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*To*: srfi-77@xxxxxxxxxxxxxxxxx*Subject*: Re: Questions about srfi-77 Generic Arithmetic*From*: William D Clinger <will@xxxxxxxxxxx>*Date*: Sat, 18 Feb 2006 12:22:59 -0500*Delivered-to*: srfi-77@xxxxxxxxxxxxxxxxx

Bradley Lucier wrote: > I'm trying to understand the R5RS-style Generic Arithmetic > and I have some questions. Those were excellent questions indeed. Thank you, Brad. You have spotted many miskates (!). I agree with you and with Mike's reply to you with regard to questions 4, 5, 6, 7, 11, and 15. I have a little more to say about the others. > 1. In the first paragraph it says "The generic operations all > implement inexact operations." What does that mean? I don't know either. I propose the following change in wording: Replace A number is inexact if it was written as an inexact constant, if it was derived using inexact ingredients, or if it was derived using inexact operations. The generic operations all implement inexact operations. Thus inexactness is a contagious property of a number. by A number is inexact if it was written as an inexact constant or was derived from inexact numbers. Thus inexactness is contagious. The generic operations generally return the correct exact result when all of their arguments are exact and the result is mathematically well-defined, but return an inexact result when any argument is inexact. As indicated in the specification of individual operations, some exceptions to this rule are allowed (but not required). One general exception to the rule above is that an implementation may return an exact result despite inexact arguments if that exact result would be the correct result for all possible substitutions of exact arguments for the inexact ones. While I'm at it, let me propose another correction. The current draft of SRFI 77 says "If two implementations produce exact results for a computation that did not involve inexact intermediate results or the results of numerical predicates, the two ultimate results will be mathematically equivalent." That is both untrue and misleading. In SRFI 77, the fixnum operations may return different exact results in different implementations. I regard that as a strong argument against having the fixnum operations in the core language; I would prefer they be banished to a library, so we can claim implementation-independence for results of exact computations that use only the core. If fixnum? is banished from the core, then the numerical predicates should return the same results in every implementation when given exact arguments, so it is misleading to mention them here. I therefore propose to revert the sentence quoted above to its R5RS form: "If two implementations produce exact results for a computation that did not involve inexact intermediate results, the two ultimate results will be mathematically equivalent." Although that sentence will not hold for the core augmented by certain libraries, e.g. the fixnum library, the reason for that is that the libraries will be implemented differently in different implementations. I think people can understand that. > 2. In the first paragraph it says "A number is inexact if it is > infinite, if it was written as an inexact constant, if it was derived > using inexact ingredients, or if it was derived using inexact > operations." > (a) I prefer some other word to "ingredients". > (b) What is an inexact operation? The in* functions? Or, do > you mean every operation in this section? In my reply to question 1 above, I proposed wording that would remove the references to "ingredients" and "inexact operations". Mike suggested an alternative fix that would involve some explanation of how "operations" are distinct from "procedures". I think Mike's fix would create more confusion that it would resolve. I prefer the wording I proposed above. > (c) If 0 denotes exact 0 and 1 denotes exact 1, I prefer > (log 1) = (sin 0)=(asin 0)=(atan 0)=(tan 0)=...=0 > and > (cos 0) = (exp 0)=1, > in the same way that I prefer (sqrt 4) = 2 (where 4 denotes > exact 4 and 2 denotes exact 2). Is this allowed in your proposal? Not as written, but I regard that as a mistake. See my answer to question 3 below. > 3. Paragraph 6 says "With the exception of inexact->exact, > the operations described in this section must return inexact > results when given any inexact arguments." I think we should just delete that sentence, and rely on the wording I proposed in my answer to question 1. > 8. Integer Divison. > (a) I think it is a really bad design to have the basic > operation of div+mod change when the second argument changes sign. I don't have much problem with that. These mathematical operations aren't defined at zero, so they have to compare against zero anyway. To quote Egner et al, "the sign of the modulus y determines which system of representatives of the residue class ring Z/yZ is being chosen, either non-negative (y > 0), symmetric around zero (y < 0), or the integers (y = 0)." Using the sign of y is ad hoc, but it's an ad hoc choice anyway. > (b) I think it is a really bad design to have > (div x1 0) => 0 > It conflicts with the requirement in quotient+remainder, > quotient, remainder, and modulo that "n2 should be nonzero". > (Unless I read "should" here in the language lawyer sense, > i.e., that it is totally nonprescriptive.) The only way I can > think of making division by exact 0 make sense is by divorcing > the meaning of "div" from any notion of mathematical division. I don't have a strong opinion about this. It is totally ad hoc, but the alternative is a principled (not ad hoc) hole in the domain of div. I'm inclined to think that passing 0 as the second argument to div is, in practice, most likely an error for which it would be more useful to signal an error than to return 0, but I'm willing to listen to an argument for returning 0. So far as I can see, no such argument has been