+++ /dev/null
-<HTML>
-<HEAD>
-<!-- Created by texi2html 1.56k from cln.texi on 28 August 2000 -->
-
-<TITLE>CLN, a Class Library for Numbers</TITLE>
-</HEAD>
-<BODY>
-<H1>CLN, a Class Library for Numbers</H1>
-<ADDRESS>by Bruno Haible</ADDRESS>
-<P>
-<P><HR><P>
-<H1>Table of Contents</H1>
-<UL>
-<LI><A NAME="TOC1" HREF="cln.html#SEC1">1. Introduction</A>
-<LI><A NAME="TOC2" HREF="cln.html#SEC2">2. Installation</A>
-<UL>
-<LI><A NAME="TOC3" HREF="cln.html#SEC3">2.1 Prerequisites</A>
-<UL>
-<LI><A NAME="TOC4" HREF="cln.html#SEC4">2.1.1 C++ compiler</A>
-<LI><A NAME="TOC5" HREF="cln.html#SEC5">2.1.2 Make utility</A>
-<LI><A NAME="TOC6" HREF="cln.html#SEC6">2.1.3 Sed utility</A>
-</UL>
-<LI><A NAME="TOC7" HREF="cln.html#SEC7">2.2 Building the library</A>
-<UL>
-<LI><A NAME="TOC8" HREF="cln.html#SEC8">2.2.1 Using the GNU MP Library</A>
-</UL>
-<LI><A NAME="TOC9" HREF="cln.html#SEC9">2.3 Installing the library</A>
-<LI><A NAME="TOC10" HREF="cln.html#SEC10">2.4 Cleaning up</A>
-</UL>
-<LI><A NAME="TOC11" HREF="cln.html#SEC11">3. Ordinary number types</A>
-<UL>
-<LI><A NAME="TOC12" HREF="cln.html#SEC12">3.1 Exact numbers</A>
-<LI><A NAME="TOC13" HREF="cln.html#SEC13">3.2 Floating-point numbers</A>
-<LI><A NAME="TOC14" HREF="cln.html#SEC14">3.3 Complex numbers</A>
-<LI><A NAME="TOC15" HREF="cln.html#SEC15">3.4 Conversions</A>
-</UL>
-<LI><A NAME="TOC16" HREF="cln.html#SEC16">4. Functions on numbers</A>
-<UL>
-<LI><A NAME="TOC17" HREF="cln.html#SEC17">4.1 Constructing numbers</A>
-<UL>
-<LI><A NAME="TOC18" HREF="cln.html#SEC18">4.1.1 Constructing integers</A>
-<LI><A NAME="TOC19" HREF="cln.html#SEC19">4.1.2 Constructing rational numbers</A>
-<LI><A NAME="TOC20" HREF="cln.html#SEC20">4.1.3 Constructing floating-point numbers</A>
-<LI><A NAME="TOC21" HREF="cln.html#SEC21">4.1.4 Constructing complex numbers</A>
-</UL>
-<LI><A NAME="TOC22" HREF="cln.html#SEC22">4.2 Elementary functions</A>
-<LI><A NAME="TOC23" HREF="cln.html#SEC23">4.3 Elementary rational functions</A>
-<LI><A NAME="TOC24" HREF="cln.html#SEC24">4.4 Elementary complex functions</A>
-<LI><A NAME="TOC25" HREF="cln.html#SEC25">4.5 Comparisons</A>
-<LI><A NAME="TOC26" HREF="cln.html#SEC26">4.6 Rounding functions</A>
-<LI><A NAME="TOC27" HREF="cln.html#SEC27">4.7 Roots</A>
-<LI><A NAME="TOC28" HREF="cln.html#SEC28">4.8 Transcendental functions</A>
-<UL>
-<LI><A NAME="TOC29" HREF="cln.html#SEC29">4.8.1 Exponential and logarithmic functions</A>
-<LI><A NAME="TOC30" HREF="cln.html#SEC30">4.8.2 Trigonometric functions</A>
-<LI><A NAME="TOC31" HREF="cln.html#SEC31">4.8.3 Hyperbolic functions</A>
-<LI><A NAME="TOC32" HREF="cln.html#SEC32">4.8.4 Euler gamma</A>
-<LI><A NAME="TOC33" HREF="cln.html#SEC33">4.8.5 Riemann zeta</A>
-</UL>
-<LI><A NAME="TOC34" HREF="cln.html#SEC34">4.9 Functions on integers</A>
-<UL>
-<LI><A NAME="TOC35" HREF="cln.html#SEC35">4.9.1 Logical functions</A>
-<LI><A NAME="TOC36" HREF="cln.html#SEC36">4.9.2 Number theoretic functions</A>
-<LI><A NAME="TOC37" HREF="cln.html#SEC37">4.9.3 Combinatorial functions</A>
-</UL>
-<LI><A NAME="TOC38" HREF="cln.html#SEC38">4.10 Functions on floating-point numbers</A>
-<LI><A NAME="TOC39" HREF="cln.html#SEC39">4.11 Conversion functions</A>
-<UL>
-<LI><A NAME="TOC40" HREF="cln.html#SEC40">4.11.1 Conversion to floating-point numbers</A>
-<LI><A NAME="TOC41" HREF="cln.html#SEC41">4.11.2 Conversion to rational numbers</A>
-</UL>
-<LI><A NAME="TOC42" HREF="cln.html#SEC42">4.12 Random number generators</A>
-<LI><A NAME="TOC43" HREF="cln.html#SEC43">4.13 Obfuscating operators</A>
-</UL>
-<LI><A NAME="TOC44" HREF="cln.html#SEC44">5. Input/Output</A>
-<UL>
-<LI><A NAME="TOC45" HREF="cln.html#SEC45">5.1 Internal and printed representation</A>
-<LI><A NAME="TOC46" HREF="cln.html#SEC46">5.2 Input functions</A>
-<LI><A NAME="TOC47" HREF="cln.html#SEC47">5.3 Output functions</A>
-</UL>
-<LI><A NAME="TOC48" HREF="cln.html#SEC48">6. Rings</A>
-<LI><A NAME="TOC49" HREF="cln.html#SEC49">7. Modular integers</A>
-<UL>
-<LI><A NAME="TOC50" HREF="cln.html#SEC50">7.1 Modular integer rings</A>
-<LI><A NAME="TOC51" HREF="cln.html#SEC51">7.2 Functions on modular integers</A>
-</UL>
-<LI><A NAME="TOC52" HREF="cln.html#SEC52">8. Symbolic data types</A>
-<UL>
-<LI><A NAME="TOC53" HREF="cln.html#SEC53">8.1 Strings</A>
-<LI><A NAME="TOC54" HREF="cln.html#SEC54">8.2 Symbols</A>
-</UL>
-<LI><A NAME="TOC55" HREF="cln.html#SEC55">9. Univariate polynomials</A>
-<UL>
-<LI><A NAME="TOC56" HREF="cln.html#SEC56">9.1 Univariate polynomial rings</A>
-<LI><A NAME="TOC57" HREF="cln.html#SEC57">9.2 Functions on univariate polynomials</A>
-<LI><A NAME="TOC58" HREF="cln.html#SEC58">9.3 Special polynomials</A>
-</UL>
-<LI><A NAME="TOC59" HREF="cln.html#SEC59">10. Internals</A>
-<UL>
-<LI><A NAME="TOC60" HREF="cln.html#SEC60">10.1 Why C++ ?</A>
-<LI><A NAME="TOC61" HREF="cln.html#SEC61">10.2 Memory efficiency</A>
-<LI><A NAME="TOC62" HREF="cln.html#SEC62">10.3 Speed efficiency</A>
-<LI><A NAME="TOC63" HREF="cln.html#SEC63">10.4 Garbage collection</A>
-</UL>
-<LI><A NAME="TOC64" HREF="cln.html#SEC64">11. Using the library</A>
-<UL>
-<LI><A NAME="TOC65" HREF="cln.html#SEC65">11.1 Compiler options</A>
-<LI><A NAME="TOC66" HREF="cln.html#SEC66">11.2 Compatibility to old CLN versions</A>
-<LI><A NAME="TOC67" HREF="cln.html#SEC67">11.3 Include files</A>
-<LI><A NAME="TOC68" HREF="cln.html#SEC68">11.4 An Example</A>
-<LI><A NAME="TOC69" HREF="cln.html#SEC69">11.5 Debugging support</A>
-</UL>
-<LI><A NAME="TOC70" HREF="cln.html#SEC70">12. Customizing</A>
-<UL>
-<LI><A NAME="TOC71" HREF="cln.html#SEC71">12.1 Error handling</A>
-<LI><A NAME="TOC72" HREF="cln.html#SEC72">12.2 Floating-point underflow</A>
-<LI><A NAME="TOC73" HREF="cln.html#SEC73">12.3 Customizing I/O</A>
-<LI><A NAME="TOC74" HREF="cln.html#SEC74">12.4 Customizing the memory allocator</A>
-</UL>
-<LI><A NAME="TOC75" HREF="cln.html#SEC75">Index</A>
-</UL>
-<P><HR><P>
-
-
-<H1><A NAME="SEC1" HREF="cln.html#TOC1">1. Introduction</A></H1>
-
-<P>
-CLN is a library for computations with all kinds of numbers.
-It has a rich set of number classes:
-
-
-
-<UL>
-<LI>
-
-Integers (with unlimited precision),
-
-<LI>
-
-Rational numbers,
-
-<LI>
-
-Floating-point numbers:
-
-
-<UL>
-<LI>
-
-Short float,
-<LI>
-
-Single float,
-<LI>
-
-Double float,
-<LI>
-
-Long float (with unlimited precision),
-</UL>
-
-<LI>
-
-Complex numbers,
-
-<LI>
-
-Modular integers (integers modulo a fixed integer),
-
-<LI>
-
-Univariate polynomials.
-</UL>
-
-<P>
-The subtypes of the complex numbers among these are exactly the
-types of numbers known to the Common Lisp language. Therefore
-<CODE>CLN</CODE> can be used for Common Lisp implementations, giving
-<SAMP>`CLN'</SAMP> another meaning: it becomes an abbreviation of
-"Common Lisp Numbers".
-
-
-<P>
-The CLN package implements
-
-
-
-<UL>
-<LI>
-
-Elementary functions (<CODE>+</CODE>, <CODE>-</CODE>, <CODE>*</CODE>, <CODE>/</CODE>, <CODE>sqrt</CODE>,
-comparisons, ...),
-
-<LI>
-
-Logical functions (logical <CODE>and</CODE>, <CODE>or</CODE>, <CODE>not</CODE>, ...),
-
-<LI>
-
-Transcendental functions (exponential, logarithmic, trigonometric, hyperbolic
-functions and their inverse functions).
-</UL>
-
-<P>
-CLN is a C++ library. Using C++ as an implementation language provides
-
-
-
-<UL>
-<LI>
-
-efficiency: it compiles to machine code,
-<LI>
-
-type safety: the C++ compiler knows about the number types and complains
-if, for example, you try to assign a float to an integer variable.
-<LI>
-
-algebraic syntax: You can use the <CODE>+</CODE>, <CODE>-</CODE>, <CODE>*</CODE>, <CODE>=</CODE>,
-<CODE>==</CODE>, ... operators as in C or C++.
-</UL>
-
-<P>
-CLN is memory efficient:
-
-
-
-<UL>
-<LI>
-
-Small integers and short floats are immediate, not heap allocated.
-<LI>
-
-Heap-allocated memory is reclaimed through an automatic, non-interruptive
-garbage collection.
-</UL>
-
-<P>
-CLN is speed efficient:
-
-
-
-<UL>
-<LI>
-
-The kernel of CLN has been written in assembly language for some CPUs
-(<CODE>i386</CODE>, <CODE>m68k</CODE>, <CODE>sparc</CODE>, <CODE>mips</CODE>, <CODE>arm</CODE>).
-<LI>
-
-<A NAME="IDX1"></A>
-On all CPUs, CLN may be configured to use the superefficient low-level
-routines from GNU GMP version 3.
-<LI>
-
-It uses Karatsuba multiplication, which is significantly faster
-for large numbers than the standard multiplication algorithm.
-<LI>
-
-For very large numbers (more than 12000 decimal digits), it uses
-Schönhage-Strassen
-<A NAME="IDX2"></A>
-multiplication, which is an asymptotically optimal multiplication
-algorithm, for multiplication, division and radix conversion.
-</UL>
-
-<P>
-CLN aims at being easily integrated into larger software packages:
-
-
-
-<UL>
-<LI>
-
-The garbage collection imposes no burden on the main application.
-<LI>
-
-The library provides hooks for memory allocation and exceptions.
-<LI>
-
-<A NAME="IDX3"></A>
-All non-macro identifiers are hidden in namespace <CODE>cln</CODE> in
-order to avoid name clashes.
-</UL>
-
-
-
-<H1><A NAME="SEC2" HREF="cln.html#TOC2">2. Installation</A></H1>
-
-<P>
-This section describes how to install the CLN package on your system.
-
-
-
-
-<H2><A NAME="SEC3" HREF="cln.html#TOC3">2.1 Prerequisites</A></H2>
-
-
-
-<H3><A NAME="SEC4" HREF="cln.html#TOC4">2.1.1 C++ compiler</A></H3>
-
-<P>
-To build CLN, you need a C++ compiler.
-Actually, you need GNU <CODE>g++ 2.90</CODE> or newer, the EGCS compilers will
-do.
-I recommend GNU <CODE>g++ 2.95</CODE> or newer.
-
-
-<P>
-The following C++ features are used:
-classes, member functions, overloading of functions and operators,
-constructors and destructors, inline, const, multiple inheritance,
-templates and namespaces.
-
-
-<P>
-The following C++ features are not used:
-<CODE>new</CODE>, <CODE>delete</CODE>, virtual inheritance, exceptions.
-
-
-<P>
-CLN relies on semi-automatic ordering of initializations
-of static and global variables, a feature which I could
-implement for GNU g++ only.
-
-
-
-
-<H3><A NAME="SEC5" HREF="cln.html#TOC5">2.1.2 Make utility</A></H3>
-<P>
-<A NAME="IDX4"></A>
-
-
-<P>
-To build CLN, you also need to have GNU <CODE>make</CODE> installed.
-
-
-
-
-<H3><A NAME="SEC6" HREF="cln.html#TOC6">2.1.3 Sed utility</A></H3>
-<P>
-<A NAME="IDX5"></A>
-
-
-<P>
-To build CLN on HP-UX, you also need to have GNU <CODE>sed</CODE> installed.
-This is because the libtool script, which creates the CLN library, relies
-on <CODE>sed</CODE>, and the vendor's <CODE>sed</CODE> utility on these systems is too
-limited.
-
-
-
-
-<H2><A NAME="SEC7" HREF="cln.html#TOC7">2.2 Building the library</A></H2>
-
-<P>
-As with any autoconfiguring GNU software, installation is as easy as this:
-
-
-
-<PRE>
-$ ./configure
-$ make
-$ make check
-</PRE>
-
-<P>
-If on your system, <SAMP>`make'</SAMP> is not GNU <CODE>make</CODE>, you have to use
-<SAMP>`gmake'</SAMP> instead of <SAMP>`make'</SAMP> above.
-
-
-<P>
-The <CODE>configure</CODE> command checks out some features of your system and
-C++ compiler and builds the <CODE>Makefile</CODE>s. The <CODE>make</CODE> command
-builds the library. This step may take 4 hours on an average workstation.
-The <CODE>make check</CODE> runs some test to check that no important subroutine
-has been miscompiled.
-
-
-<P>
-The <CODE>configure</CODE> command accepts options. To get a summary of them, try
-
-
-
-<PRE>
-$ ./configure --help
-</PRE>
-
-<P>
-Some of the options are explained in detail in the <SAMP>`INSTALL.generic'</SAMP> file.
-
-
-<P>
-You can specify the C compiler, the C++ compiler and their options through
-the following environment variables when running <CODE>configure</CODE>:
-
-
-<DL COMPACT>
-
-<DT><CODE>CC</CODE>
-<DD>
-Specifies the C compiler.
-
-<DT><CODE>CFLAGS</CODE>
-<DD>
-Flags to be given to the C compiler when compiling programs (not when linking).
-
-<DT><CODE>CXX</CODE>
-<DD>
-Specifies the C++ compiler.
-
-<DT><CODE>CXXFLAGS</CODE>
-<DD>
-Flags to be given to the C++ compiler when compiling programs (not when linking).
-</DL>
-
-<P>
-Examples:
-
-
-
-<PRE>
-$ CC="gcc" CFLAGS="-O" CXX="g++" CXXFLAGS="-O" ./configure
-$ CC="gcc -V egcs-2.91.60" CFLAGS="-O -g" \
- CXX="g++ -V egcs-2.91.60" CXXFLAGS="-O -g" ./configure
-$ CC="gcc -V 2.95.2" CFLAGS="-O2 -fno-exceptions" \
- CXX="g++ -V 2.95.2" CFLAGS="-O2 -fno-exceptions" ./configure
-</PRE>
-
-<P>
-Note that for these environment variables to take effect, you have to set
-them (assuming a Bourne-compatible shell) on the same line as the
-<CODE>configure</CODE> command. If you made the settings in earlier shell
-commands, you have to <CODE>export</CODE> the environment variables before
-calling <CODE>configure</CODE>. In a <CODE>csh</CODE> shell, you have to use the
-<SAMP>`setenv'</SAMP> command for setting each of the environment variables.
-
-
-<P>
-Currently CLN works only with the GNU <CODE>g++</CODE> compiler, and only in
-optimizing mode. So you should specify at least <CODE>-O</CODE> in the CXXFLAGS,
-or no CXXFLAGS at all. (If CXXFLAGS is not set, CLN will use <CODE>-O</CODE>.)
-
-
-<P>
-If you use <CODE>g++</CODE> version 2.8.x or egcs-2.91.x (a.k.a. egcs-1.1) or
-gcc-2.95.x, I recommend adding <SAMP>`-fno-exceptions'</SAMP> to the CXXFLAGS.
-This will likely generate better code.
-
-
-<P>
-If you use <CODE>g++</CODE> version egcs-2.91.x (egcs-1.1) or gcc-2.95.x on Sparc,
-add either <SAMP>`-O'</SAMP>, <SAMP>`-O1'</SAMP> or <SAMP>`-O2 -fno-schedule-insns'</SAMP> to the
-CXXFLAGS. With full <SAMP>`-O2'</SAMP>, <CODE>g++</CODE> miscompiles the division routines.
-Also, if you have <CODE>g++</CODE> version egcs-1.1.1 or older on Sparc, you must
-specify <SAMP>`--disable-shared'</SAMP> because <CODE>g++</CODE> would miscompile parts of
-the library.
-
-
-<P>
-By default, both a shared and a static library are built. You can build
-CLN as a static (or shared) library only, by calling <CODE>configure</CODE> with
-the option <SAMP>`--disable-shared'</SAMP> (or <SAMP>`--disable-static'</SAMP>). While
-shared libraries are usually more convenient to use, they may not work
-on all architectures. Try disabling them if you run into linker
-problems. Also, they are generally somewhat slower than static
-libraries so runtime-critical applications should be linked statically.
-
-
-
-
-<H3><A NAME="SEC8" HREF="cln.html#TOC8">2.2.1 Using the GNU MP Library</A></H3>
-<P>
-<A NAME="IDX6"></A>
-
-
-<P>
-Starting with version 1.1, CLN may be configured to make use of a
-preinstalled <CODE>gmp</CODE> library. Please make sure that you have at
-least <CODE>gmp</CODE> version 3.0 installed since earlier versions are
-unsupported and likely not to work. Enabling this feature by calling
-<CODE>configure</CODE> with the option <SAMP>`--with-gmp'</SAMP> is known to be quite
-a boost for CLN's performance.
-
-
-<P>
-If you have installed the <CODE>gmp</CODE> library and its header file in
-some place where your compiler cannot find it by default, you must help
-<CODE>configure</CODE> by setting <CODE>CPPFLAGS</CODE> and <CODE>LDFLAGS</CODE>. Here is
-an example:
-
-
-
-<PRE>
-$ CC="gcc" CFLAGS="-O2" CXX="g++" CXXFLAGS="-O2 -fno-exceptions" \
- CPPFLAGS="-I/opt/gmp/include" LDFLAGS="-L/opt/gmp/lib" ./configure --with-gmp
-</PRE>
-
-
-
-<H2><A NAME="SEC9" HREF="cln.html#TOC9">2.3 Installing the library</A></H2>
-<P>
-<A NAME="IDX7"></A>
-
-
-<P>
-As with any autoconfiguring GNU software, installation is as easy as this:
-
-
-
-<PRE>
-$ make install
-</PRE>
-
-<P>
-The <SAMP>`make install'</SAMP> command installs the library and the include files
-into public places (<TT>`/usr/local/lib/'</TT> and <TT>`/usr/local/include/'</TT>,
-if you haven't specified a <CODE>--prefix</CODE> option to <CODE>configure</CODE>).
-This step may require superuser privileges.
-
-
-<P>
-If you have already built the library and wish to install it, but didn't
-specify <CODE>--prefix=...</CODE> at configure time, just re-run
-<CODE>configure</CODE>, giving it the same options as the first time, plus
-the <CODE>--prefix=...</CODE> option.
-
-
-
-
-<H2><A NAME="SEC10" HREF="cln.html#TOC10">2.4 Cleaning up</A></H2>
-
-<P>
-You can remove system-dependent files generated by <CODE>make</CODE> through
-
-
-
-<PRE>
-$ make clean
-</PRE>
-
-<P>
-You can remove all files generated by <CODE>make</CODE>, thus reverting to a
-virgin distribution of CLN, through
-
-
-
-<PRE>
-$ make distclean
-</PRE>
-
-
-
-<H1><A NAME="SEC11" HREF="cln.html#TOC11">3. Ordinary number types</A></H1>
-
-<P>
-CLN implements the following class hierarchy:
-
-
-
-<PRE>
- Number
- cl_number
- <cln/number.h>
- |
- |
- Real or complex number
- cl_N
- <cln/complex.h>
- |
- |
- Real number
- cl_R
- <cln/real.h>
- |
- +-------------------+-------------------+
- | |
-Rational number Floating-point number
- cl_RA cl_F
-<cln/rational.h> <cln/float.h>
- | |
- | +--------------+--------------+--------------+
- Integer | | | |
- cl_I Short-Float Single-Float Double-Float Long-Float
-<cln/integer.h> cl_SF cl_FF cl_DF cl_LF
- <cln/sfloat.h> <cln/ffloat.h> <cln/dfloat.h> <cln/lfloat.h>
-</PRE>
-
-<P>
-<A NAME="IDX8"></A>
-<A NAME="IDX9"></A>
-The base class <CODE>cl_number</CODE> is an abstract base class.
-It is not useful to declare a variable of this type except if you want
-to completely disable compile-time type checking and use run-time type
-checking instead.
-
-
-<P>
-<A NAME="IDX10"></A>
-<A NAME="IDX11"></A>
-<A NAME="IDX12"></A>
-The class <CODE>cl_N</CODE> comprises real and complex numbers. There is
-no special class for complex numbers since complex numbers with imaginary
-part <CODE>0</CODE> are automatically converted to real numbers.
-
-
-<P>
-<A NAME="IDX13"></A>
-The class <CODE>cl_R</CODE> comprises real numbers of different kinds. It is an
-abstract class.
-
-
-<P>
-<A NAME="IDX14"></A>
-<A NAME="IDX15"></A>
-<A NAME="IDX16"></A>
-The class <CODE>cl_RA</CODE> comprises exact real numbers: rational numbers, including
-integers. There is no special class for non-integral rational numbers
-since rational numbers with denominator <CODE>1</CODE> are automatically converted
-to integers.
-
-
-<P>
-<A NAME="IDX17"></A>
-The class <CODE>cl_F</CODE> implements floating-point approximations to real numbers.
-It is an abstract class.
-
-
-
-
-<H2><A NAME="SEC12" HREF="cln.html#TOC12">3.1 Exact numbers</A></H2>
-<P>
-<A NAME="IDX18"></A>
-
-
-<P>
-Some numbers are represented as exact numbers: there is no loss of information
-when such a number is converted from its mathematical value to its internal
-representation. On exact numbers, the elementary operations (<CODE>+</CODE>,
-<CODE>-</CODE>, <CODE>*</CODE>, <CODE>/</CODE>, comparisons, ...) compute the completely
-correct result.
-
-
-<P>
-In CLN, the exact numbers are:
-
-
-
-<UL>
-<LI>
-
-rational numbers (including integers),
-<LI>
-
-complex numbers whose real and imaginary parts are both rational numbers.
-</UL>
-
-<P>
-Rational numbers are always normalized to the form
-<CODE><VAR>numerator</VAR>/<VAR>denominator</VAR></CODE> where the numerator and denominator
-are coprime integers and the denominator is positive. If the resulting
-denominator is <CODE>1</CODE>, the rational number is converted to an integer.
-
-
-<P>
-Small integers (typically in the range <CODE>-2^30</CODE>...<CODE>2^30-1</CODE>,
-for 32-bit machines) are especially efficient, because they consume no heap
-allocation. Otherwise the distinction between these immediate integers
-(called "fixnums") and heap allocated integers (called "bignums")
-is completely transparent.
-
-
-
-
-<H2><A NAME="SEC13" HREF="cln.html#TOC13">3.2 Floating-point numbers</A></H2>
-<P>
-<A NAME="IDX19"></A>
-
-
-<P>
-Not all real numbers can be represented exactly. (There is an easy mathematical
-proof for this: Only a countable set of numbers can be stored exactly in
-a computer, even if one assumes that it has unlimited storage. But there
-are uncountably many real numbers.) So some approximation is needed.
-CLN implements ordinary floating-point numbers, with mantissa and exponent.
-
-
-<P>
-<A NAME="IDX20"></A>
-The elementary operations (<CODE>+</CODE>, <CODE>-</CODE>, <CODE>*</CODE>, <CODE>/</CODE>, ...)
-only return approximate results. For example, the value of the expression
-<CODE>(cl_F) 0.3 + (cl_F) 0.4</CODE> prints as <SAMP>`0.70000005'</SAMP>, not as
-<SAMP>`0.7'</SAMP>. Rounding errors like this one are inevitable when computing
-with floating-point numbers.
-
-
-<P>
-Nevertheless, CLN rounds the floating-point results of the operations <CODE>+</CODE>,
-<CODE>-</CODE>, <CODE>*</CODE>, <CODE>/</CODE>, <CODE>sqrt</CODE> according to the "round-to-even"
-rule: It first computes the exact mathematical result and then returns the
-floating-point number which is nearest to this. If two floating-point numbers
-are equally distant from the ideal result, the one with a <CODE>0</CODE> in its least
-significant mantissa bit is chosen.
-
-
-<P>
-Similarly, testing floating point numbers for equality <SAMP>`x == y'</SAMP>
-is gambling with random errors. Better check for <SAMP>`abs(x - y) < epsilon'</SAMP>
-for some well-chosen <CODE>epsilon</CODE>.
-
-
-<P>
-Floating point numbers come in four flavors:
-
-
-
-<UL>
-<LI>
-
-<A NAME="IDX21"></A>
-Short floats, type <CODE>cl_SF</CODE>.
-They have 1 sign bit, 8 exponent bits (including the exponent's sign),
-and 17 mantissa bits (including the "hidden" bit).
-They don't consume heap allocation.
-
-<LI>
-
-<A NAME="IDX22"></A>
-Single floats, type <CODE>cl_FF</CODE>.
-They have 1 sign bit, 8 exponent bits (including the exponent's sign),
-and 24 mantissa bits (including the "hidden" bit).
-In CLN, they are represented as IEEE single-precision floating point numbers.
-This corresponds closely to the C/C++ type <SAMP>`float'</SAMP>.
-
-<LI>
-
-<A NAME="IDX23"></A>
-Double floats, type <CODE>cl_DF</CODE>.
-They have 1 sign bit, 11 exponent bits (including the exponent's sign),
-and 53 mantissa bits (including the "hidden" bit).
-In CLN, they are represented as IEEE double-precision floating point numbers.
-This corresponds closely to the C/C++ type <SAMP>`double'</SAMP>.
-
-<LI>
-
-<A NAME="IDX24"></A>
-Long floats, type <CODE>cl_LF</CODE>.
-They have 1 sign bit, 32 exponent bits (including the exponent's sign),
-and n mantissa bits (including the "hidden" bit), where n >= 64.
-The precision of a long float is unlimited, but once created, a long float
-has a fixed precision. (No "lazy recomputation".)
-</UL>
-
-<P>
-Of course, computations with long floats are more expensive than those
-with smaller floating-point formats.
-
-
-<P>
-CLN does not implement features like NaNs, denormalized numbers and
-gradual underflow. If the exponent range of some floating-point type
-is too limited for your application, choose another floating-point type
-with larger exponent range.
-
-
-<P>
-<A NAME="IDX25"></A>
-As a user of CLN, you can forget about the differences between the
-four floating-point types and just declare all your floating-point
-variables as being of type <CODE>cl_F</CODE>. This has the advantage that
-when you change the precision of some computation (say, from <CODE>cl_DF</CODE>
-to <CODE>cl_LF</CODE>), you don't have to change the code, only the precision
-of the initial values. Also, many transcendental functions have been
-declared as returning a <CODE>cl_F</CODE> when the argument is a <CODE>cl_F</CODE>,
-but such declarations are missing for the types <CODE>cl_SF</CODE>, <CODE>cl_FF</CODE>,
-<CODE>cl_DF</CODE>, <CODE>cl_LF</CODE>. (Such declarations would be wrong if
-the floating point contagion rule happened to change in the future.)
-
-
-
-
-<H2><A NAME="SEC14" HREF="cln.html#TOC14">3.3 Complex numbers</A></H2>
-<P>
-<A NAME="IDX26"></A>
-
-
-<P>
-Complex numbers, as implemented by the class <CODE>cl_N</CODE>, have a real
-part and an imaginary part, both real numbers. A complex number whose
-imaginary part is the exact number <CODE>0</CODE> is automatically converted
-to a real number.
-
-
-<P>
-Complex numbers can arise from real numbers alone, for example
-through application of <CODE>sqrt</CODE> or transcendental functions.
-
-
-
-
-<H2><A NAME="SEC15" HREF="cln.html#TOC15">3.4 Conversions</A></H2>
-<P>
-<A NAME="IDX27"></A>
-
-
-<P>
-Conversions from any class to any its superclasses ("base classes" in
-C++ terminology) is done automatically.
-
-
-<P>
-Conversions from the C built-in types <SAMP>`long'</SAMP> and <SAMP>`unsigned long'</SAMP>
-are provided for the classes <CODE>cl_I</CODE>, <CODE>cl_RA</CODE>, <CODE>cl_R</CODE>,
-<CODE>cl_N</CODE> and <CODE>cl_number</CODE>.
-
-
-<P>
-Conversions from the C built-in types <SAMP>`int'</SAMP> and <SAMP>`unsigned int'</SAMP>
-are provided for the classes <CODE>cl_I</CODE>, <CODE>cl_RA</CODE>, <CODE>cl_R</CODE>,
-<CODE>cl_N</CODE> and <CODE>cl_number</CODE>. However, these conversions emphasize
-efficiency. Their range is therefore limited:
-
-
-
-<UL>
-<LI>
-
-The conversion from <SAMP>`int'</SAMP> works only if the argument is < 2^29 and > -2^29.
-<LI>
-
-The conversion from <SAMP>`unsigned int'</SAMP> works only if the argument is < 2^29.
-</UL>
-
-<P>
-In a declaration like <SAMP>`cl_I x = 10;'</SAMP> the C++ compiler is able to
-do the conversion of <CODE>10</CODE> from <SAMP>`int'</SAMP> to <SAMP>`cl_I'</SAMP> at compile time
-already. On the other hand, code like <SAMP>`cl_I x = 1000000000;'</SAMP> is
-in error.
-So, if you want to be sure that an <SAMP>`int'</SAMP> whose magnitude is not guaranteed
-to be < 2^29 is correctly converted to a <SAMP>`cl_I'</SAMP>, first convert it to a
-<SAMP>`long'</SAMP>. Similarly, if a large <SAMP>`unsigned int'</SAMP> is to be converted to a
-<SAMP>`cl_I'</SAMP>, first convert it to an <SAMP>`unsigned long'</SAMP>.
-
-
-<P>
-Conversions from the C built-in type <SAMP>`float'</SAMP> are provided for the classes
-<CODE>cl_FF</CODE>, <CODE>cl_F</CODE>, <CODE>cl_R</CODE>, <CODE>cl_N</CODE> and <CODE>cl_number</CODE>.
-
-
-<P>
-Conversions from the C built-in type <SAMP>`double'</SAMP> are provided for the classes
-<CODE>cl_DF</CODE>, <CODE>cl_F</CODE>, <CODE>cl_R</CODE>, <CODE>cl_N</CODE> and <CODE>cl_number</CODE>.
-
-
-<P>
-Conversions from <SAMP>`const char *'</SAMP> are provided for the classes
-<CODE>cl_I</CODE>, <CODE>cl_RA</CODE>,
-<CODE>cl_SF</CODE>, <CODE>cl_FF</CODE>, <CODE>cl_DF</CODE>, <CODE>cl_LF</CODE>, <CODE>cl_F</CODE>,
-<CODE>cl_R</CODE>, <CODE>cl_N</CODE>.
-The easiest way to specify a value which is outside of the range of the
-C++ built-in types is therefore to specify it as a string, like this:
-<A NAME="IDX28"></A>
-
-<PRE>
- cl_I order_of_rubiks_cube_group = "43252003274489856000";
-</PRE>
-
-<P>
-Note that this conversion is done at runtime, not at compile-time.
-
-
-<P>
-Conversions from <CODE>cl_I</CODE> to the C built-in types <SAMP>`int'</SAMP>,
-<SAMP>`unsigned int'</SAMP>, <SAMP>`long'</SAMP>, <SAMP>`unsigned long'</SAMP> are provided through
-the functions
-
-
-<DL COMPACT>
-
-<DT><CODE>int cl_I_to_int (const cl_I& x)</CODE>
-<DD>
-<A NAME="IDX29"></A>
-<DT><CODE>unsigned int cl_I_to_uint (const cl_I& x)</CODE>
-<DD>
-<A NAME="IDX30"></A>
-<DT><CODE>long cl_I_to_long (const cl_I& x)</CODE>
-<DD>
-<A NAME="IDX31"></A>
-<DT><CODE>unsigned long cl_I_to_ulong (const cl_I& x)</CODE>
-<DD>
-<A NAME="IDX32"></A>
-Returns <CODE>x</CODE> as element of the C type <VAR>ctype</VAR>. If <CODE>x</CODE> is not
-representable in the range of <VAR>ctype</VAR>, a runtime error occurs.
-</DL>
-
-<P>
-Conversions from the classes <CODE>cl_I</CODE>, <CODE>cl_RA</CODE>,
-<CODE>cl_SF</CODE>, <CODE>cl_FF</CODE>, <CODE>cl_DF</CODE>, <CODE>cl_LF</CODE>, <CODE>cl_F</CODE> and
-<CODE>cl_R</CODE>
-to the C built-in types <SAMP>`float'</SAMP> and <SAMP>`double'</SAMP> are provided through
-the functions
-
-
-<DL COMPACT>
-
-<DT><CODE>float float_approx (const <VAR>type</VAR>& x)</CODE>
-<DD>
-<A NAME="IDX33"></A>
-<DT><CODE>double double_approx (const <VAR>type</VAR>& x)</CODE>
-<DD>
-<A NAME="IDX34"></A>
-Returns an approximation of <CODE>x</CODE> of C type <VAR>ctype</VAR>.
