Issue REAL-NUMBER-TYPE Writeup

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

	      05-APR-89, Version 4 by Pitman (changes per X3J13)

Status:	      Accepted v3 Mar-89 by X3J13 (on a 12-3 vote) with 

	      amendments. The proposal as amended is v4.


Problem Description:


  There is no standard type specifier symbol for the CL type

  '(OR RATIONAL FLOAT). 


Proposal (REAL-NUMBER-TYPE:X3J13-MAR-89):


  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).


Rationale:


  Mathematics has a name for (OR RATIONAL FLOAT) -- it is "real".

  This class is important because it is all the numbers which can be

  ordered.


  Throughout the "Numbers" chapter, the phrase "non-complex number" is

  used.

  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

  users.


Cost of Non-Adoption:


  Occasional inconvenience and/or inefficiency.


Benefits:


  Mathematical clarity.


  Ability to do CLOS method dispatch on the type.


Aesthetics:


  As mentioned under "rationale," this would be a more concise way to

  express a common programming idiom.


Discussion:


  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

  accurate.


  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.