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Funcons-beta : Floats.cbs | PLAIN | PDF

OUTLINE

Floats

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

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

See:

\[\begin{align*} \KEY{Datatype} \quad \NAMEDECL{float-formats} \ ::= \ & \NAMEDECL{binary32} \mid \NAMEDECL{binary64} \mid \NAMEDECL{binary128} \mid \NAMEDECL{decimal64} \mid \NAMEDECL{decimal128} \end{align*}\] \[\begin{align*} \KEY{Built-in Type} \quad & \NAMEDECL{floats}( \_ : \NAMEREF{float-formats}) \end{align*}\]

Note that for distinct formats \(\SHADE{\VAR{FF}\SUB{1}}\), \(\SHADE{\VAR{FF}\SUB{2}}\), the types \(\SHADE{\NAMEREF{floats} ( \VAR{FF}\SUB{1} )}\) and \(\SHADE{\NAMEREF{floats} ( \VAR{FF}\SUB{2} )}\) are not necessarily disjoint.

\[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{float}(\\&\quad \VAR{FF} : \NAMEREF{float-formats}, \\&\quad \_ : \NAMEHYPER{../.}{Integers}{bounded-integers} ( 0, 1 ), \_ : \NAMEHYPER{../.}{Integers}{natural-numbers}, \_ : \NAMEHYPER{../.}{Integers}{integers}) \\&\quad : \TO \NAMEREF{floats} ( \VAR{FF} ) \end{align*}\]

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 \(\SHADE{\NAMEREF{float} ( \VAR{FF}, \VAR{S}, \VAR{C}, \VAR{Q} )}\) is not a 1-1 operation.

\[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{quiet-not-a-number}( \VAR{FF} : \NAMEREF{float-formats}) : \NAMEREF{floats} ( \VAR{FF} ) \\ \KEY{Alias} \quad & \NAMEDECL{qNaN} = \NAMEREF{quiet-not-a-number} \end{align*}\] \[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{signaling-not-a-number}( \VAR{FF} : \NAMEREF{float-formats}) : \NAMEREF{floats} ( \VAR{FF} ) \\ \KEY{Alias} \quad & \NAMEDECL{sNaN} = \NAMEREF{signaling-not-a-number} \end{align*}\] \[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{positive-infinity}( \VAR{FF} : \NAMEREF{float-formats}) : \NAMEREF{floats} ( \VAR{FF} ) \\ \KEY{Alias} \quad & \NAMEDECL{pos-inf} = \NAMEREF{positive-infinity} \end{align*}\] \[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{negative-infinity}( \VAR{FF} : \NAMEREF{float-formats}) : \NAMEREF{floats} ( \VAR{FF} ) \\ \KEY{Alias} \quad & \NAMEDECL{neg-inf} = \NAMEREF{negative-infinity} \end{align*}\]

Conversions

\[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{float-convert}(\\&\quad \VAR{FF}\SUB{1} : \NAMEREF{float-formats}, \\&\quad \VAR{FF}\SUB{2} : \NAMEREF{float-formats}, \VAR{F} : \NAMEREF{floats} ( \VAR{FF}\SUB{1} )) \\&\quad : \TO \NAMEREF{floats} ( \VAR{FF}\SUB{2} ) \end{align*}\] \[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{decimal-float}(\\&\quad \VAR{FF} : \NAMEREF{float-formats}, \\&\quad \_ : \NAMEHYPER{../../Composite}{Strings}{strings}, \_ : \NAMEHYPER{../../Composite}{Strings}{strings}, \_ : \NAMEHYPER{../../Composite}{Strings}{strings}) \\&\quad : \TO \NAMEREF{floats} ( \VAR{FF} ) \end{align*}\]

\(\SHADE{\NAMEREF{decimal-float} ( \VAR{F}, \STRING{M}, \STRING{N}, \STRING{E} )}\) is an approximation in \(\SHADE{\NAMEREF{floats} ( \VAR{FF} )}\) 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 \(\SHADE{\NAMEREF{quiet-not-a-number} ( \VAR{F} )}\).