-If <CODE>abs(x)</CODE> is too close to 0 (underflow), 0 is returned.
-If <CODE>abs(x)</CODE> is too large (overflow), an IEEE infinity is returned.
-</DL>
-
-<P>
-Conversions from any class to any of its subclasses ("derived classes" in
-C++ terminology) are not provided. Instead, you can assert and check
-that a value belongs to a certain subclass, and return it as element of that
-class, using the <SAMP>`As'</SAMP> and <SAMP>`The'</SAMP> macros.
-<A NAME="IDX35"></A>
-<CODE>As(<VAR>type</VAR>)(<VAR>value</VAR>)</CODE> checks that <VAR>value</VAR> belongs to
-<VAR>type</VAR> and returns it as such.
-<A NAME="IDX36"></A>
-<CODE>The(<VAR>type</VAR>)(<VAR>value</VAR>)</CODE> assumes that <VAR>value</VAR> belongs to
-<VAR>type</VAR> and returns it as such. It is your responsibility to ensure
-that this assumption is valid.
-Example:
-
-
-
-<PRE>
- cl_I x = ...;
- if (!(x >= 0)) abort();
- cl_I ten_x = The(cl_I)(expt(10,x)); // If x >= 0, 10^x is an integer.
- // In general, it would be a rational number.
-</PRE>
-
-
-
-<H1><A NAME="SEC16" HREF="cln.html#TOC16">4. Functions on numbers</A></H1>
-
-<P>
-Each of the number classes declares its mathematical operations in the
-corresponding include file. For example, if your code operates with
-objects of type <CODE>cl_I</CODE>, it should <CODE>#include <cln/integer.h></CODE>.
-
-
-
-
-<H2><A NAME="SEC17" HREF="cln.html#TOC17">4.1 Constructing numbers</A></H2>
-
-<P>
-Here is how to create number objects "from nothing".
-
-
-
-
-<H3><A NAME="SEC18" HREF="cln.html#TOC18">4.1.1 Constructing integers</A></H3>
-
-<P>
-<CODE>cl_I</CODE> objects are most easily constructed from C integers and from
-strings. See section <A HREF="cln.html#SEC15">3.4 Conversions</A>.
-
-
-
-
-<H3><A NAME="SEC19" HREF="cln.html#TOC19">4.1.2 Constructing rational numbers</A></H3>
-
-<P>
-<CODE>cl_RA</CODE> objects can be constructed from strings. The syntax
-for rational numbers is described in section <A HREF="cln.html#SEC45">5.1 Internal and printed representation</A>.
-Another standard way to produce a rational number is through application
-of <SAMP>`operator /'</SAMP> or <SAMP>`recip'</SAMP> on integers.
-
-
-
-
-<H3><A NAME="SEC20" HREF="cln.html#TOC20">4.1.3 Constructing floating-point numbers</A></H3>
-
-<P>
-<CODE>cl_F</CODE> objects with low precision are most easily constructed from
-C <SAMP>`float'</SAMP> and <SAMP>`double'</SAMP>. See section <A HREF="cln.html#SEC15">3.4 Conversions</A>.
-
-
-<P>
-To construct a <CODE>cl_F</CODE> with high precision, you can use the conversion
-from <SAMP>`const char *'</SAMP>, but you have to specify the desired precision
-within the string. (See section <A HREF="cln.html#SEC45">5.1 Internal and printed representation</A>.)
-Example:
-
-<PRE>
- cl_F e = "0.271828182845904523536028747135266249775724709369996e+1_40";
-</PRE>
-
-<P>
-will set <SAMP>`e'</SAMP> to the given value, with a precision of 40 decimal digits.
-
-
-<P>
-The programmatic way to construct a <CODE>cl_F</CODE> with high precision is
-through the <CODE>cl_float</CODE> conversion function, see
-section <A HREF="cln.html#SEC40">4.11.1 Conversion to floating-point numbers</A>. For example, to compute
-<CODE>e</CODE> to 40 decimal places, first construct 1.0 to 40 decimal places
-and then apply the exponential function:
-
-<PRE>
- cl_float_format_t precision = cl_float_format(40);
- cl_F e = exp(cl_float(1,precision));
-</PRE>
-
-
-
-<H3><A NAME="SEC21" HREF="cln.html#TOC21">4.1.4 Constructing complex numbers</A></H3>
-
-<P>
-Non-real <CODE>cl_N</CODE> objects are normally constructed through the function
-
-<PRE>
- cl_N complex (const cl_R& realpart, const cl_R& imagpart)
-</PRE>
-
-<P>
-See section <A HREF="cln.html#SEC24">4.4 Elementary complex functions</A>.
-
-
-
-
-<H2><A NAME="SEC22" HREF="cln.html#TOC22">4.2 Elementary functions</A></H2>
-
-<P>
-Each of the classes <CODE>cl_N</CODE>, <CODE>cl_R</CODE>, <CODE>cl_RA</CODE>, <CODE>cl_I</CODE>,
-<CODE>cl_F</CODE>, <CODE>cl_SF</CODE>, <CODE>cl_FF</CODE>, <CODE>cl_DF</CODE>, <CODE>cl_LF</CODE>
-defines the following operations:
-
-
-<DL COMPACT>
-
-<DT><CODE><VAR>type</VAR> operator + (const <VAR>type</VAR>&, const <VAR>type</VAR>&)</CODE>
-<DD>
-<A NAME="IDX37"></A>
-Addition.
-
-<DT><CODE><VAR>type</VAR> operator - (const <VAR>type</VAR>&, const <VAR>type</VAR>&)</CODE>
-<DD>
-<A NAME="IDX38"></A>
-Subtraction.
-
-<DT><CODE><VAR>type</VAR> operator - (const <VAR>type</VAR>&)</CODE>
-<DD>
-Returns the negative of the argument.
-
-<DT><CODE><VAR>type</VAR> plus1 (const <VAR>type</VAR>& x)</CODE>
-<DD>
-<A NAME="IDX39"></A>
-Returns <CODE>x + 1</CODE>.
-
-<DT><CODE><VAR>type</VAR> minus1 (const <VAR>type</VAR>& x)</CODE>
-<DD>
-<A NAME="IDX40"></A>
-Returns <CODE>x - 1</CODE>.
-
-<DT><CODE><VAR>type</VAR> operator * (const <VAR>type</VAR>&, const <VAR>type</VAR>&)</CODE>
-<DD>
-<A NAME="IDX41"></A>
-Multiplication.
-
-<DT><CODE><VAR>type</VAR> square (const <VAR>type</VAR>& x)</CODE>
-<DD>
-<A NAME="IDX42"></A>
-Returns <CODE>x * x</CODE>.
-</DL>
-
-<P>
-Each of the classes <CODE>cl_N</CODE>, <CODE>cl_R</CODE>, <CODE>cl_RA</CODE>,
-<CODE>cl_F</CODE>, <CODE>cl_SF</CODE>, <CODE>cl_FF</CODE>, <CODE>cl_DF</CODE>, <CODE>cl_LF</CODE>
-defines the following operations:
-
-
-<DL COMPACT>
-
-<DT><CODE><VAR>type</VAR> operator / (const <VAR>type</VAR>&, const <VAR>type</VAR>&)</CODE>
-<DD>
-<A NAME="IDX43"></A>
-Division.
-
-<DT><CODE><VAR>type</VAR> recip (const <VAR>type</VAR>&)</CODE>
-<DD>
-<A NAME="IDX44"></A>
-Returns the reciprocal of the argument.
-</DL>
-
-<P>
-The class <CODE>cl_I</CODE> doesn't define a <SAMP>`/'</SAMP> operation because
-in the C/C++ language this operator, applied to integral types,
-denotes the <SAMP>`floor'</SAMP> or <SAMP>`truncate'</SAMP> operation (which one of these,
-is implementation dependent). (See section <A HREF="cln.html#SEC26">4.6 Rounding functions</A>.)
-Instead, <CODE>cl_I</CODE> defines an "exact quotient" function:
-
-
-<DL COMPACT>
-
-<DT><CODE>cl_I exquo (const cl_I& x, const cl_I& y)</CODE>
-<DD>
-<A NAME="IDX45"></A>
-Checks that <CODE>y</CODE> divides <CODE>x</CODE>, and returns the quotient <CODE>x</CODE>/<CODE>y</CODE>.
-</DL>
-
-<P>
-The following exponentiation functions are defined:
-
-
-<DL COMPACT>
-
-<DT><CODE>cl_I expt_pos (const cl_I& x, const cl_I& y)</CODE>
-<DD>
-<A NAME="IDX46"></A>
-<DT><CODE>cl_RA expt_pos (const cl_RA& x, const cl_I& y)</CODE>
-<DD>
-<CODE>y</CODE> must be > 0. Returns <CODE>x^y</CODE>.
-
-<DT><CODE>cl_RA expt (const cl_RA& x, const cl_I& y)</CODE>
-<DD>
-<A NAME="IDX47"></A>
-<DT><CODE>cl_R expt (const cl_R& x, const cl_I& y)</CODE>
-<DD>
-<DT><CODE>cl_N expt (const cl_N& x, const cl_I& y)</CODE>
-<DD>
-Returns <CODE>x^y</CODE>.
-</DL>
-
-<P>
-Each of the classes <CODE>cl_R</CODE>, <CODE>cl_RA</CODE>, <CODE>cl_I</CODE>,
-<CODE>cl_F</CODE>, <CODE>cl_SF</CODE>, <CODE>cl_FF</CODE>, <CODE>cl_DF</CODE>, <CODE>cl_LF</CODE>
-defines the following operation:
-
-
-<DL COMPACT>
-
-<DT><CODE><VAR>type</VAR> abs (const <VAR>type</VAR>& x)</CODE>
-<DD>
-<A NAME="IDX48"></A>
-Returns the absolute value of <CODE>x</CODE>.
-This is <CODE>x</CODE> if <CODE>x >= 0</CODE>, and <CODE>-x</CODE> if <CODE>x <= 0</CODE>.
-</DL>
-
-<P>
-The class <CODE>cl_N</CODE> implements this as follows:
-
-
-<DL COMPACT>
-
-<DT><CODE>cl_R abs (const cl_N x)</CODE>
-<DD>
-Returns the absolute value of <CODE>x</CODE>.
-</DL>
-
-<P>
-Each of the classes <CODE>cl_N</CODE>, <CODE>cl_R</CODE>, <CODE>cl_RA</CODE>, <CODE>cl_I</CODE>,
-<CODE>cl_F</CODE>, <CODE>cl_SF</CODE>, <CODE>cl_FF</CODE>, <CODE>cl_DF</CODE>, <CODE>cl_LF</CODE>
-defines the following operation:
-
-
-<DL COMPACT>
-
-<DT><CODE><VAR>type</VAR> signum (const <VAR>type</VAR>& x)</CODE>
-<DD>
-<A NAME="IDX49"></A>
-Returns the sign of <CODE>x</CODE>, in the same number format as <CODE>x</CODE>.
-This is defined as <CODE>x / abs(x)</CODE> if <CODE>x</CODE> is non-zero, and
-<CODE>x</CODE> if <CODE>x</CODE> is zero. If <CODE>x</CODE> is real, the value is either
-0 or 1 or -1.
-</DL>
-
-
-
-<H2><A NAME="SEC23" HREF="cln.html#TOC23">4.3 Elementary rational functions</A></H2>
-
-<P>
-Each of the classes <CODE>cl_RA</CODE>, <CODE>cl_I</CODE> defines the following operations:
-
-
-<DL COMPACT>
-
-<DT><CODE>cl_I numerator (const <VAR>type</VAR>& x)</CODE>
-<DD>
-<A NAME="IDX50"></A>
-Returns the numerator of <CODE>x</CODE>.
-
-<DT><CODE>cl_I denominator (const <VAR>type</VAR>& x)</CODE>
-<DD>
-<A NAME="IDX51"></A>
-Returns the denominator of <CODE>x</CODE>.
-</DL>
-
-<P>
-The numerator and denominator of a rational number are normalized in such
-a way that they have no factor in common and the denominator is positive.
-
-
-
-
-<H2><A NAME="SEC24" HREF="cln.html#TOC24">4.4 Elementary complex functions</A></H2>
-
-<P>
-The class <CODE>cl_N</CODE> defines the following operation:
-
-
-<DL COMPACT>
-
-<DT><CODE>cl_N complex (const cl_R& a, const cl_R& b)</CODE>
-<DD>
-<A NAME="IDX52"></A>
-Returns the complex number <CODE>a+bi</CODE>, that is, the complex number with
-real part <CODE>a</CODE> and imaginary part <CODE>b</CODE>.
-</DL>
-
-<P>
-Each of the classes <CODE>cl_N</CODE>, <CODE>cl_R</CODE> defines the following operations:
-
-
-<DL COMPACT>
-
-<DT><CODE>cl_R realpart (const <VAR>type</VAR>& x)</CODE>
-<DD>
-<A NAME="IDX53"></A>
-Returns the real part of <CODE>x</CODE>.
-
-<DT><CODE>cl_R imagpart (const <VAR>type</VAR>& x)</CODE>
-<DD>
-<A NAME="IDX54"></A>
-Returns the imaginary part of <CODE>x</CODE>.
-
-<DT><CODE><VAR>type</VAR> conjugate (const <VAR>type</VAR>& x)</CODE>
-<DD>
-<A NAME="IDX55"></A>
-Returns the complex conjugate of <CODE>x</CODE>.
-</DL>
-
-<P>
-We have the relations
-
-
-
-<UL>
-<LI>
-
-<CODE>x = complex(realpart(x), imagpart(x))</CODE>
-<LI>
-
-<CODE>conjugate(x) = complex(realpart(x), -imagpart(x))</CODE>
-</UL>
-
-
-
-<H2><A NAME="SEC25" HREF="cln.html#TOC25">4.5 Comparisons</A></H2>
-<P>
-<A NAME="IDX56"></A>
-
-
-<P>
-Each of the classes <CODE>cl_N</CODE>, <CODE>cl_R</CODE>, <CODE>cl_RA</CODE>, <CODE>cl_I</CODE>,
-<CODE>cl_F</CODE>, <CODE>cl_SF</CODE>, <CODE>cl_FF</CODE>, <CODE>cl_DF</CODE>, <CODE>cl_LF</CODE>
-defines the following operations:
-
-
-<DL COMPACT>
-
-<DT><CODE>bool operator == (const <VAR>type</VAR>&, const <VAR>type</VAR>&)</CODE>
-<DD>
-<A NAME="IDX57"></A>
-<DT><CODE>bool operator != (const <VAR>type</VAR>&, const <VAR>type</VAR>&)</CODE>
-<DD>
-<A NAME="IDX58"></A>
-Comparison, as in C and C++.
-
-<DT><CODE>uint32 equal_hashcode (const <VAR>type</VAR>&)</CODE>
-<DD>
-<A NAME="IDX59"></A>
-Returns a 32-bit hash code that is the same for any two numbers which are
-the same according to <CODE>==</CODE>. This hash code depends on the number's value,
-not its type or precision.
-
-<DT><CODE>cl_boolean zerop (const <VAR>type</VAR>& x)</CODE>
-<DD>
-<A NAME="IDX60"></A>
-Compare against zero: <CODE>x == 0</CODE>
-</DL>
-
-<P>
-Each of the classes <CODE>cl_R</CODE>, <CODE>cl_RA</CODE>, <CODE>cl_I</CODE>,
-<CODE>cl_F</CODE>, <CODE>cl_SF</CODE>, <CODE>cl_FF</CODE>, <CODE>cl_DF</CODE>, <CODE>cl_LF</CODE>
-defines the following operations:
-
-
-<DL COMPACT>
-
-<DT><CODE>cl_signean compare (const <VAR>type</VAR>& x, const <VAR>type</VAR>& y)</CODE>
-<DD>
-<A NAME="IDX61"></A>
-Compares <CODE>x</CODE> and <CODE>y</CODE>. Returns +1 if <CODE>x</CODE>><CODE>y</CODE>,
--1 if <CODE>x</CODE><<CODE>y</CODE>, 0 if <CODE>x</CODE>=<CODE>y</CODE>.
-
-<DT><CODE>bool operator <= (const <VAR>type</VAR>&, const <VAR>type</VAR>&)</CODE>
-<DD>
-<A NAME="IDX62"></A>
-<DT><CODE>bool operator < (const <VAR>type</VAR>&, const <VAR>type</VAR>&)</CODE>
-<DD>
-<A NAME="IDX63"></A>
-<DT><CODE>bool operator >= (const <VAR>type</VAR>&, const <VAR>type</VAR>&)</CODE>
-<DD>
-<A NAME="IDX64"></A>
-<DT><CODE>bool operator > (const <VAR>type</VAR>&, const <VAR>type</VAR>&)</CODE>
-<DD>
-<A NAME="IDX65"></A>
-Comparison, as in C and C++.
-
-<DT><CODE>cl_boolean minusp (const <VAR>type</VAR>& x)</CODE>
-<DD>
-<A NAME="IDX66"></A>
-Compare against zero: <CODE>x < 0</CODE>
-
-<DT><CODE>cl_boolean plusp (const <VAR>type</VAR>& x)</CODE>
-<DD>
-<A NAME="IDX67"></A>
-Compare against zero: <CODE>x > 0</CODE>
-
-<DT><CODE><VAR>type</VAR> max (const <VAR>type</VAR>& x, const <VAR>type</VAR>& y)</CODE>
-<DD>
-<A NAME="IDX68"></A>
-Return the maximum of <CODE>x</CODE> and <CODE>y</CODE>.
-
-<DT><CODE><VAR>type</VAR> min (const <VAR>type</VAR>& x, const <VAR>type</VAR>& y)</CODE>
-<DD>
-<A NAME="IDX69"></A>
-Return the minimum of <CODE>x</CODE> and <CODE>y</CODE>.
-</DL>
-
-<P>
-When a floating point number and a rational number are compared, the float
-is first converted to a rational number using the function <CODE>rational</CODE>.
-Since a floating point number actually represents an interval of real numbers,
-the result might be surprising.
-For example, <CODE>(cl_F)(cl_R)"1/3" == (cl_R)"1/3"</CODE> returns false because
-there is no floating point number whose value is exactly <CODE>1/3</CODE>.
-
-
-
-
-<H2><A NAME="SEC26" HREF="cln.html#TOC26">4.6 Rounding functions</A></H2>
-<P>
-<A NAME="IDX70"></A>
-
-
-<P>
-When a real number is to be converted to an integer, there is no "best"
-rounding. The desired rounding function depends on the application.
-The Common Lisp and ISO Lisp standards offer four rounding functions:
-
-
-<DL COMPACT>
-
-<DT><CODE>floor(x)</CODE>
-<DD>
-This is the largest integer <=<CODE>x</CODE>.
-
-<DT><CODE>ceiling(x)</CODE>
-<DD>
-This is the smallest integer >=<CODE>x</CODE>.
-
-<DT><CODE>truncate(x)</CODE>
-<DD>
-Among the integers between 0 and <CODE>x</CODE> (inclusive) the one nearest to <CODE>x</CODE>.
-
-<DT><CODE>round(x)</CODE>
-<DD>
-The integer nearest to <CODE>x</CODE>. If <CODE>x</CODE> is exactly halfway between two
-integers, choose the even one.
-</DL>
-
-<P>
-These functions have different advantages:
-
-
-<P>
-<CODE>floor</CODE> and <CODE>ceiling</CODE> are translation invariant:
-<CODE>floor(x+n) = floor(x) + n</CODE> and <CODE>ceiling(x+n) = ceiling(x) + n</CODE>
-for every <CODE>x</CODE> and every integer <CODE>n</CODE>.
-
-
-<P>
-On the other hand, <CODE>truncate</CODE> and <CODE>round</CODE> are symmetric:
-<CODE>truncate(-x) = -truncate(x)</CODE> and <CODE>round(-x) = -round(x)</CODE>,
-and furthermore <CODE>round</CODE> is unbiased: on the "average", it rounds
-down exactly as often as it rounds up.
-
-
-<P>
-The functions are related like this:
-
-
-
-<UL>
-<LI>
-
-<CODE>ceiling(m/n) = floor((m+n-1)/n) = floor((m-1)/n)+1</CODE>
-for rational numbers <CODE>m/n</CODE> (<CODE>m</CODE>, <CODE>n</CODE> integers, <CODE>n</CODE>>0), and
-<LI>
-
-<CODE>truncate(x) = sign(x) * floor(abs(x))</CODE>
-</UL>
-
-<P>
-Each of the classes <CODE>cl_R</CODE>, <CODE>cl_RA</CODE>,
-<CODE>cl_F</CODE>, <CODE>cl_SF</CODE>, <CODE>cl_FF</CODE>, <CODE>cl_DF</CODE>, <CODE>cl_LF</CODE>
-defines the following operations:
-
-
-<DL COMPACT>
-
-<DT><CODE>cl_I floor1 (const <VAR>type</VAR>& x)</CODE>
-<DD>
-<A NAME="IDX71"></A>
-Returns <CODE>floor(x)</CODE>.
-<DT><CODE>cl_I ceiling1 (const <VAR>type</VAR>& x)</CODE>
-<DD>
-<A NAME="IDX72"></A>
-Returns <CODE>ceiling(x)</CODE>.
-<DT><CODE>cl_I truncate1 (const <VAR>type</VAR>& x)</CODE>
-<DD>
-<A NAME="IDX73"></A>
-Returns <CODE>truncate(x)</CODE>.
-<DT><CODE>cl_I round1 (const <VAR>type</VAR>& x)</CODE>
-<DD>
-<A NAME="IDX74"></A>
-Returns <CODE>round(x)</CODE>.
-</DL>
-
-<P>
-Each of the classes <CODE>cl_R</CODE>, <CODE>cl_RA</CODE>, <CODE>cl_I</CODE>,
-<CODE>cl_F</CODE>, <CODE>cl_SF</CODE>, <CODE>cl_FF</CODE>, <CODE>cl_DF</CODE>, <CODE>cl_LF</CODE>
-defines the following operations:
-
-
-<DL COMPACT>
-
-<DT><CODE>cl_I floor1 (const <VAR>type</VAR>& x, const <VAR>type</VAR>& y)</CODE>
-<DD>
-Returns <CODE>floor(x/y)</CODE>.
-<DT><CODE>cl_I ceiling1 (const <VAR>type</VAR>& x, const <VAR>type</VAR>& y)</CODE>
-<DD>
-Returns <CODE>ceiling(x/y)</CODE>.
-<DT><CODE>cl_I truncate1 (const <VAR>type</VAR>& x, const <VAR>type</VAR>& y)</CODE>
-<DD>
-Returns <CODE>truncate(x/y)</CODE>.
-<DT><CODE>cl_I round1 (const <VAR>type</VAR>& x, const <VAR>type</VAR>& y)</CODE>
-<DD>
-Returns <CODE>round(x/y)</CODE>.
-</DL>
-
-<P>
-These functions are called <SAMP>`floor1'</SAMP>, ... here instead of
-<SAMP>`floor'</SAMP>, ..., because on some systems, system dependent include
-files define <SAMP>`floor'</SAMP> and <SAMP>`ceiling'</SAMP> as macros.
-
-
-<P>
-In many cases, one needs both the quotient and the remainder of a division.
-It is more efficient to compute both at the same time than to perform
-two divisions, one for quotient and the next one for the remainder.
-The following functions therefore return a structure containing both
-the quotient and the remainder. The suffix <SAMP>`2'</SAMP> indicates the number
-of "return values". The remainder is defined as follows:
-
-
-
-<UL>
-<LI>
-
-for the computation of <CODE>quotient = floor(x)</CODE>,
-<CODE>remainder = x - quotient</CODE>,
-<LI>
-
-for the computation of <CODE>quotient = floor(x,y)</CODE>,
-<CODE>remainder = x - quotient*y</CODE>,
-</UL>
-
-<P>
-and similarly for the other three operations.
-
-
-<P>
-Each of the classes <CODE>cl_R</CODE>, <CODE>cl_RA</CODE>,
-<CODE>cl_F</CODE>, <CODE>cl_SF</CODE>, <CODE>cl_FF</CODE>, <CODE>cl_DF</CODE>, <CODE>cl_LF</CODE>
-defines the following operations:
-
-
-<DL COMPACT>
-
-<DT><CODE>struct <VAR>type</VAR>_div_t { cl_I quotient; <VAR>type</VAR> remainder; };</CODE>
-<DD>
-<DT><CODE><VAR>type</VAR>_div_t floor2 (const <VAR>type</VAR>& x)</CODE>
-<DD>
-<DT><CODE><VAR>type</VAR>_div_t ceiling2 (const <VAR>type</VAR>& x)</CODE>
-<DD>
-<DT><CODE><VAR>type</VAR>_div_t truncate2 (const <VAR>type</VAR>& x)</CODE>
-<DD>
-<DT><CODE><VAR>type</VAR>_div_t round2 (const <VAR>type</VAR>& x)</CODE>
-<DD>
-</DL>
-
-<P>
-Each of the classes <CODE>cl_R</CODE>, <CODE>cl_RA</CODE>, <CODE>cl_I</CODE>,
-<CODE>cl_F</CODE>, <CODE>cl_SF</CODE>, <CODE>cl_FF</CODE>, <CODE>cl_DF</CODE>, <CODE>cl_LF</CODE>
-defines the following operations:
-
-
-<DL COMPACT>
-
-<DT><CODE>struct <VAR>type</VAR>_div_t { cl_I quotient; <VAR>type</VAR> remainder; };</CODE>
-<DD>
-<DT><CODE><VAR>type</VAR>_div_t floor2 (const <VAR>type</VAR>& x, const <VAR>type</VAR>& y)</CODE>
-<DD>
-<A NAME="IDX75"></A>
-<DT><CODE><VAR>type</VAR>_div_t ceiling2 (const <VAR>type</VAR>& x, const <VAR>type</VAR>& y)</CODE>
-<DD>
-<A NAME="IDX76"></A>
-<DT><CODE><VAR>type</VAR>_div_t truncate2 (const <VAR>type</VAR>& x, const <VAR>type</VAR>& y)</CODE>
-<DD>
-<A NAME="IDX77"></A>
-<DT><CODE><VAR>type</VAR>_div_t round2 (const <VAR>type</VAR>& x, const <VAR>type</VAR>& y)</CODE>
-<DD>
-<A NAME="IDX78"></A>
-</DL>
-
-<P>
-Sometimes, one wants the quotient as a floating-point number (of the
-same format as the argument, if the argument is a float) instead of as
-an integer. The prefix <SAMP>`f'</SAMP> indicates this.
-
-
-<P>
-Each of the classes
-<CODE>cl_F</CODE>, <CODE>cl_SF</CODE>, <CODE>cl_FF</CODE>, <CODE>cl_DF</CODE>, <CODE>cl_LF</CODE>
-defines the following operations:
-
-
-<DL COMPACT>
-
-<DT><CODE><VAR>type</VAR> ffloor (const <VAR>type</VAR>& x)</CODE>
-<DD>
-<A NAME="IDX79"></A>
-<DT><CODE><VAR>type</VAR> fceiling (const <VAR>type</VAR>& x)</CODE>
-<DD>
-<A NAME="IDX80"></A>
-<DT><CODE><VAR>type</VAR> ftruncate (const <VAR>type</VAR>& x)</CODE>
-<DD>
-<A NAME="IDX81"></A>
-<DT><CODE><VAR>type</VAR> fround (const <VAR>type</VAR>& x)</CODE>
-<DD>
-<A NAME="IDX82"></A>
-</DL>
-
-<P>
-and similarly for class <CODE>cl_R</CODE>, but with return type <CODE>cl_F</CODE>.
-
-
-<P>
-The class <CODE>cl_R</CODE> defines the following operations:
-
-
-<DL COMPACT>
-
-<DT><CODE>cl_F ffloor (const <VAR>type</VAR>& x, const <VAR>type</VAR>& y)</CODE>
-<DD>
-<DT><CODE>cl_F fceiling (const <VAR>type</VAR>& x, const <VAR>type</VAR>& y)</CODE>
-<DD>
-<DT><CODE>cl_F ftruncate (const <VAR>type</VAR>& x, const <VAR>type</VAR>& y)</CODE>
-<DD>
-<DT><CODE>cl_F fround (const <VAR>type</VAR>& x, const <VAR>type</VAR>& y)</CODE>
-<DD>
-</DL>
-
-<P>
-These functions also exist in versions which return both the quotient
-and the remainder. The suffix <SAMP>`2'</SAMP> indicates this.
-
-
-<P>
-Each of the classes
-<CODE>cl_F</CODE>, <CODE>cl_SF</CODE>, <CODE>cl_FF</CODE>, <CODE>cl_DF</CODE>, <CODE>cl_LF</CODE>
-defines the following operations:
-<A NAME="IDX83"></A>
-<A NAME="IDX84"></A>
-<A NAME="IDX85"></A>
-<A NAME="IDX86"></A>
-<A NAME="IDX87"></A>
-
-
-<DL COMPACT>
-
-<DT><CODE>struct <VAR>type</VAR>_fdiv_t { <VAR>type</VAR> quotient; <VAR>type</VAR> remainder; };</CODE>
-<DD>
-<DT><CODE><VAR>type</VAR>_fdiv_t ffloor2 (const <VAR>type</VAR>& x)</CODE>
-<DD>
-<A NAME="IDX88"></A>
-<DT><CODE><VAR>type</VAR>_fdiv_t fceiling2 (const <VAR>type</VAR>& x)</CODE>
-<DD>
-<A NAME="IDX89"></A>
-<DT><CODE><VAR>type</VAR>_fdiv_t ftruncate2 (const <VAR>type</VAR>& x)</CODE>
-<DD>
-<A NAME="IDX90"></A>
-<DT><CODE><VAR>type</VAR>_fdiv_t fround2 (const <VAR>type</VAR>& x)</CODE>
-<DD>
-<A NAME="IDX91"></A>
-</DL>
-<P>
-and similarly for class <CODE>cl_R</CODE>, but with quotient type <CODE>cl_F</CODE>.
-<A NAME="IDX92"></A>
-
-
-<P>
-The class <CODE>cl_R</CODE> defines the following operations:
-
-
-<DL COMPACT>
-
-<DT><CODE>struct <VAR>type</VAR>_fdiv_t { cl_F quotient; cl_R remainder; };</CODE>
-<DD>
-<DT><CODE><VAR>type</VAR>_fdiv_t ffloor2 (const <VAR>type</VAR>& x, const <VAR>type</VAR>& y)</CODE>
-<DD>
-<DT><CODE><VAR>type</VAR>_fdiv_t fceiling2 (const <VAR>type</VAR>& x, const <VAR>type</VAR>& y)</CODE>
-<DD>
-<DT><CODE><VAR>type</VAR>_fdiv_t ftruncate2 (const <VAR>type</VAR>& x, const <VAR>type</VAR>& y)</CODE>
-<DD>
-<DT><CODE><VAR>type</VAR>_fdiv_t fround2 (const <VAR>type</VAR>& x, const <VAR>type</VAR>& y)</CODE>
-<DD>
-</DL>
-
-<P>
-Other applications need only the remainder of a division.
-The remainder of <SAMP>`floor'</SAMP> and <SAMP>`ffloor'</SAMP> is called <SAMP>`mod'</SAMP>
-(abbreviation of "modulo"). The remainder <SAMP>`truncate'</SAMP> and
-<SAMP>`ftruncate'</SAMP> is called <SAMP>`rem'</SAMP> (abbreviation of "remainder").
-
-
-
-<UL>
-<LI>
-
-<CODE>mod(x,y) = floor2(x,y).remainder = x - floor(x/y)*y</CODE>
-<LI>
-
-<CODE>rem(x,y) = truncate2(x,y).remainder = x - truncate(x/y)*y</CODE>
-</UL>
-
-<P>
-If <CODE>x</CODE> and <CODE>y</CODE> are both >= 0, <CODE>mod(x,y) = rem(x,y) >= 0</CODE>.
-In general, <CODE>mod(x,y)</CODE> has the sign of <CODE>y</CODE> or is zero,
-and <CODE>rem(x,y)</CODE> has the sign of <CODE>x</CODE> or is zero.
-
-
-<P>
-The classes <CODE>cl_R</CODE>, <CODE>cl_I</CODE> define the following operations:
-
-
-<DL COMPACT>
-
-<DT><CODE><VAR>type</VAR> mod (const <VAR>type</VAR>& x, const <VAR>type</VAR>& y)</CODE>
-<DD>
-<A NAME="IDX93"></A>
-<DT><CODE><VAR>type</VAR> rem (const <VAR>type</VAR>& x, const <VAR>type</VAR>& y)</CODE>
-<DD>
-<A NAME="IDX94"></A>
-</DL>
-
-
-
-<H2><A NAME="SEC27" HREF="cln.html#TOC27">4.7 Roots</A></H2>
-
-<P>
-Each of the classes <CODE>cl_R</CODE>,
-<CODE>cl_F</CODE>, <CODE>cl_SF</CODE>, <CODE>cl_FF</CODE>, <CODE>cl_DF</CODE>, <CODE>cl_LF</CODE>
-defines the following operation:
-
-
-<DL COMPACT>
-
-<DT><CODE><VAR>type</VAR> sqrt (const <VAR>type</VAR>& x)</CODE>
-<DD>
-<A NAME="IDX95"></A>
-<CODE>x</CODE> must be >= 0. This function returns the square root of <CODE>x</CODE>,
-normalized to be >= 0. If <CODE>x</CODE> is the square of a rational number,
-<CODE>sqrt(x)</CODE> will be a rational number, else it will return a
-floating-point approximation.
-</DL>
-
-<P>
-The classes <CODE>cl_RA</CODE>, <CODE>cl_I</CODE> define the following operation:
-
-
-<DL COMPACT>
-
-<DT><CODE>cl_boolean sqrtp (const <VAR>type</VAR>& x, <VAR>type</VAR>* root)</CODE>
-<DD>
-<A NAME="IDX96"></A>
-This tests whether <CODE>x</CODE> is a perfect square. If so, it returns true
-and the exact square root in <CODE>*root</CODE>, else it returns false.
-</DL>
-
-<P>
-Furthermore, for integers, similarly:
-
-
-<DL COMPACT>
-
-<DT><CODE>cl_boolean isqrt (const <VAR>type</VAR>& x, <VAR>type</VAR>* root)</CODE>
-<DD>
-<A NAME="IDX97"></A>
-<CODE>x</CODE> should be >= 0. This function sets <CODE>*root</CODE> to
-<CODE>floor(sqrt(x))</CODE> and returns the same value as <CODE>sqrtp</CODE>:
-the boolean value <CODE>(expt(*root,2) == x)</CODE>.
-</DL>
-
-<P>
-For <CODE>n</CODE>th roots, the classes <CODE>cl_RA</CODE>, <CODE>cl_I</CODE>
-define the following operation:
-
-
-<DL COMPACT>
-
-<DT><CODE>cl_boolean rootp (const <VAR>type</VAR>& x, const cl_I& n, <VAR>type</VAR>* root)</CODE>
-<DD>
-<A NAME="IDX98"></A>
-<CODE>x</CODE> must be >= 0. <CODE>n</CODE> must be > 0.
-This tests whether <CODE>x</CODE> is an <CODE>n</CODE>th power of a rational number.
