19 Floats 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, Float exists, and is based on the arbitrary-precision routines in mpfr. 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 ((1.0e-100+1.0)-1.0 is not the same as 1.0e-100+(1.0-1.0), in the default precision setting). 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?) 19.1 A sample run Floating-point numbers can be input into GAP in the standard floating-point notation:  Example  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  Floating-point numbers can also be created using Float, from strings or rational numbers; and can be converted back using String,Rat,Int. GAP allows rational and floating-point numbers to be mixed in the elementary operations +,-,*,/. However, floating-point numbers and rational numbers may not be compared. Conversions are performed using the creator Float:  Example  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  19.2 Methods 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 .1, .1e1, -.999 etc. are all valid GAP inputs. 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. Floating-point numbers may be followed by an underscore, as in 1._. 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 'l' as in 1._l. Higher-precision floating-point computations is available via external packages; float for example. A record, FLOAT (19.2-5), contains all relevant constants for the current floating-point format; see its documentation for details. Typical fields are FLOAT.MANT_DIG=53, the constant FLOAT.VIEW_DIG=6 specifying the number of digits to view, and FLOAT.PI for the constant π. The constants have the same name as their C counterparts, except for the missing initial DBL_ or M_. Floating-point numbers may be created using the single function Float (19.2-1), which accepts as arguments rational, string, or floating-point numbers. Floating-point numbers may also be created, in any floating-point representation, using NewFloat (19.2-1) as in NewFloat(IsIEEE754FloatRep,355/113), by supplying the category filter of the desired new floating-point number; or using MakeFloat (19.2-1) as in MakeFloat(1.0,355/113), by supplying a sample floating-point number. Floating-point numbers may also be converted to other GAP formats using the usual commands Int (14.2-3), Rat (17.2-6), String (27.7-6). Exact conversion to and from floating-point format may be done using external representations. The "external representation" of a floating-point number x is a pair [m,e] of integers, such that x=m*2^(-1+e-LogInt(AbsInt(m),2)). Conversion to and from external representation is performed as usual using ExtRepOfObj (79.8-1) and ObjByExtRep (79.8-1):  Example  gap> ExtRepOfObj(3.14); [ 7070651414971679, 2 ] gap> ObjByExtRep(IEEE754FloatsFamily,last); 3.14  Computations with floating-point numbers never raise any error. Division by zero is allowed, and produces a signed infinity. Illegal operations, such as 0./0., produce NaN's (not-a-number); this is the only floating-point number x such that not EqFloat(x+0.0,x). The IEEE754 standard requires NaN 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 EqFloat (19.2-6) should be used instead of =. The category a floating-point belongs to can be checked using the filters IsFinite (30.4-2), IsPInfinity (19.2-9), IsNInfinity (19.2-9), IsXInfinity (19.2-9), IsNaN (19.2-9). 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  gap> 1.0/3.0 = Float(1/3); true gap> (1.0/3.0)^5 = Float((1/3)^5); false  19.2-1 Float creators Float( obj )  function NewFloat( filter, obj )  constructor MakeFloat( sample, obj, obj )  operation Returns: A new floating-point number, based on obj This function creates a new floating-point number. If obj is a rational number, the created number is created with sufficient precision so that the number can (usually) be converted back to the original number (see Rat (Reference: Rat) and Rat (17.2-6)). For an integer, the precision, if unspecified, is chosen sufficient so that Int(Float(obj))=obj always holds, but at least 64 bits. obj may also be a string, which may be of the form "3.14e0" or ".314e1" or ".314@1" etc. An option may be passed to specify, it bits, a desired precision. The format is Float("3.14":PrecisionFloat:=1000) to create a 1000-bit approximation of 3.14. In particular, if obj is already a floating-point number, then Float(obj:PrecisionFloat:=prec) creates a copy of obj with a new precision. prec 19.2-2 Rat Rat( f )  attribute Returns: A rational approximation to f This command constructs a rational approximation to the floating-point number f. Of course, it is not guaranteed to return the original rational number f was created from, though it returns the most `reasonable' one given the precision of f. Two options control the precision of the rational approximation: In the form Rat(f:maxdenom:=md,maxpartial:=mp), the rational returned is such that the denominator is at most md and the partials in its continued fraction expansion are at most mp. The default values are maxpartial:=10000 and maxdenom:=2^(precision/2). 19.2-3 Cyc Cyc( f[, degree] )  operation Returns: A cyclotomic approximation to f This command constructs a cyclotomic approximation to the floating-point number f. Of course, it is not guaranteed to return the original rational number f was created from, though it returns the most `reasonable' one given the precision of f. An optional argument degree specifies the maximal degree of the cyclotomic to be constructed. The method used is LLL lattice reduction. 19.2-4 SetFloats SetFloats( rec[, bits][, install] )  function Installs a new interface to floating-point numbers in GAP, optionally with a desired precision bits in binary digits. The last optional argument install is a boolean value; if false, it only installs the eager handler and the precision for the floateans, without making them the default. 