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 feature groups 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 are of little use.
So here are ten possible numeric towers. I use + if a feature is present and - if it is absent, and give the numeric properties of the tower is.
|
exactness-preserving |
ratios |
inexact |
complex |
Description |
|
- |
- |
- |
- |
A "toy" Scheme with fixnums (bounded exact integers) only. Appropriate where no numerical work will be done. |
|
- |
- |
+ |
- |
A Scheme where all numbers are real and all arithmetic happens in essentially constant time. Most non-Lisp languages are like this, except that their fixnums wrap instead of switching to flonums. Chicken uses this. |
|
- |
- |
+ |
+ |
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 uses this. Many people seem to like this design. |
|
+ |
- |
+ |
+ |
The same as the preceding, but with inexact/inexact non-real numbers as well. |
|
+ |
+ |
- |
- |
Exact numbers only. 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. |
|
+ |
+ |
+ |
+ |
Full numeric tower. R6RS requires this. |