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A Semifield is a field with no subtraction.

Since: 0.12

A Field is a set on which addition, subtraction, multiplication, and division are defined. It is also assumed that multiplication is distributive over addition.

A summary of the rules inherited from super-classes of Field:

zero + a == a
a + zero == a
((a + b) + c) (a + (b + c))
a + b == b + a
a - a == zero
negate a == zero - a
negate a + a == zero
a + negate a == zero
one * a == a
a * one == a
((a * b) * c) == (a * (b * c))
(a * (b + c)) == (a * b + a * c)
((a + b) * c) == (a * c + b * c)
a * zero == zero
zero * a == zero
a / a == one || a == zero
recip a == one / a || a == zero
recip a * a == one || a == zero
a * recip a == one || a == zero

A hyperbolic field class

a < zero || (sqrt . (**2)) a == a
a < zero || (log . exp) a ~= a
b < zero || a <= zero || a == 1 || abs (a ** logBase a b - b) < 10 * epsilon

>>> (sqrt . (**2)) (4.0 :: Double) == 4.0
True

>>> (log . exp) (1.0 :: Double) ~= 1.0
True

>>> 2 ** logBase 2 8 ~= (8 :: Double)
True

Quotienting of a Field into a Ring

See Field of fractions

\a -> a - one < floor a <= a <= ceiling a < a + one

>>> properFraction (1.5 :: Double)
Quotient {whole = 1, fraction = 0.5}

>>> fromQuotient (properFraction (1.5 :: Double))
1.5

Whole-and-fraction pair. properFraction is the normalizer (Prelude toward-zero). The constructor is raw; the range is not enforced.

properFraction as the named normalizer into Quotient.

The canonical range is the Prelude toward-zero split: fraction ∈ (-1, 1) with the sign of the input, and fromQuotient (toQuotient a) == a.

Recombine a Quotient. Law: fromQuotient (toQuotient a) == a.

Bilinear multiplication of quotients: recombine, multiply, re-split.

(w1 + f1) * (w2 + f2) expands to the four terms w1*w2 + w1*f2 + f1*w2 + f1*f2, and toQuotient pushes any overflow back into whole. The same bilinear-then-normalize pattern as complex multiplication.

>>> mulQuotient (toQuotient (0.5 :: Double)) (toQuotient (3.0 :: Double))
Quotient {whole = 1, fraction = 0.5}

>>> mulQuotient (toQuotient (1.5 :: Double)) (toQuotient (2.0 :: Double)) == toQuotient (1.5 * 2.0 :: Double)
True

infinity is defined for any Field.

>>> one / zero + infinity
Infinity

>>> infinity + 1
Infinity

negative infinity

>>> negInfinity + infinity
NaN

nan is defined as zero/zero

but note the (social) law:

>>> nan == zero / zero
False

Trigonometric Field

The list of laws is quite long: trigonometric identities

A half of one

>>> half :: Double
0.5

Approximate modulo for fields

>>> modF 1.5 1.2
0.30000000000000004

Since: 0.13

Approximate diviso for fields.

Compared with div, divF returns the original type rather than the Whole type.

>>> divF 1.5 1.2
1.0

Since: 0.13

Approximate divMod for fields.

>>> divModF 1.5 1.2
DivMod {div_ = 1.0, mod_ = 0.30000000000000004}

Since: 0.13