FloatingPointSystemΒΆ

sf.spad line 78

This category is intended as a model for floating point systems. A floating point system is a model for the real numbers. In fact, it is an approximation in the sense that not all real numbers are exactly representable by floating point numbers. A floating point system is characterized by the following: 1: base of the exponent. (actual implemenations are usually binary or decimal) 2: precision of the mantissa (arbitrary or fixed) 3: rounding error for operations Because a Float is an approximation to the real numbers, even though it is defined to be a join of a Field and OrderedRing, some of the attributes do not hold. In particular associative("+") does not hold. Algorithms defined over a field need special considerations when the field is a floating point system.

0: %
from AbelianMonoid
1: %
from MagmaWithUnit
*: (%, %) -> %
from Magma
*: (%, Fraction Integer) -> %
from RightModule Fraction Integer
*: (Fraction Integer, %) -> %
from LeftModule Fraction Integer
*: (Integer, %) -> %
from AbelianGroup
*: (NonNegativeInteger, %) -> %
from AbelianMonoid
*: (PositiveInteger, %) -> %
from AbelianSemiGroup
+: (%, %) -> %
from AbelianSemiGroup
-: % -> %
from AbelianGroup
-: (%, %) -> %
from AbelianGroup
/: (%, %) -> %
from Field
<: (%, %) -> Boolean
from PartialOrder
<=: (%, %) -> Boolean
from PartialOrder
=: (%, %) -> Boolean
from BasicType
>: (%, %) -> Boolean
from PartialOrder
>=: (%, %) -> Boolean
from PartialOrder
^: (%, Fraction Integer) -> %
from RadicalCategory
^: (%, Integer) -> %
from DivisionRing
^: (%, NonNegativeInteger) -> %
from MagmaWithUnit
^: (%, PositiveInteger) -> %
from Magma
~=: (%, %) -> Boolean
from BasicType
abs: % -> %
from OrderedRing
annihilate?: (%, %) -> Boolean
from Rng
antiCommutator: (%, %) -> %
from NonAssociativeSemiRng
associates?: (%, %) -> Boolean
from EntireRing
associator: (%, %, %) -> %
from NonAssociativeRng
base: () -> PositiveInteger
base() returns the base of the exponent.
bits: () -> PositiveInteger
bits() returns ceiling's precision in bits.
ceiling: % -> %
from RealNumberSystem
characteristic: () -> NonNegativeInteger
from NonAssociativeRing
coerce: % -> %
from Algebra %
coerce: % -> OutputForm
from CoercibleTo OutputForm
coerce: Fraction Integer -> %
from RetractableTo Fraction Integer
coerce: Integer -> %
from NonAssociativeRing
commutator: (%, %) -> %
from NonAssociativeRng
convert: % -> DoubleFloat
from ConvertibleTo DoubleFloat
convert: % -> Float
from ConvertibleTo Float
convert: % -> Pattern Float
from ConvertibleTo Pattern Float
digits: () -> PositiveInteger
digits() returns ceiling's precision in decimal digits.
divide: (%, %) -> Record(quotient: %, remainder: %)
from EuclideanDomain
euclideanSize: % -> NonNegativeInteger
from EuclideanDomain
exponent: % -> Integer
exponent(x) returns the exponent part of x.
expressIdealMember: (List %, %) -> Union(List %, failed)
from PrincipalIdealDomain
exquo: (%, %) -> Union(%, failed)
from EntireRing
extendedEuclidean: (%, %) -> Record(coef1: %, coef2: %, generator: %)
from EuclideanDomain
extendedEuclidean: (%, %, %) -> Union(Record(coef1: %, coef2: %), failed)
from EuclideanDomain
factor: % -> Factored %
from UniqueFactorizationDomain
float: (Integer, Integer) -> %
float(a, e) returns a * base() ^ e.
float: (Integer, Integer, PositiveInteger) -> %
float(a, e, b) returns a * b ^ e.
floor: % -> %
from RealNumberSystem
fractionPart: % -> %
from RealNumberSystem
gcd: (%, %) -> %
from GcdDomain
gcd: List % -> %
from GcdDomain
gcdPolynomial: (SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> SparseUnivariatePolynomial %
from GcdDomain
hash: % -> SingleInteger
from SetCategory
hashUpdate!: (HashState, %) -> HashState
from SetCategory
inv: % -> %
from DivisionRing
latex: % -> String
from SetCategory
lcm: (%, %) -> %
from GcdDomain
lcm: List % -> %
from GcdDomain
