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Along with the usual one-argument and two-argument exponential and
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logarithm functions, <tt>sqrt</tt> is considered to be an exponential
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function, because it raises a number to the power 1/2.
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<P>
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<BR><b>[Function]</b><BR>
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<tt>exp <i>number</i></tt><P>Returns <i>e</i> raised to the power <i>number</i>,
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where <i>e</i> is the base of the natural logarithms.
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<P>
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<BR><b>[Function]</b><BR>
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<tt>expt <i>base-number</i> <i>power-number</i></tt><P>Returns <i>base-number</i> raised to the power <i>power-number</i>.
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If the <i>base-number</i> is of type <tt>rational</tt> and the <i>power-number</i> is
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an <tt>integer</tt>,
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the calculation will be exact and the result will be of type <tt>rational</tt>;
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otherwise a floating-point approximation may result.
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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=11788> </A>
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to clarify that provisions similar to those of the previous paragraph apply to complex
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numbers. If the <i>base-number</i> is of type <tt>(complex rational)</tt>
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and the <i>power-number</i> is
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an <tt>integer</tt>,
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the calculation will also be exact and the result will be of type
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<tt>(or rational (complex rational))</tt>;
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otherwise a floating-point or complex floating-point approximation may result.
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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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When <i>power-number</i> is <tt>0</tt> (a zero of type integer),
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then the result is always the value 1 in the type of <i>base-number</i>,
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even if the <i>base-number</i> is zero (of any type). That is:
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<P><pre>
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(expt <i>x</i> 0) == (coerce 1 (type-of <i>x</i>))
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</pre><P>
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If the <i>power-number</i> is a zero of any other data type,
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then the result is also the value 1, in the type of the arguments
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after the application of the contagion rules, with one exception:
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it is an error if <i>base-number</i> is zero when the <i>power-number</i>
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is a zero not of type integer.
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<P>
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Implementations of <tt>expt</tt> are permitted to use different algorithms
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for the cases of a rational <i>power-number</i> and a floating-point
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<i>power-number</i>; the motivation is that in many cases greater accuracy
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can be achieved for the case of a rational <i>power-number</i>.
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For example, <tt>(expt pi 16)</tt> and <tt>(expt pi 16.0)</tt> may yield
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slightly different results if the first case is computed by repeated squaring
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and the second by the use of logarithms. Similarly, an implementation
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might choose to compute <tt>(expt x 3/2)</tt> as if it had
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been written <tt>(sqrt (expt x 3))</tt>, perhaps producing a more accurate
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result than would <tt>(expt x 1.5)</tt>. It is left to the implementor
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to determine the best strategies.
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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 January 1989
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(EXPT-RATIO) <A NAME=11816> </A>
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to clarify that the preceding remark is in
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error, because <tt>(sqrt (expt x 3))</tt> does not produce the same value
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as <tt>(expt x 3/2)</tt> in most cases, and to specify that the
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specification of the principal value of <tt>expt</tt> as given in section <A HREF="node129.html#BRANCHCUTSSECTION">12.5.3</A>
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should be regarded as definitive.
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<P>
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As an example of the difficulty, let
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<IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap40793.gif">.
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Then <IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap40794.gif">, but
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<IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap40795.gif">.
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Another example is <b><i>x</i>=-1</b>; then <IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap40796.gif">, but
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<IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap40797.gif">.
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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 result of <tt>expt</tt> can be a complex number, even when neither argument
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is complex, if <i>base-number</i> is negative and <i>power-number</i>
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is not an integer. The result is always the principal complex value.
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Note that <tt>(expt -8 1/3)</tt> is not permitted to return <tt>-2</tt>;
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while <tt>-2</tt> is indeed one of the cube roots of <tt>-8</tt>, it is
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not the principal cube root, which is a complex number
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approximately equal to <tt>#C(1.0 1.73205)</tt>.
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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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<i>Notice of correction.</i> The first edition gave the incorrect value
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<tt>#C(0.5 1.73205)</tt> for the principal cube root of <tt>-8</tt>. The correct
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value is <tt>#C(1.0 1.73205)</tt>, that is, 1+SQRT(3)<i>i</i>. I simply don't know what
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I was thinking of!
