<Chapter Label="Floats">
<Heading>Floats</Heading>

Starting with version 4.5, &GAP; has built-in support for
floating-point numbers in machine format, and allows package to
implement arbitrary-precision floating-point arithmetic in a uniform
manner. For now, one such package, <Package>Float</Package> exists,
and is based on the arbitrary-precision routines
in <Package>mpfr</Package>.

<P/> A word of caution: &GAP; deals primarily with algebraic objects,
which can be represented exactly in a computer. Numerical imprecision
means that floating-point numbers do not form a ring in the strict
&GAP; sense, because addition is in general not associative
(<C>(1.0e-100+1.0)-1.0</C> is not the same
as <C>1.0e-100+(1.0-1.0)</C>, in the default precision setting).

<P/> Most algorithms in &GAP; which require ring elements will
therefore not be applicable to floating-point elements. In some cases,
such a notion would not even make any sense (what is the greatest
common divisor of two floating-point numbers?)

<Section><Heading>A sample run</Heading>

Floating-point numbers can be input into &GAP; in the standard
floating-point notation:
<P/>
<Example><![CDATA[
gap> 3.14;
3.14
gap> last^2/6;
1.64327
gap> h := 6.62606896e-34;
6.62607e-34
gap> pi := 4*Atan(1.0);
3.14159
gap> hbar := h/(2*pi);
1.05457e-34
]]></Example>

<P/> Floating-point numbers can also be created using <C>Float</C>,
from strings or rational numbers; and can be converted back
using <C>String,Rat,Int</C>.

<P/> &GAP; allows rational and floating-point numbers to be mixed in
the elementary operations <C>+,-,*,/</C>. However, floating-point
numbers and rational numbers may not be compared. Conversions are
performed using the creator <C>Float</C>:
<P/>
<Example><![CDATA[
gap> Float("3.1416");
3.1416
gap> Float(355/113);
3.14159
gap> Rat(last);
355/113
gap> Rat(0.33333);
1/3
gap> Int(1.e10);
10000000000
gap> Int(1.e20);
100000000000000000000
gap> Int(1.e30);
1000000000000000019884624838656
]]></Example>
</Section>

<Section><Heading>Methods</Heading>

Floating-point numbers may be directly input, as in any usual
mathematical software or language; with the exception that every
floating-point number must contain a decimal
digit. Therefore <C>.1</C>, <C>.1e1</C>, <C>-.999</C> etc. are all valid
&GAP; inputs.
<P/>

Floating-point numbers so entered in &GAP; are stored as strings. They
are converted to floating-point when they are first used. This means that,
if the floating-point precision is increased, the constants are
reevaluated to fit the new format.
<P/>

Floating-point numbers may be followed by an underscore, as
in <C>1._</C>. This means that they are to be immediately converted to
the current floating-point format. The underscore may be followed by a
single letter, which specifies which format/precision to use. By
default, &GAP; has a single floating-point handler, with fixed (53
bits) precision, and its format specifier is <C>'l'</C> as
in <C>1._l</C>. Higher-precision floating-point computations is
available via external packages; <Package>float</Package> for example.
<P/>

A record, <Ref Var="FLOAT" Label="constants"/>,
contains all relevant constants for the
current floating-point format; see its documentation for details.
Typical fields are <C>FLOAT.MANT_DIG=53</C>, the
constant <C>FLOAT.VIEW_DIG=6</C> specifying the number of digits to
view, and <C>FLOAT.PI</C> for the constant <M>\pi</M>. The constants
have the same name as their C counterparts, except for the missing
initial <C>DBL_</C> or <C>M_</C>.
<P/>

Floating-point numbers may be created using the single function
<Ref Func="Float"/>, which accepts as arguments rational, string,
or floating-point numbers. Floating-point numbers may
also be created, in any floating-point representation, using
<Ref Func="NewFloat"/> as in <C>NewFloat(IsIEEE754FloatRep,355/113)</C>,
by supplying the category filter of the desired new floating-point number;
or using <Ref Func="MakeFloat"/> as in <C>NewFloat(1.0,355/113)</C>,
by supplying a sample floating-point number.
<P/>

Floating-point numbers may also be converted to other &GAP; formats
using the usual commands <Ref Oper="Int"/>, <Ref Oper="Rat"/>,
<Ref Oper="String"/>.
<P/>

Exact conversion to and from floating-point format may be done using
external representations. The "external representation" of a
floating-point number <C>x</C> is a pair <C>[m,e]</C> of integers,
such that <C>x=m*2^(-1+e-LogInt(AbsInt(m),2))</C>. Conversion to and from
external representation is performed as usual using
<Ref Oper="ExtRepOfObj"/> and <Ref Oper="ObjByExtRep"/>:
<Example><![CDATA[
gap> ExtRepOfObj(3.14);
[ 7070651414971679, 2 ]
gap> ObjByExtRep(IEEE754FloatsFamily,last);
3.14
]]></Example>
<P/>

Computations with floating-point numbers never raise any
error. Division by zero is allowed, and produces a signed
infinity. Illegal operations, such as <C>0./0.</C>,
produce <K>NaN</K>'s (not-a-number); this is the only floating-point
number <C>x</C> such that <C>not EqFloat(x+0.0,x)</C>.
<P/>

The IEEE754 standard requires <K>NaN</K> to be non-equal to itself. On
the other hand, &GAP; requires every object to be equal to itself. To
respect the IEEE754 standard, the function <Ref Oper="EqFloat"/>
should be used instead of <C>=</C>.
<P/>

