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*To*: CL-Cleanup@Sail.Stanford.EDU*Subject*: Issue: REAL-NUMBER-TYPE (version 3)*From*: Robert A. Cassels <Cassels@STONY-BROOK.SCRC.Symbolics.COM>*Date*: Fri, 13 Jan 89 13:50 EST*Cc*: DySak@STONY-BROOK.SCRC.Symbolics.COM, JGA@STONY-BROOK.SCRC.Symbolics.COM, Common-Lisp-Implementors@STONY-BROOK.SCRC.Symbolics.COM*Supersedes*: <19890106223654.4.CASSELS@GROUSE.SCRC.Symbolics.COM>, <19890108162936.5.CASSELS@GROUSE.SCRC.Symbolics.COM>

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 '(OR RATIONAL FLOAT). Proposal (REAL-NUMBER-TYPE:REAL): 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. Proposal (REAL-NUMBER-TYPE:REALP): 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.

**Follow-Ups**:**Issue: REAL-NUMBER-TYPE (version 3)***From:*David A. Moon <Moon@STONY-BROOK.SCRC.Symbolics.COM>

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