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<TITLE>CLN, a Class Library for Numbers - 3. Ordinary number types</TITLE></HEAD><BODY>Go to the <A HREF="cln_1.html">first</A>, <A HREF="cln_2.html">previous</A>, <A HREF="cln_4.html">next</A>, <A HREF="cln_13.html">last</A> section, <A HREF="cln_toc.html">table of contents</A>.<P><HR><P>
<H1><A NAME="SEC11" HREF="cln_toc.html#TOC11">3. Ordinary number types</A></H1>
<P>CLN implements the following class hierarchy:
<PRE> Number cl_number <cl_number.h> | | Real or complex number cl_N <cl_complex.h> | | Real number cl_R <cl_real.h> | +-------------------+-------------------+ | |Rational number Floating-point number cl_RA cl_F<cl_rational.h> <cl_float.h> | | | +-------------+-------------+-------------+ Integer | | | | cl_I Short-Float Single-Float Double-Float Long-Float <cl_integer.h> cl_SF cl_FF cl_DF cl_LF <cl_sfloat.h> <cl_ffloat.h> <cl_dfloat.h> <cl_lfloat.h></PRE>
<P><A NAME="IDX7"></A><A NAME="IDX8"></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 wantto completely disable compile-time type checking and use run-time typechecking instead.
<P><A NAME="IDX9"></A><A NAME="IDX10"></A><A NAME="IDX11"></A>The class <CODE>cl_N</CODE> comprises real and complex numbers. There isno special class for complex numbers since complex numbers with imaginarypart <CODE>0</CODE> are automatically converted to real numbers.
<P><A NAME="IDX12"></A>The class <CODE>cl_R</CODE> comprises real numbers of different kinds. It is anabstract class.
<P><A NAME="IDX13"></A><A NAME="IDX14"></A><A NAME="IDX15"></A>The class <CODE>cl_RA</CODE> comprises exact real numbers: rational numbers, includingintegers. There is no special class for non-integral rational numberssince rational numbers with denominator <CODE>1</CODE> are automatically convertedto integers.
<P><A NAME="IDX16"></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_toc.html#TOC12">3.1 Exact numbers</A></H2><P><A NAME="IDX17"></A>
<P>Some numbers are represented as exact numbers: there is no loss of informationwhen such a number is converted from its mathematical value to its internalrepresentation. On exact numbers, the elementary operations (<CODE>+</CODE>,<CODE>-</CODE>, <CODE>*</CODE>, <CODE>/</CODE>, comparisons, ...) compute the completelycorrect 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 denominatorare coprime integers and the denominator is positive. If the resultingdenominator 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 heapallocation. Otherwise the distinction between these immediate integers(called "fixnums") and heap allocated integers (called "bignums")is completely transparent.
<H2><A NAME="SEC13" HREF="cln_toc.html#TOC13">3.2 Floating-point numbers</A></H2><P><A NAME="IDX18"></A>
<P>Not all real numbers can be represented exactly. (There is an easy mathematicalproof for this: Only a countable set of numbers can be stored exactly ina computer, even if one assumes that it has unlimited storage. But thereare uncountably many real numbers.) So some approximation is needed.CLN implements ordinary floating-point numbers, with mantissa and exponent.
<P><A NAME="IDX19"></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 computingwith 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 thefloating-point number which is nearest to this. If two floating-point numbersare equally distant from the ideal result, the one with a <CODE>0</CODE> in its leastsignificant 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="IDX20"></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="IDX21"></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="IDX22"></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="IDX23"></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 floathas a fixed precision. (No "lazy recomputation".)</UL>
<P>Of course, computations with long floats are more expensive than thosewith smaller floating-point formats.
<P>CLN does not implement features like NaNs, denormalized numbers andgradual underflow. If the exponent range of some floating-point typeis too limited for your application, choose another floating-point typewith larger exponent range.
<P><A NAME="IDX24"></A>As a user of CLN, you can forget about the differences between thefour floating-point types and just declare all your floating-pointvariables as being of type <CODE>cl_F</CODE>. This has the advantage thatwhen 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 precisionof the initial values. Also, many transcendental functions have beendeclared 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 ifthe floating point contagion rule happened to change in the future.)
<H2><A NAME="SEC14" HREF="cln_toc.html#TOC14">3.3 Complex numbers</A></H2><P><A NAME="IDX25"></A>
<P>Complex numbers, as implemented by the class <CODE>cl_N</CODE>, have a realpart and an imaginary part, both real numbers. A complex number whoseimaginary part is the exact number <CODE>0</CODE> is automatically convertedto a real number.
<P>Complex numbers can arise from real numbers alone, for examplethrough application of <CODE>sqrt</CODE> or transcendental functions.
<H2><A NAME="SEC15" HREF="cln_toc.html#TOC15">3.4 Conversions</A></H2><P><A NAME="IDX26"></A>
<P>Conversions from any class to any its superclasses ("base classes" inC++ 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 emphasizeefficiency. 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 todo the conversion of <CODE>10</CODE> from <SAMP>`int'</SAMP> to <SAMP>`cl_I'</SAMP> at compile timealready. On the other hand, code like <SAMP>`cl_I x = 1000000000;'</SAMP> isin error.So, if you want to be sure that an <SAMP>`int'</SAMP> whose magnitude is not guaranteedto 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 theC++ built-in types is therefore to specify it as a string, like this:<A NAME="IDX27"></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 throughthe functions
<DL COMPACT>
<DT><CODE>int cl_I_to_int (const cl_I& x)</CODE><DD><A NAME="IDX28"></A><DT><CODE>unsigned int cl_I_to_uint (const cl_I& x)</CODE><DD><A NAME="IDX29"></A><DT><CODE>long cl_I_to_long (const cl_I& x)</CODE><DD><A NAME="IDX30"></A><DT><CODE>unsigned long cl_I_to_ulong (const cl_I& x)</CODE><DD><A NAME="IDX31"></A>Returns <CODE>x</CODE> as element of the C type <VAR>ctype</VAR>. If <CODE>x</CODE> is notrepresentable 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 throughthe functions
<DL COMPACT>
<DT><CODE>float cl_float_approx (const <VAR>type</VAR>& x)</CODE><DD><A NAME="IDX32"></A><DT><CODE>double cl_double_approx (const <VAR>type</VAR>& x)</CODE><DD><A NAME="IDX33"></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" inC++ terminology) are not provided. Instead, you can assert and checkthat a value belongs to a certain subclass, and return it as element of thatclass, using the <SAMP>`As'</SAMP> and <SAMP>`The'</SAMP> macros.<A NAME="IDX34"></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="IDX35"></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 ensurethat 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>
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