The category a floating-point belongs to can be checked using the
filters <Ref Filt="IsFinite"/>, <Ref Filt="IsPInfinity"/>,
<Ref Filt="IsNInfinity"/>, <Ref Filt="IsXInfinity"/>,
<Ref Filt="IsNaN"/>.
<P/>

Comparisons between floating-point numbers and rationals are
explicitly forbidden. The rationale is that objects belonging to
different families should in general not be comparable in
&GAP;. Floating-point numbers are also approximations of real numbers,
and don't follow the same rules; consider for example, using the
default &GAP; implementation of floating-point numbers,
<Example><![CDATA[
gap> 1.0/3.0 = Float(1/3);
true
gap> (1.0/3.0)^5 = Float((1/3)^5);
false
]]></Example>
<P/>

<#Include Label="Float">

<ManSection>
  <Var Name="FLOAT" Label="constants"/>
  <Description>
    This record contains useful floating-point constants: <List>
    <Mark>DECIMAL_DIG</Mark> <Item>Maximal number of useful digits;</Item>
    <Mark>DIG</Mark> <Item>Number of significant digits;</Item>
    <Mark>VIEW_DIG</Mark> <Item>Number of digits to print in short view;</Item>
    <Mark>EPSILON</Mark> <Item>Smallest number such that <M>1\neq1+\epsilon</M>;</Item>
    <Mark>MANT_DIG</Mark> <Item>Number of bits in the mantissa;</Item>
    <Mark>MAX</Mark> <Item>Maximal representable number;</Item>
    <Mark>MAX_10_EXP</Mark> <Item>Maximal decimal exponent;</Item>
    <Mark>MAX_EXP</Mark> <Item>Maximal binary exponent;</Item>
    <Mark>MIN</Mark> <Item>Minimal positive representable number;</Item>
    <Mark>MIN_10_EXP</Mark> <Item>Minimal decimal exponent;</Item>
    <Mark>MIN_EXP</Mark> <Item>Minimal exponent;</Item>
    <Mark>INFINITY</Mark> <Item>Positive infinity;</Item>
    <Mark>NINFINITY</Mark> <Item>Negative infinity;</Item>
    <Mark>NAN</Mark> <Item>Not-a-number,</Item>
    </List>
    as well as mathematical constants <C>E</C>, <C>LOG2E</C>, <C>LOG10E</C>,
    <C>LN2</C>, <C>LN10</C>, <C>PI</C>, <C>PI_2</C>, <C>PI_4</C>,
    <C>1_PI</C>, <C>2_PI</C>, <C>2_SQRTPI</C>, <C>SQRT2</C>, <C>SQRT1_2</C>.
  </Description>
</ManSection>

<#Include Label="Float-Extra">
<#Include Label="Float-Infinities">
<#Include Label="Float-Math-Commands">

</Section>

<Section><Heading>High-precision-specific methods</Heading>

&GAP; provides a mechanism for packages to implement new
floating-point numerical interfaces. The following describes that
mechanism, actual examples of packages are documented separately.
<P/>

A package must create a record with fields (all optional) <List>
<Mark>creator</Mark> <Item>a function converting strings to floating-point;</Item>
<Mark>eager</Mark> <Item>a character allowing immediate conversion
  to floating-point;</Item>
<Mark>objbyextrep</Mark> <Item>a function creating a floating-point
  number out of a list <C>[mantissa,exponent]</C>;</Item>
<Mark>filter</Mark> <Item>a filter for the new floating-point objects;</Item>
<Mark>constants</Mark> <Item>a record containing numerical constants,
  such as <C>MANT_DIG</C>, <C>MAX</C>, <C>MIN</C>, <C>NAN</C>.</Item>
</List>
<P/>

The package must install methods <C>Int</C>, <C>Rat</C>, <C>String</C>
for its objects, and
creators <C>NewFloat(filter,IsRat)</C>, <C>NewFloat(IsString)</C>.
<P/>

It must then install methods for all arithmetic and numerical
operations: <C>PLUS</C>, <C>Exp</C>, ...
<P/>

The user chooses that implementation by calling
<Ref Func="SetFloats"/> with the record as argument, and with an
optional second argument requesting a precision in binary digits.

</Section>

<Section><Heading>Complex arithmetic</Heading>

Complex arithmetic may be implemented in packages, and is present in
<Package>float</Package>. Complex numbers are treated as usual
numbers; they may be input with an extra "i" as in
<C>-0.5+0.866i</C>. They may also be created using <Ref
Oper="NewFloat"/> with three arguments: the float filter, the real
part, and the imaginary part.
<P/>

Methods should then be implemented for <C>Norm</C>, <C>RealPart</C>,
<C>ImaginaryPart</C>, <C>ComplexConjugate</C>, ...

<#Include Label="Float-Complex">

</Section>

<Section><Heading>Interval-specific methods</Heading>

Interval arithmetic may also be implemented in packages. Intervals
are in fact efficient implementations of sets of real numbers. The
only non-trivial issue is how they should be compared. The standard
<C>EQ</C> tests if the intervals are equal; however, it is usually
more useful to know if intervals overlap, or are disjoint, or are
contained in each other.
<P/>

Note the usual convention that intervals are compared as
in <M>[a,b]\le[c,d]</M> if and only if <M>a\le c</M> and <M>b\le
d</M>.

<#Include Label="Float-Intervals">

</Section>

</Chapter>
