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[1X19 [33X[0;0YFloats[133X[101X
[33X[0;0YStarting with version 4.5, [5XGAP[105X 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, [5XFloat[105X exists, and is based on the arbitrary-precision
routines in [5Xmpfr[105X.[133X
[33X[0;0YA word of caution: [5XGAP[105X 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 [5XGAP[105X sense, because
addition is in general not associative ([10X(1.0e-100+1.0)-1.0[110X is not the same
as [10X1.0e-100+(1.0-1.0)[110X, in the default precision setting).[133X
[33X[0;0YMost algorithms in [5XGAP[105X 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?)[133X
[1X19.1 [33X[0;0YA sample run[133X[101X
[33X[0;0YFloating-point numbers can be input into [5XGAP[105X in the standard floating-point
notation:[133X
[4X[32X Example [32X[104X
[4X[25Xgap>[125X [27X3.14;[127X[104X
[4X[28X3.14[128X[104X
[4X[25Xgap>[125X [27Xlast^2/6;[127X[104X
[4X[28X1.64327[128X[104X
[4X[25Xgap>[125X [27Xh := 6.62606896e-34;[127X[104X
[4X[28X6.62607e-34[128X[104X
[4X[25Xgap>[125X [27Xpi := 4*Atan(1.0);[127X[104X
[4X[28X3.14159[128X[104X
[4X[25Xgap>[125X [27Xhbar := h/(2*pi);[127X[104X
[4X[28X1.05457e-34[128X[104X
[4X[32X[104X
[33X[0;0YFloating-point numbers can also be created using [10XFloat[110X, from strings or
rational numbers; and can be converted back using [10XString,Rat,Int[110X.[133X
[33X[0;0Y[5XGAP[105X allows rational and floating-point numbers to be mixed in the elementary
operations [10X+,-,*,/[110X. However, floating-point numbers and rational numbers may
not be compared. Conversions are performed using the creator [10XFloat[110X:[133X
[4X[32X Example [32X[104X
[4X[25Xgap>[125X [27XFloat("3.1416");[127X[104X
[4X[28X3.1416[128X[104X
[4X[25Xgap>[125X [27XFloat(355/113);[127X[104X
[4X[28X3.14159[128X[104X
[4X[25Xgap>[125X [27XRat(last);[127X[104X
[4X[28X355/113[128X[104X
[4X[25Xgap>[125X [27XRat(0.33333);[127X[104X
[4X[28X1/3[128X[104X
[4X[25Xgap>[125X [27XInt(1.e10);[127X[104X
[4X[28X10000000000[128X[104X
[4X[25Xgap>[125X [27XInt(1.e20);[127X[104X
[4X[28X100000000000000000000[128X[104X
[4X[25Xgap>[125X [27XInt(1.e30);[127X[104X
[4X[28X1000000000000000019884624838656[128X[104X
[4X[32X[104X
[1X19.2 [33X[0;0YMethods[133X[101X
[33X[0;0YFloating-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 [10X.1[110X, [10X.1e1[110X, [10X-.999[110X etc. are all valid
[5XGAP[105X inputs.[133X
[33X[0;0YFloating-point numbers so entered in [5XGAP[105X 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.[133X
[33X[0;0YFloating-point numbers may be followed by an underscore, as in [10X1._[110X. 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, [5XGAP[105X has a single
floating-point handler, with fixed (53 bits) precision, and its format
specifier is [10X'l'[110X as in [10X1._l[110X. Higher-precision floating-point computations is
available via external packages; [5Xfloat[105X for example.[133X
[33X[0;0YA record, [2XFLOAT[102X ([14X19.2-6[114X), contains all relevant constants for the current
floating-point format; see its documentation for details. Typical fields are
[10XFLOAT.MANT_DIG=53[110X, the constant [10XFLOAT.VIEW_DIG=6[110X specifying the number of
digits to view, and [10XFLOAT.PI[110X for the constant [22Xπ[122X. The constants have the same
name as their C counterparts, except for the missing initial [10XDBL_[110X or [10XM_[110X.[133X
