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, which refer to the closure of exact numbers under rational operations (except /), exact non-integer values, inexact rationals, and non-real numbers, respectively. A priori, any combination of these features might be implemented, except that ratios makes no sense without exactness-preserving. I also assume that complex makes no sense without either ratios or inexact: Gaussian integers by themselves seem to be of little use.
So here are ten possible numeric towers. I use + if a feature is present and - if it is absent, and give a general description of the resulting tower.
|
exactness-preserving |
ratios |
inexact |
complex |
Description |
|
- |
- |
- |
- |
A "toy" tower with fixnums (bounded exact integers) only. Possibly appropriate where no numerical work will be done. SSCM, SigScheme provide this. |
|
- |
- |
+ |
- |
A Scheme where all numbers are real and all arithmetic happens in essentially constant time. Chicken provides this unless you import the numbers package. Sixx, Sizzle provide this. Most non-Lisp languages are like this, except that their fixnums wrap on overflow instead of switching to flonums, producing grossly incorrect results. |
|
- |
- |
+ |
+ |
The complex-number equivalent of the preceding. Non-real numbers are typically inexact/inexact. |
|
+ |
- |
- |
- |
Unlimited-precision integers only. Possibly usable for crypto work. Fortunately, ratios are fairly cheap given bignums. |
|
+ |
- |
+ |
- |
A good all-round compromise, providing exact integers and inexact real numbers. ISLisp requires this. Bigloo, Chibi, SCM, Scheme 9 provide this. Many people seem to like this tower. |
|
+ |
- |
+ |
+ |
The same as the preceding, but with inexact/inexact non-real numbers as well. |
|
+ |
+ |
- |
- |
Exact rational numbers only. Dream provides this. Reasonable for some purposes, but will often run very slowly without inexact support, so unsuitable for scientific-type work. |
|
+ |
- |
+ |
+ |
The same as the preceding, but with exact general complex numbers. Even less suitable for scientific-type work, given that in practice non-real numbers are usually inexact/inexact. |
|
+ |
+ |
+ |
- |
Full numeric tower for real numbers only. Psyche provides this. |
|
+ |
+ |
+ |
+ |
Full numeric tower. R6RS and CL require this. Racket, Gauche, MIT, Gambit, Chicken with the numbers package, Scheme48/scsh, Kawa, Chez, STklos provide this. Guile, SISC provide this, but don't have exact complex numbers. Many people seem to like this tower. |