non-generic arithmetic

[tmb@idiap.ch (Thomas M. Breuel)]

=> PROPOSAL

Several people (including myself) have in the past asked for a
standardization of non-generic arithmetic procedures. Several
implementations provide such procedures already, and it would be
useful to standardize them by adding them as a non-essential feature
to the standard, or at least by agreeing informally to a common naming
scheme.

=> RATIONALE

The reasons why people want to use such procedures are two-fold: (1)
efficiency (many compilers simply aren't smart enough to optimize the
Scheme number system), and (2) a different view of arithmetic (in
which, say, there are "real" functions that only take real arguments
and return real values and flag errors if they get an argument of the
wrong type or cannot represent the result as a real number).

=> GOALS

I think such procedures

 * should be defined such that applying the procedures to an
   object of the wrong type (e.g., "(flonum-sin 1)") results
   in an error (*)

 * should be "arithmetically safe", i.e., overflows and other
   arithmetic errors should be detected

 * should have names that allow people to use them conveniently and
   regularly in their code

=> PORTABLE IMPLEMENTATIONS

A reasonably portable implementation might define "flonum-+" in terms
of type-checks, coercions, and the "+" function. Such an
implementation would allow code written in terms of the non-generic
primitives to be executed on implementations that do not support
non-generic arithmetic.

Of course, I would hope that many compilers actually try to do a
better job at generating code for such procedures.

=> NAMING

Prefix schemes are commonly used for naming arithmetic operations of
the kind proposed here. That is, the name of a non-generic arithmetic
function is derived by putting a prefix in front of the name of the
corresponding generic function.

People would use these operations primarily with fixnums and flonums;
hence, the prefixes for those functions should be short:

    DATATYPE	PREFIX

    fixnum	x or i (as in fi[x]num or f[i]xnum)
    flonum	f

    rational	rational
    integer	integer
    real	real
    complex	complex

For functions that begin with a non-alphanumeric letter, the prefix is
used directly. For other functions, it is separated by a "-" from the
name of the corresponding generic procedure.

Examples:

	f+
	x*
	f-sin
	rational+
	rational-quotient
	complex-sin

=> STANDARD

I'm not sufficiently familiar with a lot of Scheme dialects to make a
more concrete proposal, and I'm not well-versed in writing sections
for the Scheme standard, so I hope that somebody else can pick this up
from here.

Of course, some care must be taken to say what is meant by "fixnum"
and "flonum" and how that relates to the rest of the number system in
Scheme. In particular, it should be made clear that this proposal in
no way constitutes a recommendation to implement "fixnum" and "flonum"
types, but merely suggests a naming convention if such types are
already present in the implementation (as they are, in many
implementations).

					Thomas.

(*) I think it is a bad idea to allow implementations not to flag an
error for something like "(flonum-sin 1)", since it makes it very
difficult to track down unportable constructs on implementations that,
say, silently convert the integer 1 to a floating point number. I
have encountered such problems frequently when trying to port Lisp
code that had type declarations in it that were never actually checked
by the implementation that the code was developed on.