HELP REAL                                   R. J. Duncan, September 1989

Functions on real numbers.


    CONTENTS - (Use <ENTER> g to access required sections)

 -- The Real Module
 -- Basic Functions
 -- Trigonometric Functions
 -- Real Constants


-- The Real Module ----------------------------------------------------

signature Real
structure Real : Real
    The structure -Real- is a built-in structure of PML which  defines a
    collection of useful functions on real numbers. It has the following
    signature:

        signature Real = sig

            (* Basic Functions *)

            exception Log10
            exception Power

            val round           : real -> int
            val intof           : real -> int
            val fracof          : real -> real
            val sign            : real -> real
            val max             : real -> real -> real
            val min             : real -> real -> real
            val log10           : real -> real
            val power           : real -> real -> real

            val **              : real * real -> real

            (* Trigonometric Functions *)

            exception ArcSin
            exception ArcCos
            exception ArcCosh
            exception ArcTanh

            val tan             : real -> real
            val arcsin          : real -> real
            val arccos          : real -> real
            val sinh            : real -> real
            val cosh            : real -> real
            val tanh            : real -> real
            val arcsinh         : real -> real
            val arccosh         : real -> real
            val arctanh         : real -> real

            val arctan2         : real -> real -> real

            (* Real Constants *)

            val pi              : real
            val most_positive   : real
            val least_positive  : real
            val most_negative   : real
            val least_negative  : real
            val add_epsilon     : real
            val sub_epsilon     : real

        end

    For the definition of the type  -real- see HELP * STDTYPES. For  the
    basic mathematical operators (+, -, etc.) see HELP * STDVALUES.  For
    a similar set of functions on integers see HELP * INT.


-- Basic Functions ----------------------------------------------------

val round (r : real) : int
    Rounds the real number -r- to an integer.


val intof (r : real) : int
    Truncates the real number -r- to an integer.


val fracof (r : real) : real
    Returns the fractional part of the real number -r-.


val sign (r : real) : real
    Returns the  sign of  the real  number -r-,  i.e., one  of the  real
    numbers ~1.0, 0.0,  or 1.0 according  to whether -r-  is less  than,
    equal to or greater than zero.


val max (r1 : real) (r2 : real) : real
    Returns the greater of the real numbers -r1- and -r2-.


val min (r1 : real) (r2 : real) : real
    Returns the lesser of the real numbers -r1- and -r2-.


exception Log10
val log10 (r : real) : real
    Returns the base  10 logarithm of  the real number  -r-. Raises  the
    exception -Log10- if -r- is non-positive.


exception Power
infix 8 **
val power (r : real) (m : real) : real
val (r : real) ** (m : real) : real
    Returns the  value  of -r-  raised  to  the power  -m-.  Raises  the
    exception -Power- if -r- is negative, or  if -r- is zero and -m-  is
    non-positive; otherwise the result is computed as

        exp(m * ln r)

    The exception -Overflow- is raised if the result is out of range.


-- Trigonometric Functions --------------------------------------------

For SIN, COS and ARCTAN functions  see HELP * STDVALUES. All angles  are
measured in radians.


val tan (r : real) : real
    Returns the tangent of the real angle -r-.


exception ArcSin
exception ArcCos
val arcsin (r : real) : real
val arccos (r : real) : real
    Return the  arcsine  and  arccosine  of the  real  number  -r-.  The
    exception -ArcSin-  or -ArcCos-  is raised  as appropriate  whenever
    abs(r) > 1.0.


val sinh (r : real) : real
val cosh (r : real) : real
val tanh (r : real) : real
    Return the hyperbolic  sine, cosine  and tangent of  the real  angle
    -r-.


exception ArcCosh
exception ArcTanh
val arcsinh (r : real) : real
val arccosh (r : real) : real
val arctanh (r : real) : real
    Return the hyperbolic arcsine, arccosine and arctangent of the  real
    number -r-. The  function -arccosh- raises  the exception  -ArcCosh-
    when r < 1.0; the function -arctanh- raises the exception  -ArcTanh-
    when abs(r) > 1.0 and the exception -Overflow- when abs(r) = 1.0.


val arctan2 (r1 : real) (r2 : real) : real
    Returns the  arctangent of  r1 /  r2,  using the  signs of  the  two
    numbers to derive quadrant information. When both -r1- and -r2-  are
    zero, the result is defined (arbitrarily) to be zero also.


-- Real Constants -----------------------------------------------------

Note:  the  values  of  these  constants  will  vary  between  different
implementations of PML, depending  on the floating-point  representation
of the host machine.


val pi : real
    The best real approximation to pi.


val most_positive : real
    The greatest positive value representable as a real number.


val least_positive : real
    The smallest  positive  (non-zero)  value representable  as  a  real
    number.


val most_negative : real
    The  most  negative  (i.e.  closest  to  negative  infinity)   value
    representable as a real number.


val least_negative : real
    The least negative (i.e. closest  to zero) value representable  as a
    real number.


val add_epsilon : real
    The smallest positive real number with the property:

        1.0 + add_epsilon <> 1.0


val sub_epsilon : real
    The smallest positive real number with the property:

        1.0 - sub_epsilon <> 1.0


--- C.all/pml/help/real
--- Copyright University of Sussex 1991. All rights reserved. ----------