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Re: Exposing more math functionality

From: Andy Wingo
Subject: Re: Exposing more math functionality
Date: Fri, 27 Aug 2010 08:44:40 -0700
User-agent: Gnus/5.13 (Gnus v5.13) Emacs/23.2 (gnu/linux)

On Thu 19 Aug 2010 09:04, Andy Wingo <address@hidden> writes:

> Regarding modf -- it seems that the R6RS extends the definition of
> `modulo' (called `mod') to be defined over the real numbers.
>   (mod 10 3) => 1
>   (mod 10 3.0) => 1.0
>   (mod 10 3.1) => 0.7
> This appears to be a compatible extension of the R5RS' `modulo', so we
> should just extend our definition. That way we can avoid adding another
> symbol.

This was a mistaken impression, something I realized while trying to
implement this. From the R6RS rationale, section 11.6.6:

    div and mod

    Given arithmetic on exact integer objects of arbitrary precision, it
    is a trivial matter to derive signed and unsigned integer types of
    finite range from it by modular reduction.  For example 32-bit
    signed two-complement arithmetic behaves like computing with the
    residue classes “mod 232 ”, where the set {−231 , . . . , 231 − 1}
    has been chosen to represent the residue classes. Likewise, unsigned
    32-bit arithmetic also behaves like computing “mod 232 ”, but with a
    different set of representatives {0, . . . , 232 − 1}.

    Unfortunately, the R5 RS operations quotient, remainder, and modulo
    are not ideal for this purpose. In the following example, remainder
    fails to transport the additive group structure of the integers over
    to the residues modulo 3.
          (remainder (+ -2 3) 3) =⇒ 1
          (remainder (+ (remainder -2 3)
                        (remainder 3 3))
                     3) =⇒ -2
    In fact, modulo should have been used, producing residues in {0, 1,
    2}.  For modular reduction with symmetric residues, i.e., in {−1, 0,
    1} in the example, it is necessary to define a more complicated
    reduction altogether.
    Therefore, quotient, remainder, and modulo have been replaced in R6
    RS by the div, mod, div0, and mod0 procedures, which are more useful
    when implementing modular reduction. The underlying mathematical
    functions div, mod, div0 , and mod0 (see report section 11.7.3) have
    been adapted from the div and mod operations by Egner et
    al. [11]. They differ in the representatives from the residue
    classes they return: div and mod always compute a nonnegative
    residue, whereas div0 and mod0 compute a residue from a set centered
    on 0. The former can be used, for example, to implement unsigned
    fixed-width arithmetic, whereas the latter correspond to
    two’s-complement arithmetic.
    These operations differ slightly from the div and mod operations
    from Egner et al. The latter make both operations available through
    a single pair of operations that distinguish between the two cases
    for residues by the sign of the divisor (as well as returning 0 for
    a zero divisor). Splitting the operations into two sets of
    procedures avoids potential confusion.
    The procedures modulo, remainder, and quotient from R5RS can easily
    be defined in terms of div and mod.
So I'm not sure what to do exactly. If Scheme people are happy with div
and mod, as in the R6RS, then we should implement them and make them the
default, implementing quotient, modulo, and remainder in terms of
them. Hmm.


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