lcmCoef: (%, %) -> Record(llcm_res: %, coeff1: %, coeff2: %)
from LeftOreRing
leftPower: (%, NonNegativeInteger) -> %
from MagmaWithUnit
leftPower: (%, PositiveInteger) -> %
from Magma
leftRecip: % -> Union(%, failed)
from MagmaWithUnit
mantissa: % -> Integer
mantissa(x) returns the mantissa part of x.
max: (%, %) -> %
from OrderedSet
max: () -> % if % hasn’t arbitraryPrecision and % hasn’t arbitraryExponent
max() returns the maximum floating point number.
min: (%, %) -> %
from OrderedSet
min: () -> % if % hasn’t arbitraryPrecision and % hasn’t arbitraryExponent
min() returns the minimum floating point number.
multiEuclidean: (List %, %) -> Union(List %, failed)
from EuclideanDomain
negative?: % -> Boolean
from OrderedRing
norm: % -> %
from RealNumberSystem
nthRoot: (%, Integer) -> %
from RadicalCategory
one?: % -> Boolean
from MagmaWithUnit
opposite?: (%, %) -> Boolean
from AbelianMonoid
order: % -> Integer
order x is the order of magnitude of x. Note: base ^ order x <= |x| < base ^ (1 + order x).
patternMatch: (%, Pattern Float, PatternMatchResult(Float, %)) -> PatternMatchResult(Float, %)
from PatternMatchable Float
positive?: % -> Boolean
from OrderedRing
precision: () -> PositiveInteger
precision() returns the precision in digits base.
prime?: % -> Boolean
from UniqueFactorizationDomain
principalIdeal: List % -> Record(coef: List %, generator: %)
from PrincipalIdealDomain
quo: (%, %) -> %
from EuclideanDomain
recip: % -> Union(%, failed)
from MagmaWithUnit
rem: (%, %) -> %
from EuclideanDomain
retract: % -> Fraction Integer
from RetractableTo Fraction Integer
retract: % -> Integer
from RetractableTo Integer
retractIfCan: % -> Union(Fraction Integer, failed)
from RetractableTo Fraction Integer
retractIfCan: % -> Union(Integer, failed)
from RetractableTo Integer
rightPower: (%, NonNegativeInteger) -> %
from MagmaWithUnit
rightPower: (%, PositiveInteger) -> %
from Magma
rightRecip: % -> Union(%, failed)
from MagmaWithUnit
round: % -> %
from RealNumberSystem
sample: %
from AbelianMonoid
sign: % -> Integer
from OrderedRing
sizeLess?: (%, %) -> Boolean
from EuclideanDomain
smaller?: (%, %) -> Boolean
from Comparable
sqrt: % -> %
from RadicalCategory
squareFree: % -> Factored %
from UniqueFactorizationDomain
squareFreePart: % -> %
from UniqueFactorizationDomain
subtractIfCan: (%, %) -> Union(%, failed)
from CancellationAbelianMonoid
toString: (%, NonNegativeInteger) -> String
toString(x, n) returns a string representation of x truncated to n decimal digits.
truncate: % -> %
from RealNumberSystem
unit?: % -> Boolean
from EntireRing
unitCanonical: % -> %
from EntireRing
unitNormal: % -> Record(unit: %, canonical: %, associate: %)
from EntireRing
wholePart: % -> Integer
from RealNumberSystem
zero?: % -> Boolean
from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianSemiGroup

Algebra %

Algebra Fraction Integer

Approximate

BasicType

BiModule(%, %)

BiModule(Fraction Integer, Fraction Integer)

CancellationAbelianMonoid

canonicalsClosed

canonicalUnitNormal

CharacteristicZero

CoercibleTo OutputForm

CommutativeRing

CommutativeStar

Comparable

ConvertibleTo DoubleFloat

ConvertibleTo Float

ConvertibleTo Pattern Float

DivisionRing

EntireRing

EuclideanDomain

Field

GcdDomain

IntegralDomain

LeftModule %

LeftModule Fraction Integer

LeftOreRing

Magma

MagmaWithUnit

Module %

Module Fraction Integer

Monoid

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors

OrderedAbelianGroup

OrderedAbelianMonoid

OrderedAbelianSemiGroup

OrderedCancellationAbelianMonoid

OrderedRing

OrderedSet

PartialOrder

PatternMatchable Float

PrincipalIdealDomain

RadicalCategory

RealConstant

RealNumberSystem

RetractableTo Fraction Integer

RetractableTo Integer

RightModule %

RightModule Fraction Integer

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

UniqueFactorizationDomain

unitsKnown