-If so, it returns true and the exact root in <CODE>*root</CODE>, else it returns
-false.
-</DL>
-
-<P>
-The only square root function which accepts negative numbers is the one
-for class <CODE>cl_N</CODE>:
-
-
-<DL COMPACT>
-
-<DT><CODE>cl_N sqrt (const cl_N& z)</CODE>
-<DD>
-<A NAME="IDX99"></A>
-Returns the square root of <CODE>z</CODE>, as defined by the formula
-<CODE>sqrt(z) = exp(log(z)/2)</CODE>. Conversion to a floating-point type
-or to a complex number are done if necessary. The range of the result is the
-right half plane <CODE>realpart(sqrt(z)) >= 0</CODE>
-including the positive imaginary axis and 0, but excluding
-the negative imaginary axis.
-The result is an exact number only if <CODE>z</CODE> is an exact number.
-</DL>
-
-
-
-<H2><A NAME="SEC28" HREF="cln.html#TOC28">4.8 Transcendental functions</A></H2>
-<P>
-<A NAME="IDX100"></A>
-
-
-<P>
-The transcendental functions return an exact result if the argument
-is exact and the result is exact as well. Otherwise they must return
-inexact numbers even if the argument is exact.
-For example, <CODE>cos(0) = 1</CODE> returns the rational number <CODE>1</CODE>.
-
-
-
-
-<H3><A NAME="SEC29" HREF="cln.html#TOC29">4.8.1 Exponential and logarithmic functions</A></H3>
-
-<DL COMPACT>
-
-<DT><CODE>cl_R exp (const cl_R& x)</CODE>
-<DD>
-<A NAME="IDX101"></A>
-<DT><CODE>cl_N exp (const cl_N& x)</CODE>
-<DD>
-Returns the exponential function of <CODE>x</CODE>. This is <CODE>e^x</CODE> where
-<CODE>e</CODE> is the base of the natural logarithms. The range of the result
-is the entire complex plane excluding 0.
-
-<DT><CODE>cl_R ln (const cl_R& x)</CODE>
-<DD>
-<A NAME="IDX102"></A>
-<CODE>x</CODE> must be > 0. Returns the (natural) logarithm of x.
-
-<DT><CODE>cl_N log (const cl_N& x)</CODE>
-<DD>
-<A NAME="IDX103"></A>
-Returns the (natural) logarithm of x. If <CODE>x</CODE> is real and positive,
-this is <CODE>ln(x)</CODE>. In general, <CODE>log(x) = log(abs(x)) + i*phase(x)</CODE>.
-The range of the result is the strip in the complex plane
-<CODE>-pi < imagpart(log(x)) <= pi</CODE>.
-
-<DT><CODE>cl_R phase (const cl_N& x)</CODE>
-<DD>
-<A NAME="IDX104"></A>
-Returns the angle part of <CODE>x</CODE> in its polar representation as a
-complex number. That is, <CODE>phase(x) = atan(realpart(x),imagpart(x))</CODE>.
-This is also the imaginary part of <CODE>log(x)</CODE>.
-The range of the result is the interval <CODE>-pi < phase(x) <= pi</CODE>.
-The result will be an exact number only if <CODE>zerop(x)</CODE> or
-if <CODE>x</CODE> is real and positive.
-
-<DT><CODE>cl_R log (const cl_R& a, const cl_R& b)</CODE>
-<DD>
-<CODE>a</CODE> and <CODE>b</CODE> must be > 0. Returns the logarithm of <CODE>a</CODE> with
-respect to base <CODE>b</CODE>. <CODE>log(a,b) = ln(a)/ln(b)</CODE>.
-The result can be exact only if <CODE>a = 1</CODE> or if <CODE>a</CODE> and <CODE>b</CODE>
-are both rational.
-
-<DT><CODE>cl_N log (const cl_N& a, const cl_N& b)</CODE>
-<DD>
-Returns the logarithm of <CODE>a</CODE> with respect to base <CODE>b</CODE>.
-<CODE>log(a,b) = log(a)/log(b)</CODE>.
-
-<DT><CODE>cl_N expt (const cl_N& x, const cl_N& y)</CODE>
-<DD>
-<A NAME="IDX105"></A>
-Exponentiation: Returns <CODE>x^y = exp(y*log(x))</CODE>.
-</DL>
-
-<P>
-The constant e = exp(1) = 2.71828... is returned by the following functions:
-
-
-<DL COMPACT>
-
-<DT><CODE>cl_F exp1 (cl_float_format_t f)</CODE>
-<DD>
-<A NAME="IDX106"></A>
-Returns e as a float of format <CODE>f</CODE>.
-
-<DT><CODE>cl_F exp1 (const cl_F& y)</CODE>
-<DD>
-Returns e in the float format of <CODE>y</CODE>.
-
-<DT><CODE>cl_F exp1 (void)</CODE>
-<DD>
-Returns e as a float of format <CODE>default_float_format</CODE>.
-</DL>
-
-
-
-<H3><A NAME="SEC30" HREF="cln.html#TOC30">4.8.2 Trigonometric functions</A></H3>
-
-<DL COMPACT>
-
-<DT><CODE>cl_R sin (const cl_R& x)</CODE>
-<DD>
-<A NAME="IDX107"></A>
-Returns <CODE>sin(x)</CODE>. The range of the result is the interval
-<CODE>-1 <= sin(x) <= 1</CODE>.
-
-<DT><CODE>cl_N sin (const cl_N& z)</CODE>
-<DD>
-Returns <CODE>sin(z)</CODE>. The range of the result is the entire complex plane.
-
-<DT><CODE>cl_R cos (const cl_R& x)</CODE>
-<DD>
-<A NAME="IDX108"></A>
-Returns <CODE>cos(x)</CODE>. The range of the result is the interval
-<CODE>-1 <= cos(x) <= 1</CODE>.
-
-<DT><CODE>cl_N cos (const cl_N& x)</CODE>
-<DD>
-Returns <CODE>cos(z)</CODE>. The range of the result is the entire complex plane.
-
-<DT><CODE>struct cos_sin_t { cl_R cos; cl_R sin; };</CODE>
-<DD>
-<A NAME="IDX109"></A>
-<DT><CODE>cos_sin_t cos_sin (const cl_R& x)</CODE>
-<DD>
-Returns both <CODE>sin(x)</CODE> and <CODE>cos(x)</CODE>. This is more efficient than
-<A NAME="IDX110"></A>
-computing them separately. The relation <CODE>cos^2 + sin^2 = 1</CODE> will
-hold only approximately.
-
-<DT><CODE>cl_R tan (const cl_R& x)</CODE>
-<DD>
-<A NAME="IDX111"></A>
-<DT><CODE>cl_N tan (const cl_N& x)</CODE>
-<DD>
-Returns <CODE>tan(x) = sin(x)/cos(x)</CODE>.
-
-<DT><CODE>cl_N cis (const cl_R& x)</CODE>
-<DD>
-<A NAME="IDX112"></A>
-<DT><CODE>cl_N cis (const cl_N& x)</CODE>
-<DD>
-Returns <CODE>exp(i*x)</CODE>. The name <SAMP>`cis'</SAMP> means "cos + i sin", because
-<CODE>e^(i*x) = cos(x) + i*sin(x)</CODE>.
-
-<A NAME="IDX113"></A>
-<A NAME="IDX114"></A>
-<DT><CODE>cl_N asin (const cl_N& z)</CODE>
-<DD>
-Returns <CODE>arcsin(z)</CODE>. This is defined as
-<CODE>arcsin(z) = log(iz+sqrt(1-z^2))/i</CODE> and satisfies
-<CODE>arcsin(-z) = -arcsin(z)</CODE>.
-The range of the result is the strip in the complex domain
-<CODE>-pi/2 <= realpart(arcsin(z)) <= pi/2</CODE>, excluding the numbers
-with <CODE>realpart = -pi/2</CODE> and <CODE>imagpart < 0</CODE> and the numbers
-with <CODE>realpart = pi/2</CODE> and <CODE>imagpart > 0</CODE>.
-
-<DT><CODE>cl_N acos (const cl_N& z)</CODE>
-<DD>
-<A NAME="IDX115"></A>
-Returns <CODE>arccos(z)</CODE>. This is defined as
-<CODE>arccos(z) = pi/2 - arcsin(z) = log(z+i*sqrt(1-z^2))/i</CODE>
-and satisfies <CODE>arccos(-z) = pi - arccos(z)</CODE>.
-The range of the result is the strip in the complex domain
-<CODE>0 <= realpart(arcsin(z)) <= pi</CODE>, excluding the numbers
-with <CODE>realpart = 0</CODE> and <CODE>imagpart < 0</CODE> and the numbers
-with <CODE>realpart = pi</CODE> and <CODE>imagpart > 0</CODE>.
-
-<A NAME="IDX116"></A>
-<A NAME="IDX117"></A>
-<DT><CODE>cl_R atan (const cl_R& x, const cl_R& y)</CODE>
-<DD>
-Returns the angle of the polar representation of the complex number
-<CODE>x+iy</CODE>. This is <CODE>atan(y/x)</CODE> if <CODE>x>0</CODE>. The range of
-the result is the interval <CODE>-pi < atan(x,y) <= pi</CODE>. The result will
-be an exact number only if <CODE>x > 0</CODE> and <CODE>y</CODE> is the exact <CODE>0</CODE>.
-WARNING: In Common Lisp, this function is called as <CODE>(atan y x)</CODE>,
-with reversed order of arguments.
-
-<DT><CODE>cl_R atan (const cl_R& x)</CODE>
-<DD>
-Returns <CODE>arctan(x)</CODE>. This is the same as <CODE>atan(1,x)</CODE>. The range
-of the result is the interval <CODE>-pi/2 < atan(x) < pi/2</CODE>. The result
-will be an exact number only if <CODE>x</CODE> is the exact <CODE>0</CODE>.
-
-<DT><CODE>cl_N atan (const cl_N& z)</CODE>
-<DD>
-Returns <CODE>arctan(z)</CODE>. This is defined as
-<CODE>arctan(z) = (log(1+iz)-log(1-iz)) / 2i</CODE> and satisfies
-<CODE>arctan(-z) = -arctan(z)</CODE>. The range of the result is
-the strip in the complex domain
-<CODE>-pi/2 <= realpart(arctan(z)) <= pi/2</CODE>, excluding the numbers
-with <CODE>realpart = -pi/2</CODE> and <CODE>imagpart >= 0</CODE> and the numbers
-with <CODE>realpart = pi/2</CODE> and <CODE>imagpart <= 0</CODE>.
-
-</DL>
-
-<P>
-<A NAME="IDX118"></A>
-<A NAME="IDX119"></A>
-Archimedes' constant pi = 3.14... is returned by the following functions:
-
-
-<DL COMPACT>
-
-<DT><CODE>cl_F pi (cl_float_format_t f)</CODE>
-<DD>
-<A NAME="IDX120"></A>
-Returns pi as a float of format <CODE>f</CODE>.
-
-<DT><CODE>cl_F pi (const cl_F& y)</CODE>
-<DD>
-Returns pi in the float format of <CODE>y</CODE>.
-
-<DT><CODE>cl_F pi (void)</CODE>
-<DD>
-Returns pi as a float of format <CODE>default_float_format</CODE>.
-</DL>
-
-
-
-<H3><A NAME="SEC31" HREF="cln.html#TOC31">4.8.3 Hyperbolic functions</A></H3>
-
-<DL COMPACT>
-
-<DT><CODE>cl_R sinh (const cl_R& x)</CODE>
-<DD>
-<A NAME="IDX121"></A>
-Returns <CODE>sinh(x)</CODE>.
-
-<DT><CODE>cl_N sinh (const cl_N& z)</CODE>
-<DD>
-Returns <CODE>sinh(z)</CODE>. The range of the result is the entire complex plane.
-
-<DT><CODE>cl_R cosh (const cl_R& x)</CODE>
-<DD>
-<A NAME="IDX122"></A>
-Returns <CODE>cosh(x)</CODE>. The range of the result is the interval
-<CODE>cosh(x) >= 1</CODE>.
-
-<DT><CODE>cl_N cosh (const cl_N& z)</CODE>
-<DD>
-Returns <CODE>cosh(z)</CODE>. The range of the result is the entire complex plane.
-
-<DT><CODE>struct cosh_sinh_t { cl_R cosh; cl_R sinh; };</CODE>
-<DD>
-<A NAME="IDX123"></A>
-<DT><CODE>cosh_sinh_t cosh_sinh (const cl_R& x)</CODE>
-<DD>
-<A NAME="IDX124"></A>
-Returns both <CODE>sinh(x)</CODE> and <CODE>cosh(x)</CODE>. This is more efficient than
-computing them separately. The relation <CODE>cosh^2 - sinh^2 = 1</CODE> will
-hold only approximately.
-
-<DT><CODE>cl_R tanh (const cl_R& x)</CODE>
-<DD>
-<A NAME="IDX125"></A>
-<DT><CODE>cl_N tanh (const cl_N& x)</CODE>
-<DD>
-Returns <CODE>tanh(x) = sinh(x)/cosh(x)</CODE>.
-
-<DT><CODE>cl_N asinh (const cl_N& z)</CODE>
-<DD>
-<A NAME="IDX126"></A>
-Returns <CODE>arsinh(z)</CODE>. This is defined as
-<CODE>arsinh(z) = log(z+sqrt(1+z^2))</CODE> and satisfies
-<CODE>arsinh(-z) = -arsinh(z)</CODE>.
-The range of the result is the strip in the complex domain
-<CODE>-pi/2 <= imagpart(arsinh(z)) <= pi/2</CODE>, excluding the numbers
-with <CODE>imagpart = -pi/2</CODE> and <CODE>realpart > 0</CODE> and the numbers
-with <CODE>imagpart = pi/2</CODE> and <CODE>realpart < 0</CODE>.
-
-<DT><CODE>cl_N acosh (const cl_N& z)</CODE>
-<DD>
-<A NAME="IDX127"></A>
-Returns <CODE>arcosh(z)</CODE>. This is defined as
-<CODE>arcosh(z) = 2*log(sqrt((z+1)/2)+sqrt((z-1)/2))</CODE>.
-The range of the result is the half-strip in the complex domain
-<CODE>-pi < imagpart(arcosh(z)) <= pi, realpart(arcosh(z)) >= 0</CODE>,
-excluding the numbers with <CODE>realpart = 0</CODE> and <CODE>-pi < imagpart < 0</CODE>.
-
-<DT><CODE>cl_N atanh (const cl_N& z)</CODE>
-<DD>
-<A NAME="IDX128"></A>
-Returns <CODE>artanh(z)</CODE>. This is defined as
-<CODE>artanh(z) = (log(1+z)-log(1-z)) / 2</CODE> and satisfies
-<CODE>artanh(-z) = -artanh(z)</CODE>. The range of the result is
-the strip in the complex domain
-<CODE>-pi/2 <= imagpart(artanh(z)) <= pi/2</CODE>, excluding the numbers
-with <CODE>imagpart = -pi/2</CODE> and <CODE>realpart <= 0</CODE> and the numbers
-with <CODE>imagpart = pi/2</CODE> and <CODE>realpart >= 0</CODE>.
-</DL>
-
-
-
-<H3><A NAME="SEC32" HREF="cln.html#TOC32">4.8.4 Euler gamma</A></H3>
-<P>
-<A NAME="IDX129"></A>
-
-
-<P>
-Euler's constant C = 0.577... is returned by the following functions:
-
-
-<DL COMPACT>
-
-<DT><CODE>cl_F eulerconst (cl_float_format_t f)</CODE>
-<DD>
-<A NAME="IDX130"></A>
-Returns Euler's constant as a float of format <CODE>f</CODE>.
-
-<DT><CODE>cl_F eulerconst (const cl_F& y)</CODE>
-<DD>
-Returns Euler's constant in the float format of <CODE>y</CODE>.
-
-<DT><CODE>cl_F eulerconst (void)</CODE>
-<DD>
-Returns Euler's constant as a float of format <CODE>default_float_format</CODE>.
-</DL>
-
-<P>
-Catalan's constant G = 0.915... is returned by the following functions:
-<A NAME="IDX131"></A>
-
-
-<DL COMPACT>
-
-<DT><CODE>cl_F catalanconst (cl_float_format_t f)</CODE>
-<DD>
-<A NAME="IDX132"></A>
-Returns Catalan's constant as a float of format <CODE>f</CODE>.
-
-<DT><CODE>cl_F catalanconst (const cl_F& y)</CODE>
-<DD>
-Returns Catalan's constant in the float format of <CODE>y</CODE>.
-
-<DT><CODE>cl_F catalanconst (void)</CODE>
-<DD>
-Returns Catalan's constant as a float of format <CODE>default_float_format</CODE>.
-</DL>
-
-
-
-<H3><A NAME="SEC33" HREF="cln.html#TOC33">4.8.5 Riemann zeta</A></H3>
-<P>
-<A NAME="IDX133"></A>
-
-
-<P>
-Riemann's zeta function at an integral point <CODE>s>1</CODE> is returned by the
-following functions:
-
-
-<DL COMPACT>
-
-<DT><CODE>cl_F zeta (int s, cl_float_format_t f)</CODE>
-<DD>
-<A NAME="IDX134"></A>
-Returns Riemann's zeta function at <CODE>s</CODE> as a float of format <CODE>f</CODE>.
-
-<DT><CODE>cl_F zeta (int s, const cl_F& y)</CODE>
-<DD>
-Returns Riemann's zeta function at <CODE>s</CODE> in the float format of <CODE>y</CODE>.
-
-<DT><CODE>cl_F zeta (int s)</CODE>
-<DD>
-Returns Riemann's zeta function at <CODE>s</CODE> as a float of format
-<CODE>default_float_format</CODE>.
-</DL>
-
-
-
-<H2><A NAME="SEC34" HREF="cln.html#TOC34">4.9 Functions on integers</A></H2>
-
-
-
-<H3><A NAME="SEC35" HREF="cln.html#TOC35">4.9.1 Logical functions</A></H3>
-
-<P>
-Integers, when viewed as in two's complement notation, can be thought as
-infinite bit strings where the bits' values eventually are constant.
-For example,
-
-<PRE>
- 17 = ......00010001
- -6 = ......11111010
-</PRE>
-
-<P>
-The logical operations view integers as such bit strings and operate
-on each of the bit positions in parallel.
-
-
-<DL COMPACT>
-
-<DT><CODE>cl_I lognot (const cl_I& x)</CODE>
-<DD>
-<A NAME="IDX135"></A>
-<DT><CODE>cl_I operator ~ (const cl_I& x)</CODE>
-<DD>
-<A NAME="IDX136"></A>
-Logical not, like <CODE>~x</CODE> in C. This is the same as <CODE>-1-x</CODE>.
-
-<DT><CODE>cl_I logand (const cl_I& x, const cl_I& y)</CODE>
-<DD>
-<A NAME="IDX137"></A>
-<DT><CODE>cl_I operator & (const cl_I& x, const cl_I& y)</CODE>
-<DD>
-<A NAME="IDX138"></A>
-Logical and, like <CODE>x & y</CODE> in C.
-
-<DT><CODE>cl_I logior (const cl_I& x, const cl_I& y)</CODE>
-<DD>
-<A NAME="IDX139"></A>
-<DT><CODE>cl_I operator | (const cl_I& x, const cl_I& y)</CODE>
-<DD>
-<A NAME="IDX140"></A>
-Logical (inclusive) or, like <CODE>x | y</CODE> in C.
-
-<DT><CODE>cl_I logxor (const cl_I& x, const cl_I& y)</CODE>
-<DD>
-<A NAME="IDX141"></A>
-<DT><CODE>cl_I operator ^ (const cl_I& x, const cl_I& y)</CODE>
-<DD>
-<A NAME="IDX142"></A>
-Exclusive or, like <CODE>x ^ y</CODE> in C.
-
-<DT><CODE>cl_I logeqv (const cl_I& x, const cl_I& y)</CODE>
-<DD>
-<A NAME="IDX143"></A>
-Bitwise equivalence, like <CODE>~(x ^ y)</CODE> in C.
-
-<DT><CODE>cl_I lognand (const cl_I& x, const cl_I& y)</CODE>
-<DD>
-<A NAME="IDX144"></A>
-Bitwise not and, like <CODE>~(x & y)</CODE> in C.
-
-<DT><CODE>cl_I lognor (const cl_I& x, const cl_I& y)</CODE>
-<DD>
-<A NAME="IDX145"></A>
-Bitwise not or, like <CODE>~(x | y)</CODE> in C.
-
-<DT><CODE>cl_I logandc1 (const cl_I& x, const cl_I& y)</CODE>
-<DD>
-<A NAME="IDX146"></A>
-Logical and, complementing the first argument, like <CODE>~x & y</CODE> in C.
-
-<DT><CODE>cl_I logandc2 (const cl_I& x, const cl_I& y)</CODE>
-<DD>
-<A NAME="IDX147"></A>
-Logical and, complementing the second argument, like <CODE>x & ~y</CODE> in C.
-
-<DT><CODE>cl_I logorc1 (const cl_I& x, const cl_I& y)</CODE>
-<DD>
-<A NAME="IDX148"></A>
-Logical or, complementing the first argument, like <CODE>~x | y</CODE> in C.
-
-<DT><CODE>cl_I logorc2 (const cl_I& x, const cl_I& y)</CODE>
-<DD>
-<A NAME="IDX149"></A>
-Logical or, complementing the second argument, like <CODE>x | ~y</CODE> in C.
-</DL>
-
-<P>
-These operations are all available though the function
-<DL COMPACT>
-
-<DT><CODE>cl_I boole (cl_boole op, const cl_I& x, const cl_I& y)</CODE>
-<DD>
-<A NAME="IDX150"></A>
-</DL>
-<P>
-where <CODE>op</CODE> must have one of the 16 values (each one stands for a function
-which combines two bits into one bit): <CODE>boole_clr</CODE>, <CODE>boole_set</CODE>,
-<CODE>boole_1</CODE>, <CODE>boole_2</CODE>, <CODE>boole_c1</CODE>, <CODE>boole_c2</CODE>,
-<CODE>boole_and</CODE>, <CODE>boole_ior</CODE>, <CODE>boole_xor</CODE>, <CODE>boole_eqv</CODE>,
-<CODE>boole_nand</CODE>, <CODE>boole_nor</CODE>, <CODE>boole_andc1</CODE>, <CODE>boole_andc2</CODE>,
-<CODE>boole_orc1</CODE>, <CODE>boole_orc2</CODE>.
-<A NAME="IDX151"></A>
-<A NAME="IDX152"></A>
-<A NAME="IDX153"></A>
-<A NAME="IDX154"></A>
-<A NAME="IDX155"></A>
-<A NAME="IDX156"></A>
-<A NAME="IDX157"></A>
-<A NAME="IDX158"></A>
-<A NAME="IDX159"></A>
-<A NAME="IDX160"></A>
-<A NAME="IDX161"></A>
-<A NAME="IDX162"></A>
-<A NAME="IDX163"></A>
-<A NAME="IDX164"></A>
-<A NAME="IDX165"></A>
-
-
-<P>
-Other functions that view integers as bit strings:
-
-
-<DL COMPACT>
-
-<DT><CODE>cl_boolean logtest (const cl_I& x, const cl_I& y)</CODE>
-<DD>
-<A NAME="IDX166"></A>
-Returns true if some bit is set in both <CODE>x</CODE> and <CODE>y</CODE>, i.e. if
-<CODE>logand(x,y) != 0</CODE>.
-
-<DT><CODE>cl_boolean logbitp (const cl_I& n, const cl_I& x)</CODE>
-<DD>
-<A NAME="IDX167"></A>
-Returns true if the <CODE>n</CODE>th bit (from the right) of <CODE>x</CODE> is set.
-Bit 0 is the least significant bit.
-
-<DT><CODE>uintL logcount (const cl_I& x)</CODE>
-<DD>
-<A NAME="IDX168"></A>
-Returns the number of one bits in <CODE>x</CODE>, if <CODE>x</CODE> >= 0, or
-the number of zero bits in <CODE>x</CODE>, if <CODE>x</CODE> < 0.
-</DL>
-
-<P>
-The following functions operate on intervals of bits in integers.
-The type
-
-<PRE>
-struct cl_byte { uintL size; uintL position; };
-</PRE>
-
-<P>
-<A NAME="IDX169"></A>
-represents the bit interval containing the bits
-<CODE>position</CODE>...<CODE>position+size-1</CODE> of an integer.
-The constructor <CODE>cl_byte(size,position)</CODE> constructs a <CODE>cl_byte</CODE>.
-
-
-<DL COMPACT>
-
-<DT><CODE>cl_I ldb (const cl_I& n, const cl_byte& b)</CODE>
-<DD>
-<A NAME="IDX170"></A>
-extracts the bits of <CODE>n</CODE> described by the bit interval <CODE>b</CODE>
-and returns them as a nonnegative integer with <CODE>b.size</CODE> bits.
-
-<DT><CODE>cl_boolean ldb_test (const cl_I& n, const cl_byte& b)</CODE>
-<DD>
-<A NAME="IDX171"></A>
-Returns true if some bit described by the bit interval <CODE>b</CODE> is set in
-<CODE>n</CODE>.
-
-<DT><CODE>cl_I dpb (const cl_I& newbyte, const cl_I& n, const cl_byte& b)</CODE>
-<DD>
-<A NAME="IDX172"></A>
-Returns <CODE>n</CODE>, with the bits described by the bit interval <CODE>b</CODE>
-replaced by <CODE>newbyte</CODE>. Only the lowest <CODE>b.size</CODE> bits of
-<CODE>newbyte</CODE> are relevant.
-</DL>
-
-<P>
-The functions <CODE>ldb</CODE> and <CODE>dpb</CODE> implicitly shift. The following
-functions are their counterparts without shifting:
-
-
-<DL COMPACT>
-
-<DT><CODE>cl_I mask_field (const cl_I& n, const cl_byte& b)</CODE>
-<DD>
-<A NAME="IDX173"></A>
-returns an integer with the bits described by the bit interval <CODE>b</CODE>
-copied from the corresponding bits in <CODE>n</CODE>, the other bits zero.
-
-<DT><CODE>cl_I deposit_field (const cl_I& newbyte, const cl_I& n, const cl_byte& b)</CODE>
-<DD>
-<A NAME="IDX174"></A>
-returns an integer where the bits described by the bit interval <CODE>b</CODE>
-come from <CODE>newbyte</CODE> and the other bits come from <CODE>n</CODE>.
-</DL>
-
-<P>
-The following relations hold:
-
-
-
-<UL>
-<LI>
-
-<CODE>ldb (n, b) = mask_field(n, b) >> b.position</CODE>,
-<LI>
-
-<CODE>dpb (newbyte, n, b) = deposit_field (newbyte << b.position, n, b)</CODE>,
-<LI>
-
-<CODE>deposit_field(newbyte,n,b) = n ^ mask_field(n,b) ^ mask_field(new_byte,b)</CODE>.
-</UL>
-
-<P>
-The following operations on integers as bit strings are efficient shortcuts
-for common arithmetic operations:
-
-
-<DL COMPACT>
-
-<DT><CODE>cl_boolean oddp (const cl_I& x)</CODE>
-<DD>
-<A NAME="IDX175"></A>
-Returns true if the least significant bit of <CODE>x</CODE> is 1. Equivalent to
-<CODE>mod(x,2) != 0</CODE>.
-
-<DT><CODE>cl_boolean evenp (const cl_I& x)</CODE>
-<DD>
-<A NAME="IDX176"></A>
-Returns true if the least significant bit of <CODE>x</CODE> is 0. Equivalent to
-<CODE>mod(x,2) == 0</CODE>.
-
-<DT><CODE>cl_I operator << (const cl_I& x, const cl_I& n)</CODE>
-<DD>
-<A NAME="IDX177"></A>
-Shifts <CODE>x</CODE> by <CODE>n</CODE> bits to the left. <CODE>n</CODE> should be >=0.
-Equivalent to <CODE>x * expt(2,n)</CODE>.
-
-<DT><CODE>cl_I operator >> (const cl_I& x, const cl_I& n)</CODE>
-<DD>
-<A NAME="IDX178"></A>
-Shifts <CODE>x</CODE> by <CODE>n</CODE> bits to the right. <CODE>n</CODE> should be >=0.
-Bits shifted out to the right are thrown away.
-Equivalent to <CODE>floor(x / expt(2,n))</CODE>.
-
-<DT><CODE>cl_I ash (const cl_I& x, const cl_I& y)</CODE>
-<DD>
-<A NAME="IDX179"></A>
-Shifts <CODE>x</CODE> by <CODE>y</CODE> bits to the left (if <CODE>y</CODE>>=0) or
-by <CODE>-y</CODE> bits to the right (if <CODE>y</CODE><=0). In other words, this
-returns <CODE>floor(x * expt(2,y))</CODE>.
-
-<DT><CODE>uintL integer_length (const cl_I& x)</CODE>
-<DD>
-<A NAME="IDX180"></A>
-Returns the number of bits (excluding the sign bit) needed to represent <CODE>x</CODE>
-in two's complement notation. This is the smallest n >= 0 such that
--2^n <= x < 2^n. If x > 0, this is the unique n > 0 such that
-2^(n-1) <= x < 2^n.
-
-<DT><CODE>uintL ord2 (const cl_I& x)</CODE>
-<DD>
-<A NAME="IDX181"></A>
-<CODE>x</CODE> must be non-zero. This function returns the number of 0 bits at the
-right of <CODE>x</CODE> in two's complement notation. This is the largest n >= 0
-such that 2^n divides <CODE>x</CODE>.
-
-<DT><CODE>uintL power2p (const cl_I& x)</CODE>
-<DD>
-<A NAME="IDX182"></A>
-<CODE>x</CODE> must be > 0. This function checks whether <CODE>x</CODE> is a power of 2.
-If <CODE>x</CODE> = 2^(n-1), it returns n. Else it returns 0.
-(See also the function <CODE>logp</CODE>.)
-</DL>
-
-
-
-<H3><A NAME="SEC36" HREF="cln.html#TOC36">4.9.2 Number theoretic functions</A></H3>
-
-<DL COMPACT>
-
-<DT><CODE>uint32 gcd (uint32 a, uint32 b)</CODE>
-<DD>
-<A NAME="IDX183"></A>
-<DT><CODE>cl_I gcd (const cl_I& a, const cl_I& b)</CODE>
-<DD>
-This function returns the greatest common divisor of <CODE>a</CODE> and <CODE>b</CODE>,
-normalized to be >= 0.
-
-<DT><CODE>cl_I xgcd (const cl_I& a, const cl_I& b, cl_I* u, cl_I* v)</CODE>
-<DD>
-<A NAME="IDX184"></A>
-This function ("extended gcd") returns the greatest common divisor <CODE>g</CODE> of
-<CODE>a</CODE> and <CODE>b</CODE> and at the same time the representation of <CODE>g</CODE>
-as an integral linear combination of <CODE>a</CODE> and <CODE>b</CODE>:
-<CODE>u</CODE> and <CODE>v</CODE> with <CODE>u*a+v*b = g</CODE>, <CODE>g</CODE> >= 0.
-<CODE>u</CODE> and <CODE>v</CODE> will be normalized to be of smallest possible absolute
-value, in the following sense: If <CODE>a</CODE> and <CODE>b</CODE> are non-zero, and
-<CODE>abs(a) != abs(b)</CODE>, <CODE>u</CODE> and <CODE>v</CODE> will satisfy the inequalities
-<CODE>abs(u) <= abs(b)/(2*g)</CODE>, <CODE>abs(v) <= abs(a)/(2*g)</CODE>.
-
-<DT><CODE>cl_I lcm (const cl_I& a, const cl_I& b)</CODE>
-<DD>
-<A NAME="IDX185"></A>
-This function returns the least common multiple of <CODE>a</CODE> and <CODE>b</CODE>,
-normalized to be >= 0.
-
-<DT><CODE>cl_boolean logp (const cl_I& a, const cl_I& b, cl_RA* l)</CODE>
-<DD>
-<A NAME="IDX186"></A>
-<DT><CODE>cl_boolean logp (const cl_RA& a, const cl_RA& b, cl_RA* l)</CODE>
-<DD>
-<CODE>a</CODE> must be > 0. <CODE>b</CODE> must be >0 and != 1. If log(a,b) is
-rational number, this function returns true and sets *l = log(a,b), else
-it returns false.
-</DL>
-
-
-
-<H3><A NAME="SEC37" HREF="cln.html#TOC37">4.9.3 Combinatorial functions</A></H3>
-
-<DL COMPACT>
-
-<DT><CODE>cl_I factorial (uintL n)</CODE>
-<DD>
-<A NAME="IDX187"></A>
-<CODE>n</CODE> must be a small integer >= 0. This function returns the factorial
-<CODE>n</CODE>! = <CODE>1*2*...*n</CODE>.
-
-<DT><CODE>cl_I doublefactorial (uintL n)</CODE>
-<DD>
-<A NAME="IDX188"></A>
-<CODE>n</CODE> must be a small integer >= 0. This function returns the
-doublefactorial <CODE>n</CODE>!! = <CODE>1*3*...*n</CODE> or
-<CODE>n</CODE>!! = <CODE>2*4*...*n</CODE>, respectively.
-
-<DT><CODE>cl_I binomial (uintL n, uintL k)</CODE>
-<DD>
-<A NAME="IDX189"></A>
-<CODE>n</CODE> and <CODE>k</CODE> must be small integers >= 0. This function returns the
-binomial coefficient
-(<CODE>n</CODE> choose <CODE>k</CODE>) = <CODE>n</CODE>! / <CODE>k</CODE>! <CODE>(n-k)</CODE>!
-for 0 <= k <= n, 0 else.
-</DL>
-
-
-
-<H2><A NAME="SEC38" HREF="cln.html#TOC38">4.10 Functions on floating-point numbers</A></H2>
-
-<P>
-Recall that a floating-point number consists of a sign <CODE>s</CODE>, an
-exponent <CODE>e</CODE> and a mantissa <CODE>m</CODE>. The value of the number is
-<CODE>(-1)^s * 2^e * m</CODE>.
-
-
-<P>
-Each of the classes
-<CODE>cl_F</CODE>, <CODE>cl_SF</CODE>, <CODE>cl_FF</CODE>, <CODE>cl_DF</CODE>, <CODE>cl_LF</CODE>
-defines the following operations.
-
-
-<DL COMPACT>
-
-<DT><CODE><VAR>type</VAR> scale_float (const <VAR>type</VAR>& x, sintL delta)</CODE>
-<DD>
-<A NAME="IDX190"></A>
-<DT><CODE><VAR>type</VAR> scale_float (const <VAR>type</VAR>& x, const cl_I& delta)</CODE>
-<DD>
-Returns <CODE>x*2^delta</CODE>. This is more efficient than an explicit multiplication
-because it copies <CODE>x</CODE> and modifies the exponent.
-</DL>
-
-<P>
-The following functions provide an abstract interface to the underlying
-representation of floating-point numbers.
-
-
-<DL COMPACT>
-
-<DT><CODE>sintL float_exponent (const <VAR>type</VAR>& x)</CODE>
-<DD>
-<A NAME="IDX191"></A>
-Returns the exponent <CODE>e</CODE> of <CODE>x</CODE>.
-For <CODE>x = 0.0</CODE>, this is 0. For <CODE>x</CODE> non-zero, this is the unique
-integer with <CODE>2^(e-1) <= abs(x) < 2^e</CODE>.
