float

absolute value using `|a|` built from a `prefix |` and `postfix |` as an operator
alias of `a.abs`

Due to the low precedence of `|`, this works also on expressions like `|a-b|`, even
with spaces `| a-b |`, `|a - b|`, `| a-b|` or `|a-b |`.

Nesting, however, does not work, e.g, `| - |a| |`, this requires parentheses `|(- |a|)|`.

NYI: CLEANUP: Due to #3081, we need `postfix |` as the first operation, should be
`prefix |` first

absolute value

convert a float value to i32 dropping any fraction.
the value must be in the range of i32

convert a float value to i64 dropping any fraction.
the value must be in the range of i64

create a String from this instance. Unless redefined, `a.as_string` will
create `"instance[T]"` where `T` is the dynamic type of `a`

this numeric value as an u8

ceiling: the smallest integer greater or equal to this

Get the dynamic type of this instance. For value instances `x`, this is
equal to `type_of x`, but for `x` with a `ref` type `x.dynamic_type` gives
the actual runtime type, while `type_of x` results in the static
compile-time type.

There is no dynamic type of a type instance since this would result in an
endless hierarchy of types. So for Type values, dynamic_type is redefined
to just return Type.type.

the `val`-th power of ℇ
i.e. ℇᵛᵃˡ

the normalized exponent of this float

the biased exponent of this float

does this float value fit into an i64? This is redefined by children
of float that support `as_i64`.

does this numeric value fit into an u8? This is redefined by children
of numeric that support `as_u8`.

floor: the greatest integer lower or equal to this

basic operations: 'infix %' (division remainder)

basic operations: 'infix *' (multiplication)

basic operations: 'infix **' (exponentiation)

basic operations: 'infix +' (addition)

basic operations: 'infix -' (subtraction)

basic operations: 'infix /' (division)

comparison

This provides comparison operators using IEEE semantics.

type.equality and type.lteq should be used for equivalence relations
and total ordering in the mathematical sense.

is not a number?

is the bit denoting the sign of the number set?
this is different from smaller than zero since
there is +0, -0, NaN, etc. in floating point numbers.

logarithm with base ℇ

logarithm with base `base`

the normalized mantissa of this float

convenience prefix operator to create a string from a value.

This permits usage of `$` as a prefix operator in a similar way both
inside and outside of constant strings: $x and "$x" will produce the
same string.

basic operations: 'prefix +' (identity)

preconditions for basic operations: true if the operation's result is
representable and defined for the given values

default implementations all return `true` such that children have to
redefine these only for partial operations such as those resulting in
an overflow or that are undefined like a division by zero for most
types.

basic operations: 'prefix -' (negation)

overflow checking operations

saturating operations

round floating point number
ties to away (0.5 => 1; -0.5 => -1; etc.)

NYI this could be made faster, see here:
https://cs.opensource.google/go/go/+/refs/tags/go1.18.1:src/math/floor.go;l=79

sign function resulting in `-1`/`0`/`+1` depending on whether `numeric.this`
is less than, equal or larger than zero

square root alias

square root

true when the absolute value
is smaller than 0.001
or greater than 9_999_999

string representation of this type to be used for debugging.

result has the form "Type of '<name>'", but this might change in the future

number of bytes required to store this value

There is no dynamic type of a type instance since this would result in an
endless hierarchy of types, so dynamic_type is redefined to just return
Type.type here.

equality

This provides an equivalence relation in the mathematical
sense and therefore breaks the IEEE semantics. infix = should
be used for IEEE semantics.

The equivalence relation is provided by the usual comparison
of floating-point numbers, with the definition of NaN = NaN.

number of bits used for exponent

convert an i64 value to a floating point value

if the value cannot be represented exactly, the result is either
the nearest higher or nearest lower value

the value corresponding to v in whatever integer implementation we have,
maximum in case of overflow

create hash code for this instance

This should satisfy the following condition:

(T.equality a b) : (T.hash_code a = T.hash_code b)

infinity

Is this type assignable to a type parameter with constraint `T`?

The result of this is a compile-time constant that can be used to specialize
code for a particular type.

is_of_integer_type(n T : numeric) => T : integer
say (is_of_integer_type 1234) # true
say (is_of_integer_type 3.14) # false

it is most useful in conjunction preconditions or `if` statements as in

pair(a,b T) is

=>

or

val(n T) is

additional restrictions on when equality is permitted,
e.g., `option T` might require `T : property.equatable`.
to implement `equality`

total order

This provides a total order in the mathematical sense and
therefore breaks the IEEE semantics. infix <= should be
used for IEEE semantics.

The total order is provided by the usual comparison of
floating-point numbers, with the definition that NaN <= x,
for any x (including x = NaN).

the amount of bits that are used to encode the mantissa

name of this type, including type parameters, e.g. 'option (list i32)'.

not a number

identity element for 'infix *'

convenience prefix operator to create a string from a value.

This permits usage of `$` as a prefix operator in a similar way both
inside and outside of constant strings: $x and "$x" will produce the
same string.

NYI: Redefinition allows the type feature to be distinguished from its normal counterpart, see #3913

monoid of numeric with infix * operation. Will create product of all elements
it is applied to.

monoid of numeric with infix *^ operation. Will create product of all elements
it is applied to, stopping at max/min value in case of overflow.

non signaling not a number

number of bits used for mantissa,
including leading '1' that is not actually stored

number of bits required to store this value

monoid of numeric with infix + operation. Will create sum of all elements it
is applied to.

monoid of numeric with infix +^ operation. Will create sum of all elements it
is applied to, stopping at max/min value in case of overflow.

the constant '10' in whatever integer implementation we have, maximum in case of overflow

the constant '2' in whatever integer implementation we have, maximum in case of overflow

Get a type as a value.

This is a feature with the effect equivalent to Fuzion's `expr.type` call tail.
It is recommended to use `expr.type` and not `expr.type_value`.

`type_value` is here to show how this can be implemented and to illustrate the
difference to `dynamic_type`.

identity element for 'infix +'

pi 3.14...

eulers number 2.72..