19.2-5 FLOAT FLOAT  global variable This record contains useful floating-point constants: DECIMAL_DIG Maximal number of useful digits; DIG Number of significant digits; VIEW_DIG Number of digits to print in short view; EPSILON Smallest number such that 1≠1+ϵ; MANT_DIG Number of bits in the mantissa; MAX Maximal representable number; MAX_10_EXP Maximal decimal exponent; MAX_EXP Maximal binary exponent; MIN Minimal positive representable number; MIN_10_EXP Minimal decimal exponent; MIN_EXP Minimal exponent; INFINITY Positive infinity; NINFINITY Negative infinity; NAN Not-a-number, as well as mathematical constants E, LOG2E, LOG10E, LN2, LN10, PI, PI_2, PI_4, 1_PI, 2_PI, 2_SQRTPI, SQRT2, SQRT1_2. 19.2-6 EqFloat EqFloat( x, y )  operation Returns: Whether the floateans x and y are equal This function compares two floating-point numbers, and returns true if they are equal, and false otherwise; with the exception that NaN is always considered to be different from itself. 19.2-7 PrecisionFloat PrecisionFloat( x )  attribute Returns: The precision of x This function returns the precision, counted in number of binary digits, of the floating-point number x. 19.2-8 SignBit SignBit( x )  attribute SignFloat( x )  attribute Returns: The sign of x. The first function SignBit returns the sign bit of the floating-point number x: true if x is negative (including -0.) and false otherwise. The second function SignFloat returns the integer -1 if x<0, 0 if x=0 and 1 if x>0. 19.2-9 Infinity testers IsPInfinity( x )  property IsNInfinity( x )  property IsXInfinity( x )  property IsFinite( x )  property IsNaN( x )  property Returns true if the floating-point number x is respectively +∞, -∞, ±∞, finite, or `not a number', such as the result of 0.0/0.0. 19.2-10 Standard mathematical operations Sin( f )  attribute Cos( f )  attribute Tan( f )  attribute Sec( f )  attribute Csc( f )  attribute Cot( f )  attribute Asin( f )  attribute Acos( f )  attribute Atan( f )  attribute Sinh( f )  attribute Cosh( f )  attribute Tanh( f )  attribute Sech( f )  attribute Csch( f )  attribute Coth( f )  attribute Asinh( f )  attribute Acosh( f )  attribute Atanh( f )  attribute Log( f )  operation Log2( f )  attribute Log10( f )  attribute Log1p( f )  attribute Exp( f )  attribute Exp2( f )  attribute Exp10( f )  attribute Expm1( f )  attribute CubeRoot( f )  attribute Square( f )  attribute Atan2( y, x )  operation Hypothenuse( x, y )  operation Ceil( f )  attribute Floor( f )  attribute Round( f )  attribute Trunc( f )  attribute FrExp( f )  attribute LdExp( f, exp )  operation AbsoluteValue( f )  attribute Norm( f )  attribute Frac( f )  attribute SinCos( f )  attribute Erf( f )  attribute Zeta( f )  attribute Gamma( f )  attribute Standard math functions. 19.3 High-precision-specific methods 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. A package must create a record with fields (all optional) creator a function converting strings to floating-point; eager a character allowing immediate conversion to floating-point; objbyextrep a function creating a floating-point number out of a list [mantissa,exponent]; filter a filter for the new floating-point objects; constants a record containing numerical constants, such as MANT_DIG, MAX, MIN, NAN. The package must install methods Int, Rat, String for its objects, and creators NewFloat(filter,IsRat), NewFloat(IsString). It must then install methods for all arithmetic and numerical operations: SUM, Exp, ... The user chooses that implementation by calling SetFloats (19.2-4) with the record as argument, and with an optional second argument requesting a precision in binary digits. 19.4 Complex arithmetic Complex arithmetic may be implemented in packages, and is present in float. Complex numbers are treated as usual numbers; they may be input with an extra "i" as in -0.5+0.866i. They may also be created using NewFloat (19.2-1) with three arguments: the float filter, the real part, and the imaginary part. Methods should then be implemented for Norm, RealPart, ImaginaryPart, ComplexConjugate, ... 19.4-1 Argument Argument( z )  attribute Returns the argument of the complex number z, namely the value Atan2(ImaginaryPart(z),RealPart(z)). 19.5 Interval-specific methods 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 EQ 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. Note the usual convention that intervals are compared as in [a,b]≤[c,d] if and only if a≤ c and b≤ d. 19.5-1 Sup Sup( x )  attribute Returns the supremum of the interval x. 19.5-2 Inf Inf( x )  attribute Returns the infimum of the interval x. 19.5-3 Mid Mid( x )  attribute Returns the midpoint of the interval x. 19.5-4 AbsoluteDiameter AbsoluteDiameter( x )  attribute Diameter( x )  operation Returns the absolute diameter of the interval x, namely the difference Sup(x)-Inf(x). 19.5-5 RelativeDiameter RelativeDiameter( x )  attribute Returns the relative diameter of the interval x, namely (Sup(x)-Inf(x))/AbsoluteValue(Min(x)). 19.5-6 IsDisjoint IsDisjoint( x1, x2 )  operation Returns true if the two intervals x1, x2 are disjoint. 19.5-7 IsSubset IsSubset( x1, x2 )  operation Returns true if the interval x1 contains x2. 19.5-8 IncreaseInterval IncreaseInterval( x, delta )  operation Returns an interval with same midpoint as x but absolute diameter increased by delta. 19.5-9 BlowupInterval BlowupInterval( x, ratio )  operation Returns an interval with same midpoint as x but relative diameter increased by ratio. 19.5-10 BisectInterval BisectInterval( x )  operation Returns a list of two intervals whose union equals the interval x.