-
-<DT><CODE>sintL float_radix (const <VAR>type</VAR>& x)</CODE>
-<DD>
-<A NAME="IDX192"></A>
-Returns the base of the floating-point representation. This is always <CODE>2</CODE>.
-
-<DT><CODE><VAR>type</VAR> float_sign (const <VAR>type</VAR>& x)</CODE>
-<DD>
-<A NAME="IDX193"></A>
-Returns the sign <CODE>s</CODE> of <CODE>x</CODE> as a float. The value is 1 for
-<CODE>x</CODE> >= 0, -1 for <CODE>x</CODE> < 0.
-
-<DT><CODE>uintL float_digits (const <VAR>type</VAR>& x)</CODE>
-<DD>
-<A NAME="IDX194"></A>
-Returns the number of mantissa bits in the floating-point representation
-of <CODE>x</CODE>, including the hidden bit. The value only depends on the type
-of <CODE>x</CODE>, not on its value.
-
-<DT><CODE>uintL float_precision (const <VAR>type</VAR>& x)</CODE>
-<DD>
-<A NAME="IDX195"></A>
-Returns the number of significant mantissa bits in the floating-point
-representation of <CODE>x</CODE>. Since denormalized numbers are not supported,
-this is the same as <CODE>float_digits(x)</CODE> if <CODE>x</CODE> is non-zero, and
-0 if <CODE>x</CODE> = 0.
-</DL>
-
-<P>
-The complete internal representation of a float is encoded in the type
-<A NAME="IDX196"></A>
-<A NAME="IDX197"></A>
-<A NAME="IDX198"></A>
-<A NAME="IDX199"></A>
-<A NAME="IDX200"></A>
-<CODE>decoded_float</CODE> (or <CODE>decoded_sfloat</CODE>, <CODE>decoded_ffloat</CODE>,
-<CODE>decoded_dfloat</CODE>, <CODE>decoded_lfloat</CODE>, respectively), defined by
-
-<PRE>
-struct decoded_<VAR>type</VAR>float {
- <VAR>type</VAR> mantissa; cl_I exponent; <VAR>type</VAR> sign;
-};
-</PRE>
-
-<P>
-and returned by the function
-
-
-<DL COMPACT>
-
-<DT><CODE>decoded_<VAR>type</VAR>float decode_float (const <VAR>type</VAR>& x)</CODE>
-<DD>
-<A NAME="IDX201"></A>
-For <CODE>x</CODE> non-zero, this returns <CODE>(-1)^s</CODE>, <CODE>e</CODE>, <CODE>m</CODE> with
-<CODE>x = (-1)^s * 2^e * m</CODE> and <CODE>0.5 <= m < 1.0</CODE>. For <CODE>x</CODE> = 0,
-it returns <CODE>(-1)^s</CODE>=1, <CODE>e</CODE>=0, <CODE>m</CODE>=0.
-<CODE>e</CODE> is the same as returned by the function <CODE>float_exponent</CODE>.
-</DL>
-
-<P>
-A complete decoding in terms of integers is provided as type
-<A NAME="IDX202"></A>
-
-<PRE>
-struct cl_idecoded_float {
- cl_I mantissa; cl_I exponent; cl_I sign;
-};
-</PRE>
-
-<P>
-by the following function:
-
-
-<DL COMPACT>
-
-<DT><CODE>cl_idecoded_float integer_decode_float (const <VAR>type</VAR>& x)</CODE>
-<DD>
-<A NAME="IDX203"></A>
-For <CODE>x</CODE> non-zero, this returns <CODE>(-1)^s</CODE>, <CODE>e</CODE>, <CODE>m</CODE> with
-<CODE>x = (-1)^s * 2^e * m</CODE> and <CODE>m</CODE> an integer with <CODE>float_digits(x)</CODE>
-bits. For <CODE>x</CODE> = 0, it returns <CODE>(-1)^s</CODE>=1, <CODE>e</CODE>=0, <CODE>m</CODE>=0.
-WARNING: The exponent <CODE>e</CODE> is not the same as the one returned by
-the functions <CODE>decode_float</CODE> and <CODE>float_exponent</CODE>.
-</DL>
-
-<P>
-Some other function, implemented only for class <CODE>cl_F</CODE>:
-
-
-<DL COMPACT>
-
-<DT><CODE>cl_F float_sign (const cl_F& x, const cl_F& y)</CODE>
-<DD>
-<A NAME="IDX204"></A>
-This returns a floating point number whose precision and absolute value
-is that of <CODE>y</CODE> and whose sign is that of <CODE>x</CODE>. If <CODE>x</CODE> is
-zero, it is treated as positive. Same for <CODE>y</CODE>.
-</DL>
-
-
-
-<H2><A NAME="SEC39" HREF="cln.html#TOC39">4.11 Conversion functions</A></H2>
-<P>
-<A NAME="IDX205"></A>
-
-
-
-
-<H3><A NAME="SEC40" HREF="cln.html#TOC40">4.11.1 Conversion to floating-point numbers</A></H3>
-
-<P>
-The type <CODE>cl_float_format_t</CODE> describes a floating-point format.
-<A NAME="IDX206"></A>
-
-
-<DL COMPACT>
-
-<DT><CODE>cl_float_format_t cl_float_format (uintL n)</CODE>
-<DD>
-<A NAME="IDX207"></A>
-Returns the smallest float format which guarantees at least <CODE>n</CODE>
-decimal digits in the mantissa (after the decimal point).
-
-<DT><CODE>cl_float_format_t cl_float_format (const cl_F& x)</CODE>
-<DD>
-Returns the floating point format of <CODE>x</CODE>.
-
-<DT><CODE>cl_float_format_t default_float_format</CODE>
-<DD>
-<A NAME="IDX208"></A>
-Global variable: the default float format used when converting rational numbers
-to floats.
-</DL>
-
-<P>
-To convert a real number to a float, each of the types
-<CODE>cl_R</CODE>, <CODE>cl_F</CODE>, <CODE>cl_I</CODE>, <CODE>cl_RA</CODE>,
-<CODE>int</CODE>, <CODE>unsigned int</CODE>, <CODE>float</CODE>, <CODE>double</CODE>
-defines the following operations:
-
-
-<DL COMPACT>
-
-<DT><CODE>cl_F cl_float (const <VAR>type</VAR>&x, cl_float_format_t f)</CODE>
-<DD>
-<A NAME="IDX209"></A>
-Returns <CODE>x</CODE> as a float of format <CODE>f</CODE>.
-<DT><CODE>cl_F cl_float (const <VAR>type</VAR>&x, const cl_F& y)</CODE>
-<DD>
-Returns <CODE>x</CODE> in the float format of <CODE>y</CODE>.
-<DT><CODE>cl_F cl_float (const <VAR>type</VAR>&x)</CODE>
-<DD>
-Returns <CODE>x</CODE> as a float of format <CODE>default_float_format</CODE> if
-it is an exact number, or <CODE>x</CODE> itself if it is already a float.
-</DL>
-
-<P>
-Of course, converting a number to a float can lose precision.
-
-
-<P>
-Every floating-point format has some characteristic numbers:
-
-
-<DL COMPACT>
-
-<DT><CODE>cl_F most_positive_float (cl_float_format_t f)</CODE>
-<DD>
-<A NAME="IDX210"></A>
-Returns the largest (most positive) floating point number in float format <CODE>f</CODE>.
-
-<DT><CODE>cl_F most_negative_float (cl_float_format_t f)</CODE>
-<DD>
-<A NAME="IDX211"></A>
-Returns the smallest (most negative) floating point number in float format <CODE>f</CODE>.
-
-<DT><CODE>cl_F least_positive_float (cl_float_format_t f)</CODE>
-<DD>
-<A NAME="IDX212"></A>
-Returns the least positive floating point number (i.e. > 0 but closest to 0)
-in float format <CODE>f</CODE>.
-
-<DT><CODE>cl_F least_negative_float (cl_float_format_t f)</CODE>
-<DD>
-<A NAME="IDX213"></A>
-Returns the least negative floating point number (i.e. < 0 but closest to 0)
-in float format <CODE>f</CODE>.
-
-<DT><CODE>cl_F float_epsilon (cl_float_format_t f)</CODE>
-<DD>
-<A NAME="IDX214"></A>
-Returns the smallest floating point number e > 0 such that <CODE>1+e != 1</CODE>.
-
-<DT><CODE>cl_F float_negative_epsilon (cl_float_format_t f)</CODE>
-<DD>
-<A NAME="IDX215"></A>
-Returns the smallest floating point number e > 0 such that <CODE>1-e != 1</CODE>.
-</DL>
-
-
-
-<H3><A NAME="SEC41" HREF="cln.html#TOC41">4.11.2 Conversion to rational numbers</A></H3>
-
-<P>
-Each of the classes <CODE>cl_R</CODE>, <CODE>cl_RA</CODE>, <CODE>cl_F</CODE>
-defines the following operation:
-
-
-<DL COMPACT>
-
-<DT><CODE>cl_RA rational (const <VAR>type</VAR>& x)</CODE>
-<DD>
-<A NAME="IDX216"></A>
-Returns the value of <CODE>x</CODE> as an exact number. If <CODE>x</CODE> is already
-an exact number, this is <CODE>x</CODE>. If <CODE>x</CODE> is a floating-point number,
-the value is a rational number whose denominator is a power of 2.
-</DL>
-
-<P>
-In order to convert back, say, <CODE>(cl_F)(cl_R)"1/3"</CODE> to <CODE>1/3</CODE>, there is
-the function
-
-
-<DL COMPACT>
-
-<DT><CODE>cl_RA rationalize (const cl_R& x)</CODE>
-<DD>
-<A NAME="IDX217"></A>
-If <CODE>x</CODE> is a floating-point number, it actually represents an interval
-of real numbers, and this function returns the rational number with
-smallest denominator (and smallest numerator, in magnitude)
-which lies in this interval.
-If <CODE>x</CODE> is already an exact number, this function returns <CODE>x</CODE>.
-</DL>
-
-<P>
-If <CODE>x</CODE> is any float, one has
-
-
-
-<UL>
-<LI>
-
-<CODE>cl_float(rational(x),x) = x</CODE>
-<LI>
-
-<CODE>cl_float(rationalize(x),x) = x</CODE>
-</UL>
-
-
-
-<H2><A NAME="SEC42" HREF="cln.html#TOC42">4.12 Random number generators</A></H2>
-
-<P>
-A random generator is a machine which produces (pseudo-)random numbers.
-The include file <CODE><cln/random.h></CODE> defines a class <CODE>random_state</CODE>
-which contains the state of a random generator. If you make a copy
-of the random number generator, the original one and the copy will produce
-the same sequence of random numbers.
-
-
-<P>
-The following functions return (pseudo-)random numbers in different formats.
-Calling one of these modifies the state of the random number generator in
-a complicated but deterministic way.
-
-
-<P>
-The global variable
-<A NAME="IDX218"></A>
-<A NAME="IDX219"></A>
-
-<PRE>
-random_state default_random_state
-</PRE>
-
-<P>
-contains a default random number generator. It is used when the functions
-below are called without <CODE>random_state</CODE> argument.
-
-
-<DL COMPACT>
-
-<DT><CODE>uint32 random32 (random_state& randomstate)</CODE>
-<DD>
-<DT><CODE>uint32 random32 ()</CODE>
-<DD>
-<A NAME="IDX220"></A>
-Returns a random unsigned 32-bit number. All bits are equally random.
-
-<DT><CODE>cl_I random_I (random_state& randomstate, const cl_I& n)</CODE>
-<DD>
-<DT><CODE>cl_I random_I (const cl_I& n)</CODE>
-<DD>
-<A NAME="IDX221"></A>
-<CODE>n</CODE> must be an integer > 0. This function returns a random integer <CODE>x</CODE>
-in the range <CODE>0 <= x < n</CODE>.
-
-<DT><CODE>cl_F random_F (random_state& randomstate, const cl_F& n)</CODE>
-<DD>
-<DT><CODE>cl_F random_F (const cl_F& n)</CODE>
-<DD>
-<A NAME="IDX222"></A>
-<CODE>n</CODE> must be a float > 0. This function returns a random floating-point
-number of the same format as <CODE>n</CODE> in the range <CODE>0 <= x < n</CODE>.
-
-<DT><CODE>cl_R random_R (random_state& randomstate, const cl_R& n)</CODE>
-<DD>
-<DT><CODE>cl_R random_R (const cl_R& n)</CODE>
-<DD>
-<A NAME="IDX223"></A>
-Behaves like <CODE>random_I</CODE> if <CODE>n</CODE> is an integer and like <CODE>random_F</CODE>
-if <CODE>n</CODE> is a float.
-</DL>
-
-
-
-<H2><A NAME="SEC43" HREF="cln.html#TOC43">4.13 Obfuscating operators</A></H2>
-<P>
-<A NAME="IDX224"></A>
-
-
-<P>
-The modifying C/C++ operators <CODE>+=</CODE>, <CODE>-=</CODE>, <CODE>*=</CODE>, <CODE>/=</CODE>,
-<CODE>&=</CODE>, <CODE>|=</CODE>, <CODE>^=</CODE>, <CODE><<=</CODE>, <CODE>>>=</CODE>
-are not available by default because their
-use tends to make programs unreadable. It is trivial to get away without
-them. However, if you feel that you absolutely need these operators
-to get happy, then add
-
-<PRE>
-#define WANT_OBFUSCATING_OPERATORS
-</PRE>
-
-<P>
-<A NAME="IDX225"></A>
-to the beginning of your source files, before the inclusion of any CLN
-include files. This flag will enable the following operators:
-
-
-<P>
-For the classes <CODE>cl_N</CODE>, <CODE>cl_R</CODE>, <CODE>cl_RA</CODE>,
-<CODE>cl_F</CODE>, <CODE>cl_SF</CODE>, <CODE>cl_FF</CODE>, <CODE>cl_DF</CODE>, <CODE>cl_LF</CODE>:
-
-
-<DL COMPACT>
-
-<DT><CODE><VAR>type</VAR>& operator += (<VAR>type</VAR>&, const <VAR>type</VAR>&)</CODE>
-<DD>
-<A NAME="IDX226"></A>
-<DT><CODE><VAR>type</VAR>& operator -= (<VAR>type</VAR>&, const <VAR>type</VAR>&)</CODE>
-<DD>
-<A NAME="IDX227"></A>
-<DT><CODE><VAR>type</VAR>& operator *= (<VAR>type</VAR>&, const <VAR>type</VAR>&)</CODE>
-<DD>
-<A NAME="IDX228"></A>
-<DT><CODE><VAR>type</VAR>& operator /= (<VAR>type</VAR>&, const <VAR>type</VAR>&)</CODE>
-<DD>
-<A NAME="IDX229"></A>
-</DL>
-
-<P>
-For the class <CODE>cl_I</CODE>:
-
-
-<DL COMPACT>
-
-<DT><CODE><VAR>type</VAR>& operator += (<VAR>type</VAR>&, const <VAR>type</VAR>&)</CODE>
-<DD>
-<DT><CODE><VAR>type</VAR>& operator -= (<VAR>type</VAR>&, const <VAR>type</VAR>&)</CODE>
-<DD>
-<DT><CODE><VAR>type</VAR>& operator *= (<VAR>type</VAR>&, const <VAR>type</VAR>&)</CODE>
-<DD>
-<DT><CODE><VAR>type</VAR>& operator &= (<VAR>type</VAR>&, const <VAR>type</VAR>&)</CODE>
-<DD>
-<A NAME="IDX230"></A>
-<DT><CODE><VAR>type</VAR>& operator |= (<VAR>type</VAR>&, const <VAR>type</VAR>&)</CODE>
-<DD>
-<A NAME="IDX231"></A>
-<DT><CODE><VAR>type</VAR>& operator ^= (<VAR>type</VAR>&, const <VAR>type</VAR>&)</CODE>
-<DD>
-<A NAME="IDX232"></A>
-<DT><CODE><VAR>type</VAR>& operator <<= (<VAR>type</VAR>&, const <VAR>type</VAR>&)</CODE>
-<DD>
-<A NAME="IDX233"></A>
-<DT><CODE><VAR>type</VAR>& operator >>= (<VAR>type</VAR>&, const <VAR>type</VAR>&)</CODE>
-<DD>
-<A NAME="IDX234"></A>
-</DL>
-
-<P>
-For the classes <CODE>cl_N</CODE>, <CODE>cl_R</CODE>, <CODE>cl_RA</CODE>, <CODE>cl_I</CODE>,
-<CODE>cl_F</CODE>, <CODE>cl_SF</CODE>, <CODE>cl_FF</CODE>, <CODE>cl_DF</CODE>, <CODE>cl_LF</CODE>:
-
-
-<DL COMPACT>
-
-<DT><CODE><VAR>type</VAR>& operator ++ (<VAR>type</VAR>& x)</CODE>
-<DD>
-<A NAME="IDX235"></A>
-The prefix operator <CODE>++x</CODE>.
-
-<DT><CODE>void operator ++ (<VAR>type</VAR>& x, int)</CODE>
-<DD>
-The postfix operator <CODE>x++</CODE>.
-
-<DT><CODE><VAR>type</VAR>& operator -- (<VAR>type</VAR>& x)</CODE>
-<DD>
-<A NAME="IDX236"></A>
-The prefix operator <CODE>--x</CODE>.
-
-<DT><CODE>void operator -- (<VAR>type</VAR>& x, int)</CODE>
-<DD>
-The postfix operator <CODE>x--</CODE>.
-</DL>
-
-<P>
-Note that by using these obfuscating operators, you wouldn't gain efficiency:
-In CLN <SAMP>`x += y;'</SAMP> is exactly the same as <SAMP>`x = x+y;'</SAMP>, not more
-efficient.
-
-
-
-
-<H1><A NAME="SEC44" HREF="cln.html#TOC44">5. Input/Output</A></H1>
-<P>
-<A NAME="IDX237"></A>
-
-
-
-
-<H2><A NAME="SEC45" HREF="cln.html#TOC45">5.1 Internal and printed representation</A></H2>
-<P>
-<A NAME="IDX238"></A>
-
-
-<P>
-All computations deal with the internal representations of the numbers.
-
-
-<P>
-Every number has an external representation as a sequence of ASCII characters.
-Several external representations may denote the same number, for example,
-"20.0" and "20.000".
-
-
-<P>
-Converting an internal to an external representation is called "printing",
-<A NAME="IDX239"></A>
-converting an external to an internal representation is called "reading".
-<A NAME="IDX240"></A>
-In CLN, it is always true that conversion of an internal to an external
-representation and then back to an internal representation will yield the
-same internal representation. Symbolically: <CODE>read(print(x)) == x</CODE>.
-This is called "print-read consistency".
-
-
-<P>
-Different types of numbers have different external representations (case
-is insignificant):
-
-
-<DL COMPACT>
-
-<DT>Integers
-<DD>
-External representation: <VAR>sign</VAR>{<VAR>digit</VAR>}+. The reader also accepts the
-Common Lisp syntaxes <VAR>sign</VAR>{<VAR>digit</VAR>}+<CODE>.</CODE> with a trailing dot
-for decimal integers
-and the <CODE>#<VAR>n</VAR>R</CODE>, <CODE>#b</CODE>, <CODE>#o</CODE>, <CODE>#x</CODE> prefixes.
-
-<DT>Rational numbers
-<DD>
-External representation: <VAR>sign</VAR>{<VAR>digit</VAR>}+<CODE>/</CODE>{<VAR>digit</VAR>}+.
-The <CODE>#<VAR>n</VAR>R</CODE>, <CODE>#b</CODE>, <CODE>#o</CODE>, <CODE>#x</CODE> prefixes are allowed
-here as well.
-
-<DT>Floating-point numbers
-<DD>
-External representation: <VAR>sign</VAR>{<VAR>digit</VAR>}*<VAR>exponent</VAR> or
-<VAR>sign</VAR>{<VAR>digit</VAR>}*<CODE>.</CODE>{<VAR>digit</VAR>}*<VAR>exponent</VAR> or
-<VAR>sign</VAR>{<VAR>digit</VAR>}*<CODE>.</CODE>{<VAR>digit</VAR>}+. A precision specifier
-of the form _<VAR>prec</VAR> may be appended. There must be at least
-one digit in the non-exponent part. The exponent has the syntax
-<VAR>expmarker</VAR> <VAR>expsign</VAR> {<VAR>digit</VAR>}+.
-The exponent marker is
-
-
-<UL>
-<LI>
-
-<SAMP>`s'</SAMP> for short-floats,
-<LI>
-
-<SAMP>`f'</SAMP> for single-floats,
-<LI>
-
-<SAMP>`d'</SAMP> for double-floats,
-<LI>
-
-<SAMP>`L'</SAMP> for long-floats,
-</UL>
-
-or <SAMP>`e'</SAMP>, which denotes a default float format. The precision specifying
-suffix has the syntax _<VAR>prec</VAR> where <VAR>prec</VAR> denotes the number of
-valid mantissa digits (in decimal, excluding leading zeroes), cf. also
-function <SAMP>`cl_float_format'</SAMP>.
-
-<DT>Complex numbers
-<DD>
-External representation:
-
-<UL>
-<LI>
-
-In algebraic notation: <CODE><VAR>realpart</VAR>+<VAR>imagpart</VAR>i</CODE>. Of course,
-if <VAR>imagpart</VAR> is negative, its printed representation begins with
-a <SAMP>`-'</SAMP>, and the <SAMP>`+'</SAMP> between <VAR>realpart</VAR> and <VAR>imagpart</VAR>
-may be omitted. Note that this notation cannot be used when the <VAR>imagpart</VAR>
-is rational and the rational number's base is >18, because the <SAMP>`i'</SAMP>
-is then read as a digit.
-<LI>
-
-In Common Lisp notation: <CODE>#C(<VAR>realpart</VAR> <VAR>imagpart</VAR>)</CODE>.
-</UL>
-
-</DL>
-
-
-
-<H2><A NAME="SEC46" HREF="cln.html#TOC46">5.2 Input functions</A></H2>
-
-<P>
-Including <CODE><cln/io.h></CODE> defines a type <CODE>cl_istream</CODE>, which is
-the type of the first argument to all input functions. <CODE>cl_istream</CODE>
-is the same as <CODE>std::istream&</CODE>.
-
-
-<P>
-The variable
-
-<UL>
-<LI>
-
-<CODE>cl_istream stdin</CODE>
-</UL>
-
-<P>
-contains the standard input stream.
-
-
-<P>
-These are the simple input functions:
-
-
-<DL COMPACT>
-
-<DT><CODE>int freadchar (cl_istream stream)</CODE>
-<DD>
-Reads a character from <CODE>stream</CODE>. Returns <CODE>cl_EOF</CODE> (not a <SAMP>`char'</SAMP>!)
-if the end of stream was encountered or an error occurred.
-
-<DT><CODE>int funreadchar (cl_istream stream, int c)</CODE>
-<DD>
-Puts back <CODE>c</CODE> onto <CODE>stream</CODE>. <CODE>c</CODE> must be the result of the
-last <CODE>freadchar</CODE> operation on <CODE>stream</CODE>.
-</DL>
-
-<P>
-Each of the classes <CODE>cl_N</CODE>, <CODE>cl_R</CODE>, <CODE>cl_RA</CODE>, <CODE>cl_I</CODE>,
-<CODE>cl_F</CODE>, <CODE>cl_SF</CODE>, <CODE>cl_FF</CODE>, <CODE>cl_DF</CODE>, <CODE>cl_LF</CODE>
-defines, in <CODE><cln/<VAR>type</VAR>_io.h></CODE>, the following input function:
-
-
-<DL COMPACT>
-
-<DT><CODE>cl_istream operator>> (cl_istream stream, <VAR>type</VAR>& result)</CODE>
-<DD>
-Reads a number from <CODE>stream</CODE> and stores it in the <CODE>result</CODE>.
-</DL>
-
-<P>
-The most flexible input functions, defined in <CODE><cln/<VAR>type</VAR>_io.h></CODE>,
-are the following:
-
-
-<DL COMPACT>
-
-<DT><CODE>cl_N read_complex (cl_istream stream, const cl_read_flags& flags)</CODE>
-<DD>
-<DT><CODE>cl_R read_real (cl_istream stream, const cl_read_flags& flags)</CODE>
-<DD>
-<DT><CODE>cl_F read_float (cl_istream stream, const cl_read_flags& flags)</CODE>
-<DD>
-<DT><CODE>cl_RA read_rational (cl_istream stream, const cl_read_flags& flags)</CODE>
-<DD>
-<DT><CODE>cl_I read_integer (cl_istream stream, const cl_read_flags& flags)</CODE>
-<DD>
-Reads a number from <CODE>stream</CODE>. The <CODE>flags</CODE> are parameters which
-affect the input syntax. Whitespace before the number is silently skipped.
-
-<DT><CODE>cl_N read_complex (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse)</CODE>
-<DD>
-<DT><CODE>cl_R read_real (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse)</CODE>
-<DD>
-<DT><CODE>cl_F read_float (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse)</CODE>
-<DD>
-<DT><CODE>cl_RA read_rational (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse)</CODE>
-<DD>
-<DT><CODE>cl_I read_integer (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse)</CODE>
-<DD>
-Reads a number from a string in memory. The <CODE>flags</CODE> are parameters which
-affect the input syntax. The string starts at <CODE>string</CODE> and ends at
-<CODE>string_limit</CODE> (exclusive limit). <CODE>string_limit</CODE> may also be
-<CODE>NULL</CODE>, denoting the entire string, i.e. equivalent to
-<CODE>string_limit = string + strlen(string)</CODE>. If <CODE>end_of_parse</CODE> is
-<CODE>NULL</CODE>, the string in memory must contain exactly one number and nothing
-more, else a fatal error will be signalled. If <CODE>end_of_parse</CODE>
-is not <CODE>NULL</CODE>, <CODE>*end_of_parse</CODE> will be assigned a pointer past
-the last parsed character (i.e. <CODE>string_limit</CODE> if nothing came after
-the number). Whitespace is not allowed.
-</DL>
-
-<P>
-The structure <CODE>cl_read_flags</CODE> contains the following fields:
-
-
-<DL COMPACT>
-
-<DT><CODE>cl_read_syntax_t syntax</CODE>
-<DD>
-The possible results of the read operation. Possible values are
-<CODE>syntax_number</CODE>, <CODE>syntax_real</CODE>, <CODE>syntax_rational</CODE>,
-<CODE>syntax_integer</CODE>, <CODE>syntax_float</CODE>, <CODE>syntax_sfloat</CODE>,
-<CODE>syntax_ffloat</CODE>, <CODE>syntax_dfloat</CODE>, <CODE>syntax_lfloat</CODE>.
-
-<DT><CODE>cl_read_lsyntax_t lsyntax</CODE>
-<DD>
-Specifies the language-dependent syntax variant for the read operation.
-Possible values are
-
-<DL COMPACT>
-
-<DT><CODE>lsyntax_standard</CODE>
-<DD>
-accept standard algebraic notation only, no complex numbers,
-<DT><CODE>lsyntax_algebraic</CODE>
-<DD>
-accept the algebraic notation <CODE><VAR>x</VAR>+<VAR>y</VAR>i</CODE> for complex numbers,
-<DT><CODE>lsyntax_commonlisp</CODE>
-<DD>
-accept the <CODE>#b</CODE>, <CODE>#o</CODE>, <CODE>#x</CODE> syntaxes for binary, octal,
-hexadecimal numbers,
-<CODE>#<VAR>base</VAR>R</CODE> for rational numbers in a given base,
-<CODE>#c(<VAR>realpart</VAR> <VAR>imagpart</VAR>)</CODE> for complex numbers,
-<DT><CODE>lsyntax_all</CODE>
-<DD>
-accept all of these extensions.
-</DL>
-
-<DT><CODE>unsigned int rational_base</CODE>
-<DD>
-The base in which rational numbers are read.
-
-<DT><CODE>cl_float_format_t float_flags.default_float_format</CODE>
-<DD>
-The float format used when reading floats with exponent marker <SAMP>`e'</SAMP>.
-
-<DT><CODE>cl_float_format_t float_flags.default_lfloat_format</CODE>
-<DD>
-The float format used when reading floats with exponent marker <SAMP>`l'</SAMP>.
-
-<DT><CODE>cl_boolean float_flags.mantissa_dependent_float_format</CODE>
-<DD>
-When this flag is true, floats specified with more digits than corresponding
-to the exponent marker they contain, but without <VAR>_nnn</VAR> suffix, will get a
-precision corresponding to their number of significant digits.
-</DL>
-
-
-
-<H2><A NAME="SEC47" HREF="cln.html#TOC47">5.3 Output functions</A></H2>
-
-<P>
-Including <CODE><cln/io.h></CODE> defines a type <CODE>cl_ostream</CODE>, which is
-the type of the first argument to all output functions. <CODE>cl_ostream</CODE>
-is the same as <CODE>std::ostream&</CODE>.
-
-
-<P>
-The variable
-
-<UL>
-<LI>
-
-<CODE>cl_ostream stdout</CODE>
-</UL>
-
-<P>
-contains the standard output stream.
-
-
-<P>
-The variable
-
-<UL>
-<LI>
-
-<CODE>cl_ostream stderr</CODE>
-</UL>
-
-<P>
-contains the standard error output stream.
-
-
-<P>
-These are the simple output functions:
-
-
-<DL COMPACT>
-
-<DT><CODE>void fprintchar (cl_ostream stream, char c)</CODE>
-<DD>
-Prints the character <CODE>x</CODE> literally on the <CODE>stream</CODE>.
-
-<DT><CODE>void fprint (cl_ostream stream, const char * string)</CODE>
-<DD>
-Prints the <CODE>string</CODE> literally on the <CODE>stream</CODE>.
-
-<DT><CODE>void fprintdecimal (cl_ostream stream, int x)</CODE>
-<DD>
-<DT><CODE>void fprintdecimal (cl_ostream stream, const cl_I& x)</CODE>
-<DD>
-Prints the integer <CODE>x</CODE> in decimal on the <CODE>stream</CODE>.
-
-<DT><CODE>void fprintbinary (cl_ostream stream, const cl_I& x)</CODE>
-<DD>
-Prints the integer <CODE>x</CODE> in binary (base 2, without prefix)
-on the <CODE>stream</CODE>.
-
-<DT><CODE>void fprintoctal (cl_ostream stream, const cl_I& x)</CODE>
-<DD>
-Prints the integer <CODE>x</CODE> in octal (base 8, without prefix)
-on the <CODE>stream</CODE>.
-
-<DT><CODE>void fprinthexadecimal (cl_ostream stream, const cl_I& x)</CODE>
-<DD>
-Prints the integer <CODE>x</CODE> in hexadecimal (base 16, without prefix)
-on the <CODE>stream</CODE>.
-</DL>
-
-<P>
-Each of the classes <CODE>cl_N</CODE>, <CODE>cl_R</CODE>, <CODE>cl_RA</CODE>, <CODE>cl_I</CODE>,
-<CODE>cl_F</CODE>, <CODE>cl_SF</CODE>, <CODE>cl_FF</CODE>, <CODE>cl_DF</CODE>, <CODE>cl_LF</CODE>
-defines, in <CODE><cln/<VAR>type</VAR>_io.h></CODE>, the following output functions:
-
-
-<DL COMPACT>
-
-<DT><CODE>void fprint (cl_ostream stream, const <VAR>type</VAR>& x)</CODE>
-<DD>
-<DT><CODE>cl_ostream operator<< (cl_ostream stream, const <VAR>type</VAR>& x)</CODE>
-<DD>
-Prints the number <CODE>x</CODE> on the <CODE>stream</CODE>. The output may depend
-on the global printer settings in the variable <CODE>default_print_flags</CODE>.
-The <CODE>ostream</CODE> flags and settings (flags, width and locale) are
-ignored.
-</DL>
-
-<P>
-The most flexible output function, defined in <CODE><cln/<VAR>type</VAR>_io.h></CODE>,
-are the following:
-
-<PRE>
-void print_complex (cl_ostream stream, const cl_print_flags& flags,
- const cl_N& z);
-void print_real (cl_ostream stream, const cl_print_flags& flags,
- const cl_R& z);
-void print_float (cl_ostream stream, const cl_print_flags& flags,
- const cl_F& z);
-void print_rational (cl_ostream stream, const cl_print_flags& flags,
- const cl_RA& z);
-void print_integer (cl_ostream stream, const cl_print_flags& flags,
- const cl_I& z);
-</PRE>
-
-<P>
-Prints the number <CODE>x</CODE> on the <CODE>stream</CODE>. The <CODE>flags</CODE> are
-parameters which affect the output.
-
-
-<P>
-The structure type <CODE>cl_print_flags</CODE> contains the following fields:
-
-
-<DL COMPACT>
-
-<DT><CODE>unsigned int rational_base</CODE>
-<DD>
-The base in which rational numbers are printed. Default is <CODE>10</CODE>.
-
-<DT><CODE>cl_boolean rational_readably</CODE>
-<DD>
-If this flag is true, rational numbers are printed with radix specifiers in
-Common Lisp syntax (<CODE>#<VAR>n</VAR>R</CODE> or <CODE>#b</CODE> or <CODE>#o</CODE> or <CODE>#x</CODE>
-prefixes, trailing dot). Default is false.
-
-<DT><CODE>cl_boolean float_readably</CODE>
-<DD>
-If this flag is true, type specific exponent markers have precedence over 'E'.
-Default is false.
-
-<DT><CODE>cl_float_format_t default_float_format</CODE>
-<DD>
-Floating point numbers of this format will be printed using the 'E' exponent
-marker. Default is <CODE>cl_float_format_ffloat</CODE>.
-
-<DT><CODE>cl_boolean complex_readably</CODE>
-<DD>
-If this flag is true, complex numbers will be printed using the Common Lisp
-syntax <CODE>#C(<VAR>realpart</VAR> <VAR>imagpart</VAR>)</CODE>. Default is false.
-
-<DT><CODE>cl_string univpoly_varname</CODE>
-<DD>
-Univariate polynomials with no explicit indeterminate name will be printed
-using this variable name. Default is <CODE>"x"</CODE>.
-</DL>
-
-<P>
-The global variable <CODE>default_print_flags</CODE> contains the default values,
-used by the function <CODE>fprint</CODE>.
-
-
-
-
-<H1><A NAME="SEC48" HREF="cln.html#TOC48">6. Rings</A></H1>
-
-<P>
-CLN has a class of abstract rings.
-
-
-
-<PRE>
- Ring
- cl_ring
- <cln/ring.h>
-</PRE>
-
-<P>
-Rings can be compared for equality:
-
-
-<DL COMPACT>
-
-<DT><CODE>bool operator== (const cl_ring&, const cl_ring&)</CODE>
-<DD>
-<DT><CODE>bool operator!= (const cl_ring&, const cl_ring&)</CODE>
-<DD>
-These compare two rings for equality.
-</DL>
-
-<P>
-Given a ring <CODE>R</CODE>, the following members can be used.