[33X[0;0YFloating-point numbers may be created using the single function [2XFloat[102X
([14X19.2-7[114X), which accepts as arguments rational, string, or floating-point
numbers. Floating-point numbers may also be created, in any floating-point
representation, using [2XNewFloat[102X ([14X19.2-7[114X) as in
[10XNewFloat(IsIEEE754FloatRep,355/113)[110X, by supplying the category filter of the
desired new floating-point number; or using [2XMakeFloat[102X ([14X19.2-7[114X) as in
[10XNewFloat(1.0,355/113)[110X, by supplying a sample floating-point number.[133X
[33X[0;0YFloating-point numbers may also be converted to other [5XGAP[105X formats using the
usual commands [2XInt[102X ([14X14.2-3[114X), [2XRat[102X ([14X17.2-6[114X), [2XString[102X ([14X27.7-6[114X).[133X
[33X[0;0YExact conversion to and from floating-point format may be done using
external representations. The "external representation" of a floating-point
number [10Xx[110X is a pair [10X[m,e][110X of integers, such that
[10Xx=m*2^(-1+e-LogInt(AbsInt(m),2))[110X. Conversion to and from external
representation is performed as usual using [2XExtRepOfObj[102X ([14X79.16-1[114X) and
[2XObjByExtRep[102X ([14X79.16-1[114X):[133X
[4X[32X Example [32X[104X
[4X[25Xgap>[125X [27XExtRepOfObj(3.14);[127X[104X
[4X[28X[ 7070651414971679, 2 ][128X[104X
[4X[25Xgap>[125X [27XObjByExtRep(IEEE754FloatsFamily,last);[127X[104X
[4X[28X3.14[128X[104X
[4X[32X[104X
[33X[0;0YComputations with floating-point numbers never raise any error. Division by
zero is allowed, and produces a signed infinity. Illegal operations, such as
[10X0./0.[110X, produce [9XNaN[109X's (not-a-number); this is the only floating-point number
[10Xx[110X such that [10Xnot EqFloat(x+0.0,x)[110X.[133X
[33X[0;0YThe IEEE754 standard requires [9XNaN[109X to be non-equal to itself. On the other
hand, [5XGAP[105X requires every object to be equal to itself. To respect the
IEEE754 standard, the function [2XEqFloat[102X ([14X19.2-2[114X) should be used instead of [10X=[110X.[133X
[33X[0;0YThe category a floating-point belongs to can be checked using the filters
[2XIsFinite[102X ([14X30.4-2[114X), [2XIsPInfinity[102X ([14X19.2-5[114X), [2XIsNInfinity[102X ([14X19.2-5[114X), [2XIsXInfinity[102X
([14X19.2-5[114X), [2XIsNaN[102X ([14X19.2-5[114X).[133X
[33X[0;0YComparisons 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 [5XGAP[105X. Floating-point numbers are also
approximations of real numbers, and don't follow the same rules; consider
for example, using the default [5XGAP[105X implementation of floating-point numbers,[133X
[4X[32X Example [32X[104X
[4X[25Xgap>[125X [27X1.0/3.0 = Float(1/3);[127X[104X
[4X[28Xtrue[128X[104X
[4X[25Xgap>[125X [27X(1.0/3.0)^5 = Float((1/3)^5);[127X[104X
[4X[28Xfalse[128X[104X
[4X[32X[104X
[1X19.2-1 [33X[0;0YMathematical operations[133X[101X
[33X[1;0Y[29X[2XCos[102X( [3Xx[103X ) [32X attribute[133X
[33X[1;0Y[29X[2XSin[102X( [3Xx[103X ) [32X attribute[133X
[33X[1;0Y[29X[2XSinCos[102X( [3Xx[103X ) [32X attribute[133X
[33X[1;0Y[29X[2XTan[102X( [3Xx[103X ) [32X attribute[133X
[33X[1;0Y[29X[2XSec[102X( [3Xx[103X ) [32X attribute[133X
[33X[1;0Y[29X[2XCsc[102X( [3Xx[103X ) [32X attribute[133X
[33X[1;0Y[29X[2XCot[102X( [3Xx[103X ) [32X attribute[133X
[33X[1;0Y[29X[2XAsin[102X( [3Xx[103X ) [32X attribute[133X
[33X[1;0Y[29X[2XAcos[102X( [3Xx[103X ) [32X attribute[133X
[33X[1;0Y[29X[2XAtan[102X( [3Xx[103X ) [32X attribute[133X