made. In particular, the paper by Egner et al contains no such argument. > (c) This is one place where the Rationale says to refer to > Egner et al. to see an argument why these procedures "are > better suited than quotient and remainder to implement > modular reduction." As a point of procedure, I would really > prefer to have the arguments included in this document; I > don't want to have to discuss my opinions of Egner et al. > just to discuss my opinions of this specific proposal. If Egner et al actually contained a rationale for this, it would be completely proper to refer to that paper by reference instead of including its rationale in the SRFI. Since Egner et al do not actually offer a rationale for defining (div x 0) to be 0, however, it is extremely misleading for this SRFI to refer readers to that paper for the rationale. The reference to Egner et al should certainly be dropped, and some rationale (if anyone can come up with one) added to this SRFI. > 9. I don't agree that +inf.0, -inf.0, and +nan.0 should be in the > domain of floor, round, ceiling, and truncate; returning these values > as results implies, to me, that they are integers, and I believe they > aren't integers, since they aren't rational. The current draft of SRFI 77 says "These procedures return inexact integers on inexact arguments that are not infinities or NaNs, and exact integers on exact rational arguments." I believe the intent of this was to exclude non-rational arguments from the domain of floor, round, ceiling, and truncate. It would be simpler to replace the x in their templates by q, thereby implying it is an error to pass them a non-rational argument. I believe the last two examples, for (floor +inf.0) and (ceiling -inf.0), are just mistakes. Mike Sperber wrote: > This is on the issues list---I'm almost convinced, but wonder what > happens in representations where the argument isn't infinite or NaN, > and the result still can't be represented as an inexact integer. That is something to worry about, all right, but that has to do with the range, not the domain. I think we should exclude non-rational arguments from the domain. Back to Brad Lucier's questions: > 12. (a) You give the examples: > (log +inf.0) => +inf.0 > (log -1.0+0.0i) => 0.0+pii > (log -1.0-0.0i) => 0.0-pii > yet > (log -inf.0) => unspecified > Given the previous examples, shouldn't > (log -inf.0) => +inf.0+pii Yes. That was a mistake, which we'll fix. > (b) You don't say what (log 0) evaluates to; I would prefer it > to be an error. Me too. In general, passing exact arguments for which the exact result is not mathematically well-defined should be an error. We shouldn't require implementations to return an inexact result (e.g. -inf.0) when given exact arguments, and we shouldn't allow implementations to return some random exact result. > 13. For sqrt: Is your intention to no longer have sqrt return exact > results when possible given exact arguments? The language I proposed in my answer to question 1 would *require* sqrt to return exact results when given exact arguments. Inasmuch as most implementations cannot represent the mathematically correct result in most cases, the description of sqrt should explicitly allow (but not require) it to return inexact results even when given exact arguments. A similar exception to the general rule should be stated for exp, log, sin, cos, tan, asin, acos, atan, expt, make-polar, magnitude, and angle. I'd argue for making this exception for make-rectangular as well, to avoid making rectangular coordinates privileged over polar. Which brings up a larger question: If neither make-polar nor make-rectangular are required to return an exact result when given exact arguments, what requires implementations to provide non-real exact numbers? I think the only requirement of that sort follows from the ability to write things like 3+4i and to compute with them. Do we actually need to require implementations to represent 3+4i exactly and to perform exact arithmetic on non-real numbers? > Or will > (sqrt 4) => 2 > be allowed? Your definition > (sqrt z) => (exp (/ (log z) 2.0)) > seems to preclude it by putting in that inexact 2. Do you mean > the right hand side of this formula literally, with exp and log being > Scheme procedures, or do you mean to represent the mathematical > formula $e^{\log(z)/2}$ (in TeX notation). When the meaning is mathematical, as here, we should use mathematical notation instead of Scheme's syntax. > 13 bis: Again for sqrt, why is > (sqrt -inf.0) => unspecified > and not > (sqrt -inf.0) => +inf.0i > You didn't have any problem with > (log 0.0) => -inf.0 That, too, was a mistake. (sqrt -inf.0) should return +inf.0i. > 14. About expt: In contrast to my comments about sqrt, I note > that your definition of expt uses mathematical notation to > define it, so > > (expt 5 3) => 125, not 125.0 > is not only allowed, but required? Is > (expt 125 1/3) => 5 > allowed? As mentioned in my answer to question 13, I think expt should be explicitly excepted from the general requirement that operations return exact results when given exact arguments. I think, however, that expt should be explicitly required to return exact results when both arguments are exact, and the second argument is an integer. Thus (expt 5 -3) would be required to evaluate to 1/125, but (expt 125 1/3) would be allowed but not required to evaluate to 5. > 16. About magnitude: Following Kahan, I think it should be required > that > (magnitude z) => +inf.0 > if either the real or imaginary part of z is infinite (even if > the other part is a NaN). Here Kahan is following the argument that, > if the real or imaginary part is infinite, you would get the same > answer no matter the (finite) value of the other part, so you should > give that answer even if the other part is infinite or NaN. I agree with you and Kahan on this. Will

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