-
-
-<DL COMPACT>
-
-<DT><CODE>void R->fprint (cl_ostream stream, const cl_ring_element& x)</CODE>
-<DD>
-<A NAME="IDX241"></A>
-<DT><CODE>cl_boolean R->equal (const cl_ring_element& x, const cl_ring_element& y)</CODE>
-<DD>
-<A NAME="IDX242"></A>
-<DT><CODE>cl_ring_element R->zero ()</CODE>
-<DD>
-<A NAME="IDX243"></A>
-<DT><CODE>cl_boolean R->zerop (const cl_ring_element& x)</CODE>
-<DD>
-<A NAME="IDX244"></A>
-<DT><CODE>cl_ring_element R->plus (const cl_ring_element& x, const cl_ring_element& y)</CODE>
-<DD>
-<A NAME="IDX245"></A>
-<DT><CODE>cl_ring_element R->minus (const cl_ring_element& x, const cl_ring_element& y)</CODE>
-<DD>
-<A NAME="IDX246"></A>
-<DT><CODE>cl_ring_element R->uminus (const cl_ring_element& x)</CODE>
-<DD>
-<A NAME="IDX247"></A>
-<DT><CODE>cl_ring_element R->one ()</CODE>
-<DD>
-<A NAME="IDX248"></A>
-<DT><CODE>cl_ring_element R->canonhom (const cl_I& x)</CODE>
-<DD>
-<A NAME="IDX249"></A>
-<DT><CODE>cl_ring_element R->mul (const cl_ring_element& x, const cl_ring_element& y)</CODE>
-<DD>
-<A NAME="IDX250"></A>
-<DT><CODE>cl_ring_element R->square (const cl_ring_element& x)</CODE>
-<DD>
-<A NAME="IDX251"></A>
-<DT><CODE>cl_ring_element R->expt_pos (const cl_ring_element& x, const cl_I& y)</CODE>
-<DD>
-<A NAME="IDX252"></A>
-</DL>
-
-<P>
-The following rings are built-in.
-
-
-<DL COMPACT>
-
-<DT><CODE>cl_null_ring cl_0_ring</CODE>
-<DD>
-The null ring, containing only zero.
-
-<DT><CODE>cl_complex_ring cl_C_ring</CODE>
-<DD>
-The ring of complex numbers. This corresponds to the type <CODE>cl_N</CODE>.
-
-<DT><CODE>cl_real_ring cl_R_ring</CODE>
-<DD>
-The ring of real numbers. This corresponds to the type <CODE>cl_R</CODE>.
-
-<DT><CODE>cl_rational_ring cl_RA_ring</CODE>
-<DD>
-The ring of rational numbers. This corresponds to the type <CODE>cl_RA</CODE>.
-
-<DT><CODE>cl_integer_ring cl_I_ring</CODE>
-<DD>
-The ring of integers. This corresponds to the type <CODE>cl_I</CODE>.
-</DL>
-
-<P>
-Type tests can be performed for any of <CODE>cl_C_ring</CODE>, <CODE>cl_R_ring</CODE>,
-<CODE>cl_RA_ring</CODE>, <CODE>cl_I_ring</CODE>:
-
-
-<DL COMPACT>
-
-<DT><CODE>cl_boolean instanceof (const cl_number& x, const cl_number_ring& R)</CODE>
-<DD>
-<A NAME="IDX253"></A>
-Tests whether the given number is an element of the number ring R.
-</DL>
-
-
-
-<H1><A NAME="SEC49" HREF="cln.html#TOC49">7. Modular integers</A></H1>
-<P>
-<A NAME="IDX254"></A>
-
-
-
-
-<H2><A NAME="SEC50" HREF="cln.html#TOC50">7.1 Modular integer rings</A></H2>
-<P>
-<A NAME="IDX255"></A>
-
-
-<P>
-CLN implements modular integers, i.e. integers modulo a fixed integer N.
-The modulus is explicitly part of every modular integer. CLN doesn't
-allow you to (accidentally) mix elements of different modular rings,
-e.g. <CODE>(3 mod 4) + (2 mod 5)</CODE> will result in a runtime error.
-(Ideally one would imagine a generic data type <CODE>cl_MI(N)</CODE>, but C++
-doesn't have generic types. So one has to live with runtime checks.)
-
-
-<P>
-The class of modular integer rings is
-
-
-
-<PRE>
- Ring
- cl_ring
- <cln/ring.h>
- |
- |
- Modular integer ring
- cl_modint_ring
- <cln/modinteger.h>
-</PRE>
-
-<P>
-<A NAME="IDX256"></A>
-
-
-<P>
-and the class of all modular integers (elements of modular integer rings) is
-
-
-
-<PRE>
- Modular integer
- cl_MI
- <cln/modinteger.h>
-</PRE>
-
-<P>
-Modular integer rings are constructed using the function
-
-
-<DL COMPACT>
-
-<DT><CODE>cl_modint_ring find_modint_ring (const cl_I& N)</CODE>
-<DD>
-<A NAME="IDX257"></A>
-This function returns the modular ring <SAMP>`Z/NZ'</SAMP>. It takes care
-of finding out about special cases of <CODE>N</CODE>, like powers of two
-and odd numbers for which Montgomery multiplication will be a win,
-<A NAME="IDX258"></A>
-and precomputes any necessary auxiliary data for computing modulo <CODE>N</CODE>.
-There is a cache table of rings, indexed by <CODE>N</CODE> (or, more precisely,
-by <CODE>abs(N)</CODE>). This ensures that the precomputation costs are reduced
-to a minimum.
-</DL>
-
-<P>
-Modular integer rings can be compared for equality:
-
-
-<DL COMPACT>
-
-<DT><CODE>bool operator== (const cl_modint_ring&, const cl_modint_ring&)</CODE>
-<DD>
-<A NAME="IDX259"></A>
-<DT><CODE>bool operator!= (const cl_modint_ring&, const cl_modint_ring&)</CODE>
-<DD>
-<A NAME="IDX260"></A>
-These compare two modular integer rings for equality. Two different calls
-to <CODE>find_modint_ring</CODE> with the same argument necessarily return the
-same ring because it is memoized in the cache table.
-</DL>
-
-
-
-<H2><A NAME="SEC51" HREF="cln.html#TOC51">7.2 Functions on modular integers</A></H2>
-
-<P>
-Given a modular integer ring <CODE>R</CODE>, the following members can be used.
-
-
-<DL COMPACT>
-
-<DT><CODE>cl_I R->modulus</CODE>
-<DD>
-<A NAME="IDX261"></A>
-This is the ring's modulus, normalized to be nonnegative: <CODE>abs(N)</CODE>.
-
-<DT><CODE>cl_MI R->zero()</CODE>
-<DD>
-<A NAME="IDX262"></A>
-This returns <CODE>0 mod N</CODE>.
-
-<DT><CODE>cl_MI R->one()</CODE>
-<DD>
-<A NAME="IDX263"></A>
-This returns <CODE>1 mod N</CODE>.
-
-<DT><CODE>cl_MI R->canonhom (const cl_I& x)</CODE>
-<DD>
-<A NAME="IDX264"></A>
-This returns <CODE>x mod N</CODE>.
-
-<DT><CODE>cl_I R->retract (const cl_MI& x)</CODE>
-<DD>
-<A NAME="IDX265"></A>
-This is a partial inverse function to <CODE>R->canonhom</CODE>. It returns the
-standard representative (<CODE>>=0</CODE>, <CODE><N</CODE>) of <CODE>x</CODE>.
-
-<DT><CODE>cl_MI R->random(random_state& randomstate)</CODE>
-<DD>
-<DT><CODE>cl_MI R->random()</CODE>
-<DD>
-<A NAME="IDX266"></A>
-This returns a random integer modulo <CODE>N</CODE>.
-</DL>
-
-<P>
-The following operations are defined on modular integers.
-
-
-<DL COMPACT>
-
-<DT><CODE>cl_modint_ring x.ring ()</CODE>
-<DD>
-<A NAME="IDX267"></A>
-Returns the ring to which the modular integer <CODE>x</CODE> belongs.
-
-<DT><CODE>cl_MI operator+ (const cl_MI&, const cl_MI&)</CODE>
-<DD>
-<A NAME="IDX268"></A>
-Returns the sum of two modular integers. One of the arguments may also
-be a plain integer.
-
-<DT><CODE>cl_MI operator- (const cl_MI&, const cl_MI&)</CODE>
-<DD>
-<A NAME="IDX269"></A>
-Returns the difference of two modular integers. One of the arguments may also
-be a plain integer.
-
-<DT><CODE>cl_MI operator- (const cl_MI&)</CODE>
-<DD>
-Returns the negative of a modular integer.
-
-<DT><CODE>cl_MI operator* (const cl_MI&, const cl_MI&)</CODE>
-<DD>
-<A NAME="IDX270"></A>
-Returns the product of two modular integers. One of the arguments may also
-be a plain integer.
-
-<DT><CODE>cl_MI square (const cl_MI&)</CODE>
-<DD>
-<A NAME="IDX271"></A>
-Returns the square of a modular integer.
-
-<DT><CODE>cl_MI recip (const cl_MI& x)</CODE>
-<DD>
-<A NAME="IDX272"></A>
-Returns the reciprocal <CODE>x^-1</CODE> of a modular integer <CODE>x</CODE>. <CODE>x</CODE>
-must be coprime to the modulus, otherwise an error message is issued.
-
-<DT><CODE>cl_MI div (const cl_MI& x, const cl_MI& y)</CODE>
-<DD>
-<A NAME="IDX273"></A>
-Returns the quotient <CODE>x*y^-1</CODE> of two modular integers <CODE>x</CODE>, <CODE>y</CODE>.
-<CODE>y</CODE> must be coprime to the modulus, otherwise an error message is issued.
-
-<DT><CODE>cl_MI expt_pos (const cl_MI& x, const cl_I& y)</CODE>
-<DD>
-<A NAME="IDX274"></A>
-<CODE>y</CODE> must be > 0. Returns <CODE>x^y</CODE>.
-
-<DT><CODE>cl_MI expt (const cl_MI& x, const cl_I& y)</CODE>
-<DD>
-<A NAME="IDX275"></A>
-Returns <CODE>x^y</CODE>. If <CODE>y</CODE> is negative, <CODE>x</CODE> must be coprime to the
-modulus, else an error message is issued.
-
-<DT><CODE>cl_MI operator<< (const cl_MI& x, const cl_I& y)</CODE>
-<DD>
-<A NAME="IDX276"></A>
-Returns <CODE>x*2^y</CODE>.
-
-<DT><CODE>cl_MI operator>> (const cl_MI& x, const cl_I& y)</CODE>
-<DD>
-<A NAME="IDX277"></A>
-Returns <CODE>x*2^-y</CODE>. When <CODE>y</CODE> is positive, the modulus must be odd,
-or an error message is issued.
-
-<DT><CODE>bool operator== (const cl_MI&, const cl_MI&)</CODE>
-<DD>
-<A NAME="IDX278"></A>
-<DT><CODE>bool operator!= (const cl_MI&, const cl_MI&)</CODE>
-<DD>
-<A NAME="IDX279"></A>
-Compares two modular integers, belonging to the same modular integer ring,
-for equality.
-
-<DT><CODE>cl_boolean zerop (const cl_MI& x)</CODE>
-<DD>
-<A NAME="IDX280"></A>
-Returns true if <CODE>x</CODE> is <CODE>0 mod N</CODE>.
-</DL>
-
-<P>
-The following output functions are defined (see also the chapter on
-input/output).
-
-
-<DL COMPACT>
-
-<DT><CODE>void fprint (cl_ostream stream, const cl_MI& x)</CODE>
-<DD>
-<A NAME="IDX281"></A>
-<DT><CODE>cl_ostream operator<< (cl_ostream stream, const cl_MI& x)</CODE>
-<DD>
-<A NAME="IDX282"></A>
-Prints the modular integer <CODE>x</CODE> on the <CODE>stream</CODE>. The output may depend
-on the global printer settings in the variable <CODE>default_print_flags</CODE>.
-</DL>
-
-
-
-<H1><A NAME="SEC52" HREF="cln.html#TOC52">8. Symbolic data types</A></H1>
-<P>
-<A NAME="IDX283"></A>
-
-
-<P>
-CLN implements two symbolic (non-numeric) data types: strings and symbols.
-
-
-
-
-<H2><A NAME="SEC53" HREF="cln.html#TOC53">8.1 Strings</A></H2>
-<P>
-<A NAME="IDX284"></A>
-<A NAME="IDX285"></A>
-
-
-<P>
-The class
-
-
-
-<PRE>
- String
- cl_string
- <cln/string.h>
-</PRE>
-
-<P>
-implements immutable strings.
-
-
-<P>
-Strings are constructed through the following constructors:
-
-
-<DL COMPACT>
-
-<DT><CODE>cl_string (const char * s)</CODE>
-<DD>
-Returns an immutable copy of the (zero-terminated) C string <CODE>s</CODE>.
-
-<DT><CODE>cl_string (const char * ptr, unsigned long len)</CODE>
-<DD>
-Returns an immutable copy of the <CODE>len</CODE> characters at
-<CODE>ptr[0]</CODE>, ..., <CODE>ptr[len-1]</CODE>. NUL characters are allowed.
-</DL>
-
-<P>
-The following functions are available on strings:
-
-
-<DL COMPACT>
-
-<DT><CODE>operator =</CODE>
-<DD>
-Assignment from <CODE>cl_string</CODE> and <CODE>const char *</CODE>.
-
-<DT><CODE>s.length()</CODE>
-<DD>
-<A NAME="IDX286"></A>
-<DT><CODE>strlen(s)</CODE>
-<DD>
-<A NAME="IDX287"></A>
-Returns the length of the string <CODE>s</CODE>.
-
-<DT><CODE>s[i]</CODE>
-<DD>
-<A NAME="IDX288"></A>
-Returns the <CODE>i</CODE>th character of the string <CODE>s</CODE>.
-<CODE>i</CODE> must be in the range <CODE>0 <= i < s.length()</CODE>.
-
-<DT><CODE>bool equal (const cl_string& s1, const cl_string& s2)</CODE>
-<DD>
-<A NAME="IDX289"></A>
-Compares two strings for equality. One of the arguments may also be a
-plain <CODE>const char *</CODE>.
-</DL>
-
-
-
-<H2><A NAME="SEC54" HREF="cln.html#TOC54">8.2 Symbols</A></H2>
-<P>
-<A NAME="IDX290"></A>
-<A NAME="IDX291"></A>
-
-
-<P>
-Symbols are uniquified strings: all symbols with the same name are shared.
-This means that comparison of two symbols is fast (effectively just a pointer
-comparison), whereas comparison of two strings must in the worst case walk
-both strings until their end.
-Symbols are used, for example, as tags for properties, as names of variables
-in polynomial rings, etc.
-
-
-<P>
-Symbols are constructed through the following constructor:
-
-
-<DL COMPACT>
-
-<DT><CODE>cl_symbol (const cl_string& s)</CODE>
-<DD>
-Looks up or creates a new symbol with a given name.
-</DL>
-
-<P>
-The following operations are available on symbols:
-
-
-<DL COMPACT>
-
-<DT><CODE>cl_string (const cl_symbol& sym)</CODE>
-<DD>
-Conversion to <CODE>cl_string</CODE>: Returns the string which names the symbol
-<CODE>sym</CODE>.
-
-<DT><CODE>bool equal (const cl_symbol& sym1, const cl_symbol& sym2)</CODE>
-<DD>
-<A NAME="IDX292"></A>
-Compares two symbols for equality. This is very fast.
-</DL>
-
-
-
-<H1><A NAME="SEC55" HREF="cln.html#TOC55">9. Univariate polynomials</A></H1>
-<P>
-<A NAME="IDX293"></A>
-<A NAME="IDX294"></A>
-
-
-
-
-<H2><A NAME="SEC56" HREF="cln.html#TOC56">9.1 Univariate polynomial rings</A></H2>
-
-<P>
-CLN implements univariate polynomials (polynomials in one variable) over an
-arbitrary ring. The indeterminate variable may be either unnamed (and will be
-printed according to <CODE>default_print_flags.univpoly_varname</CODE>, which
-defaults to <SAMP>`x'</SAMP>) or carry a given name. The base ring and the
-indeterminate are explicitly part of every polynomial. CLN doesn't allow you to
-(accidentally) mix elements of different polynomial rings, e.g.
-<CODE>(a^2+1) * (b^3-1)</CODE> will result in a runtime error. (Ideally this should
-return a multivariate polynomial, but they are not yet implemented in CLN.)
-
-
-<P>
-The classes of univariate polynomial rings are
-
-
-
-<PRE>
- Ring
- cl_ring
- <cln/ring.h>
- |
- |
- Univariate polynomial ring
- cl_univpoly_ring
- <cln/univpoly.h>
- |
- +----------------+-------------------+
- | | |
- Complex polynomial ring | Modular integer polynomial ring
- cl_univpoly_complex_ring | cl_univpoly_modint_ring
- <cln/univpoly_complex.h> | <cln/univpoly_modint.h>
- |
- +----------------+
- | |
- Real polynomial ring |
- cl_univpoly_real_ring |
- <cln/univpoly_real.h> |
- |
- +----------------+
- | |
- Rational polynomial ring |
- cl_univpoly_rational_ring |
- <cln/univpoly_rational.h> |
- |
- +----------------+
- |
- Integer polynomial ring
- cl_univpoly_integer_ring
- <cln/univpoly_integer.h>
-</PRE>
-
-<P>
-and the corresponding classes of univariate polynomials are
-
-
-
-<PRE>
- Univariate polynomial
- cl_UP
- <cln/univpoly.h>
- |
- +----------------+-------------------+
- | | |
- Complex polynomial | Modular integer polynomial
- cl_UP_N | cl_UP_MI
- <cln/univpoly_complex.h> | <cln/univpoly_modint.h>
- |
- +----------------+
- | |
- Real polynomial |
- cl_UP_R |
- <cln/univpoly_real.h> |
- |
- +----------------+
- | |
- Rational polynomial |
- cl_UP_RA |
- <cln/univpoly_rational.h> |
- |
- +----------------+
- |
- Integer polynomial
- cl_UP_I
- <cln/univpoly_integer.h>
-</PRE>
-
-<P>
-Univariate polynomial rings are constructed using the functions
-
-
-<DL COMPACT>
-
-<DT><CODE>cl_univpoly_ring find_univpoly_ring (const cl_ring& R)</CODE>
-<DD>
-<DT><CODE>cl_univpoly_ring find_univpoly_ring (const cl_ring& R, const cl_symbol& varname)</CODE>
-<DD>
-This function returns the polynomial ring <SAMP>`R[X]'</SAMP>, unnamed or named.
-<CODE>R</CODE> may be an arbitrary ring. This function takes care of finding out
-about special cases of <CODE>R</CODE>, such as the rings of complex numbers,
-real numbers, rational numbers, integers, or modular integer rings.
-There is a cache table of rings, indexed by <CODE>R</CODE> and <CODE>varname</CODE>.
-This ensures that two calls of this function with the same arguments will
-return the same polynomial ring.
-
-<DT><CODE>cl_univpoly_complex_ring find_univpoly_ring (const cl_complex_ring& R)</CODE>
-<DD>
-<A NAME="IDX295"></A>
-<DT><CODE>cl_univpoly_complex_ring find_univpoly_ring (const cl_complex_ring& R, const cl_symbol& varname)</CODE>
-<DD>
-<DT><CODE>cl_univpoly_real_ring find_univpoly_ring (const cl_real_ring& R)</CODE>
-<DD>
-<DT><CODE>cl_univpoly_real_ring find_univpoly_ring (const cl_real_ring& R, const cl_symbol& varname)</CODE>
-<DD>
-<DT><CODE>cl_univpoly_rational_ring find_univpoly_ring (const cl_rational_ring& R)</CODE>
-<DD>
-<DT><CODE>cl_univpoly_rational_ring find_univpoly_ring (const cl_rational_ring& R, const cl_symbol& varname)</CODE>
-<DD>
-<DT><CODE>cl_univpoly_integer_ring find_univpoly_ring (const cl_integer_ring& R)</CODE>
-<DD>
-<DT><CODE>cl_univpoly_integer_ring find_univpoly_ring (const cl_integer_ring& R, const cl_symbol& varname)</CODE>
-<DD>
-<DT><CODE>cl_univpoly_modint_ring find_univpoly_ring (const cl_modint_ring& R)</CODE>
-<DD>
-<DT><CODE>cl_univpoly_modint_ring find_univpoly_ring (const cl_modint_ring& R, const cl_symbol& varname)</CODE>
-<DD>
-These functions are equivalent to the general <CODE>find_univpoly_ring</CODE>,
-only the return type is more specific, according to the base ring's type.
-</DL>
-
-
-
-<H2><A NAME="SEC57" HREF="cln.html#TOC57">9.2 Functions on univariate polynomials</A></H2>
-
-<P>
-Given a univariate polynomial ring <CODE>R</CODE>, the following members can be used.
-
-
-<DL COMPACT>
-
-<DT><CODE>cl_ring R->basering()</CODE>
-<DD>
-<A NAME="IDX296"></A>
-This returns the base ring, as passed to <SAMP>`find_univpoly_ring'</SAMP>.
-
-<DT><CODE>cl_UP R->zero()</CODE>
-<DD>
-<A NAME="IDX297"></A>
-This returns <CODE>0 in R</CODE>, a polynomial of degree -1.
-
-<DT><CODE>cl_UP R->one()</CODE>
-<DD>
-<A NAME="IDX298"></A>
-This returns <CODE>1 in R</CODE>, a polynomial of degree <= 0.
-
-<DT><CODE>cl_UP R->canonhom (const cl_I& x)</CODE>
-<DD>
-<A NAME="IDX299"></A>
-This returns <CODE>x in R</CODE>, a polynomial of degree <= 0.
-
-<DT><CODE>cl_UP R->monomial (const cl_ring_element& x, uintL e)</CODE>
-<DD>
-<A NAME="IDX300"></A>
-This returns a sparse polynomial: <CODE>x * X^e</CODE>, where <CODE>X</CODE> is the
-indeterminate.
-
-<DT><CODE>cl_UP R->create (sintL degree)</CODE>
-<DD>
-<A NAME="IDX301"></A>
-Creates a new polynomial with a given degree. The zero polynomial has degree
-<CODE>-1</CODE>. After creating the polynomial, you should put in the coefficients,
-using the <CODE>set_coeff</CODE> member function, and then call the <CODE>finalize</CODE>
-member function.
-</DL>
-
-<P>
-The following are the only destructive operations on univariate polynomials.
-
-
-<DL COMPACT>
-
-<DT><CODE>void set_coeff (cl_UP& x, uintL index, const cl_ring_element& y)</CODE>
-<DD>
-<A NAME="IDX302"></A>
-This changes the coefficient of <CODE>X^index</CODE> in <CODE>x</CODE> to be <CODE>y</CODE>.
-After changing a polynomial and before applying any "normal" operation on it,
-you should call its <CODE>finalize</CODE> member function.
-
-<DT><CODE>void finalize (cl_UP& x)</CODE>
-<DD>
-<A NAME="IDX303"></A>
-This function marks the endpoint of destructive modifications of a polynomial.
-It normalizes the internal representation so that subsequent computations have
-less overhead. Doing normal computations on unnormalized polynomials may
-produce wrong results or crash the program.
-</DL>
-
-<P>
-The following operations are defined on univariate polynomials.
-
-
-<DL COMPACT>
-
-<DT><CODE>cl_univpoly_ring x.ring ()</CODE>
-<DD>
-<A NAME="IDX304"></A>
-Returns the ring to which the univariate polynomial <CODE>x</CODE> belongs.
-
-<DT><CODE>cl_UP operator+ (const cl_UP&, const cl_UP&)</CODE>
-<DD>
-<A NAME="IDX305"></A>
-Returns the sum of two univariate polynomials.
-
-<DT><CODE>cl_UP operator- (const cl_UP&, const cl_UP&)</CODE>
-<DD>
-<A NAME="IDX306"></A>
-Returns the difference of two univariate polynomials.
-
-<DT><CODE>cl_UP operator- (const cl_UP&)</CODE>
-<DD>
-Returns the negative of a univariate polynomial.
-
-<DT><CODE>cl_UP operator* (const cl_UP&, const cl_UP&)</CODE>
-<DD>
-<A NAME="IDX307"></A>
-Returns the product of two univariate polynomials. One of the arguments may
-also be a plain integer or an element of the base ring.
-
-<DT><CODE>cl_UP square (const cl_UP&)</CODE>
-<DD>
-<A NAME="IDX308"></A>
-Returns the square of a univariate polynomial.
-
-<DT><CODE>cl_UP expt_pos (const cl_UP& x, const cl_I& y)</CODE>
-<DD>
-<A NAME="IDX309"></A>
-<CODE>y</CODE> must be > 0. Returns <CODE>x^y</CODE>.
-
-<DT><CODE>bool operator== (const cl_UP&, const cl_UP&)</CODE>
-<DD>
-<A NAME="IDX310"></A>
-<DT><CODE>bool operator!= (const cl_UP&, const cl_UP&)</CODE>
-<DD>
-<A NAME="IDX311"></A>
-Compares two univariate polynomials, belonging to the same univariate
-polynomial ring, for equality.
-
-<DT><CODE>cl_boolean zerop (const cl_UP& x)</CODE>
-<DD>
-<A NAME="IDX312"></A>
-Returns true if <CODE>x</CODE> is <CODE>0 in R</CODE>.
-
-<DT><CODE>sintL degree (const cl_UP& x)</CODE>
-<DD>
-<A NAME="IDX313"></A>
-Returns the degree of the polynomial. The zero polynomial has degree <CODE>-1</CODE>.
-
-<DT><CODE>cl_ring_element coeff (const cl_UP& x, uintL index)</CODE>
-<DD>
-<A NAME="IDX314"></A>
-Returns the coefficient of <CODE>X^index</CODE> in the polynomial <CODE>x</CODE>.
-
-<DT><CODE>cl_ring_element x (const cl_ring_element& y)</CODE>
-<DD>
-<A NAME="IDX315"></A>
-Evaluation: If <CODE>x</CODE> is a polynomial and <CODE>y</CODE> belongs to the base ring,
-then <SAMP>`x(y)'</SAMP> returns the value of the substitution of <CODE>y</CODE> into
-<CODE>x</CODE>.
-
-<DT><CODE>cl_UP deriv (const cl_UP& x)</CODE>
-<DD>
-<A NAME="IDX316"></A>
-Returns the derivative of the polynomial <CODE>x</CODE> with respect to the
-indeterminate <CODE>X</CODE>.
-</DL>
-
-<P>
-The following output functions are defined (see also the chapter on
-input/output).
-
-
-<DL COMPACT>
-
-<DT><CODE>void fprint (cl_ostream stream, const cl_UP& x)</CODE>
-<DD>
-<A NAME="IDX317"></A>
-<DT><CODE>cl_ostream operator<< (cl_ostream stream, const cl_UP& x)</CODE>
-<DD>
-<A NAME="IDX318"></A>
-Prints the univariate polynomial <CODE>x</CODE> on the <CODE>stream</CODE>. The output may
-depend on the global printer settings in the variable
-<CODE>default_print_flags</CODE>.
-</DL>
-
-
-
-<H2><A NAME="SEC58" HREF="cln.html#TOC58">9.3 Special polynomials</A></H2>
-
-<P>
-The following functions return special polynomials.
-
-
-<DL COMPACT>
-
-<DT><CODE>cl_UP_I tschebychev (sintL n)</CODE>
-<DD>
-<A NAME="IDX319"></A>
-<A NAME="IDX320"></A>
-Returns the n-th Chebyshev polynomial (n >= 0).
-
-<DT><CODE>cl_UP_I hermite (sintL n)</CODE>
-<DD>
-<A NAME="IDX321"></A>
-<A NAME="IDX322"></A>
-Returns the n-th Hermite polynomial (n >= 0).
-
-<DT><CODE>cl_UP_RA legendre (sintL n)</CODE>
-<DD>
-<A NAME="IDX323"></A>
-<A NAME="IDX324"></A>
-Returns the n-th Legendre polynomial (n >= 0).
-
-<DT><CODE>cl_UP_I laguerre (sintL n)</CODE>
-<DD>
-<A NAME="IDX325"></A>
-<A NAME="IDX326"></A>
-Returns the n-th Laguerre polynomial (n >= 0).
-</DL>
-
-<P>
-Information how to derive the differential equation satisfied by each
-of these polynomials from their definition can be found in the
-<CODE>doc/polynomial/</CODE> directory.
-
-
-
-
-<H1><A NAME="SEC59" HREF="cln.html#TOC59">10. Internals</A></H1>
-
-
-
-<H2><A NAME="SEC60" HREF="cln.html#TOC60">10.1 Why C++ ?</A></H2>
-<P>
-<A NAME="IDX327"></A>
-
-
-<P>
-Using C++ as an implementation language provides
-
-
-
-<UL>
-<LI>
-
-Efficiency: It compiles to machine code.
-
-<LI>
-
-<A NAME="IDX328"></A>
-Portability: It runs on all platforms supporting a C++ compiler. Because
-of the availability of GNU C++, this includes all currently used 32-bit and
-64-bit platforms, independently of the quality of the vendor's C++ compiler.
-
-<LI>
-
-Type safety: The C++ compilers knows about the number types and complains if,
-for example, you try to assign a float to an integer variable. However,
-a drawback is that C++ doesn't know about generic types, hence a restriction
-like that <CODE>operator+ (const cl_MI&, const cl_MI&)</CODE> requires that both
-arguments belong to the same modular ring cannot be expressed as a compile-time
-information.
-
-<LI>
-
-Algebraic syntax: The elementary operations <CODE>+</CODE>, <CODE>-</CODE>, <CODE>*</CODE>,
-<CODE>=</CODE>, <CODE>==</CODE>, ... can be used in infix notation, which is more
-convenient than Lisp notation <SAMP>`(+ x y)'</SAMP> or C notation <SAMP>`add(x,y,&z)'</SAMP>.
-</UL>
-
-<P>
-With these language features, there is no need for two separate languages,
-one for the implementation of the library and one in which the library's users
-can program. This means that a prototype implementation of an algorithm
-can be integrated into the library immediately after it has been tested and
-debugged. No need to rewrite it in a low-level language after having prototyped
-in a high-level language.
-
-
-
-
-<H2><A NAME="SEC61" HREF="cln.html#TOC61">10.2 Memory efficiency</A></H2>
-
-<P>
-In order to save memory allocations, CLN implements:
-
-
-
-<UL>
-<LI>
-
-Object sharing: An operation like <CODE>x+0</CODE> returns <CODE>x</CODE> without copying
-it.
-<LI>
-
-<A NAME="IDX329"></A>
-<A NAME="IDX330"></A>
-Garbage collection: A reference counting mechanism makes sure that any
-number object's storage is freed immediately when the last reference to the
-object is gone.
-<LI>
-
-Small integers are represented as immediate values instead of pointers
-to heap allocated storage. This means that integers <CODE>> -2^29</CODE>,
-<CODE>< 2^29</CODE> don't consume heap memory, unless they were explicitly allocated
-on the heap.
-</UL>
-
-
-
-<H2><A NAME="SEC62" HREF="cln.html#TOC62">10.3 Speed efficiency</A></H2>
-
-<P>
-Speed efficiency is obtained by the combination of the following tricks
-and algorithms:
-
-
-
-<UL>
-<LI>
-
-Small integers, being represented as immediate values, don't require
-memory access, just a couple of instructions for each elementary operation.
-<LI>
-
-The kernel of CLN has been written in assembly language for some CPUs
-(<CODE>i386</CODE>, <CODE>m68k</CODE>, <CODE>sparc</CODE>, <CODE>mips</CODE>, <CODE>arm</CODE>).
-<LI>
-
-On all CPUs, CLN may be configured to use the superefficient low-level
-routines from GNU GMP version 3.
-<LI>
-
-For large numbers, CLN uses, instead of the standard <CODE>O(N^2)</CODE>
-algorithm, the Karatsuba multiplication, which is an
-<CODE>O(N^1.6)</CODE>
-algorithm.
-<LI>
-
-For very large numbers (more than 12000 decimal digits), CLN uses
-Schönhage-Strassen
-<A NAME="IDX331"></A>
-multiplication, which is an asymptotically optimal multiplication
-algorithm.
-<LI>
-
-These fast multiplication algorithms also give improvements in the speed
-of division and radix conversion.
-</UL>
-
-
-
-<H2><A NAME="SEC63" HREF="cln.html#TOC63">10.4 Garbage collection</A></H2>
-<P>
-<A NAME="IDX332"></A>
-
-
-<P>
-All the number classes are reference count classes: They only contain a pointer
-to an object in the heap. Upon construction, assignment and destruction of
-number objects, only the objects' reference count are manipulated.
-
-
-<P>
-Memory occupied by number objects are automatically reclaimed as soon as
-their reference count drops to zero.
-
-
-<P>
-For number rings, another strategy is implemented: There is a cache of,
-for example, the modular integer rings. A modular integer ring is destroyed
-only if its reference count dropped to zero and the cache is about to be
-resized. The effect of this strategy is that recently used rings remain
-cached, whereas undue memory consumption through cached rings is avoided.
-
-
-
-
-<H1><A NAME="SEC64" HREF="cln.html#TOC64">11. Using the library</A></H1>
-
-<P>
-For the following discussion, we will assume that you have installed
-the CLN source in <CODE>$CLN_DIR</CODE> and built it in <CODE>$CLN_TARGETDIR</CODE>.
-For example, for me it's <CODE>CLN_DIR="$HOME/cln"</CODE> and
-<CODE>CLN_TARGETDIR="$HOME/cln/linuxelf"</CODE>. You might define these as
-environment variables, or directly substitute the appropriate values.
-
-
-
-
-<H2><A NAME="SEC65" HREF="cln.html#TOC65">11.1 Compiler options</A></H2>
-<P>
-<A NAME="IDX333"></A>
-
-
-<P>
-Until you have installed CLN in a public place, the following options are
-needed:
-
-
-<P>
-When you compile CLN application code, add the flags
-
-<PRE>
- -I$CLN_DIR/include -I$CLN_TARGETDIR/include
-</PRE>
-
-<P>
-to the C++ compiler's command line (<CODE>make</CODE> variable CFLAGS or CXXFLAGS).
-When you link CLN application code to form an executable, add the flags
-
-<PRE>
- $CLN_TARGETDIR/src/libcln.a
-</PRE>
-
-<P>
-to the C/C++ compiler's command line (<CODE>make</CODE> variable LIBS).
-
-
-<P>
-If you did a <CODE>make install</CODE>, the include files are installed in a
-public directory (normally <CODE>/usr/local/include</CODE>), hence you don't
-need special flags for compiling. The library has been installed to a
-public directory as well (normally <CODE>/usr/local/lib</CODE>), hence when
-linking a CLN application it is sufficient to give the flag <CODE>-lcln</CODE>.
-
-
-
-
-<H2><A NAME="SEC66" HREF="cln.html#TOC66">11.2 Compatibility to old CLN versions</A></H2>
-<P>
-<A NAME="IDX334"></A>
-<A NAME="IDX335"></A>
-
-
-<P>
-As of CLN version 1.1 all non-macro identifiers were hidden in namespace
-<CODE>cln</CODE> in order to avoid potential name clashes with other C++
-libraries. If you have an old application, you will have to manually
-port it to the new scheme. The following principles will help during
-the transition:
-
-<UL>
-<LI>
-
-All headers are now in a separate subdirectory. Instead of including
-<CODE>cl_</CODE><VAR>something</VAR><CODE>.h</CODE>, include
-<CODE>cln/</CODE><VAR>something</VAR><CODE>.h</CODE> now.