Comparison

\[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{float-equal}(\\&\quad \VAR{FF} : \NAMEREF{float-formats}, \\&\quad \_ : \NAMEREF{floats} ( \VAR{FF} ), \_ : \NAMEREF{floats} ( \VAR{FF} )) \\&\quad : \TO \NAMEHYPER{../.}{Booleans}{booleans} \end{align*}\] \[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{float-is-less}(\\&\quad \VAR{FF} : \NAMEREF{float-formats}, \\&\quad \_ : \NAMEREF{floats} ( \VAR{FF} ), \_ : \NAMEREF{floats} ( \VAR{FF} )) \\&\quad : \TO \NAMEHYPER{../.}{Booleans}{booleans} \end{align*}\] \[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{float-is-less-or-equal}(\\&\quad \VAR{FF} : \NAMEREF{float-formats}, \\&\quad \_ : \NAMEREF{floats} ( \VAR{FF} ), \_ : \NAMEREF{floats} ( \VAR{FF} )) \\&\quad : \TO \NAMEHYPER{../.}{Booleans}{booleans} \end{align*}\] \[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{float-is-greater}(\\&\quad \VAR{FF} : \NAMEREF{float-formats}, \\&\quad \_ : \NAMEREF{floats} ( \VAR{FF} ), \_ : \NAMEREF{floats} ( \VAR{FF} )) \\&\quad : \TO \NAMEHYPER{../.}{Booleans}{booleans} \end{align*}\] \[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{float-is-greater-or-equal}(\\&\quad \VAR{FF} : \NAMEREF{float-formats}, \\&\quad \_ : \NAMEREF{floats} ( \VAR{FF} ), \_ : \NAMEREF{floats} ( \VAR{FF} )) \\&\quad : \TO \NAMEHYPER{../.}{Booleans}{booleans} \end{align*}\]

Arithmetic

\[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{float-negate}( \VAR{FF} : \NAMEREF{float-formats}, \_ : \NAMEREF{floats} ( \VAR{FF} )) : \TO \NAMEREF{floats} ( \VAR{FF} ) \end{align*}\] \[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{float-absolute-value}( \VAR{FF} : \NAMEREF{float-formats}, \_ : \NAMEREF{floats} ( \VAR{FF} )) : \TO \NAMEREF{floats} ( \VAR{FF} ) \end{align*}\] \[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{float-add}( \VAR{FF} : \NAMEREF{float-formats}, \_ : \NAMEREF{floats} ( \VAR{FF} ), \_ : \NAMEREF{floats} ( \VAR{FF} )) : \TO \NAMEREF{floats} ( \VAR{FF} ) \end{align*}\] \[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{float-subtract}( \VAR{FF} : \NAMEREF{float-formats}, \_ : \NAMEREF{floats} ( \VAR{FF} ), \_ : \NAMEREF{floats} ( \VAR{FF} )) : \TO \NAMEREF{floats} ( \VAR{FF} ) \end{align*}\] \[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{float-multiply}( \VAR{FF} : \NAMEREF{float-formats}, \_ : \NAMEREF{floats} ( \VAR{FF} ), \_ : \NAMEREF{floats} ( \VAR{FF} )) : \TO \NAMEREF{floats} ( \VAR{FF} ) \end{align*}\] \[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{float-multiply-add}(\\&\quad \VAR{FF} : \NAMEREF{float-formats}, \\&\quad \_ : \NAMEREF{floats} ( \VAR{FF} ), \_ : \NAMEREF{floats} ( \VAR{FF} ), \_ : \NAMEREF{floats} ( \VAR{FF} )) \\&\quad : \TO \NAMEREF{floats} ( \VAR{FF} ) \end{align*}\] \[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{float-divide}( \VAR{FF} : \NAMEREF{float-formats}, \_ : \NAMEREF{floats} ( \VAR{FF} ), \_ : \NAMEREF{floats} ( \VAR{FF} )) : \TO \NAMEREF{floats} ( \VAR{FF} ) \end{align*}\] \[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{float-remainder}( \VAR{FF} : \NAMEREF{float-formats}, \_ : \NAMEREF{floats} ( \VAR{FF} ), \_ : \NAMEREF{floats} ( \VAR{FF} )) : \TO \NAMEREF{floats} ( \VAR{FF} ) \end{align*}\] \[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{float-sqrt}( \VAR{FF} : \NAMEREF{float-formats}, \_ : \NAMEREF{floats} ( \VAR{FF} )) : \TO \NAMEREF{floats} ( \VAR{FF} ) \end{align*}\] \[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{float-integer-power}( \VAR{FF} : \NAMEREF{float-formats}, \_ : \NAMEREF{floats} ( \VAR{FF} ), \_ : \NAMEHYPER{../.}{Integers}{integers}) : \TO \NAMEREF{floats} ( \VAR{FF} ) \end{align*}\] \[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{float-float-power}( \VAR{FF} : \NAMEREF{float-formats}, \_ : \NAMEREF{floats} ( \VAR{FF} ), \_ : \NAMEREF{floats} ( \VAR{FF} )) : \TO \NAMEREF{floats} ( \VAR{FF} ) \end{align*}\]

Rounding

\[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{float-round-ties-to-even}( \VAR{FF} : \NAMEREF{float-formats}, \_ : \NAMEREF{floats} ( \VAR{FF} )) : \TO \NAMEHYPER{../.}{Integers}{integers} \end{align*}\] \[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{float-round-ties-to-infinity}( \VAR{FF} : \NAMEREF{float-formats}, \_ : \NAMEREF{floats} ( \VAR{FF} )) : \TO \NAMEHYPER{../.}{Integers}{integers} \end{align*}\] \[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{float-floor}( \VAR{FF} : \NAMEREF{float-formats}, \_ : \NAMEREF{floats} ( \VAR{FF} )) : \TO \NAMEHYPER{../.}{Integers}{integers} \end{align*}\] \[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{float-ceiling}( \VAR{FF} : \NAMEREF{float-formats}, \_ : \NAMEREF{floats} ( \VAR{FF} )) : \TO \NAMEHYPER{../.}{Integers}{integers} \end{align*}\] \[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{float-truncate}( \VAR{FF} : \NAMEREF{float-formats}, \_ : \NAMEREF{floats} ( \VAR{FF} )) : \TO \NAMEHYPER{../.}{Integers}{integers} \end{align*}\]

