This is an analysis of the R5RS provision that the full numeric tower may be subsetted. Four boolean feature settings characterize different kinds of numeric towers: exactness-preserving, ratios, inexact, and complex. These refer respectively to the closure of exact numbers under rational operations (except /), exact non-integer values, inexact rationals, and non-real numbers.
There are therefore 16 possible numeric towers. 9 of them are known to have implementations, as shown below. I write + if a feature is present and - if it is absent, and give a general description of the resulting tower and some Scheme implementations that provide it.
||exactness-preserving||ratios||inexact||complex||Description||Implementations
|
- |
- |
- |
- |
A "toy" tower with bounded exact integers only (possibly appropriate where no numerical work will be done) |
SigScheme* |
|
- |
- |
+ |
- |
Fixnums and flonums (all arithmetic runs in essentially constant time) |
Plain Chicken, Shoe*, TinyScheme*, RScheme, JScheme*, BDC*, XLisp*, Schemik*, VX, SXM*, Inlab, Llava, Sixx, Sizzle, Dfsch*, Stalin (also Elisp*, C*) |
|
- |
+ |
+ |
1 |
Limited-range exact and inexact real numbers |
NexJ (also Java) |
|
- |
+ |
+ |
+ |
Limited-range exact and inexact numbers |
S7, Wraith |
|
+ |
- |
+ |
- |
Exact integers and inexact real numbers |
Bigloo, Scheme 9, Elk (also ISLisp) |
|
+ |
- |
+ |
+ |
Exact integers, inexact real numbers, and inexact complex numbers |
SCM |
|
+ |
+ |
- |
- |
Exact rational numbers only (reasonable for some purposes, but will often run very slowly without inexact support, so unsuitable for scientific-type work) |
Dream, Oaklisp, Owl Lisp |
|
+ |
+ |
+ |
- |
Real numbers only |
Psyche, Ikarus, Rep, Dfsch (also Clojure) |
|
+ |
+ |
+ |
+ |
Full numeric tower |
Racket, Gauche, MIT, Gambit, Chicken with the numbers egg, Scheme48/scsh, Kawa, SISC, Chibi, Guile, Chez, Vicare, Larceny, Ypsilon, Mosh, IronScheme, STklos, KSi, UMB, Spark (also R6RS, Common Lisp, Pure) |
*These systems are technically exactness-preserving, but silently return the wrong answers when their arithmetic operations overflow.
In my opinion, the reasonable general-purpose towers are --+- (fixnums and flonums), +-+- (fixnums, bignums, and flonums), and ++++ (full).
See NumericTowerManisComments for Vincent Manis's comments on an earlier version of this page.
See also ComplexRepresentations for information on which Schemes support exact, inexact, and mixed-exactness complex numbers.
See also FloatPrecision for information on the different precisions of inexact numbers that some Schemes support.