[33X[1;0Y[29X[2XAtan2[102X( [3Xy[103X, [3Xx[103X ) [32X operation[133X
[33X[1;0Y[29X[2XCosh[102X( [3Xx[103X ) [32X attribute[133X
[33X[1;0Y[29X[2XSinh[102X( [3Xx[103X ) [32X attribute[133X
[33X[1;0Y[29X[2XTanh[102X( [3Xx[103X ) [32X attribute[133X
[33X[1;0Y[29X[2XSech[102X( [3Xx[103X ) [32X attribute[133X
[33X[1;0Y[29X[2XCsch[102X( [3Xx[103X ) [32X attribute[133X
[33X[1;0Y[29X[2XCoth[102X( [3Xx[103X ) [32X attribute[133X
[33X[1;0Y[29X[2XAsinh[102X( [3Xx[103X ) [32X attribute[133X
[33X[1;0Y[29X[2XAcosh[102X( [3Xx[103X ) [32X attribute[133X
[33X[1;0Y[29X[2XAtanh[102X( [3Xx[103X ) [32X attribute[133X
[33X[1;0Y[29X[2XLog[102X( [3Xx[103X ) [32X operation[133X
[33X[1;0Y[29X[2XLog2[102X( [3Xx[103X ) [32X attribute[133X
[33X[1;0Y[29X[2XLog10[102X( [3Xx[103X ) [32X attribute[133X
[33X[1;0Y[29X[2XLog1p[102X( [3Xx[103X ) [32X attribute[133X
[33X[1;0Y[29X[2XExp[102X( [3Xx[103X ) [32X attribute[133X
[33X[1;0Y[29X[2XExp2[102X( [3Xx[103X ) [32X attribute[133X
[33X[1;0Y[29X[2XExp10[102X( [3Xx[103X ) [32X attribute[133X
[33X[1;0Y[29X[2XExpm1[102X( [3Xx[103X ) [32X attribute[133X
[33X[1;0Y[29X[2XCubeRoot[102X( [3Xx[103X ) [32X attribute[133X
[33X[1;0Y[29X[2XSquare[102X( [3Xx[103X ) [32X attribute[133X
[33X[1;0Y[29X[2XHypothenuse[102X( [3Xx[103X, [3Xy[103X ) [32X operation[133X
[33X[1;0Y[29X[2XCeil[102X( [3Xx[103X ) [32X attribute[133X
[33X[1;0Y[29X[2XFloor[102X( [3Xx[103X ) [32X attribute[133X
[33X[1;0Y[29X[2XRound[102X( [3Xx[103X ) [32X attribute[133X
[33X[1;0Y[29X[2XTrunc[102X( [3Xx[103X ) [32X attribute[133X
[33X[1;0Y[29X[2XFrac[102X( [3Xx[103X ) [32X attribute[133X
[33X[1;0Y[29X[2XSignFloat[102X( [3Xx[103X ) [32X attribute[133X
[33X[1;0Y[29X[2XArgument[102X( [3Xx[103X ) [32X attribute[133X
[33X[1;0Y[29X[2XErf[102X( [3Xx[103X ) [32X attribute[133X
[33X[1;0Y[29X[2XZeta[102X( [3Xx[103X ) [32X attribute[133X
[33X[1;0Y[29X[2XGamma[102X( [3Xx[103X ) [32X attribute[133X
[33X[1;0Y[29X[2XComplexI[102X( [3Xx[103X ) [32X attribute[133X
[33X[0;0YUsual mathematical functions.[133X
[1X19.2-2 EqFloat[101X
[33X[1;0Y[29X[2XEqFloat[102X( [3Xx[103X, [3Xy[103X ) [32X operation[133X
[6XReturns:[106X [33X[0;10YWhether the floateans [3Xx[103X and [3Xy[103X are equal[133X
[33X[0;0YThis function compares two floating-point numbers, and returns [9Xtrue[109X if they
are equal, and [9Xfalse[109X otherwise; with the exception that [9XNaN[109X is always
considered to be different from itself.[133X
[1X19.2-3 PrecisionFloat[101X
[33X[1;0Y[29X[2XPrecisionFloat[102X( [3Xx[103X ) [32X attribute[133X
[6XReturns:[106X [33X[0;10YThe precision of [3Xx[103X[133X
[33X[0;0YThis function returns the precision, counted in number of binary digits, of
the floating-point number [3Xx[103X.[133X
[1X19.2-4 [33X[0;0YInterval operations[133X[101X
[33X[1;0Y[29X[2XSup[102X( [3Xinterval[103X ) [32X attribute[133X
[33X[1;0Y[29X[2XInf[102X( [3Xinterval[103X ) [32X attribute[133X
[33X[1;0Y[29X[2XMid[102X( [3Xinterval[103X ) [32X attribute[133X
[33X[1;0Y[29X[2XAbsoluteDiameter[102X( [3Xinterval[103X ) [32X attribute[133X
[33X[1;0Y[29X[2XRelativeDiameter[102X( [3Xinterval[103X ) [32X attribute[133X
[33X[1;0Y[29X[2XOverlaps[102X( [3Xinterval1[103X, [3Xinterval2[103X ) [32X operation[133X
[33X[1;0Y[29X[2XIsDisjoint[102X( [3Xinterval1[103X, [3Xinterval2[103X ) [32X operation[133X
[33X[1;0Y[29X[2XIncreaseInterval[102X( [3Xinterval[103X, [3Xdelta[103X ) [32X operation[133X
[33X[1;0Y[29X[2XBlowupInterval[102X( [3Xinterval[103X, [3Xratio[103X ) [32X operation[133X
