Floats

### Floats


[
  Datatype float-formats
  Funcon   binary32
  Funcon   binary64
  Funcon   binary128
  Funcon   decimal64
  Funcon   decimal128
  Type     floats
  Funcon   float
  Funcon   quiet-not-a-number       Alias qNaN
  Funcon   signaling-not-a-number   Alias sNaN
  Funcon   positive-infinity        Alias pos-inf
  Funcon   negative-infinity        Alias neg-inf
  Funcon   float-convert
  Funcon   decimal-float
  Funcon   float-equal
  Funcon   float-is-less
  Funcon   float-is-less-or-equal
  Funcon   float-is-greater
  Funcon   float-is-greater-or-equal
  Funcon   float-negate
  Funcon   float-absolute-value
  Funcon   float-add
  Funcon   float-subtract
  Funcon   float-multiply
  Funcon   float-multiply-add
  Funcon   float-divide
  Funcon   float-remainder
  Funcon   float-sqrt
  Funcon   float-integer-power
  Funcon   float-float-power
  Funcon   float-round-ties-to-even
  Funcon   float-round-ties-to-infinity
  Funcon   float-floor
  Funcon   float-ceiling
  Funcon   float-truncate
  Funcon   float-pi
  Funcon   float-e
  Funcon   float-log
  Funcon   float-log10
  Funcon   float-exp
  Funcon   float-sin
  Funcon   float-cos
  Funcon   float-tan
  Funcon   float-asin
  Funcon   float-acos
  Funcon   float-atan
  Funcon   float-sinh
  Funcon   float-cosh
  Funcon   float-tanh
  Funcon   float-asinh
  Funcon   float-acosh
  Funcon   float-atanh
  Funcon   float-atan2
]

/*

  Floating-point numbers according to the IEEE 754 Standard (2008).

  See:
    - http://doi.org/10.1109/IEEESTD.2008.4610935
    - https://en.wikipedia.org/wiki/IEEE_754

*/

Datatype
  float-formats ::= binary32 | binary64 | binary128 | decimal64 | decimal128
Built-in Type
  floats(_:float-formats)

/*

  Note that for distinct formats `FF₁`, `FF₂`, the types `floats(FF₁)` and
  `floats(FF₂)` are not necessarily disjoint.

*/

Built-in Funcon
  float(FF:float-formats, 
        _:bounded-integers(0, 1), _:natural-numbers, _:integers) : =>floats(FF)

/*

  Each finite number is described by three integers: 
  * s = a sign (zero or one), 
  * c = a significand (or 'coefficient'), 
  * q = an exponent. 
  The numerical value of a finite number is (-1)^s * c * b^q
  where b is the base (2 or 10), also called radix.
  
  The possible finite values that can be represented in a format
  are determined by the base b, the number of digits in the significand 
  (precision p), and the exponent parameter emax:
  * c must be an integer in the range zero through (b^p)-1
    (e.g., if b=10 and p=7 then c is 0 through 9999999);
  * q must be an integer such that 1-emax <= q+p-1 <= emax
    (e.g., if p=7 and emax=96 then q is -101 through 90).

  Note that `float(FF, S, C, Q)` is not a 1-1 operation.

*/

Built-in Funcon
  quiet-not-a-number(FF:float-formats) : floats(FF)
Alias
  qNaN = quiet-not-a-number
Built-in Funcon
  signaling-not-a-number(FF:float-formats) : floats(FF)
Alias
  sNaN = signaling-not-a-number
Built-in Funcon
  positive-infinity(FF:float-formats) : floats(FF)
Alias
  pos-inf = positive-infinity
Built-in Funcon
  negative-infinity(FF:float-formats) : floats(FF)
Alias
  neg-inf = negative-infinity

#### Conversions


Built-in Funcon
  float-convert(FF₁:float-formats, 
                FF₂:float-formats, F:floats(FF₁)) : =>floats(FF₂)
Built-in Funcon
  decimal-float(FF:float-formats, 
                _:strings, _:strings, _:strings) : =>floats(FF)

/*

  `decimal-float(F, "M", "N", "E")` is an approximation in `floats(FF)` to the
  value of 'M.N' times 10 to the power 'E', where "M.N" is decimal notation
  (optionally-signed) for a fixed-point number and "E" is decimal notation
  (optionally signed) for an integer. When any argument string is invalid,
  the result is `quiet-not-a-number(F)`.

