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Issue: REAL-NUMBER-TYPE (version 3)

Issue:        REAL-NUMBER-TYPE
Forum:	      CLEANUP
References:   Table 4-1.
Category:     ADDITION
Edit history: 04-JAN-89, Version 1 by Bob Cassels, Don Sakahara, Kent Pitman,
                         and John Aspinall
              08-JAN-89, Version 2 by Bob Cassels -- incorporate
                         Masinter's suggestion and make REAL a CLOS class
              13-JAN-89, Version 3 by Cassels and Aspinall -- incorporate Marc LeBrun's
                         suggestions clarifying the relationship between CL
                         numeric type names and mathematical names
Status:	      For Internal Discussion

Problem Description:

  There is no standard type specifier symbol for the CL type


  Make REAL be a CL data type:

  p.13 "Numbers"

    Add:     The NUMBER data type encompasses all of these kinds of
             numbers.  For convenience, there are names for some
             subclasses of numbers.  @i[Integers] and @i[ratios] are of
             type RATIONAL.  @i[Rational numbers] and @[floating-point
             numbers] are of type REAL.  @i[Real numbers] and @i[complex
             numbers] are of type NUMBER.

	     Although the names of these types were chosen with the
	     terminology of mathematics in mind, the correspondences
	     are not always exact.  Integers and ratios model the
	     corresponding mathematical concepts directly.  Numbers
	     of the FLOAT type may be used to approximate real
	     numbers, both rational and irrational.  The REAL type
	     includes all Common Lisp numbers which represent
	     mathematical real numbers, though there are
	     mathematical real numbers (irrational numbers)
	     which do not have an exact Common Lisp representation.
	     Only REAL numbers may be ordered using the <, >, <=,
	     and >= functions.

             Compatibility note:  The Fortran standard defines the term
             "real datum" to mean "a processor approximation to the value
             of a real number."  In practice the Fortran "basic real" type
             is the floating-point data type Common Lisp calls
             SINGLE-FLOAT.  The Fortran "double precision" type is
             Common Lisp's DOUBLE-FLOAT.  The Pascal "real" data type is
             an "implementation-defined subset of the real numbers."  In
             practice this is usually a floating-point type, often what
             Common Lisp calls DOUBLE-FLOAT.

             A translation of an algorithm written in Fortran or Pascal
             which uses "real" data usually will use some appropriate
             precision of Common Lisp's FLOAT type.  Some algorithms may
             gain accuracy and/or flexibility by using Common Lisp's
             RATIONAL or REAL types instead.

  p.33 "Overlap, Inclusion, and Disjointness of Types":

    Remove:  The types RATIONAL, FLOAT, and COMPLEX are pairwise
             disjoint subtypes of NUMBER.

             Rationale: It might be thought that INTEGER and RATIO ...

             Rationale: It might be thought that FIXNUM and BIGNUM ...

    Add:     The types RATIONAL and FLOAT are pairwise disjoint subtypes
             of REAL.

             The types REAL and COMPLEX are pairwise disjoint subtypes
             of NUMBER.

             Rationale: It might be thought that FIXNUM and BIGNUM should 
             form an exhaustive partition of the type INTEGER, that INTEGER
             and RATIO should form an exhaustive partition of RATIONAL,
             that RATIONAL and FLOAT should form an exhaustive partition of 
             REAL, and that REAL and COMPLEX should form an exhaustive
             partition of NUMBER.  These are all purposely avoided in order 
             to permit compatible experimentation with extensions to the
             Common Lisp number system, such as the idea of adding explicit 
             representations of infinity or of positive and negative infinity.

   p.43 Table 4-1 "Standard Type Specifier Symbols"

    Add:     REAL

   p.49 "Type Specifiers that Abbreviate"

     Add:    (REAL low high)
             Denotes the set of real numbers between low and high.  ...
             [As with RATIONAL and FLOAT.]

  Make REAL a built-in CLOS class.


  Add a specific data type predicate REALP which tests for membership in
  this type.  [By analogy with NUMBERP.]

Test Case:

  If a programmer wishes to test for "a number between 1 and 10", the
  only current CL types would be '(or (rational 1 10) (float 1 10)) or
  something like '(and numberp (not complexp) (satisfies range-1-10))
  with (defun range-1-10 (real) (<= 1 real 10)).  Both of these are
  likely less efficient, and certainly less expressive than '(real 1 10).


  Mathematics has a name for (OR RATIONAL FLOAT) -- it is "real".
  This class is important because it is all the numbers which can be

  Throughout the "Numbers" chapter, the phrase "non-complex number" is
  MAX, MIN, p. 198 "The arguments may be any non-complex numbers."
  CIS p. 207 "The argument ... may be any non-complex number."

Current Practice:

  Probably nobody does this.
Cost to Implementors:

  Some work is necessary to add this name.  But since the underlying
  type already exists the amount of work should be minimal.
Cost to Users:

  Since this is an upward-compatible extension, it may be ignored by

Cost of Non-Adoption:

  Occasional inconvenience and/or inefficiency.


  Mathematical clarity.

  Ability to do CLOS method dispatch on the type.


  As mentioned under "rationale," this would be a more concise way to
  express a common programming idiom.


  The name "non-complex number" is incorrect because future
  implementations may wish to include numerical types which are neither
  complex nor real.  [e.g. pure imaginary numbers or quaternions]
  The name "scalar" is incorrect because the mathematical concept of
  scalar may indeed include complex numbers.

  Fortran and Pascal use the name "real" to mean what CL calls
  SINGLE-FLOAT.  That should cause no significant problem, since a Lisp
  program written using the type REAL will do mathematically what the
  equivalent Fortran program would do.  This is because Fortran's "real"
  data-type is a subtype of the CL REAL type.  The only differences
  might be that the Lisp program could be less efficient and/or more

  A survey of several Fortran and Pascal books shows that the distinction
  between INTEGER and REAL is that REAL numbers may have fractional
  parts, while INTEGERs do not.  Later discussions explain that REALs
  cover a greater range.  Much later discussions cover precision
  considerations, over/underflow, etc.  So the average Fortran or Pascal
  programmer should be completely comfortable with the proposed Lisp
  concept of REAL.