Documentation

A Semifield is chosen here to be a Field without an Additive Inverse

A Field is a Ring plus additive invertible and multiplicative invertible operations.

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 * b == b * a
a / a = one
recip a = one / a
recip a * a = one
a * recip a = one

A hyperbolic field class

sqrt . (**2) == identity
log . exp == identity
for +ive b, a != 0,1: a ** logBase a b ≈ b

quotient fields explode constraints if they allow for polymorphic integral types

a - one < floor a <= a <= ceiling a < a + one
round a == floor (a + one/(one+one))

fixme: had to redefine Signed operators here because of the Field import in Metric, itself due to Complex being defined there

A bounded field includes the concepts of infinity and NaN, thus moving away from error throwing.

one / zero + infinity == infinity
infinity + a == infinity
zero / zero != nan

Note the tricky law that, although nan is assigned to zero/zero, they are never-the-less not equal. A committee decided this.

Trigonometric Field