-<LI>
-
-All public identifiers (typenames and functions) have lost their
-<CODE>cl_</CODE> prefix. Exceptions are all the typenames of number types,
-(cl_N, cl_I, cl_MI, ...), rings, symbolic types (cl_string,
-cl_symbol) and polynomials (cl_UP_<VAR>type</VAR>). (This is because their
-names would not be mnemonic enough once the namespace <CODE>cln</CODE> is
-imported. Even in a namespace we favor <CODE>cl_N</CODE> over <CODE>N</CODE>.)
-<LI>
-
-All public <EM>functions</EM> that had by a <CODE>cl_</CODE> in their name still
-carry that <CODE>cl_</CODE> if it is intrinsic part of a typename (as in
-<CODE>cl_I_to_int ()</CODE>).
-</UL>
-
-<P>
-When developing other libraries, please keep in mind not to import the
-namespace <CODE>cln</CODE> in one of your public header files by saying
-<CODE>using namespace cln;</CODE>. This would propagate to other applications
-and can cause name clashes there.
-
-
-
-
-<H2><A NAME="SEC67" HREF="cln.html#TOC67">11.3 Include files</A></H2>
-<P>
-<A NAME="IDX336"></A>
-<A NAME="IDX337"></A>
-
-
-<P>
-Here is a summary of the include files and their contents.
-
-
-<DL COMPACT>
-
-<DT><CODE><cln/object.h></CODE>
-<DD>
-General definitions, reference counting, garbage collection.
-<DT><CODE><cln/number.h></CODE>
-<DD>
-The class cl_number.
-<DT><CODE><cln/complex.h></CODE>
-<DD>
-Functions for class cl_N, the complex numbers.
-<DT><CODE><cln/real.h></CODE>
-<DD>
-Functions for class cl_R, the real numbers.
-<DT><CODE><cln/float.h></CODE>
-<DD>
-Functions for class cl_F, the floats.
-<DT><CODE><cln/sfloat.h></CODE>
-<DD>
-Functions for class cl_SF, the short-floats.
-<DT><CODE><cln/ffloat.h></CODE>
-<DD>
-Functions for class cl_FF, the single-floats.
-<DT><CODE><cln/dfloat.h></CODE>
-<DD>
-Functions for class cl_DF, the double-floats.
-<DT><CODE><cln/lfloat.h></CODE>
-<DD>
-Functions for class cl_LF, the long-floats.
-<DT><CODE><cln/rational.h></CODE>
-<DD>
-Functions for class cl_RA, the rational numbers.
-<DT><CODE><cln/integer.h></CODE>
-<DD>
-Functions for class cl_I, the integers.
-<DT><CODE><cln/io.h></CODE>
-<DD>
-Input/Output.
-<DT><CODE><cln/complex_io.h></CODE>
-<DD>
-Input/Output for class cl_N, the complex numbers.
-<DT><CODE><cln/real_io.h></CODE>
-<DD>
-Input/Output for class cl_R, the real numbers.
-<DT><CODE><cln/float_io.h></CODE>
-<DD>
-Input/Output for class cl_F, the floats.
-<DT><CODE><cln/sfloat_io.h></CODE>
-<DD>
-Input/Output for class cl_SF, the short-floats.
-<DT><CODE><cln/ffloat_io.h></CODE>
-<DD>
-Input/Output for class cl_FF, the single-floats.
-<DT><CODE><cln/dfloat_io.h></CODE>
-<DD>
-Input/Output for class cl_DF, the double-floats.
-<DT><CODE><cln/lfloat_io.h></CODE>
-<DD>
-Input/Output for class cl_LF, the long-floats.
-<DT><CODE><cln/rational_io.h></CODE>
-<DD>
-Input/Output for class cl_RA, the rational numbers.
-<DT><CODE><cln/integer_io.h></CODE>
-<DD>
-Input/Output for class cl_I, the integers.
-<DT><CODE><cln/input.h></CODE>
-<DD>
-Flags for customizing input operations.
-<DT><CODE><cln/output.h></CODE>
-<DD>
-Flags for customizing output operations.
-<DT><CODE><cln/malloc.h></CODE>
-<DD>
-<CODE>malloc_hook</CODE>, <CODE>free_hook</CODE>.
-<DT><CODE><cln/abort.h></CODE>
-<DD>
-<CODE>cl_abort</CODE>.
-<DT><CODE><cln/condition.h></CODE>
-<DD>
-Conditions/exceptions.
-<DT><CODE><cln/string.h></CODE>
-<DD>
-Strings.
-<DT><CODE><cln/symbol.h></CODE>
-<DD>
-Symbols.
-<DT><CODE><cln/proplist.h></CODE>
-<DD>
-Property lists.
-<DT><CODE><cln/ring.h></CODE>
-<DD>
-General rings.
-<DT><CODE><cln/null_ring.h></CODE>
-<DD>
-The null ring.
-<DT><CODE><cln/complex_ring.h></CODE>
-<DD>
-The ring of complex numbers.
-<DT><CODE><cln/real_ring.h></CODE>
-<DD>
-The ring of real numbers.
-<DT><CODE><cln/rational_ring.h></CODE>
-<DD>
-The ring of rational numbers.
-<DT><CODE><cln/integer_ring.h></CODE>
-<DD>
-The ring of integers.
-<DT><CODE><cln/numtheory.h></CODE>
-<DD>
-Number threory functions.
-<DT><CODE><cln/modinteger.h></CODE>
-<DD>
-Modular integers.
-<DT><CODE><cln/V.h></CODE>
-<DD>
-Vectors.
-<DT><CODE><cln/GV.h></CODE>
-<DD>
-General vectors.
-<DT><CODE><cln/GV_number.h></CODE>
-<DD>
-General vectors over cl_number.
-<DT><CODE><cln/GV_complex.h></CODE>
-<DD>
-General vectors over cl_N.
-<DT><CODE><cln/GV_real.h></CODE>
-<DD>
-General vectors over cl_R.
-<DT><CODE><cln/GV_rational.h></CODE>
-<DD>
-General vectors over cl_RA.
-<DT><CODE><cln/GV_integer.h></CODE>
-<DD>
-General vectors over cl_I.
-<DT><CODE><cln/GV_modinteger.h></CODE>
-<DD>
-General vectors of modular integers.
-<DT><CODE><cln/SV.h></CODE>
-<DD>
-Simple vectors.
-<DT><CODE><cln/SV_number.h></CODE>
-<DD>
-Simple vectors over cl_number.
-<DT><CODE><cln/SV_complex.h></CODE>
-<DD>
-Simple vectors over cl_N.
-<DT><CODE><cln/SV_real.h></CODE>
-<DD>
-Simple vectors over cl_R.
-<DT><CODE><cln/SV_rational.h></CODE>
-<DD>
-Simple vectors over cl_RA.
-<DT><CODE><cln/SV_integer.h></CODE>
-<DD>
-Simple vectors over cl_I.
-<DT><CODE><cln/SV_ringelt.h></CODE>
-<DD>
-Simple vectors of general ring elements.
-<DT><CODE><cln/univpoly.h></CODE>
-<DD>
-Univariate polynomials.
-<DT><CODE><cln/univpoly_integer.h></CODE>
-<DD>
-Univariate polynomials over the integers.
-<DT><CODE><cln/univpoly_rational.h></CODE>
-<DD>
-Univariate polynomials over the rational numbers.
-<DT><CODE><cln/univpoly_real.h></CODE>
-<DD>
-Univariate polynomials over the real numbers.
-<DT><CODE><cln/univpoly_complex.h></CODE>
-<DD>
-Univariate polynomials over the complex numbers.
-<DT><CODE><cln/univpoly_modint.h></CODE>
-<DD>
-Univariate polynomials over modular integer rings.
-<DT><CODE><cln/timing.h></CODE>
-<DD>
-Timing facilities.
-<DT><CODE><cln/cln.h></CODE>
-<DD>
-Includes all of the above.
-</DL>
-
-
-
-<H2><A NAME="SEC68" HREF="cln.html#TOC68">11.4 An Example</A></H2>
-
-<P>
-A function which computes the nth Fibonacci number can be written as follows.
-<A NAME="IDX338"></A>
-
-
-
-<PRE>
-#include <cln/integer.h>
-#include <cln/real.h>
-using namespace cln;
-
-// Returns F_n, computed as the nearest integer to
-// ((1+sqrt(5))/2)^n/sqrt(5). Assume n>=0.
-const cl_I fibonacci (int n)
-{
- // Need a precision of ((1+sqrt(5))/2)^-n.
- cl_float_format_t prec = cl_float_format((int)(0.208987641*n+5));
- cl_R sqrt5 = sqrt(cl_float(5,prec));
- cl_R phi = (1+sqrt5)/2;
- return round1( expt(phi,n)/sqrt5 );
-}
-</PRE>
-
-<P>
-Let's explain what is going on in detail.
-
-
-<P>
-The include file <CODE><cln/integer.h></CODE> is necessary because the type
-<CODE>cl_I</CODE> is used in the function, and the include file <CODE><cln/real.h></CODE>
-is needed for the type <CODE>cl_R</CODE> and the floating point number functions.
-The order of the include files does not matter. In order not to write out
-<CODE>cln::</CODE><VAR>foo</VAR> we can safely import the whole namespace <CODE>cln</CODE>.
-
-
-<P>
-Then comes the function declaration. The argument is an <CODE>int</CODE>, the
-result an integer. The return type is defined as <SAMP>`const cl_I'</SAMP>, not
-simply <SAMP>`cl_I'</SAMP>, because that allows the compiler to detect typos like
-<SAMP>`fibonacci(n) = 100'</SAMP>. It would be possible to declare the return
-type as <CODE>const cl_R</CODE> (real number) or even <CODE>const cl_N</CODE> (complex
-number). We use the most specialized possible return type because functions
-which call <SAMP>`fibonacci'</SAMP> will be able to profit from the compiler's type
-analysis: Adding two integers is slightly more efficient than adding the
-same objects declared as complex numbers, because it needs less type
-dispatch. Also, when linking to CLN as a non-shared library, this minimizes
-the size of the resulting executable program.
-
-
-<P>
-The result will be computed as expt(phi,n)/sqrt(5), rounded to the nearest
-integer. In order to get a correct result, the absolute error should be less
-than 1/2, i.e. the relative error should be less than sqrt(5)/(2*expt(phi,n)).
-To this end, the first line computes a floating point precision for sqrt(5)
-and phi.
-
-
-<P>
-Then sqrt(5) is computed by first converting the integer 5 to a floating point
-number and than taking the square root. The converse, first taking the square
-root of 5, and then converting to the desired precision, would not work in
-CLN: The square root would be computed to a default precision (normally
-single-float precision), and the following conversion could not help about
-the lacking accuracy. This is because CLN is not a symbolic computer algebra
-system and does not represent sqrt(5) in a non-numeric way.
-
-
-<P>
-The type <CODE>cl_R</CODE> for sqrt5 and, in the following line, phi is the only
-possible choice. You cannot write <CODE>cl_F</CODE> because the C++ compiler can
-only infer that <CODE>cl_float(5,prec)</CODE> is a real number. You cannot write
-<CODE>cl_N</CODE> because a <SAMP>`round1'</SAMP> does not exist for general complex
-numbers.
-
-
-<P>
-When the function returns, all the local variables in the function are
-automatically reclaimed (garbage collected). Only the result survives and
-gets passed to the caller.
-
-
-<P>
-The file <CODE>fibonacci.cc</CODE> in the subdirectory <CODE>examples</CODE>
-contains this implementation together with an even faster algorithm.
-
-
-
-
-<H2><A NAME="SEC69" HREF="cln.html#TOC69">11.5 Debugging support</A></H2>
-<P>
-<A NAME="IDX339"></A>
-
-
-<P>
-When debugging a CLN application with GNU <CODE>gdb</CODE>, two facilities are
-available from the library:
-
-
-
-<UL>
-<LI>The library does type checks, range checks, consistency checks at
-
-many places. When one of these fails, the function <CODE>cl_abort()</CODE> is
-called. Its default implementation is to perform an <CODE>exit(1)</CODE>, so
-you won't have a core dump. But for debugging, it is best to set a
-breakpoint at this function:
-
-<PRE>
-(gdb) break cl_abort
-</PRE>
-
-When this breakpoint is hit, look at the stack's backtrace:
-
-<PRE>
-(gdb) where
-</PRE>
-
-<LI>The debugger's normal <CODE>print</CODE> command doesn't know about
-
-CLN's types and therefore prints mostly useless hexadecimal addresses.
-CLN offers a function <CODE>cl_print</CODE>, callable from the debugger,
-for printing number objects. In order to get this function, you have
-to define the macro <SAMP>`CL_DEBUG'</SAMP> and then include all the header files
-for which you want <CODE>cl_print</CODE> debugging support. For example:
-<A NAME="IDX340"></A>
-
-<PRE>
-#define CL_DEBUG
-#include <cln/string.h>
-</PRE>
-
-Now, if you have in your program a variable <CODE>cl_string s</CODE>, and
-inspect it under <CODE>gdb</CODE>, the output may look like this:
-
-<PRE>
-(gdb) print s
-$7 = {<cl_gcpointer> = { = {pointer = 0x8055b60, heappointer = 0x8055b60,
- word = 134568800}}, }
-(gdb) call cl_print(s)
-(cl_string) ""
-$8 = 134568800
-</PRE>
-
-Note that the output of <CODE>cl_print</CODE> goes to the program's error output,
-not to gdb's standard output.
-
-Note, however, that the above facility does not work with all CLN types,
-only with number objects and similar. Therefore CLN offers a member function
-<CODE>debug_print()</CODE> on all CLN types. The same macro <SAMP>`CL_DEBUG'</SAMP>
-is needed for this member function to be implemented. Under <CODE>gdb</CODE>,
-you call it like this:
-<A NAME="IDX341"></A>
-
-<PRE>
-(gdb) print s
-$7 = {<cl_gcpointer> = { = {pointer = 0x8055b60, heappointer = 0x8055b60,
- word = 134568800}}, }
-(gdb) call s.debug_print()
-(cl_string) ""
-(gdb) define cprint
->call ($1).debug_print()
->end
-(gdb) cprint s
-(cl_string) ""
-</PRE>
-
-Unfortunately, this feature does not seem to work under all circumstances.
-</UL>
-
-
-
-<H1><A NAME="SEC70" HREF="cln.html#TOC70">12. Customizing</A></H1>
-<P>
-<A NAME="IDX342"></A>
-
-
-
-
-<H2><A NAME="SEC71" HREF="cln.html#TOC71">12.1 Error handling</A></H2>
-
-<P>
-When a fatal error occurs, an error message is output to the standard error
-output stream, and the function <CODE>cl_abort</CODE> is called. The default
-version of this function (provided in the library) terminates the application.
-To catch such a fatal error, you need to define the function <CODE>cl_abort</CODE>
-yourself, with the prototype
-
-<PRE>
-#include <cln/abort.h>
-void cl_abort (void);
-</PRE>
-
-<P>
-<A NAME="IDX343"></A>
-This function must not return control to its caller.
-
-
-
-
-<H2><A NAME="SEC72" HREF="cln.html#TOC72">12.2 Floating-point underflow</A></H2>
-<P>
-<A NAME="IDX344"></A>
-
-
-<P>
-Floating point underflow denotes the situation when a floating-point number
-is to be created which is so close to <CODE>0</CODE> that its exponent is too
-low to be represented internally. By default, this causes a fatal error.
-If you set the global variable
-
-<PRE>
-cl_boolean cl_inhibit_floating_point_underflow
-</PRE>
-
-<P>
-to <CODE>cl_true</CODE>, the error will be inhibited, and a floating-point zero
-will be generated instead. The default value of
-<CODE>cl_inhibit_floating_point_underflow</CODE> is <CODE>cl_false</CODE>.
-
-
-
-
-<H2><A NAME="SEC73" HREF="cln.html#TOC73">12.3 Customizing I/O</A></H2>
-
-<P>
-The output of the function <CODE>fprint</CODE> may be customized by changing the
-value of the global variable <CODE>default_print_flags</CODE>.
-<A NAME="IDX345"></A>
-
-
-
-
-<H2><A NAME="SEC74" HREF="cln.html#TOC74">12.4 Customizing the memory allocator</A></H2>
-
-<P>
-Every memory allocation of CLN is done through the function pointer
-<CODE>malloc_hook</CODE>. Freeing of this memory is done through the function
-pointer <CODE>free_hook</CODE>. The default versions of these functions,
-provided in the library, call <CODE>malloc</CODE> and <CODE>free</CODE> and check
-the <CODE>malloc</CODE> result against <CODE>NULL</CODE>.
-If you want to provide another memory allocator, you need to define
-the variables <CODE>malloc_hook</CODE> and <CODE>free_hook</CODE> yourself,
-like this:
-
-<PRE>
-#include <cln/malloc.h>
-namespace cln {
- void* (*malloc_hook) (size_t size) = ...;
- void (*free_hook) (void* ptr) = ...;
-}
-</PRE>
-
-<P>
-<A NAME="IDX346"></A>
-<A NAME="IDX347"></A>
-The <CODE>cl_malloc_hook</CODE> function must not return a <CODE>NULL</CODE> pointer.
-
-
-<P>
-It is not possible to change the memory allocator at runtime, because
-it is already called at program startup by the constructors of some
-global variables.
-
-
-
-
-<H1><A NAME="SEC75" HREF="cln.html#TOC75">Index</A></H1>
-
-<P>
-Jump to:
-<P>
-
-
-<P><HR><P>
-This document was generated on 28 August 2000 using
-<A HREF="http://wwwinfo.cern.ch/dis/texi2html/">texi2html</A> 1.56k.
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-b Fp(operator)28 b(-)i(\(const)e Fl(t)m(yp)s(e)5 b Fp(&,)31
-b(const)d Fl(t)m(yp)s(e)5 b Fp(&\))450 1139 y Fr(Subtraction.)-30
-1298 y Fl(t)m(yp)s(e)36 b Fp(operator)28 b(-)i(\(const)e
-Fl(t)m(yp)s(e)5 b Fp(&\))450 1407 y Fr(Returns)29 b(the)i(negativ)m(e)i
-(of)d(the)h(argumen)m(t.)-30 1567 y Fl(t)m(yp)s(e)36
-b Fp(plus1)29 b(\(const)f Fl(t)m(yp)s(e)5 b Fp(&)31 b(x\))450
-1676 y Fr(Returns)e Fp(x)h(+)h(1)p Fr(.)-30 1835 y Fl(t)m(yp)s(e)36
-b Fp(minus1)28 b(\(const)h Fl(t)m(yp)s(e)5 b Fp(&)30
-b(x\))450 1945 y Fr(Returns)f Fp(x)h(-)h(1)p Fr(.)-30
-2104 y Fl(t)m(yp)s(e)36 b Fp(operator)28 b(*)i(\(const)e
-Fl(t)m(yp)s(e)5 b Fp(&,)31 b(const)d Fl(t)m(yp)s(e)5
-b Fp(&\))450 2214 y Fr(Multiplication.)-30 2373 y Fl(t)m(yp)s(e)36
-b Fp(square)28 b(\(const)h Fl(t)m(yp)s(e)5 b Fp(&)30
-b(x\))450 2483 y Fr(Returns)f Fp(x)h(*)h(x)p Fr(.)-30
-2642 y(Eac)m(h)43 b(of)g(the)f(classes)i Fp(cl_N)p Fr(,)h
-Fp(cl_R)p Fr(,)f Fp(cl_RA)p Fr(,)h Fp(cl_F)p Fr(,)f Fp(cl_SF)p
-Fr(,)g Fp(cl_FF)p Fr(,)h Fp(cl_DF)p Fr(,)f Fp(cl_LF)d
-Fr(de\014nes)h(the)g(follo)m(wing)-30 2751 y(op)s(erations:)-30
-2911 y Fl(t)m(yp)s(e)36 b Fp(operator)28 b(/)i(\(const)e
-Fl(t)m(yp)s(e)5 b Fp(&,)31 b(const)d Fl(t)m(yp)s(e)5
-b Fp(&\))450 3020 y Fr(Division.)-30 3179 y Fl(t)m(yp)s(e)36
-b Fp(recip)29 b(\(const)f Fl(t)m(yp)s(e)5 b Fp(&\))450
-3289 y Fr(Returns)29 b(the)i(recipro)s(cal)g(of)g(the)f(argumen)m(t.)
--30 3448 y(The)h(class)h Fp(cl_I)d Fr(do)s(esn't)i(de\014ne)g(a)g(`)p
-Fp(/)p Fr(')h(op)s(eration)f(b)s(ecause)g(in)g(the)g(C/C)p
-Fp(++)g Fr(language)h(this)f(op)s(erator,)h(applied)-30
-3558 y(to)j(in)m(tegral)h(t)m(yp)s(es,)g(denotes)f(the)g(`)p
-Fp(floor)p Fr(')e(or)h(`)p Fp(truncate)p Fr(')f(op)s(eration)i(\(whic)m
-(h)g(one)f(of)h(these,)h(is)f(implemen)m(ta-)-30 3667
-y(tion)g(dep)s(enden)m(t\).)51 b(\(See)35 b(Section)g(4.6)g([Rounding)f
-(functions],)h(page)g(13.\))53 b(Instead,)36 b Fp(cl_I)d
-Fr(de\014nes)g(an)h(\\exact)-30 3777 y(quotien)m(t")e(function:)-30
-3936 y Fp(cl_I)d(exquo)g(\(const)g(cl_I&)g(x,)h(const)e(cl_I&)h(y\))450
-4046 y Fr(Chec)m(ks)i(that)g Fp(y)f Fr(divides)g Fp(x)p
-Fr(,)g(and)g(returns)f(the)h(quotien)m(t)i Fp(x)p Fr(/)p
-Fp(y)p Fr(.)-30 4205 y(The)e(follo)m(wing)i(exp)s(onen)m(tiation)f
-(functions)f(are)h(de\014ned:)-30 4364 y Fp(cl_I)e(expt_pos)f(\(const)h
-(cl_I&)g(x,)h(const)f(cl_I&)g(y\))-30 4474 y(cl_RA)g(expt_pos)f
-(\(const)h(cl_RA&)f(x,)i(const)f(cl_I&)g(y\))450 4583
-y(y)h Fr(m)m(ust)f(b)s(e)h Fp(>)g Fr(0.)41 b(Returns)30
-b Fp(x^y)p Fr(.)-30 4742 y Fp(cl_RA)f(expt)g(\(const)g(cl_RA&)f(x,)i
-(const)f(cl_I&)g(y\))-30 4852 y(cl_R)g(expt)g(\(const)g(cl_R&)g(x,)h
-(const)f(cl_I&)g(y\))-30 4962 y(cl_N)g(expt)g(\(const)g(cl_N&)g(x,)h
-(const)f(cl_I&)g(y\))450 5071 y Fr(Returns)g Fp(x^y)p
-Fr(.)-30 5230 y(Eac)m(h)43 b(of)g(the)f(classes)i Fp(cl_R)p
-Fr(,)h Fp(cl_RA)p Fr(,)f Fp(cl_I)p Fr(,)h Fp(cl_F)p Fr(,)f
-Fp(cl_SF)p Fr(,)g Fp(cl_FF)p Fr(,)h Fp(cl_DF)p Fr(,)f
-Fp(cl_LF)d Fr(de\014nes)h(the)g(follo)m(wing)-30 5340
-y(op)s(eration:)p eop
-%%Page: 12 14
-12 13 bop -30 -116 a Fr(Chapter)30 b(4:)41 b(F)-8 b(unctions)31
-b(on)f(n)m(um)m(b)s(ers)2523 b(12)-30 299 y Fl(t)m(yp)s(e)36
-b Fp(abs)29 b(\(const)g Fl(t)m(yp)s(e)5 b Fp(&)30 b(x\))450
-408 y Fr(Returns)f(the)i(absolute)g(v)-5 b(alue)31 b(of)f
-Fp(x)p Fr(.)41 b(This)29 b(is)i Fp(x)f Fr(if)g Fp(x)g(>=)g(0)p
-Fr(,)g(and)g Fp(-x)g Fr(if)g Fp(x)g(<=)g(0)p Fr(.)-30
-588 y(The)g(class)h Fp(cl_N)e Fr(implemen)m(ts)g(this)h(as)h(follo)m
-(ws:)-30 767 y Fp(cl_R)e(abs)h(\(const)e(cl_N)i(x\))450
-877 y Fr(Returns)f(the)i(absolute)g(v)-5 b(alue)31 b(of)f
-Fp(x)p Fr(.)-30 1057 y(Eac)m(h)d(of)f(the)h(classes)g
-Fp(cl_N)p Fr(,)f Fp(cl_R)p Fr(,)g Fp(cl_RA)p Fr(,)g Fp(cl_I)p
-Fr(,)g Fp(cl_F)p Fr(,)g Fp(cl_SF)p Fr(,)g Fp(cl_FF)p
-Fr(,)g Fp(cl_DF)p Fr(,)g Fp(cl_LF)f Fr(de\014nes)g(the)h(follo)m(wing)
--30 1166 y(op)s(eration:)-30 1346 y Fl(t)m(yp)s(e)36
-b Fp(signum)28 b(\(const)h Fl(t)m(yp)s(e)5 b Fp(&)30
-b(x\))450 1455 y Fr(Returns)f(the)h(sign)g(of)g Fp(x)p
-Fr(,)g(in)f(the)i(same)e(n)m(um)m(b)s(er)e(format)j(as)g
-Fp(x)p Fr(.)40 b(This)29 b(is)h(de\014ned)f(as)h Fp(x)g(/)g(abs\(x\))e
-Fr(if)450 1565 y Fp(x)i Fr(is)g(non-zero,)i(and)d Fp(x)h
-Fr(if)h Fp(x)f Fr(is)g(zero.)42 b(If)30 b Fp(x)g Fr(is)g(real,)h(the)g
-(v)-5 b(alue)31 b(is)f(either)h(0)g(or)f(1)h(or)f(-1.)-30
-1856 y Fs(4.3)68 b(Elemen)l(tary)47 b(rational)f(functions)-30
-2062 y Fr(Eac)m(h)31 b(of)g(the)f(classes)i Fp(cl_RA)p
-Fr(,)d Fp(cl_I)g Fr(de\014nes)g(the)i(follo)m(wing)h(op)s(erations:)-30
-2241 y Fp(cl_I)d(numerator)f(\(const)h Fl(t)m(yp)s(e)5
-b Fp(&)30 b(x\))450 2351 y Fr(Returns)f(the)i(n)m(umerator)e(of)i
-Fp(x)p Fr(.)-30 2524 y Fp(cl_I)e(denominator)f(\(const)g
-Fl(t)m(yp)s(e)5 b Fp(&)31 b(x\))450 2633 y Fr(Returns)e(the)i
-(denominator)f(of)g Fp(x)p Fr(.)-30 2813 y(The)g(n)m(umerator)g(and)g
-(denominator)g(of)h(a)g(rational)h(n)m(um)m(b)s(er)c(are)k(normalized)e
-(in)g(suc)m(h)h(a)g(w)m(a)m(y)g(that)h(they)f(ha)m(v)m(e)-30
-2922 y(no)f(factor)i(in)e(common)e(and)i(the)h(denominator)e(is)i(p)s
-(ositiv)m(e.)-30 3213 y Fs(4.4)68 b(Elemen)l(tary)47
-b(complex)e(functions)-30 3419 y Fr(The)30 b(class)h
-Fp(cl_N)e Fr(de\014nes)h(the)g(follo)m(wing)i(op)s(eration:)-30
-3599 y Fp(cl_N)d(complex)g(\(const)f(cl_R&)h(a,)h(const)f(cl_R&)g(b\))
-450 3708 y Fr(Returns)35 b(the)h(complex)g(n)m(um)m(b)s(er)e
-Fp(a+bi)p Fr(,)i(that)h(is,)g(the)f(complex)g(n)m(um)m(b)s(er)e(with)h
-(real)i(part)f Fp(a)g Fr(and)450 3818 y(imaginary)30
-b(part)g Fp(b)p Fr(.)-30 3997 y(Eac)m(h)h(of)g(the)f(classes)i
-Fp(cl_N)p Fr(,)d Fp(cl_R)g Fr(de\014nes)h(the)g(follo)m(wing)i(op)s
-(erations:)-30 4177 y Fp(cl_R)d(realpart)f(\(const)h
-Fl(t)m(yp)s(e)5 b Fp(&)30 b(x\))450 4286 y Fr(Returns)f(the)i(real)g
-(part)f(of)h Fp(x)p Fr(.)-30 4459 y Fp(cl_R)e(imagpart)f(\(const)h
-Fl(t)m(yp)s(e)5 b Fp(&)30 b(x\))450 4569 y Fr(Returns)f(the)i
-(imaginary)f(part)g(of)h Fp(x)p Fr(.)-30 4742 y Fl(t)m(yp)s(e)36
-b Fp(conjugate)27 b(\(const)i Fl(t)m(yp)s(e)5 b Fp(&)31
-b(x\))450 4851 y Fr(Returns)e(the)i(complex)f(conjugate)i(of)e
-Fp(x)p Fr(.)-30 5031 y(W)-8 b(e)32 b(ha)m(v)m(e)f(the)g(relations)150
-5179 y Fp(x)f(=)g(complex\(realpart\(x\),)25 b(imagpart\(x\)\))150
-5320 y(conjugate\(x\))i(=)j(complex\(realpart\(x\),)25
-b(-imagpart\(x\)\))p eop
-%%Page: 13 15
-13 14 bop -30 -116 a Fr(Chapter)30 b(4:)41 b(F)-8 b(unctions)31