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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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<BR><b>[Function]</b><BR>
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<tt>log <i>number</i> &optional <i>base</i></tt><P>Returns the logarithm of <i>number</i> in the base <i>base</i>,
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which defaults to <i>e</i>, the base of the natural logarithms.
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For example:
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<P><pre>
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(log 8.0 2) => 3.0
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(log 100.0 10) => 2.0
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</pre><P>
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The result of <tt>(log 8 2)</tt> may be either <tt>3</tt> or <tt>3.0</tt>, depending on the
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implementation.
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<P>
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Note that <tt>log</tt> may return a complex result when given a non-complex
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argument if the argument is negative. For example:
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<P><pre>
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(log -1.0) == (complex 0.0 (float pi 0.0))
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</pre><P>
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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 January 1989
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(IEEE-ATAN-BRANCH-CUT) <A NAME=11878> </A>
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to specify certain floating-point behavior when minus zero is supported.
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As a part of that vote it approved a mathematical definition of complex logarithm
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in terms of real logarithm, absolute value,
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arc tangent of two real arguments, and the phase function as
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<PRE><TT>
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Logarithm log|<i>z</i>| + <i>i</i> phase <i>z</i>
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<P>
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</TT></PRE>
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This specifies the branch cuts precisely whether minus zero is supported or not;
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see <tt>phase</tt> and <tt>atan</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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<BR><b>[Function]</b><BR>
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<tt>sqrt <i>number</i></tt><P>Returns the principal square root of <i>number</i>.
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If the <i>number</i> is not complex but is negative, then the result
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will be a complex number.
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For example:
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<P><pre>
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(sqrt 9.0) => 3.0
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(sqrt -9.0) => #c(0.0 3.0)
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</pre><P>
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The result of <tt>(sqrt 9)</tt> may be either <tt>3</tt> or <tt>3.0</tt>, depending on the
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implementation. The result of <tt>(sqrt -9)</tt> may be either <tt>#c(0 3)</tt>
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or <tt>#c(0.0 3.0)</tt>.
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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 January 1989
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(IEEE-ATAN-BRANCH-CUT) <A NAME=11900> </A>
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to specify certain floating-point behavior when minus zero is supported.
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As a part of that vote it approved a mathematical definition of complex square root
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in terms of complex logarithm and exponential functions as
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<PRE><TT>
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Square root <IMG ALIGN=BOTTOM ALT="" SRC="_24769_tex2html_wrap40799.gif">
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<P>
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</TT></PRE>
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This specifies the branch cuts precisely whether minus zero is supported or not;
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see <tt>phase</tt> and <tt>atan</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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<BR><b>[Function]</b><BR>
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<tt>isqrt <i>integer</i></tt><P>Integer square root: the argument must be a non-negative integer, and the
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result is the greatest integer less than or equal to the exact positive
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square root of the argument.
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For example:
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<P><pre>
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(isqrt 9) => 3
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(isqrt 12) => 3
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(isqrt 300) => 17
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(isqrt 325) => 18
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</pre><P>
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<P>
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<BR> <HR><A NAME=tex2html3117 HREF="node128.html"><IMG ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif"></A> <A NAME=tex2html3115 HREF="node126.html"><IMG ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif"></A> <A NAME=tex2html3109 HREF="node126.html"><IMG ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif"></A> <A NAME=tex2html3119 HREF="node1.html"><IMG ALIGN=BOTTOM ALT="contents" SRC="icons/contents_motif.gif"></A> <A NAME=tex2html3120 HREF="index.html"><IMG ALIGN=BOTTOM ALT="index" SRC="icons/index_motif.gif"></A> <BR>
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<B> Next:</B> <A NAME=tex2html3118 HREF="node128.html"> Trigonometric and Related </A>
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<B>Up:</B> <A NAME=tex2html3116 HREF="node126.html"> Irrational and Transcendental </A>
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<B> Previous:</B> <A NAME=tex2html3110 HREF="node126.html"> Irrational and Transcendental </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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