[33X[1;0Y[29X[2XBisectInterval[102X( [3Xinterval[103X ) [32X operation[133X
[33X[0;0YMost are self-explanatory. [10XBlowupInterval[110X returns an interval with same
midpoint but relative diameter increased by [3Xratio[103X; [10XIncreaseInterval[110X returns
an interval with same midpoint but absolute diameter increased by [3Xdelta[103X;
[10XBisectInterval[110X returns a list of two intervals whose union equals [3Xinterval[103X.[133X
[1X19.2-5 IsPInfinity[101X
[33X[1;0Y[29X[2XIsPInfinity[102X( [3Xx[103X ) [32X property[133X
[33X[1;0Y[29X[2XIsNInfinity[102X( [3Xx[103X ) [32X property[133X
[33X[1;0Y[29X[2XIsXInfinity[102X( [3Xx[103X ) [32X property[133X
[33X[1;0Y[29X[2XIsFinite[102X( [3Xx[103X ) [32X property[133X
[33X[1;0Y[29X[2XIsNaN[102X( [3Xx[103X ) [32X property[133X
[33X[0;0YReturns [9Xtrue[109X if the floating-point number [3Xx[103X is respectively [22X+∞[122X, [22X-∞[122X, [22X±∞[122X,
finite, or `not a number', such as the result of [10X0.0/0.0[110X.[133X
[1X19.2-6 FLOAT[101X
[33X[1;0Y[29X[2XFLOAT[102X[32X global variable[133X
[33X[0;0YThis record contains useful floating-point constants:[133X
[8XDECIMAL_DIG[108X
[33X[0;6YMaximal number of useful digits;[133X
[8XDIG[108X
[33X[0;6YNumber of significant digits;[133X
[8XVIEW_DIG[108X
[33X[0;6YNumber of digits to print in short view;[133X
[8XEPSILON[108X
[33X[0;6YSmallest number such that [22X1≠1+ϵ[122X;[133X
[8XMANT_DIG[108X
[33X[0;6YNumber of bits in the mantissa;[133X
[8XMAX[108X
[33X[0;6YMaximal representable number;[133X
[8XMAX_10_EXP[108X
[33X[0;6YMaximal decimal exponent;[133X
[8XMAX_EXP[108X
[33X[0;6YMaximal binary exponent;[133X
[8XMIN[108X
[33X[0;6YMinimal positive representable number;[133X
[8XMIN_10_EXP[108X
[33X[0;6YMinimal decimal exponent;[133X
[8XMIN_EXP[108X
[33X[0;6YMinimal exponent;[133X
[8XINFINITY[108X
[33X[0;6YPositive infinity;[133X
[8XNINFINITY[108X
[33X[0;6YNegative infinity;[133X
[8XNAN[108X
[33X[0;6YNot-a-number,[133X
[33X[0;0Yas well as mathematical constants [10XE[110X, [10XLOG2E[110X, [10XLOG10E[110X, [10XLN2[110X, [10XLN10[110X, [10XPI[110X, [10XPI_2[110X,
[10XPI_4[110X, [10X1_PI[110X, [10X2_PI[110X, [10X2_SQRTPI[110X, [10XSQRT2[110X, [10XSQRT1_2[110X.[133X
[1X19.2-7 Float[101X
[33X[1;0Y[29X[2XFloat[102X( [3Xobj[103X ) [32X function[133X
[33X[1;0Y[29X[2XNewFloat[102X( [3Xfilter[103X, [3Xobj[103X ) [32X operation[133X
[33X[1;0Y[29X[2XMakeFloat[102X( [3Xsample[103X, [3Xobj[103X, [3Xobj[103X ) [32X operation[133X
[6XReturns:[106X [33X[0;10YA new floating-point number, based on [3Xobj[103X[133X
[33X[0;0YThis function creates a new floating-point number.[133X
[33X[0;0YIf [3Xobj[103X 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 [2XRat[102X ([14XReference: Rat[114X) and [2XRat[102X ([14X17.2-6[114X)). For an integer, the
precision, if unspecified, is chosen sufficient so that [10XInt(Float(obj))=obj[110X
always holds, but at least 64 bits.[133X
[33X[0;0Y[3Xobj[103X may also be a string, which may be of the form [10X"3.14e0"[110X or [10X".314e1"[110X or
[10X".314@1"[110X etc.[133X
[33X[0;0YAn option may be passed to specify, it bits, a desired precision. The format
is [10XFloat("3.14":PrecisionFloat:=1000)[110X to create a 1000-bit approximation of
[22X3.14[122X.[133X
[33X[0;0YIn particular, if [3Xobj[103X is already a floating-point number, then
[10XFloat(obj:PrecisionFloat:=prec)[110X creates a copy of [3Xobj[103X with a new precision.