-b(on)f(n)m(um)m(b)s(ers)2523 b(13)-30 299 y Fs(4.5)68
-b(Comparisons)-30 489 y Fr(Eac)m(h)27 b(of)f(the)h(classes)g
-Fp(cl_N)p Fr(,)f Fp(cl_R)p Fr(,)g Fp(cl_RA)p Fr(,)g Fp(cl_I)p
-Fr(,)g Fp(cl_F)p Fr(,)g Fp(cl_SF)p Fr(,)g Fp(cl_FF)p
-Fr(,)g Fp(cl_DF)p Fr(,)g Fp(cl_LF)f Fr(de\014nes)g(the)h(follo)m(wing)
--30 598 y(op)s(erations:)-30 752 y Fp(bool)j(operator)f(==)i(\(const)f
-Fl(t)m(yp)s(e)5 b Fp(&,)30 b(const)f Fl(t)m(yp)s(e)5
-b Fp(&\))-30 862 y(bool)29 b(operator)f(!=)i(\(const)f
-Fl(t)m(yp)s(e)5 b Fp(&,)30 b(const)f Fl(t)m(yp)s(e)5
-b Fp(&\))450 971 y Fr(Comparison,)29 b(as)i(in)f(C)g(and)f(C)p
-Fp(++)p Fr(.)-30 1125 y Fp(uint32)g(equal_hashcode)d(\(const)j
-Fl(t)m(yp)s(e)5 b Fp(&\))450 1235 y Fr(Returns)35 b(a)h(32-bit)h(hash)e
-(co)s(de)h(that)g(is)g(the)g(same)f(for)h(an)m(y)g(t)m(w)m(o)h(n)m(um)m
-(b)s(ers)c(whic)m(h)i(are)i(the)f(same)450 1345 y(according)24
-b(to)h Fp(==)p Fr(.)38 b(This)22 b(hash)h(co)s(de)h(dep)s(ends)e(on)h
-(the)h(n)m(um)m(b)s(er's)d(v)-5 b(alue,)26 b(not)d(its)h(t)m(yp)s(e)g
-(or)g(precision.)-30 1499 y Fp(cl_boolean)k(zerop)h(\(const)f
-Fl(t)m(yp)s(e)5 b Fp(&)31 b(x\))450 1608 y Fr(Compare)e(against)j
-(zero:)41 b Fp(x)30 b(==)g(0)-30 1762 y Fr(Eac)m(h)43
-b(of)g(the)f(classes)i Fp(cl_R)p Fr(,)h Fp(cl_RA)p Fr(,)f
-Fp(cl_I)p Fr(,)h Fp(cl_F)p Fr(,)f Fp(cl_SF)p Fr(,)g Fp(cl_FF)p
-Fr(,)h Fp(cl_DF)p Fr(,)f Fp(cl_LF)d Fr(de\014nes)h(the)g(follo)m(wing)
--30 1872 y(op)s(erations:)-30 2026 y Fp(cl_signean)28
-b(compare)g(\(const)h Fl(t)m(yp)s(e)5 b Fp(&)30 b(x,)g(const)f
-Fl(t)m(yp)s(e)5 b Fp(&)30 b(y\))450 2135 y Fr(Compares)f
-Fp(x)h Fr(and)g Fp(y)p Fr(.)40 b(Returns)30 b Fp(+)p
-Fr(1)g(if)g Fp(x>y)p Fr(,)g(-1)h(if)g Fp(x<y)p Fr(,)e(0)i(if)f
-Fp(x)p Fr(=)p Fp(y)p Fr(.)-30 2289 y Fp(bool)f(operator)f(<=)i(\(const)
-f Fl(t)m(yp)s(e)5 b Fp(&,)30 b(const)f Fl(t)m(yp)s(e)5
-b Fp(&\))-30 2399 y(bool)29 b(operator)f(<)i(\(const)f
-Fl(t)m(yp)s(e)5 b Fp(&,)30 b(const)f Fl(t)m(yp)s(e)5
-b Fp(&\))-30 2508 y(bool)29 b(operator)f(>=)i(\(const)f
-Fl(t)m(yp)s(e)5 b Fp(&,)30 b(const)f Fl(t)m(yp)s(e)5
-b Fp(&\))-30 2618 y(bool)29 b(operator)f(>)i(\(const)f
-Fl(t)m(yp)s(e)5 b Fp(&,)30 b(const)f Fl(t)m(yp)s(e)5
-b Fp(&\))450 2727 y Fr(Comparison,)29 b(as)i(in)f(C)g(and)f(C)p
-Fp(++)p Fr(.)-30 2881 y Fp(cl_boolean)f(minusp)g(\(const)h
-Fl(t)m(yp)s(e)5 b Fp(&)30 b(x\))450 2991 y Fr(Compare)f(against)j
-(zero:)41 b Fp(x)30 b(<)g(0)-30 3145 y(cl_boolean)e(plusp)h(\(const)f
-Fl(t)m(yp)s(e)5 b Fp(&)31 b(x\))450 3255 y Fr(Compare)e(against)j
-(zero:)41 b Fp(x)30 b(>)g(0)-30 3408 y Fl(t)m(yp)s(e)36
-b Fp(max)29 b(\(const)g Fl(t)m(yp)s(e)5 b Fp(&)30 b(x,)g(const)f
-Fl(t)m(yp)s(e)5 b Fp(&)30 b(y\))450 3518 y Fr(Return)g(the)g(maxim)m
-(um)d(of)k Fp(x)f Fr(and)g Fp(y)p Fr(.)-30 3672 y Fl(t)m(yp)s(e)36
-b Fp(min)29 b(\(const)g Fl(t)m(yp)s(e)5 b Fp(&)30 b(x,)g(const)f
-Fl(t)m(yp)s(e)5 b Fp(&)30 b(y\))450 3782 y Fr(Return)g(the)g(minim)m
-(um)d(of)j Fp(x)g Fr(and)g Fp(y)p Fr(.)-30 3936 y(When)35
-b(a)g(\015oating)h(p)s(oin)m(t)f(n)m(um)m(b)s(er)d(and)i(a)i(rational)g
-(n)m(um)m(b)s(er)c(are)k(compared,)f(the)g(\015oat)h(is)f(\014rst)f
-(con)m(v)m(erted)i(to)-30 4045 y(a)f(rational)h(n)m(um)m(b)s(er)c
-(using)i(the)h(function)f Fp(rational)p Fr(.)51 b(Since)35
-b(a)g(\015oating)g(p)s(oin)m(t)g(n)m(um)m(b)s(er)d(actually)k(represen)
-m(ts)-30 4155 y(an)h(in)m(terv)-5 b(al)39 b(of)e(real)h(n)m(um)m(b)s
-(ers,)f(the)h(result)f(migh)m(t)g(b)s(e)g(surprising.)60
-b(F)-8 b(or)38 b(example,)h Fp(\(cl_F\)\(cl_R\)"1/3")26
-b(==)-30 4264 y(\(cl_R\)"1/3")h Fr(returns)i(false)i(b)s(ecause)g
-(there)f(is)h(no)f(\015oating)h(p)s(oin)m(t)g(n)m(um)m(b)s(er)d(whose)i
-(v)-5 b(alue)31 b(is)f(exactly)i Fp(1/3)p Fr(.)-30 4513
-y Fs(4.6)68 b(Rounding)45 b(functions)-30 4703 y Fr(When)40
-b(a)g(real)h(n)m(um)m(b)s(er)d(is)i(to)h(b)s(e)f(con)m(v)m(erted)h(to)g
-(an)f(in)m(teger,)45 b(there)40 b(is)g(no)g(\\b)s(est")h(rounding.)69
-b(The)39 b(desired)-30 4813 y(rounding)27 b(function)g(dep)s(ends)f(on)
-i(the)g(application.)41 b(The)28 b(Common)d(Lisp)i(and)g(ISO)g(Lisp)g
-(standards)g(o\013er)i(four)-30 4923 y(rounding)g(functions:)-30
-5076 y Fp(floor\(x\))96 b Fr(This)30 b(is)g(the)h(largest)g(in)m(teger)
-h Fp(<)p Fr(=)p Fp(x)p Fr(.)-30 5230 y Fp(ceiling\(x\))450
-5340 y Fr(This)e(is)g(the)h(smallest)f(in)m(teger)i Fp(>)p
-Fr(=)p Fp(x)p Fr(.)p eop
-%%Page: 14 16
-14 15 bop -30 -116 a Fr(Chapter)30 b(4:)41 b(F)-8 b(unctions)31
-b(on)f(n)m(um)m(b)s(ers)2523 b(14)-30 299 y Fp(truncate\(x\))450
-408 y Fr(Among)30 b(the)g(in)m(tegers)i(b)s(et)m(w)m(een)f(0)g(and)e
-Fp(x)h Fr(\(inclusiv)m(e\))i(the)f(one)g(nearest)f(to)i
-Fp(x)p Fr(.)-30 566 y Fp(round\(x\))96 b Fr(The)32 b(in)m(teger)i
-(nearest)g(to)f Fp(x)p Fr(.)48 b(If)32 b Fp(x)g Fr(is)h(exactly)i
-(halfw)m(a)m(y)e(b)s(et)m(w)m(een)h(t)m(w)m(o)g(in)m(tegers,)h(c)m(ho)s
-(ose)e(the)g(ev)m(en)450 676 y(one.)-30 833 y(These)d(functions)g(ha)m
-(v)m(e)i(di\013eren)m(t)e(adv)-5 b(an)m(tages:)-30 967
-y Fp(floor)44 b Fr(and)i Fp(ceiling)e Fr(are)i(translation)h(in)m(v)-5
-b(arian)m(t:)73 b Fp(floor\(x+n\))27 b(=)j(floor\(x\))e(+)i(n)46
-b Fr(and)f Fp(ceiling\(x+n\))27 b(=)-30 1076 y(ceiling\(x\))h(+)i(n)g
-Fr(for)g(ev)m(ery)h Fp(x)f Fr(and)g(ev)m(ery)h(in)m(teger)h
-Fp(n)p Fr(.)-30 1210 y(On)51 b(the)h(other)h(hand,)j
-Fp(truncate)50 b Fr(and)h Fp(round)g Fr(are)h(symmetric:)83
-b Fp(truncate\(-x\))27 b(=)j(-truncate\(x\))49 b Fr(and)-30
-1319 y Fp(round\(-x\))28 b(=)i(-round\(x\))p Fr(,)44
-b(and)f(furthermore)e Fp(round)g Fr(is)j(un)m(biased:)65
-b(on)44 b(the)f(\\a)m(v)m(erage",)50 b(it)44 b(rounds)d(do)m(wn)-30
-1429 y(exactly)32 b(as)f(often)g(as)f(it)h(rounds)e(up.)-30
-1563 y(The)h(functions)g(are)h(related)g(lik)m(e)h(this:)150
-1696 y Fp(ceiling\(m/n\))27 b(=)j(floor\(\(m+n-1\)/n\))c(=)k
-(floor\(\(m-1\)/n\)+1)f Fr(for)j(rational)j(n)m(um)m(b)s(ers)30
-b Fp(m/n)i Fr(\()p Fp(m)p Fr(,)i Fp(n)f Fr(in)m(te-)150
-1806 y(gers,)e Fp(n>)p Fr(0\),)g(and)150 1939 y Fp(truncate\(x\))c(=)j
-(sign\(x\))f(*)h(floor\(abs\(x\)\))-30 2097 y Fr(Eac)m(h)f(of)f(the)g
-(classes)h Fp(cl_R)p Fr(,)e Fp(cl_RA)p Fr(,)h Fp(cl_F)p
-Fr(,)f Fp(cl_SF)p Fr(,)h Fp(cl_FF)p Fr(,)f Fp(cl_DF)p
-Fr(,)g Fp(cl_LF)g Fr(de\014nes)g(the)h(follo)m(wing)h(op)s(erations:)
--30 2254 y Fp(cl_I)g(floor1)g(\(const)g Fl(t)m(yp)s(e)5
-b Fp(&)30 b(x\))450 2364 y Fr(Returns)f Fp(floor\(x\))p
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-1168 y Fr(This)h(function)g(returns)f(the)h(greatest)j(common)28
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-f Fp(>)p Fr(=)f(0.)-30 1323 y Fp(cl_I)g(xgcd)g(\(const)g(cl_I&)g(a,)h
-(const)f(cl_I&)g(b,)g(cl_I*)g(u,)h(cl_I*)f(v\))450 1433
-y Fr(This)34 b(function)h(\(\\extended)h(gcd"\))g(returns)e(the)h
-(greatest)i(common)d(divisor)h Fp(g)f Fr(of)i Fp(a)e
-Fr(and)h Fp(b)g Fr(and)450 1542 y(at)30 b(the)g(same)f(time)g(the)h
-(represen)m(tation)g(of)g Fp(g)f Fr(as)h(an)g(in)m(tegral)h(linear)f
-(com)m(bination)g(of)f Fp(a)g Fr(and)g Fp(b)p Fr(:)40
-b Fp(u)450 1652 y Fr(and)32 b Fp(v)g Fr(with)g Fp(u*a+v*b)d(=)h(g)p
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-Fr(will)h(b)s(e)f(normalized)g(to)h(b)s(e)f(of)h(smallest)f(p)s
-(ossible)450 1761 y(absolute)d(v)-5 b(alue,)29 b(in)e(the)h(follo)m
-(wing)h(sense:)40 b(If)27 b Fp(a)h Fr(and)f Fp(b)g Fr(are)i(non-zero,)g
-(and)e Fp(abs\(a\))i(!=)g(abs\(b\))p Fr(,)e Fp(u)450
-1871 y Fr(and)j Fp(v)g Fr(will)h(satisfy)f(the)h(inequalities)h
-Fp(abs\(u\))c(<=)i(abs\(b\)/\(2*g\))p Fr(,)d Fp(abs\(v\))i(<=)h
-(abs\(a\)/\(2*g\))p Fr(.)-30 2026 y Fp(cl_I)f(lcm)h(\(const)e(cl_I&)h
-(a,)h(const)f(cl_I&)g(b\))450 2136 y Fr(This)h(function)g(returns)f
-(the)h(least)i(common)d(m)m(ultiple)h(of)g Fp(a)g Fr(and)g
-Fp(b)p Fr(,)g(normalized)g(to)h(b)s(e)f Fp(>)p Fr(=)g(0.)-30
-2291 y Fp(cl_boolean)e(logp)h(\(const)f(cl_I&)h(a,)h(const)f(cl_I&)g
-(b,)h(cl_RA*)e(l\))-30 2400 y(cl_boolean)g(logp)h(\(const)f(cl_RA&)h
-(a,)h(const)f(cl_RA&)f(b,)i(cl_RA*)f(l\))450 2510 y(a)38
-b Fr(m)m(ust)g(b)s(e)f Fp(>)i Fr(0.)65 b Fp(b)38 b Fr(m)m(ust)f(b)s(e)h
-Fp(>)p Fr(0)h(and)f(!=)g(1.)66 b(If)38 b(log\(a,b\))i(is)f(rational)g
-(n)m(um)m(b)s(er,)g(this)f(function)450 2619 y(returns)29
-b(true)h(and)g(sets)h(*l)g(=)f(log\(a,b\),)j(else)e(it)g(returns)e
-(false.)-30 2837 y Fn(4.9.3)63 b(Com)m(binatorial)40
-b(functions)-30 3050 y Fp(cl_I)29 b(factorial)f(\(uintL)h(n\))450
-3160 y(n)h Fr(m)m(ust)f(b)s(e)h(a)h(small)f(in)m(teger)h
-Fp(>)p Fr(=)f(0.)41 b(This)30 b(function)g(returns)f(the)i(factorial)h
-Fp(n)p Fr(!)40 b(=)30 b Fp(1*2*...)n(*n)p Fr(.)-30 3315
-y Fp(cl_I)f(doublefactorial)d(\(uintL)j(n\))450 3425
-y(n)48 b Fr(m)m(ust)e(b)s(e)i(a)g(small)f(in)m(teger)j
-Fp(>)p Fr(=)d(0.)94 b(This)47 b(function)h(returns)e(the)i
-(doublefactorial)i Fp(n)p Fr(!!)93 b(=)450 3534 y Fp(1*3*...)n(*n)30
-b Fr(or)g Fp(n)p Fr(!!)41 b(=)30 b Fp(2*4*...)n(*n)p
-Fr(,)g(resp)s(ectiv)m(ely)-8 b(.)-30 3689 y Fp(cl_I)29
-b(binomial)f(\(uintL)h(n,)h(uintL)f(k\))450 3799 y(n)40
-b Fr(and)g Fp(k)g Fr(m)m(ust)g(b)s(e)f(small)h(in)m(tegers)i
-Fp(>)p Fr(=)e(0.)71 b(This)40 b(function)g(returns)g(the)g(binomial)g
-(co)s(e\016cien)m(t)450 3839 y Fk(\000)488 3868 y Fj(n)490
-3940 y(k)529 3839 y Fk(\001)592 3908 y Fr(=)798 3873
-y Fj(n)p Fi(!)p 698 3888 261 4 v 698 3940 a Fj(n)p Fi(!\()p
-Fj(n)p Fh(\000)p Fj(k)q Fi(\)!)999 3908 y Fr(for)30 b(0)h
-Fp(<)p Fr(=)e(k)i Fp(<)p Fr(=)e(n,)h(0)h(else.)-30 4159
-y Fs(4.10)68 b(F)-11 b(unctions)44 b(on)h(\015oating-p)t(oin)l(t)h(n)l
-(um)l(b)t(ers)-30 4350 y Fr(Recall)31 b(that)f(a)f(\015oating-p)s(oin)m
-(t)i(n)m(um)m(b)s(er)c(consists)j(of)f(a)h(sign)f Fp(s)p
-Fr(,)h(an)f(exp)s(onen)m(t)g Fp(e)g Fr(and)g(a)g(man)m(tissa)h
-Fp(m)p Fr(.)40 b(The)28 b(v)-5 b(alue)-30 4459 y(of)31
-b(the)f(n)m(um)m(b)s(er)e(is)j Fp(\(-1\)^s)d(*)i(2^e)g(*)g(m)p
-Fr(.)-30 4592 y(Eac)m(h)h(of)g(the)f(classes)i Fp(cl_F)p
-Fr(,)d Fp(cl_SF)p Fr(,)g Fp(cl_FF)p Fr(,)g Fp(cl_DF)p
-Fr(,)h Fp(cl_LF)e Fr(de\014nes)i(the)h(follo)m(wing)g(op)s(erations.)
--30 4747 y Fl(t)m(yp)s(e)36 b Fp(scale_float)27 b(\(const)i
-Fl(t)m(yp)s(e)5 b Fp(&)30 b(x,)g(sintL)f(delta\))-30
-4856 y Fl(t)m(yp)s(e)36 b Fp(scale_float)27 b(\(const)i
-Fl(t)m(yp)s(e)5 b Fp(&)30 b(x,)g(const)f(cl_I&)g(delta\))450
-4966 y Fr(Returns)40 b Fp(x*2^delta)p Fr(.)70 b(This)40
-b(is)h(more)f(e\016cien)m(t)j(than)d(an)h(explicit)i(m)m(ultiplication)
-e(b)s(ecause)g(it)450 5075 y(copies)31 b Fp(x)f Fr(and)g(mo)s(di\014es)
-f(the)h(exp)s(onen)m(t.)-30 5230 y(The)36 b(follo)m(wing)j(functions)d
-(pro)m(vide)h(an)g(abstract)h(in)m(terface)g(to)g(the)f(underlying)f
-(represen)m(tation)i(of)f(\015oating-)-30 5340 y(p)s(oin)m(t)30
-b(n)m(um)m(b)s(ers.)p eop
-%%Page: 24 26
-24 25 bop -30 -116 a Fr(Chapter)30 b(4:)41 b(F)-8 b(unctions)31
-b(on)f(n)m(um)m(b)s(ers)2523 b(24)-30 299 y Fp(sintL)29
-b(float_exponent)d(\(const)j Fl(t)m(yp)s(e)5 b Fp(&)30
-b(x\))450 408 y Fr(Returns)j(the)g(exp)s(onen)m(t)h Fp(e)f
-Fr(of)h Fp(x)p Fr(.)50 b(F)-8 b(or)34 b Fp(x)c(=)g(0.0)p
-Fr(,)k(this)f(is)h(0.)50 b(F)-8 b(or)35 b Fp(x)e Fr(non-zero,)i(this)f
-(is)f(the)h(unique)450 518 y(in)m(teger)e(with)e Fp(2^\(e-1\))e(<=)i
-(abs\(x\))f(<)h(2^e)p Fr(.)-30 701 y Fp(sintL)f(float_radix)e(\(const)i
-Fl(t)m(yp)s(e)5 b Fp(&)30 b(x\))450 811 y Fr(Returns)f(the)i(base)g(of)
-f(the)h(\015oating-p)s(oin)m(t)g(represen)m(tation.)42
-b(This)30 b(is)g(alw)m(a)m(ys)i Fp(2)p Fr(.)-30 994 y
-Fl(t)m(yp)s(e)k Fp(float_sign)27 b(\(const)i Fl(t)m(yp)s(e)5
-b Fp(&)30 b(x\))450 1103 y Fr(Returns)f(the)i(sign)f
-Fp(s)g Fr(of)h Fp(x)f Fr(as)h(a)f(\015oat.)42 b(The)30
-b(v)-5 b(alue)31 b(is)f(1)h(for)f Fp(x)g(>)p Fr(=)g(0,)h(-1)g(for)f
-Fp(x)g(<)g Fr(0.)-30 1287 y Fp(uintL)f(float_digits)e(\(const)i
-Fl(t)m(yp)s(e)5 b Fp(&)30 b(x\))450 1396 y Fr(Returns)f(the)h(n)m(um)m
-(b)s(er)e(of)i(man)m(tissa)g(bits)g(in)g(the)g(\015oating-p)s(oin)m(t)h
-(represen)m(tation)g(of)f Fp(x)p Fr(,)g(including)450
-1506 y(the)h(hidden)e(bit.)40 b(The)30 b(v)-5 b(alue)31
-b(only)g(dep)s(ends)d(on)i(the)h(t)m(yp)s(e)f(of)h Fp(x)p
-Fr(,)f(not)h(on)f(its)h(v)-5 b(alue.)-30 1689 y Fp(uintL)29
-b(float_precision)d(\(const)j Fl(t)m(yp)s(e)5 b Fp(&)30
-b(x\))450 1798 y Fr(Returns)c(the)h(n)m(um)m(b)s(er)d(of)j
-(signi\014can)m(t)h(man)m(tissa)f(bits)g(in)f(the)h(\015oating-p)s(oin)
-m(t)h(represen)m(tation)g(of)f Fp(x)p Fr(.)450 1908 y(Since)j
-(denormalized)e(n)m(um)m(b)s(ers)g(are)h(not)h(supp)s(orted,)e(this)i
-(is)f(the)h(same)f(as)h Fp(float_digits\(x\))25 b Fr(if)450
-2018 y Fp(x)30 b Fr(is)g(non-zero,)i(and)d(0)i(if)f Fp(x)h
-Fr(=)f(0.)-30 2213 y(The)f(complete)g(in)m(ternal)h(represen)m(tation)h
-(of)e(a)g(\015oat)h(is)g(enco)s(ded)e(in)h(the)h(t)m(yp)s(e)f
-Fp(decoded_float)d Fr(\(or)j Fp(decoded_)-30 2322 y(sfloat)p
-Fr(,)g Fp(decoded_ffloat)p Fr(,)e Fp(decoded_dfloat)p
-Fr(,)f Fp(decoded_lfloat)p Fr(,)h(resp)s(ectiv)m(ely\),)32
-b(de\014ned)e(b)m(y)210 2475 y Fp(struct)46 b(decoded_)p
-Fl(t)m(yp)s(e)5 b Fp(float)45 b({)592 2578 y Fl(t)m(yp)s(e)53
-b Fp(mantissa;)45 b(cl_I)i(exponent;)e Fl(t)m(yp)s(e)53
-b Fp(sign;)210 2682 y(};)-30 2840 y Fr(and)30 b(returned)f(b)m(y)h(the)
-h(function)-30 3035 y Fp(decoded_)p Fl(t)m(yp)s(e)5 b
-Fp(float)27 b(decode_float)g(\(const)i Fl(t)m(yp)s(e)5
-b Fp(&)30 b(x\))450 3145 y Fr(F)-8 b(or)26 b Fp(x)e Fr(non-zero,)j
-(this)d(returns)g Fp(\(-1\)^s)p Fr(,)h Fp(e)p Fr(,)g
-Fp(m)g Fr(with)f Fp(x)30 b(=)g(\(-1\)^s)f(*)h(2^e)g(*)g(m)24
-b Fr(and)g Fp(0.5)30 b(<=)g(m)g(<)g(1.0)p Fr(.)450 3255
-y(F)-8 b(or)35 b Fp(x)f Fr(=)h(0,)h(it)f(returns)e Fp(\(-1\)^s)p
-Fr(=1,)h Fp(e)p Fr(=0,)i Fp(m)p Fr(=0.)53 b Fp(e)34 b
-Fr(is)g(the)h(same)f(as)h(returned)e(b)m(y)i(the)f(function)450
-3364 y Fp(float_exponent)p Fr(.)-30 3559 y(A)c(complete)h(deco)s(ding)g
-(in)f(terms)f(of)i(in)m(tegers)g(is)g(pro)m(vided)f(as)g(t)m(yp)s(e)210
-3712 y Fp(struct)46 b(cl_idecoded_float)d({)592 3815
-y(cl_I)j(mantissa;)g(cl_I)g(exponent;)g(cl_I)g(sign;)210
-3919 y(};)-30 4077 y Fr(b)m(y)30 b(the)h(follo)m(wing)h(function:)-30
-4272 y Fp(cl_idecoded_float)26 b(integer_decode_float)e(\(const)29
-b Fl(t)m(yp)s(e)5 b Fp(&)31 b(x\))450 4382 y Fr(F)-8
-b(or)38 b Fp(x)e Fr(non-zero,)k(this)d(returns)f Fp(\(-1\)^s)p
-Fr(,)h Fp(e)p Fr(,)h Fp(m)f Fr(with)g Fp(x)30 b(=)g(\(-1\)^s)e(*)i(2^e)
-g(*)g(m)37 b Fr(and)f Fp(m)h Fr(an)g(in)m(teger)450 4491
-y(with)e Fp(float_digits\(x\))30 b Fr(bits.)54 b(F)-8
-b(or)36 b Fp(x)e Fr(=)h(0,)h(it)g(returns)d Fp(\(-1\)^s)p
-Fr(=1,)i Fp(e)p Fr(=0,)h Fp(m)p Fr(=0.)54 b(W)-10 b(ARNING:)450
-4601 y(The)27 b(exp)s(onen)m(t)h Fp(e)f Fr(is)h(not)f(the)h(same)f(as)h
-(the)g(one)g(returned)e(b)m(y)h(the)h(functions)f Fp(decode_float)e
-Fr(and)450 4711 y Fp(float_exponent)p Fr(.)-30 4906 y(Some)k(other)i
-(function,)f(implemen)m(ted)f(only)i(for)f(class)h Fp(cl_F)p
-Fr(:)-30 5101 y Fp(cl_F)e(float_sign)f(\(const)g(cl_F&)h(x,)h(const)f
-(cl_F&)g(y\))450 5210 y Fr(This)d(returns)g(a)i(\015oating)g(p)s(oin)m
-(t)f(n)m(um)m(b)s(er)e(whose)h(precision)i(and)e(absolute)i(v)-5
-b(alue)28 b(is)f(that)h(of)f Fp(y)g Fr(and)450 5320 y(whose)j(sign)h
-(is)f(that)h(of)f Fp(x)p Fr(.)41 b(If)30 b Fp(x)g Fr(is)g(zero,)i(it)f
-(is)f(treated)h(as)g(p)s(ositiv)m(e.)42 b(Same)29 b(for)h
-Fp(y)p Fr(.)p eop
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-25 26 bop -30 -116 a Fr(Chapter)30 b(4:)41 b(F)-8 b(unctions)31
-b(on)f(n)m(um)m(b)s(ers)2523 b(25)-30 299 y Fs(4.11)68
-b(Con)l(v)l(ersion)46 b(functions)-30 612 y Fn(4.11.1)63
-b(Con)m(v)m(ersion)41 b(to)g(\015oating-p)s(oin)m(t)h(n)m(um)m(b)s(ers)
--30 821 y Fr(The)30 b(t)m(yp)s(e)g Fp(cl_float_format_t)c
-Fr(describ)s(es)k(a)g(\015oating-p)s(oin)m(t)i(format.)-30
-1003 y Fp(cl_float_format_t)26 b(cl_float_format)g(\(uintL)j(n\))450
-1113 y Fr(Returns)45 b(the)g(smallest)h(\015oat)g(format)f(whic)m(h)h
-(guaran)m(tees)g(at)h(least)g Fp(n)e Fr(decimal)g(digits)h(in)g(the)450
-1222 y(man)m(tissa)30 b(\(after)i(the)e(decimal)g(p)s(oin)m(t\).)-30
-1397 y Fp(cl_float_format_t)c(cl_float_format)g(\(const)j(cl_F&)f(x\))
-450 1507 y Fr(Returns)h(the)i(\015oating)g(p)s(oin)m(t)g(format)e(of)i
-Fp(x)p Fr(.)-30 1682 y Fp(cl_float_format_t)26 b(default_float_format)
-450 1792 y Fr(Global)d(v)-5 b(ariable:)37 b(the)23 b(default)f(\015oat)
-g(format)g(used)f(when)g(con)m(v)m(erting)j(rational)f(n)m(um)m(b)s
-(ers)c(to)k(\015oats.)-30 1974 y(T)-8 b(o)30 b(con)m(v)m(ert)h(a)f
-(real)g(n)m(um)m(b)s(er)d(to)j(a)g(\015oat,)g(eac)m(h)h(of)e(the)h(t)m
-(yp)s(es)f Fp(cl_R)p Fr(,)g Fp(cl_F)p Fr(,)f Fp(cl_I)p
-Fr(,)h Fp(cl_RA)p Fr(,)g Fp(int)p Fr(,)f Fp(unsigned)h(int)p
-Fr(,)-30 2084 y Fp(float)p Fr(,)g Fp(double)g Fr(de\014nes)g(the)i
-(follo)m(wing)h(op)s(erations:)-30 2267 y Fp(cl_F)d(cl_float)f(\(const)
-h Fl(t)m(yp)s(e)5 b Fp(&x,)30 b(cl_float_format_t)25
-b(f\))450 2376 y Fr(Returns)k Fp(x)h Fr(as)h(a)g(\015oat)g(of)f(format)
-g Fp(f)p Fr(.)-30 2551 y Fp(cl_F)f(cl_float)f(\(const)h
-Fl(t)m(yp)s(e)5 b Fp(&x,)30 b(const)f(cl_F&)g(y\))450
-2661 y Fr(Returns)g Fp(x)h Fr(in)h(the)f(\015oat)h(format)f(of)g
-Fp(y)p Fr(.)-30 2836 y Fp(cl_F)f(cl_float)f(\(const)h
-Fl(t)m(yp)s(e)5 b Fp(&x\))450 2945 y Fr(Returns)24 b
-Fp(x)g Fr(as)h(a)g(\015oat)h(of)f(format)f Fp(default_float_format)19
-b Fr(if)24 b(it)i(is)e(an)h(exact)h(n)m(um)m(b)s(er,)e(or)h
-Fp(x)f Fr(itself)450 3055 y(if)30 b(it)h(is)g(already)g(a)f(\015oat.)
--30 3238 y(Of)g(course,)h(con)m(v)m(erting)h(a)f(n)m(um)m(b)s(er)d(to)j
-(a)f(\015oat)i(can)e(lose)h(precision.)-30 3388 y(Ev)m(ery)g
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-(um)m(b)s(ers:)-30 3571 y Fp(cl_F)c(most_positive_float)c
-(\(cl_float_format_t)h(f\))450 3680 y Fr(Returns)j(the)i(largest)h
-(\(most)e(p)s(ositiv)m(e\))i(\015oating)f(p)s(oin)m(t)f(n)m(um)m(b)s
-(er)e(in)i(\015oat)h(format)f Fp(f)p Fr(.)-30 3855 y
-Fp(cl_F)f(most_negative_float)c(\(cl_float_format_t)h(f\))450
-3965 y Fr(Returns)j(the)i(smallest)g(\(most)f(negativ)m(e\))j
-(\015oating)e(p)s(oin)m(t)f(n)m(um)m(b)s(er)e(in)i(\015oat)h(format)f
-Fp(f)p Fr(.)-30 4140 y Fp(cl_F)f(least_positive_float)c
-(\(cl_float_format_t)g(f\))450 4249 y Fr(Returns)g(the)h(least)h(p)s
-(ositiv)m(e)g(\015oating)g(p)s(oin)m(t)e(n)m(um)m(b)s(er)f(\(i.e.)40
-b Fp(>)26 b Fr(0)g(but)f(closest)i(to)g(0\))g(in)e(\015oat)i(format)450
-4359 y Fp(f)p Fr(.)-30 4534 y Fp(cl_F)i(least_negative_float)c
-(\(cl_float_format_t)g(f\))450 4644 y Fr(Returns)e(the)i(least)h
-(negativ)m(e)g(\015oating)g(p)s(oin)m(t)e(n)m(um)m(b)s(er)e(\(i.e.)40
-b Fp(<)24 b Fr(0)h(but)e(closest)j(to)f(0\))g(in)g(\015oat)g(format)450
-4753 y Fp(f)p Fr(.)-30 4928 y Fp(cl_F)k(float_epsilon)e
-(\(cl_float_format_t)e(f\))450 5038 y Fr(Returns)k(the)i(smallest)g
-(\015oating)g(p)s(oin)m(t)f(n)m(um)m(b)s(er)e(e)j Fp(>)f
-Fr(0)h(suc)m(h)f(that)h Fp(1+e)e(!=)h(1)p Fr(.)-30 5213
-y Fp(cl_F)f(float_negative_epsilon)24 b(\(cl_float_format_t)i(f\))450
-5322 y Fr(Returns)j(the)i(smallest)g(\015oating)g(p)s(oin)m(t)f(n)m(um)
-m(b)s(er)e(e)j Fp(>)f Fr(0)h(suc)m(h)f(that)h Fp(1-e)e(!=)h(1)p
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--30 1129 y(These)h(are)h(the)f(simple)g(input)f(functions:)-30
-1276 y Fp(int)g(freadchar)f(\(cl_istream)g(stream\))450
-1386 y Fr(Reads)37 b(a)f(c)m(haracter)j(from)c Fp(stream)p
-Fr(.)57 b(Returns)35 b Fp(cl_EOF)g Fr(\(not)i(a)g(`)p
-Fp(char)p Fr('!\))59 b(if)36 b(the)h(end)f(of)h(stream)450
-1495 y(w)m(as)31 b(encoun)m(tered)g(or)f(an)g(error)g(o)s(ccurred.)-30
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-1753 y Fr(Puts)41 b(bac)m(k)h Fp(c)f Fr(on)m(to)i Fp(stream)p
-Fr(.)72 b Fp(c)41 b Fr(m)m(ust)f(b)s(e)h(the)h(result)f(of)g(the)h
-(last)g Fp(freadchar)d Fr(op)s(eration)j(on)450 1862
-y Fp(stream)p Fr(.)-30 2010 y(Eac)m(h)53 b(of)f(the)g(classes)i
-Fp(cl_N)p Fr(,)i Fp(cl_R)p Fr(,)h Fp(cl_RA)p Fr(,)f Fp(cl_I)p
-Fr(,)h Fp(cl_F)p Fr(,)f Fp(cl_SF)p Fr(,)h Fp(cl_FF)p
-Fr(,)f Fp(cl_DF)p Fr(,)h Fp(cl_LF)50 b Fr(de\014nes,)57
-b(in)-30 2119 y Fp(<cln/)p Fl(t)m(yp)s(e)5 b Fp(_io.h>)p
-Fr(,)28 b(the)i(follo)m(wing)i(input)e(function:)-30
-2267 y Fp(cl_istream)e(operator>>)f(\(cl_istream)g(stream,)i
-Fl(t)m(yp)s(e)5 b Fp(&)30 b(result\))450 2377 y Fr(Reads)g(a)h(n)m(um)m
-(b)s(er)d(from)h Fp(stream)g Fr(and)h(stores)g(it)h(in)f(the)h
-Fp(result)p Fr(.)-30 2524 y(The)f(most)g(\015exible)g(input)g
-(functions,)g(de\014ned)f(in)h Fp(<cln/)p Fl(t)m(yp)s(e)5
-b Fp(_io.h>)p Fr(,)28 b(are)j(the)f(follo)m(wing:)-30
-2672 y Fp(cl_N)f(read_complex)e(\(cl_istream)h(stream,)g(const)h
-(cl_read_flags&)d(flags\))-30 2781 y(cl_R)j(read_real)f(\(cl_istream)f
-(stream,)i(const)g(cl_read_flags&)d(flags\))-30 2891
-y(cl_F)j(read_float)f(\(cl_istream)f(stream,)h(const)h(cl_read_flags&)e
-(flags\))-30 3001 y(cl_RA)i(read_rational)e(\(cl_istream)g(stream,)h
-(const)h(cl_read_flags&)e(flags\))-30 3110 y(cl_I)i(read_integer)e
-(\(cl_istream)h(stream,)g(const)h(cl_read_flags&)d(flags\))450
-3220 y Fr(Reads)i(a)g(n)m(um)m(b)s(er)e(from)g Fp(stream)p
-Fr(.)38 b(The)27 b Fp(flags)g Fr(are)h(parameters)f(whic)m(h)h
-(a\013ect)h(the)f(input)f(syn)m(tax.)450 3329 y(Whitespace)32
-b(b)s(efore)e(the)g(n)m(um)m(b)s(er)e(is)j(silen)m(tly)h(skipp)s(ed.)