*/

#### Comparison


Built-in Funcon
  float-equal(FF:float-formats,
              _:floats(FF), _:floats(FF)) : =>booleans
Built-in Funcon
  float-is-less(FF:float-formats,
                _:floats(FF), _:floats(FF)) : =>booleans
Built-in Funcon
  float-is-less-or-equal(FF:float-formats,
                         _:floats(FF), _:floats(FF)) : =>booleans
Built-in Funcon
  float-is-greater(FF:float-formats,
                   _:floats(FF), _:floats(FF)) : =>booleans
Built-in Funcon
  float-is-greater-or-equal(FF:float-formats,
                            _:floats(FF), _:floats(FF)) : =>booleans

#### Arithmetic


Built-in Funcon
  float-negate(FF:float-formats, _:floats(FF)) : =>floats(FF)
Built-in Funcon
  float-absolute-value(FF:float-formats, _:floats(FF)) : =>floats(FF)
Built-in Funcon
  float-add(FF:float-formats, _:floats(FF), _:floats(FF)) : =>floats(FF)
Built-in Funcon
  float-subtract(FF:float-formats, _:floats(FF), _:floats(FF)) : =>floats(FF)
Built-in Funcon
  float-multiply(FF:float-formats, _:floats(FF), _:floats(FF)) : =>floats(FF)
Built-in Funcon
  float-multiply-add(FF:float-formats,
                     _:floats(FF), _:floats(FF), _:floats(FF)) : =>floats(FF)
Built-in Funcon
  float-divide(FF:float-formats, _:floats(FF), _:floats(FF)) : =>floats(FF)
Built-in Funcon
  float-remainder(FF:float-formats, _:floats(FF), _:floats(FF)) : =>floats(FF)
Built-in Funcon
  float-sqrt(FF:float-formats, _:floats(FF)) : =>floats(FF)
Built-in Funcon
  float-integer-power(FF:float-formats, _:floats(FF), _:integers) : =>floats(FF)
Built-in Funcon
  float-float-power(FF:float-formats, _:floats(FF), _:floats(FF)) : =>floats(FF)

#### Rounding


Built-in Funcon
  float-round-ties-to-even(FF:float-formats, _:floats(FF)) : =>integers
Built-in Funcon
  float-round-ties-to-infinity(FF:float-formats, _:floats(FF)) : =>integers
Built-in Funcon
  float-floor(FF:float-formats, _:floats(FF)) : =>integers
Built-in Funcon
  float-ceiling(FF:float-formats, _:floats(FF)) : =>integers
Built-in Funcon
  float-truncate(FF:float-formats, _:floats(FF)) : =>integers

#### Miscellaneous


Built-in Funcon
  float-pi(FF:float-formats) : =>floats(FF)
Built-in Funcon
  float-e(FF:float-formats) : =>floats(FF)
Built-in Funcon
  float-log(FF:float-formats, _:floats(FF)) : =>floats(FF)
Built-in Funcon
  float-log10(FF:float-formats, _:floats(FF)) : =>floats(FF)
Built-in Funcon
  float-exp(FF:float-formats, _:floats(FF)) : =>floats(FF)
Built-in Funcon
  float-sin(FF:float-formats, _:floats(FF)) : =>floats(FF)
Built-in Funcon
  float-cos(FF:float-formats, _:floats(FF)) : =>floats(FF)
Built-in Funcon
  float-tan(FF:float-formats, _:floats(FF)) : =>floats(FF)
Built-in Funcon
  float-asin(FF:float-formats, _:floats(FF)) : =>floats(FF)
Built-in Funcon
  float-acos(FF:float-formats, _:floats(FF)) : =>floats(FF)
Built-in Funcon
  float-atan(FF:float-formats, _:floats(FF)) : =>floats(FF)
Built-in Funcon
  float-sinh(FF:float-formats, _:floats(FF)) : =>floats(FF)
Built-in Funcon
  float-cosh(FF:float-formats, _:floats(FF)) : =>floats(FF)
Built-in Funcon
  float-tanh(FF:float-formats, _:floats(FF)) : =>floats(FF)
Built-in Funcon
  float-asinh(FF:float-formats, _:floats(FF)) : =>floats(FF)
Built-in Funcon
  float-acosh(FF:float-formats, _:floats(FF)) : =>floats(FF)
Built-in Funcon
  float-atanh(FF:float-formats, _:floats(FF)) : =>floats(FF)
Built-in Funcon
  float-atan2(FF:float-formats, _:floats(FF), _:floats(FF)) : =>floats(FF)