Miscellaneous

\[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{float-pi}( \VAR{FF} : \NAMEREF{float-formats}) : \TO \NAMEREF{floats} ( \VAR{FF} ) \end{align*}\] \[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{float-e}( \VAR{FF} : \NAMEREF{float-formats}) : \TO \NAMEREF{floats} ( \VAR{FF} ) \end{align*}\] \[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{float-log}( \VAR{FF} : \NAMEREF{float-formats}, \_ : \NAMEREF{floats} ( \VAR{FF} )) : \TO \NAMEREF{floats} ( \VAR{FF} ) \end{align*}\] \[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{float-log10}( \VAR{FF} : \NAMEREF{float-formats}, \_ : \NAMEREF{floats} ( \VAR{FF} )) : \TO \NAMEREF{floats} ( \VAR{FF} ) \end{align*}\] \[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{float-exp}( \VAR{FF} : \NAMEREF{float-formats}, \_ : \NAMEREF{floats} ( \VAR{FF} )) : \TO \NAMEREF{floats} ( \VAR{FF} ) \end{align*}\] \[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{float-sin}( \VAR{FF} : \NAMEREF{float-formats}, \_ : \NAMEREF{floats} ( \VAR{FF} )) : \TO \NAMEREF{floats} ( \VAR{FF} ) \end{align*}\] \[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{float-cos}( \VAR{FF} : \NAMEREF{float-formats}, \_ : \NAMEREF{floats} ( \VAR{FF} )) : \TO \NAMEREF{floats} ( \VAR{FF} ) \end{align*}\] \[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{float-tan}( \VAR{FF} : \NAMEREF{float-formats}, \_ : \NAMEREF{floats} ( \VAR{FF} )) : \TO \NAMEREF{floats} ( \VAR{FF} ) \end{align*}\] \[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{float-asin}( \VAR{FF} : \NAMEREF{float-formats}, \_ : \NAMEREF{floats} ( \VAR{FF} )) : \TO \NAMEREF{floats} ( \VAR{FF} ) \end{align*}\] \[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{float-acos}( \VAR{FF} : \NAMEREF{float-formats}, \_ : \NAMEREF{floats} ( \VAR{FF} )) : \TO \NAMEREF{floats} ( \VAR{FF} ) \end{align*}\] \[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{float-atan}( \VAR{FF} : \NAMEREF{float-formats}, \_ : \NAMEREF{floats} ( \VAR{FF} )) : \TO \NAMEREF{floats} ( \VAR{FF} ) \end{align*}\] \[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{float-sinh}( \VAR{FF} : \NAMEREF{float-formats}, \_ : \NAMEREF{floats} ( \VAR{FF} )) : \TO \NAMEREF{floats} ( \VAR{FF} ) \end{align*}\] \[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{float-cosh}( \VAR{FF} : \NAMEREF{float-formats}, \_ : \NAMEREF{floats} ( \VAR{FF} )) : \TO \NAMEREF{floats} ( \VAR{FF} ) \end{align*}\] \[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{float-tanh}( \VAR{FF} : \NAMEREF{float-formats}, \_ : \NAMEREF{floats} ( \VAR{FF} )) : \TO \NAMEREF{floats} ( \VAR{FF} ) \end{align*}\] \[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{float-asinh}( \VAR{FF} : \NAMEREF{float-formats}, \_ : \NAMEREF{floats} ( \VAR{FF} )) : \TO \NAMEREF{floats} ( \VAR{FF} ) \end{align*}\] \[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{float-acosh}( \VAR{FF} : \NAMEREF{float-formats}, \_ : \NAMEREF{floats} ( \VAR{FF} )) : \TO \NAMEREF{floats} ( \VAR{FF} ) \end{align*}\] \[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{float-atanh}( \VAR{FF} : \NAMEREF{float-formats}, \_ : \NAMEREF{floats} ( \VAR{FF} )) : \TO \NAMEREF{floats} ( \VAR{FF} ) \end{align*}\] \[\begin{align*} \KEY{Built-in Funcon} \quad & \NAMEDECL{float-atan2}( \VAR{FF} : \NAMEREF{float-formats}, \_ : \NAMEREF{floats} ( \VAR{FF} ), \_ : \NAMEREF{floats} ( \VAR{FF} )) : \TO \NAMEREF{floats} ( \VAR{FF} ) \end{align*}\]

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