--30 3477 y Fp(cl_N)d(read_complex)e(\(const)i(cl_read_flags&)d(flags,)j
-(const)g(char)g(*)h(string,)f(const)g(char)g(*)-30 3587
-y(string_limit,)e(const)i(char)g(*)h(*)g(end_of_parse\))-30
-3696 y(cl_R)f(read_real)f(\(const)h(cl_read_flags&)d(flags,)j(const)g
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-(const)i(char)g(*)h(*)g(end_of_parse\))-30 3915 y(cl_F)f(read_float)f
-(\(const)g(cl_read_flags&)f(flags,)i(const)g(char)g(*)h(string,)e
-(const)h(char)g(*)-30 4025 y(string_limit,)e(const)i(char)g(*)h(*)g
-(end_of_parse\))-30 4134 y(cl_RA)f(read_rational)e(\(const)h
-(cl_read_flags&)f(flags,)h(const)h(char)h(*)g(string,)e(const)h(char)g
-(*)-30 4244 y(string_limit,)e(const)i(char)g(*)h(*)g(end_of_parse\))-30
-4354 y(cl_I)f(read_integer)e(\(const)i(cl_read_flags&)d(flags,)j(const)
-g(char)g(*)h(string,)f(const)g(char)g(*)-30 4463 y(string_limit,)e
-(const)i(char)g(*)h(*)g(end_of_parse\))450 4573 y Fr(Reads)k(a)h(n)m
-(um)m(b)s(er)d(from)g(a)j(string)f(in)g(memory)-8 b(.)50
-b(The)34 b Fp(flags)e Fr(are)j(parameters)e(whic)m(h)h(a\013ect)i(the)
-450 4682 y(input)31 b(syn)m(tax.)46 b(The)31 b(string)h(starts)g(at)h
-Fp(string)d Fr(and)i(ends)f(at)h Fp(string_limit)d Fr(\(exclusiv)m(e)34
-b(limit\).)450 4792 y Fp(string_limit)29 b Fr(ma)m(y)i(also)i(b)s(e)f
-Fp(NULL)p Fr(,)f(denoting)i(the)f(en)m(tire)h(string,)g(i.e.)47
-b(equiv)-5 b(alen)m(t)33 b(to)g Fp(string_)450 4902 y(limit)c(=)h
-(string)f(+)h(strlen\(string\))p Fr(.)75 b(If)42 b Fp(end_of_parse)e
-Fr(is)j Fp(NULL)p Fr(,)j(the)d(string)g(in)g(memory)450
-5011 y(m)m(ust)30 b(con)m(tain)j(exactly)g(one)f(n)m(um)m(b)s(er)d(and)
-i(nothing)g(more,)g(else)i(a)f(fatal)g(error)f(will)h(b)s(e)f
-(signalled.)450 5121 y(If)37 b Fp(end_of_parse)d Fr(is)k(not)g
-Fp(NULL)p Fr(,)g Fp(*end_of_parse)c Fr(will)k(b)s(e)f(assigned)h(a)g(p)
-s(oin)m(ter)g(past)f(the)h(last)450 5230 y(parsed)32
-b(c)m(haracter)h(\(i.e.)48 b Fp(string_limit)29 b Fr(if)j(nothing)g
-(came)g(after)h(the)g(n)m(um)m(b)s(er\).)44 b(Whitespace)34
-b(is)450 5340 y(not)d(allo)m(w)m(ed.)p eop
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-30 31 bop -30 -116 a Fr(Chapter)30 b(5:)41 b(Input/Output)2844
-b(30)-30 299 y(The)30 b(structure)g Fp(cl_read_flags)d
-Fr(con)m(tains)k(the)g(follo)m(wing)g(\014elds:)-30 461
-y Fp(cl_read_syntax_t)26 b(syntax)450 570 y Fr(The)i(p)s(ossible)f
-(results)h(of)g(the)h(read)f(op)s(eration.)40 b(P)m(ossible)29
-b(v)-5 b(alues)28 b(are)h Fp(syntax_number)p Fr(,)c Fp(syntax_)450
-680 y(real)p Fr(,)49 b Fp(syntax_rational)p Fr(,)e Fp(syntax_integer)p
-Fr(,)g Fp(syntax_float)p Fr(,)g Fp(syntax_sfloat)p Fr(,)g
-Fp(syntax_)450 789 y(ffloat)p Fr(,)29 b Fp(syntax_dfloat)p
-Fr(,)e Fp(syntax_lfloat)p Fr(.)-30 950 y Fp(cl_read_lsyntax_t)f
-(lsyntax)450 1060 y Fr(Sp)s(eci\014es)31 b(the)g(language-dep)s(enden)m
-(t)h(syn)m(tax)g(v)-5 b(arian)m(t)32 b(for)f(the)h(read)f(op)s
-(eration.)43 b(P)m(ossible)33 b(v)-5 b(alues)450 1169
-y(are)450 1330 y Fp(lsyntax_standard)930 1440 y Fr(accept)32
-b(standard)d(algebraic)j(notation)g(only)-8 b(,)31 b(no)g(complex)f(n)m
-(um)m(b)s(ers,)450 1601 y Fp(lsyntax_algebraic)930 1710
-y Fr(accept)i(the)e(algebraic)i(notation)g Fl(x)6 b Fp(+)p
-Fl(y)i Fp(i)30 b Fr(for)g(complex)g(n)m(um)m(b)s(ers,)450
-1871 y Fp(lsyntax_commonlisp)930 1981 y Fr(accept)49
-b(the)g Fp(#b)p Fr(,)j Fp(#o)p Fr(,)g Fp(#x)47 b Fr(syn)m(taxes)i(for)f
-(binary)-8 b(,)52 b(o)s(ctal,)i(hexadecimal)48 b(n)m(um)m(b)s(ers,)930
-2090 y Fp(#)p Fl(base)5 b Fp(R)25 b Fr(for)f(rational)i(n)m(um)m(b)s
-(ers)d(in)h(a)i(giv)m(en)g(base,)g Fp(#c\()p Fl(realpart)32
-b(imagpart)r Fp(\))24 b Fr(for)h(com-)930 2200 y(plex)30
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-y Fr(accept)i(all)f(of)g(these)f(extensions.)-30 2631
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-2902 y Fp(cl_float_format_t)26 b(float_flags.default_flo)o(at_f)o(orm)o
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-(\015oats)h(with)f(exp)s(onen)m(t)g(mark)m(er)g(`)p Fp(e)p
-Fr('.)-30 3172 y Fp(cl_float_format_t)c(float_flags.default_lfl)o(oat_)
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-v 62 w Fl(nnn)e Fr(su\016x,)i(will)h(get)h(a)e(precision)h(corresp)s
-(onding)e(to)450 3772 y(their)i(n)m(um)m(b)s(er)e(of)j(signi\014can)m
-(t)g(digits.)-30 4033 y Fs(5.3)68 b(Output)45 b(functions)-30
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-f Fp(cl_ostream)p Fr(,)g(whic)m(h)g(is)h(the)f(t)m(yp)s(e)g(of)h(the)f
-(\014rst)f(argumen)m(t)h(to)h(all)-30 4337 y(output)30
-b(functions.)40 b Fp(cl_ostream)28 b Fr(is)i(the)h(same)f(as)g
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-4609 y Fp(cl_ostream)28 b(stdout)-30 4770 y Fr(con)m(tains)k(the)e
-(standard)g(output)g(stream.)-30 4906 y(The)g(v)-5 b(ariable)150
-5042 y Fp(cl_ostream)28 b(stderr)-30 5204 y Fr(con)m(tains)k(the)e
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-(simple)g(output)g(functions:)p eop
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-31 32 bop -30 -116 a Fr(Chapter)30 b(5:)41 b(Input/Output)2844
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-i(c\))450 408 y Fr(Prin)m(ts)g(the)h(c)m(haracter)h Fp(x)e
-Fr(literally)i(on)e(the)h Fp(stream)p Fr(.)-30 574 y
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-(string\))450 683 y Fr(Prin)m(ts)g(the)h Fp(string)d
-Fr(literally)33 b(on)d(the)g Fp(stream)p Fr(.)-30 848
-y Fp(void)f(fprintdecimal)e(\(cl_ostream)g(stream,)i(int)g(x\))-30
-958 y(void)g(fprintdecimal)e(\(cl_ostream)g(stream,)i(const)g(cl_I&)g
-(x\))450 1068 y Fr(Prin)m(ts)h(the)h(in)m(teger)h Fp(x)e
-Fr(in)g(decimal)g(on)g(the)h Fp(stream)p Fr(.)-30 1233
-y Fp(void)e(fprintbinary)e(\(cl_ostream)h(stream,)g(const)h(cl_I&)g
-(x\))450 1342 y Fr(Prin)m(ts)h(the)h(in)m(teger)h Fp(x)e
-Fr(in)g(binary)f(\(base)i(2,)g(without)g(pre\014x\))e(on)h(the)h
-Fp(stream)p Fr(.)-30 1508 y Fp(void)e(fprintoctal)f(\(cl_ostream)f
-(stream,)h(const)h(cl_I&)g(x\))450 1617 y Fr(Prin)m(ts)h(the)h(in)m
-(teger)h Fp(x)e Fr(in)g(o)s(ctal)h(\(base)g(8,)g(without)g(pre\014x\))e
-(on)i(the)f Fp(stream)p Fr(.)-30 1782 y Fp(void)f(fprinthexadecimal)d
-(\(cl_ostream)h(stream,)i(const)f(cl_I&)h(x\))450 1892
-y Fr(Prin)m(ts)h(the)h(in)m(teger)h Fp(x)e Fr(in)g(hexadecimal)g
-(\(base)h(16,)h(without)e(pre\014x\))g(on)g(the)h Fp(stream)p
-Fr(.)-30 2060 y(Eac)m(h)53 b(of)f(the)g(classes)i Fp(cl_N)p
-Fr(,)i Fp(cl_R)p Fr(,)h Fp(cl_RA)p Fr(,)f Fp(cl_I)p Fr(,)h
-Fp(cl_F)p Fr(,)f Fp(cl_SF)p Fr(,)h Fp(cl_FF)p Fr(,)f
-Fp(cl_DF)p Fr(,)h Fp(cl_LF)50 b Fr(de\014nes,)57 b(in)-30
-2170 y Fp(<cln/)p Fl(t)m(yp)s(e)5 b Fp(_io.h>)p Fr(,)28
-b(the)i(follo)m(wing)i(output)e(functions:)-30 2338 y
-Fp(void)f(fprint)g(\(cl_ostream)e(stream,)i(const)f Fl(t)m(yp)s(e)5
-b Fp(&)31 b(x\))-30 2447 y(cl_ostream)d(operator<<)f(\(cl_ostream)g
-(stream,)i(const)g Fl(t)m(yp)s(e)5 b Fp(&)30 b(x\))450
-2557 y Fr(Prin)m(ts)42 b(the)g(n)m(um)m(b)s(er)e Fp(x)i
-Fr(on)g(the)g Fp(stream)p Fr(.)75 b(The)41 b(output)h(ma)m(y)g(dep)s
-(end)e(on)i(the)h(global)g(prin)m(ter)450 2667 y(settings)36
-b(in)g(the)f(v)-5 b(ariable)37 b Fp(default_print_flags)p
-Fr(.)51 b(The)35 b Fp(ostream)e Fr(\015ags)j(and)f(settings)h
-(\(\015ags,)450 2776 y(width)30 b(and)f(lo)s(cale\))k(are)e(ignored.)
--30 2944 y(The)f(most)g(\015exible)g(output)g(function,)g(de\014ned)g
-(in)g Fp(<cln/)p Fl(t)m(yp)s(e)5 b Fp(_io.h>)p Fr(,)28
-b(are)i(the)h(follo)m(wing:)210 3079 y Fp(void)47 b(print_complex)92
-b(\(cl_ostream)44 b(stream,)i(const)g(cl_print_flags&)e(flags,)1212
-3183 y(const)j(cl_N&)f(z\);)210 3286 y(void)h(print_real)236
-b(\(cl_ostream)44 b(stream,)i(const)g(cl_print_flags&)e(flags,)1212
-3390 y(const)j(cl_R&)f(z\);)210 3494 y(void)h(print_float)188
-b(\(cl_ostream)44 b(stream,)i(const)g(cl_print_flags&)e(flags,)1212
-3598 y(const)j(cl_F&)f(z\);)210 3702 y(void)h(print_rational)d
-(\(cl_ostream)g(stream,)i(const)g(cl_print_flags&)e(flags,)1212
-3805 y(const)j(cl_RA&)f(z\);)210 3909 y(void)h(print_integer)92
-b(\(cl_ostream)44 b(stream,)i(const)g(cl_print_flags&)e(flags,)1212
-4013 y(const)j(cl_I&)f(z\);)-30 4153 y Fr(Prin)m(ts)30
-b(the)h(n)m(um)m(b)s(er)d Fp(x)i Fr(on)g(the)h Fp(stream)p
-Fr(.)39 b(The)30 b Fp(flags)f Fr(are)h(parameters)g(whic)m(h)g
-(a\013ect)i(the)f(output.)-30 4293 y(The)f(structure)g(t)m(yp)s(e)g
-Fp(cl_print_flags)d Fr(con)m(tains)k(the)g(follo)m(wing)h(\014elds:)-30
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-(eci\014c)g(exp)s(onen)m(t)h(mark)m(ers)e(ha)m(v)m(e)j(precedence)f(o)m
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-y Fr(If)41 b(this)g(\015ag)g(is)h(true,)i(complex)c(n)m(um)m(b)s(ers)f
-(will)j(b)s(e)e(prin)m(ted)h(using)g(the)g(Common)e(Lisp)h(syn)m(tax)
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-(of)g(abstract)g(rings.)1403 662 y Fp(Ring)1308 765 y(cl_ring)1212
-869 y(<cln/ring.h>)-30 1004 y Fr(Rings)f(can)h(b)s(e)f(compared)f(for)h
-(equalit)m(y:)-30 1163 y Fp(bool)f(operator==)f(\(const)g(cl_ring&,)g
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-(\(const)h(cl_ring_element&)d(x,)k(const)f(cl_ring_element&)d(y\))-30
-2359 y(cl_ring_element)g(R->uminus)i(\(const)h(cl_ring_element&)d(x\))
--30 2468 y(cl_ring_element)g(R->one)j(\(\))-30 2578 y(cl_ring_element)d
-(R->canonhom)h(\(const)i(cl_I&)g(x\))-30 2687 y(cl_ring_element)d
-(R->mul)j(\(const)g(cl_ring_element&)c(x,)30 b(const)f
-(cl_ring_element&)d(y\))-30 2797 y(cl_ring_element)g(R->square)i
-(\(const)h(cl_ring_element&)d(x\))-30 2907 y(cl_ring_element)g
-(R->expt_pos)h(\(const)i(cl_ring_element&)d(x,)k(const)f(cl_I&)g(y\))
--30 3041 y Fr(The)h(follo)m(wing)i(rings)e(are)g(built-in.)-30
-3200 y Fp(cl_null_ring)d(cl_0_ring)450 3310 y Fr(The)j(n)m(ull)g(ring,)
-h(con)m(taining)g(only)g(zero.)-30 3469 y Fp(cl_complex_ring)26
-b(cl_C_ring)450 3579 y Fr(The)k(ring)g(of)h(complex)f(n)m(um)m(b)s
-(ers.)38 b(This)29 b(corresp)s(onds)g(to)j(the)e(t)m(yp)s(e)h
-Fp(cl_N)p Fr(.)-30 3738 y Fp(cl_real_ring)c(cl_R_ring)450
-3848 y Fr(The)j(ring)g(of)h(real)g(n)m(um)m(b)s(ers.)38
-b(This)29 b(corresp)s(onds)g(to)i(the)g(t)m(yp)s(e)g
-Fp(cl_R)p Fr(.)-30 4007 y Fp(cl_rational_ring)26 b(cl_RA_ring)450
-4117 y Fr(The)k(ring)g(of)h(rational)g(n)m(um)m(b)s(ers.)38
-b(This)30 b(corresp)s(onds)f(to)i(the)g(t)m(yp)s(e)f
-Fp(cl_RA)p Fr(.)-30 4276 y Fp(cl_integer_ring)c(cl_I_ring)450
-4386 y Fr(The)k(ring)g(of)h(in)m(tegers.)41 b(This)30
-b(corresp)s(onds)f(to)i(the)g(t)m(yp)s(e)f Fp(cl_I)p
-Fr(.)-30 4545 y(T)m(yp)s(e)g(tests)h(can)g(b)s(e)e(p)s(erformed)f(for)i
-(an)m(y)h(of)g Fp(cl_C_ring)p Fr(,)d Fp(cl_R_ring)p Fr(,)g
-Fp(cl_RA_ring)p Fr(,)g Fp(cl_I_ring)p Fr(:)-30 4705 y
-Fp(cl_boolean)g(instanceof)f(\(const)i(cl_number&)e(x,)j(const)f
-(cl_number_ring&)d(R\))450 4814 y Fr(T)-8 b(ests)31 b(whether)f(the)g
-(giv)m(en)i(n)m(um)m(b)s(er)c(is)i(an)g(elemen)m(t)h(of)g(the)f(n)m(um)
-m(b)s(er)e(ring)i(R.)p eop
-%%Page: 34 36
-34 35 bop -30 -116 a Fr(Chapter)30 b(7:)41 b(Mo)s(dular)30
-b(in)m(tegers)2730 b(34)-30 299 y Fo(7)80 b(Mo)t(dular)55
-b(in)l(tegers)-30 657 y Fs(7.1)68 b(Mo)t(dular)45 b(in)l(teger)h(rings)
--30 850 y Fr(CLN)29 b(implemen)m(ts)f(mo)s(dular)f(in)m(tegers,)k(i.e.)
-41 b(in)m(tegers)31 b(mo)s(dulo)c(a)j(\014xed)e(in)m(teger)j(N.)f(The)e
-(mo)s(dulus)f(is)i(explicitly)-30 960 y(part)37 b(of)h(ev)m(ery)g(mo)s
-(dular)e(in)m(teger.)63 b(CLN)37 b(do)s(esn't)g(allo)m(w)i(y)m(ou)f(to)
-g(\(acciden)m(tally\))j(mix)36 b(elemen)m(ts)i(of)f(di\013eren)m(t)-30
-1069 y(mo)s(dular)25 b(rings,)j(e.g.)41 b Fp(\(3)30 b(mod)f(4\))h(+)g
-(\(2)g(mod)f(5\))e Fr(will)h(result)f(in)g(a)h(run)m(time)e(error.)40
-b(\(Ideally)28 b(one)g(w)m(ould)f(imagine)-30 1179 y(a)k(generic)h
-(data)f(t)m(yp)s(e)g Fp(cl_MI\(N\))p Fr(,)e(but)h(C)p
-Fp(++)g Fr(do)s(esn't)h(ha)m(v)m(e)h(generic)g(t)m(yp)s(es.)42
-b(So)30 b(one)h(has)g(to)h(liv)m(e)g(with)e(run)m(time)-30
-1288 y(c)m(hec)m(ks.\))-30 1423 y(The)g(class)h(of)g(mo)s(dular)d(in)m
-(teger)k(rings)e(is)1403 1552 y Fp(Ring)1308 1656 y(cl_ring)1212
-1760 y(<cln/ring.h>)1451 1863 y(|)1451 1967 y(|)1021
-2071 y(Modular)46 b(integer)g(ring)1165 2175 y(cl_modint_ring)1069
-2279 y(<cln/modinteger.h>)-30 2413 y Fr(and)30 b(the)g(class)h(of)g
-(all)g(mo)s(dular)e(in)m(tegers)i(\(elemen)m(ts)g(of)g(mo)s(dular)d(in)
-m(teger)k(rings\))e(is)1165 2542 y Fp(Modular)45 b(integer)1403
-2646 y(cl_MI)1117 2750 y(<cln/modinteger.h>)-30 2885
-y Fr(Mo)s(dular)30 b(in)m(teger)i(rings)e(are)g(constructed)h(using)f
-(the)h(function)-30 3044 y Fp(cl_modint_ring)26 b(find_modint_ring)g
-(\(const)j(cl_I&)g(N\))450 3154 y Fr(This)f(function)g(returns)f(the)h
-(mo)s(dular)e(ring)j(`)p Fp(Z/NZ)p Fr('.)39 b(It)28 b(tak)m(es)i(care)f
-(of)g(\014nding)e(out)h(ab)s(out)g(sp)s(ecial)450 3264
-y(cases)e(of)e Fp(N)p Fr(,)i(lik)m(e)g(p)s(o)m(w)m(ers)f(of)g(t)m(w)m
-(o)h(and)e(o)s(dd)f(n)m(um)m(b)s(ers)g(for)h(whic)m(h)h(Mon)m(tgomery)g
-(m)m(ultiplication)h(will)450 3373 y(b)s(e)g(a)h(win,)g(and)f
-(precomputes)f(an)m(y)i(necessary)g(auxiliary)h(data)f(for)f(computing)
-g(mo)s(dulo)f Fp(N)p Fr(.)39 b(There)450 3483 y(is)27
-b(a)h(cac)m(he)g(table)g(of)g(rings,)f(indexed)g(b)m(y)g
-Fp(N)g Fr(\(or,)h(more)e(precisely)-8 b(,)29 b(b)m(y)f
-Fp(abs\(N\))p Fr(\).)38 b(This)26 b(ensures)g(that)450
-3592 y(the)31 b(precomputation)e(costs)j(are)e(reduced)g(to)h(a)g
-(minim)m(um.)-30 3752 y(Mo)s(dular)f(in)m(teger)i(rings)e(can)g(b)s(e)g
-(compared)f(for)i(equalit)m(y:)-30 3912 y Fp(bool)e(operator==)f
-(\(const)g(cl_modint_ring&,)e(const)j(cl_modint_ring&\))-30
-4022 y(bool)g(operator!=)f(\(const)g(cl_modint_ring&,)e(const)j
-(cl_modint_ring&\))450 4131 y Fr(These)42 b(compare)g(t)m(w)m(o)i(mo)s
-(dular)c(in)m(teger)k(rings)e(for)g(equalit)m(y)-8 b(.)78
-b(Tw)m(o)43 b(di\013eren)m(t)g(calls)g(to)g Fp(find_)450
-4241 y(modint_ring)c Fr(with)i(the)h(same)f(argumen)m(t)g(necessarily)h
-(return)f(the)h(same)f(ring)g(b)s(ecause)h(it)g(is)450
-4350 y(memoized)29 b(in)h(the)h(cac)m(he)h(table.)-30
-4608 y Fs(7.2)68 b(F)-11 b(unctions)44 b(on)h(mo)t(dular)g(in)l(tegers)
--30 4801 y Fr(Giv)m(en)31 b(a)g(mo)s(dular)d(in)m(teger)k(ring)e
-Fp(R)p Fr(,)h(the)f(follo)m(wing)i(mem)m(b)s(ers)c(can)i(b)s(e)g(used.)
--30 4961 y Fp(cl_I)f(R->modulus)450 5071 y Fr(This)h(is)g(the)h(ring's)
-f(mo)s(dulus,)e(normalized)i(to)h(b)s(e)e(nonnegativ)m(e:)43
-b Fp(abs\(N\))p Fr(.)-30 5230 y Fp(cl_MI)29 b(R->zero\(\))450
-5340 y Fr(This)h(returns)f Fp(0)h(mod)f(N)p Fr(.)p eop
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-35 36 bop -30 -116 a Fr(Chapter)30 b(7:)41 b(Mo)s(dular)30
-b(in)m(tegers)2730 b(35)-30 299 y Fp(cl_MI)29 b(R->one\(\))450
-408 y Fr(This)h(returns)f Fp(1)h(mod)f(N)p Fr(.)-30 576
-y Fp(cl_MI)g(R->canonhom)e(\(const)i(cl_I&)g(x\))450
-686 y Fr(This)h(returns)f Fp(x)h(mod)f(N)p Fr(.)-30 853
-y Fp(cl_I)g(R->retract)f(\(const)g(cl_MI&)h(x\))450 963
-y Fr(This)c(is)h(a)h(partial)f(in)m(v)m(erse)h(function)f(to)h
-Fp(R->canonhom)p Fr(.)36 b(It)26 b(returns)f(the)h(standard)f(represen)
-m(tativ)m(e)450 1072 y(\()p Fp(>=0)p Fr(,)30 b Fp(<N)p
-Fr(\))g(of)h Fp(x)p Fr(.)-30 1240 y Fp(cl_MI)e
-(R->random\(random_state&)24 b(randomstate\))-30 1349
-y(cl_MI)29 b(R->random\(\))450 1459 y Fr(This)h(returns)f(a)h(random)f
-(in)m(teger)j(mo)s(dulo)d Fp(N)p Fr(.)-30 1630 y(The)h(follo)m(wing)i
-(op)s(erations)e(are)h(de\014ned)e(on)i(mo)s(dular)d(in)m(tegers.)-30
-1802 y Fp(cl_modint_ring)e(x.ring)j(\(\))450 1912 y Fr(Returns)g(the)i
-(ring)f(to)h(whic)m(h)f(the)h(mo)s(dular)d(in)m(teger)k
-Fp(x)e Fr(b)s(elongs.)-30 2079 y Fp(cl_MI)f(operator+)f(\(const)g
-(cl_MI&,)h(const)g(cl_MI&\))450 2189 y Fr(Returns)36
-b(the)h(sum)d(of)j(t)m(w)m(o)h(mo)s(dular)d(in)m(tegers.)60
-b(One)37 b(of)f(the)h(argumen)m(ts)f(ma)m(y)g(also)i(b)s(e)e(a)h(plain)
-450 2298 y(in)m(teger.)-30 2466 y Fp(cl_MI)29 b(operator-)f(\(const)g
-(cl_MI&,)h(const)g(cl_MI&\))450 2575 y Fr(Returns)37
-b(the)h(di\013erence)g(of)g(t)m(w)m(o)h(mo)s(dular)d(in)m(tegers.)65
-b(One)37 b(of)h(the)g(argumen)m(ts)f(ma)m(y)h(also)g(b)s(e)g(a)450
-2685 y(plain)30 b(in)m(teger.)-30 2852 y Fp(cl_MI)f(operator-)f
-(\(const)g(cl_MI&\))450 2962 y Fr(Returns)h(the)i(negativ)m(e)i(of)d(a)
-h(mo)s(dular)d(in)m(teger.)-30 3130 y Fp(cl_MI)h(operator*)f(\(const)g
-(cl_MI&,)h(const)g(cl_MI&\))450 3239 y Fr(Returns)e(the)g(pro)s(duct)f
-(of)i(t)m(w)m(o)h(mo)s(dular)c(in)m(tegers.)41 b(One)27
-b(of)h(the)f(argumen)m(ts)g(ma)m(y)g(also)h(b)s(e)f(a)h(plain)450
-3349 y(in)m(teger.)-30 3516 y Fp(cl_MI)h(square)g(\(const)f(cl_MI&\))
-450 3626 y Fr(Returns)h(the)i(square)f(of)h(a)f(mo)s(dular)f(in)m
-(teger.)-30 3793 y Fp(cl_MI)g(recip)g(\(const)g(cl_MI&)f(x\))450
-3903 y Fr(Returns)g(the)i(recipro)s(cal)g Fp(x^-1)e Fr(of)i(a)f(mo)s
-(dular)f(in)m(teger)i Fp(x)p Fr(.)40 b Fp(x)29 b Fr(m)m(ust)g(b)s(e)f
-(coprime)h(to)h(the)f(mo)s(dulus,)450 4012 y(otherwise)i(an)f(error)g
-(message)h(is)f(issued.)-30 4180 y Fp(cl_MI)f(div)g(\(const)g(cl_MI&)g
-(x,)h(const)e(cl_MI&)h(y\))450 4290 y Fr(Returns)36 b(the)g(quotien)m
-(t)i Fp(x*y^-1)d Fr(of)h(t)m(w)m(o)i(mo)s(dular)c(in)m(tegers)k
-Fp(x)p Fr(,)g Fp(y)p Fr(.)59 b Fp(y)36 b Fr(m)m(ust)f(b)s(e)h(coprime)g
-(to)h(the)450 4399 y(mo)s(dulus,)28 b(otherwise)j(an)f(error)g(message)
-g(is)h(issued.)-30 4567 y Fp(cl_MI)e(expt_pos)f(\(const)h(cl_MI&)f(x,)i
-(const)f(cl_I&)g(y\))450 4676 y(y)h Fr(m)m(ust)f(b)s(e)h
-Fp(>)g Fr(0.)41 b(Returns)30 b Fp(x^y)p Fr(.)-30 4844
-y Fp(cl_MI)f(expt)g(\(const)g(cl_MI&)f(x,)i(const)f(cl_I&)g(y\))450
-4953 y Fr(Returns)h Fp(x^y)p Fr(.)42 b(If)30 b Fp(y)h
-Fr(is)g(negativ)m(e,)i Fp(x)e Fr(m)m(ust)f(b)s(e)g(coprime)g(to)i(the)f
-(mo)s(dulus,)e(else)i(an)g(error)g(message)450 5063 y(is)f(issued.)-30
-5230 y Fp(cl_MI)f(operator<<)f(\(const)g(cl_MI&)h(x,)h(const)f(cl_I&)f
-(y\))450 5340 y Fr(Returns)h Fp(x*2^y)p Fr(.)p eop
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-36 37 bop -30 -116 a Fr(Chapter)30 b(7:)41 b(Mo)s(dular)30
-b(in)m(tegers)2730 b(36)-30 299 y Fp(cl_MI)29 b(operator>>)f(\(const)g
-(cl_MI&)h(x,)h(const)f(cl_I&)f(y\))450 408 y Fr(Returns)k
-Fp(x*2^-y)p Fr(.)45 b(When)33 b Fp(y)f Fr(is)g(p)s(ositiv)m(e,)j(the)e
-(mo)s(dulus)c(m)m(ust)j(b)s(e)g(o)s(dd,)g(or)h(an)f(error)g(message)h
-(is)450 518 y(issued.)-30 677 y Fp(bool)c(operator==)f(\(const)g
-(cl_MI&,)h(const)g(cl_MI&\))-30 787 y(bool)g(operator!=)f(\(const)g
-(cl_MI&,)h(const)g(cl_MI&\))450 897 y Fr(Compares)21
-b(t)m(w)m(o)j(mo)s(dular)c(in)m(tegers,)25 b(b)s(elonging)e(to)g(the)f
-(same)g(mo)s(dular)e(in)m(teger)k(ring,)g(for)e(equalit)m(y)-8
-b(.)-30 1056 y Fp(cl_boolean)28 b(zerop)h(\(const)f(cl_MI&)h(x\))450
-1166 y Fr(Returns)g(true)i(if)f Fp(x)g Fr(is)g Fp(0)g(mod)g(N)p
-Fr(.)-30 1325 y(The)g(follo)m(wing)i(output)e(functions)g(are)g
-(de\014ned)g(\(see)h(also)g(the)g(c)m(hapter)g(on)f(input/output\).)-30
-1484 y Fp(void)f(fprint)g(\(cl_ostream)e(stream,)i(const)f(cl_MI&)h
-(x\))-30 1594 y(cl_ostream)f(operator<<)f(\(cl_ostream)g(stream,)i
-(const)g(cl_MI&)f(x\))450 1704 y Fr(Prin)m(ts)40 b(the)h(mo)s(dular)d
-(in)m(teger)k Fp(x)d Fr(on)i(the)f Fp(stream)p Fr(.)69
-b(The)40 b(output)g(ma)m(y)f(dep)s(end)g(on)h(the)h(global)450
-1813 y(prin)m(ter)30 b(settings)h(in)f(the)h(v)-5 b(ariable)31
-b Fp(default_print_flags)p Fr(.)p eop
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-37 38 bop -30 -116 a Fr(Chapter)30 b(8:)41 b(Sym)m(b)s(olic)29
-b(data)i(t)m(yp)s(es)2596 b(37)-30 299 y Fo(8)80 b(Sym)l(b)t(olic)54
-b(data)g(t)l(yp)t(es)-30 513 y Fr(CLN)30 b(implemen)m(ts)f(t)m(w)m(o)j
-(sym)m(b)s(olic)d(\(non-n)m(umeric\))h(data)h(t)m(yp)s(es:)41
-b(strings)30 b(and)g(sym)m(b)s(ols.)-30 758 y Fs(8.1)68
-b(Strings)-30 946 y Fr(The)30 b(class)1260 1071 y Fp(String)1212
-1175 y(cl_string)1117 1278 y(<cln/string.h>)-30 1409
-y Fr(implemen)m(ts)f(imm)m(utable)g(strings.)-30 1539
-y(Strings)h(are)h(constructed)f(through)g(the)h(follo)m(wing)g
-(constructors:)-30 1691 y Fp(cl_string)d(\(const)h(char)g(*)h(s\))450
-1800 y Fr(Returns)f(an)i(imm)m(utable)e(cop)m(y)i(of)f(the)h
-(\(zero-terminated\))h(C)e(string)g Fp(s)p Fr(.)-30 1951
-y Fp(cl_string)e(\(const)h(char)g(*)h(ptr,)f(unsigned)f(long)h(len\))
-450 2061 y Fr(Returns)36 b(an)h(imm)m(utable)f(cop)m(y)i(of)f(the)g
-Fp(len)f Fr(c)m(haracters)j(at)e Fp(ptr[0])p Fr(,)43
-b(.)22 b(.)h(.)11 b(,)39 b Fp(ptr[len-1])p Fr(.)58 b(NUL)450
-2171 y(c)m(haracters)32 b(are)f(allo)m(w)m(ed.)-30 2322
-y(The)f(follo)m(wing)i(functions)e(are)g(a)m(v)-5 b(ailable)33
-b(on)d(strings:)-30 2473 y Fp(operator)e(=)450 2583 y
-Fr(Assignmen)m(t)i(from)f Fp(cl_string)f Fr(and)h Fp(const)g(char)h(*)p
-Fr(.)-30 2734 y Fp(s.length\(\))-30 2844 y(strlen\(s\))450
-2953 y Fr(Returns)f(the)i(length)g(of)f(the)h(string)f
-Fp(s)p Fr(.)-30 3104 y Fp(s[i])288 b Fr(Returns)28 b(the)i
-Fp(i)p Fr(th)f(c)m(haracter)i(of)e(the)g(string)h Fp(s)p
-Fr(.)40 b Fp(i)29 b Fr(m)m(ust)f(b)s(e)h(in)g(the)g(range)h
-Fp(0)g(<=)f(i)h(<)g(s.length\(\))p Fr(.)-30 3256 y Fp(bool)f(equal)g
-(\(const)g(cl_string&)e(s1,)j(const)f(cl_string&)e(s2\))450
-3365 y Fr(Compares)38 b(t)m(w)m(o)i(strings)f(for)f(equalit)m(y)-8
-b(.)69 b(One)38 b(of)h(the)g(argumen)m(ts)g(ma)m(y)f(also)i(b)s(e)e(a)i
-(plain)e Fp(const)450 3475 y(char)29 b(*)p Fr(.)-30 3720
-y Fs(8.2)68 b(Sym)l(b)t(ols)-30 3908 y Fr(Sym)m(b)s(ols)31
-b(are)i(uniqui\014ed)d(strings:)45 b(all)34 b(sym)m(b)s(ols)d(with)h
-(the)g(same)g(name)g(are)h(shared.)46 b(This)32 b(means)f(that)i(com-)
--30 4018 y(parison)38 b(of)g(t)m(w)m(o)i(sym)m(b)s(ols)d(is)h(fast)h
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-(whereas)f(comparison)e(of)i(t)m(w)m(o)-30 4128 y(strings)32
-b(m)m(ust)f(in)g(the)i(w)m(orst)f(case)h(w)m(alk)g(b)s(oth)e(strings)h
-(un)m(til)g(their)g(end.)45 b(Sym)m(b)s(ols)30 b(are)j(used,)f(for)f
-(example,)i(as)-30 4237 y(tags)e(for)g(prop)s(erties,)f(as)g(names)f
-(of)i(v)-5 b(ariables)31 b(in)f(p)s(olynomial)g(rings,)g(etc.)-30
-4368 y(Sym)m(b)s(ols)f(are)h(constructed)h(through)f(the)g(follo)m
-(wing)i(constructor:)-30 4519 y Fp(cl_symbol)c(\(const)h(cl_string&)e
-(s\))450 4629 y Fr(Lo)s(oks)j(up)g(or)g(creates)i(a)f(new)e(sym)m(b)s
-(ol)h(with)g(a)g(giv)m(en)i(name.)-30 4780 y(The)e(follo)m(wing)i(op)s
-(erations)e(are)h(a)m(v)-5 b(ailable)33 b(on)d(sym)m(b)s(ols:)-30
-4931 y Fp(cl_string)e(\(const)h(cl_symbol&)e(sym\))450
-5041 y Fr(Con)m(v)m(ersion)k(to)g Fp(cl_string)p Fr(:)38
-b(Returns)30 b(the)g(string)h(whic)m(h)f(names)f(the)i(sym)m(b)s(ol)e
-Fp(sym)p Fr(.)-30 5192 y Fp(bool)g(equal)g(\(const)g(cl_symbol&)e
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-(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
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-(.)f(.)35 b Fb(19)-30 2180 y Fs(B)-30 2303 y Fd(basering)27
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-(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
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-b Fb(21)-30 3498 y Fd(boole_nor)21 b Fc(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g
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-b Fb(21)-30 3590 y Fd(boole_orc1)17 b Fc(.)e(.)d(.)h(.)f(.)g(.)h(.)f(.)
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-b Fb(21)-30 3682 y Fd(boole_orc2)17 b Fc(.)e(.)d(.)h(.)f(.)g(.)h(.)f(.)
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-b Fb(21)-30 3774 y Fd(boole_set)21 b Fc(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g
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-b Fb(21)-30 3866 y Fd(boole_xor)21 b Fc(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g
-(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
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-b Fb(21)-30 4114 y Fs(C)-30 4236 y Fd(canonhom)27 b(\(\))8
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-b Fb(20)-30 4420 y Fd(catalanconst)28 b(\(\))17 b Fc(.)c(.)f(.)h(.)f(.)
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-(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)38 b Fb(18)-30 4880
-y Fd(cl_abort)27 b(\(\))d Fc(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
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-y Fd(cl_byte)23 b Fc(.)13 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
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-(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
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-(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)43 b Fr(10)568
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-(.)49 b Fr(12)269 3779 y(4.5)92 b(Comparisons)21 b Fe(.)14
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-(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
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-b Fr(13)269 3889 y(4.6)92 b(Rounding)29 b(functions)18
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-(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)47 b Fr(13)269
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-(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)
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-(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
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-(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
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-(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
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-(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
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-(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)49
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-(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)50
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-(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)71 b Fs(28)269 789
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-(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)58
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-(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
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-(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)72
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-(rings)c Fe(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
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-(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)58
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-(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)43 b Fr(42)-30
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