prec[133X
[1X19.2-8 Rat[101X
[33X[1;0Y[29X[2XRat[102X( [3Xf[103X ) [32X attribute[133X
[6XReturns:[106X [33X[0;10YA rational approximation to [3Xf[103X[133X
[33X[0;0YThis command constructs a rational approximation to the floating-point
number [3Xf[103X. Of course, it is not guaranteed to return the original rational
number [3Xf[103X was created from, though it returns the most `reasonable' one given
the precision of [3Xf[103X.[133X
[33X[0;0YTwo options control the precision of the rational approximation: In the form
[10XRat(f:maxdenom:=md,maxpartial:=mp)[110X, the rational returned is such that the
denominator is at most [3Xmd[103X and the partials in its continued fraction
expansion are at most [3Xmp[103X. The default values are [10Xmaxpartial:=10000[110X and
[10Xmaxdenom:=2^(precision/2)[110X.[133X
[1X19.2-9 SetFloats[101X
[33X[1;0Y[29X[2XSetFloats[102X( [3Xrec[103X[, [3Xbits[103X][, [3Xinstall[103X] ) [32X function[133X
[33X[0;0YInstalls a new interface to floating-point numbers in [5XGAP[105X, optionally with a
desired precision [3Xbits[103X in binary digits. The last optional argument [3Xinstall[103X
is a boolean value; if false, it only installs the eager handler and the
precision for the floateans, without making them the default.[133X
[1X19.3 [33X[0;0YHigh-precision-specific methods[133X[101X
[33X[0;0Y[5XGAP[105X provides a mechanism for packages to implement new floating-point
numerical interfaces. The following describes that mechanism, actual
examples of packages are documented separately.[133X
[33X[0;0YA package must create a record with fields (all optional)[133X
[8Xcreator[108X
[33X[0;6Ya function converting strings to floating-point;[133X
[8Xeager[108X
[33X[0;6Ya character allowing immediate conversion to floating-point;[133X
[8Xobjbyextrep[108X
[33X[0;6Ya function creating a floating-point number out of a list
[10X[mantissa,exponent][110X;[133X
[8Xfilter[108X
[33X[0;6Ya filter for the new floating-point objects;[133X
[8Xconstants[108X
[33X[0;6Ya record containing numerical constants, such as [10XMANT_DIG[110X, [10XMAX[110X, [10XMIN[110X,
[10XNAN[110X.[133X
[33X[0;0YThe package must install methods [10XInt[110X, [10XRat[110X, [10XString[110X for its objects, and
creators [10XNewFloat(filter,IsRat)[110X, [10XNewFloat(IsString)[110X.[133X
[33X[0;0YIt must then install methods for all arithmetic and numerical operations:
[10XPLUS[110X, [10XExp[110X, ...[133X
[33X[0;0YThe user chooses that implementation by calling [2XSetFloats[102X ([14X19.2-9[114X) with the
record as argument, and with an optional second argument requesting a
precision in binary digits.[133X
[1X19.4 [33X[0;0YComplex arithmetic[133X[101X
[33X[0;0YComplex arithmetic may be implemented in packages, and is present in [5Xfloat[105X.
Complex numbers are treated as usual numbers; they may be input with an
extra "i" as in [10X-0.5+0.866i[110X.[133X
[33X[0;0YMethods should then be implemented for [10XNorm[110X, [10XRealPart[110X, [10XImaginaryPart[110X,
[10XComplexConjugate[110X, ...[133X
[1X19.5 [33X[0;0YInterval-specific methods[133X[101X
[33X[0;0YInterval 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 [10XEQ[110X 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. The methods provided by the
package should include
[10XSup,Inf,Mid,DiameterOfInterval,Overlaps,IsSubset,IsDisjoint[110X.[133X
[33X[0;0YNote the usual convention that intervals are compared as in [22X[a,b]le[c,d][122X if
and only if [22Xale c[122X and [22Xble d[122X.[133X
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