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inexactness vs. exactness



 | Date: Mon, 18 Jul 2005 08:50:50 -0700 (PDT)
 | From: bear <bear@xxxxxxxxx>
 | 
 | This draft effectively eliminates certain properties of
 | inexactness vs. exactness,

SRFI-70 sharpens the distinction between inexacts and exacts.  From
the R5RS notation of inexacts it deduces their nature as real
neighborhoods.  Because exact numbers in Scheme designate single
mathematical numbers, the idea of exactness as a superficial attribute
of numbers must be jettisoned.

SRFI-70 promotes the exactness property of functions over the
exactness property of numbers:

  Rationale

  Inexactness

  Inexactness is as much a property of calculation as it is of data
  sources.  A language having latent types does not obviate the
  programmer's expectation that a calculation involving transcendental
  functions should return an inexact result.  To this end, two
  sentences are struck from the 6.2.2 "Exactness" section:

      An operation may, however, return an exact result if it can
      prove that the value of the result is unaffected by the
      inexactness of its arguments.  For example, multiplication of
      any number by an exact zero may produce an exact zero result,
      even if the other argument is inexact.

  By those sentences, inexactness is a contagious property of all
  numbers; except 0, the only number whose exactness is contagious!
  This conflicts with the statement earlier in 6.2.2:

      ... Thus inexactness is a contagious property of a number. 

 | and in doing so it creates a type distinction where none was
 | before.

Is exactness this "type distinction"?

 | In the dialect spec'd by R5RS, there was never any situation
 | where an inexact number was required and an exact one would
 | not do; thus it was entirely correct for implementations to
 | return exact numbers whenever the opportunity presented
 | itself.

Implementations like Elk, Gambit, and Mzscheme signal errors for
division by exact zero, but return infinities for division by inexact
zero.  This is a "situation where an inexact number was required and
an exact one would not do".

 | An implementation with exact roots in its numeric
 | system, for example, could return an exact square root of
 | two, or multiply two such numbers to get an exact two, or
 | multiply an inexact whatever by an exact zero to get an exact
 | zero, or subtract a boxed inexact number from itself to get
 | an exact zero, or whatever.  Exactness, where available, was
 | an unequivocal good.

This created the contradiction noted above.  With exact 0 being
stronger than inexact 0.0, it also leads to the opaque practice of
controlling the behavior of EXPT and `/' at 0 by coercing the
exactness of its arguments.

SQRT is not transcendental; but dividing by SQRT of something has the
same potential for lossage as discussed near the end of this message.
One of the SRFI-70 "Issues" is whether there should be exact-only
versions of SQRT and EXPT.

 | Compliance with this SRFI requires that implementations
 | do *NOT* extend exact numerics in any way;

How is that?  I deliberately left unspecified cases where exact
infinities could be returned.  Notice, for instance, that (/ k 0) does
not appear in any of the SRFI-70 examples; yet all combinations with
an inexact divisor are examples.

 | rather than pretending to revise the standard itself,

SRFI-70 is clearly a proposal to replace section 6.2 of R6RS.  Its
abstract reads:

  This SRFI reworks section 6.2 "Numbers" of R5RS to:

      * include inexact real positive and negative infinities,
      * define complete semantics for `expt',
      * extend `gcd' and `lcm' to exact rational numbers,
      * extend `quotient', `modulo', and `remainder' to finite real numbers,
      * remove a contradiction related to exactness,
      * add examples, and
      * make small improvements to the text. 

 | it should simply point out that the standard permits such extension
 | but also allows compliance with the present SRFI, which forbids it.

SRFI-70 proposes changes to the Scheme Report.  For instance, R5RS
specifies that (expt 0 -1) ==> 0.  That is wrong and SRFI-70 corrects
it.  Compliance with R5RS would not allow this correction.

 | This is an example of what I meant when I had misgivings
 | about SRFI-75 (a SRFI about R6RS) being discussed in this
 | forum (which is NOT about RnRS);

"The Revised R6RS Status Report"
(http://www.schemers.org/Documents/Standards/Charter/2004-10-13.pdf)
states:

  In general, the committee's design procedure consists in tackling
  each issue separately: gathering proposal(s) for solutions (from
  existing implementations, SRFI's, old RnRS proposals, new ideas,
  etc), discussing and modifying the proposals, and then taking a
  preliminary decision on the course of action ...

So SRFI is an endorsed channel for input to R6RS.

 | it confuses the two
 | processes to an intolerable degree, and leads the authors
 | of SRFI's to believe, erroneously, that they have the
 | authority to revise the standard.

I have authority to propose whatever I want.  And the R6RS editors
have the authority to accept it, modify it, or ignore it.

 | But, process quibbles aside, this reveals a deeper
 | issue.
 | 
 | This draft intends to constrain implementations from
 | opportunistically providing exact results or seeking
 | to provide exact results where other implementations
 | or representations cannot.  This amounts to treating
 | inexactness in some circumstances as a *desirable*
 | property of numbers.

I do a lot of numerical computing (near-field optics lately).
Division by zero happens.  If SIN of some complicated expression
returns an exact zero .001% of the time, then .001% of the time I will
get an exception from the division taking that expression as a divisor
rather than an infinity which lets the program continue.

That is broken behavior; but that is what R5RS-compliant Mzscheme 205
does; and probably other implementations also.

So yes, I prefer inexact numbers from transcendental functions!

 | I strongly suspect that if
 | inexactness has become a desirable property, then it
 | is most likely because of one or more design mistakes.
 | 
 | This is like deriving a known-false result in mathematics;
 | it indicates that at least one of the steps along the
 | way was wrong, and subsequent steps, including the
 | present SRFI in dealing with inexactness, therefore
 | invalid (in proceeding, perhaps correctly, from
 | erroneous premises).

Strong suspicions are not the same known-false results.  But it is
demagoguery for you to deprecate my program sight unseen.

http://savannah.gnu.org/cgi-bin/viewcvs/freesnell/freesnell/fresneleq.scm?rev=HEAD&content-type=text/vnd.viewcvs-markup

;; Optics is all wavelength based; so this is also.  These routines
;; work out the complex voltages in the forward and reverse directions
;; to find the transmitted and reflected amplitudes.  This works only
;; for intensities where the layers act linearly (superposition).
;; Each layer 0:n has an index of refraction and height.

;; Square the absolute value of numbers returned to get the power
;; ratios.

;; Because the P and S polarizations are independent, calculate them
;; separately.

;;; Fresnel's equations from
;;; http://www.ifm.liu.se/~boser/elma/Lect13.pdf

;;; The transmitted voltage (E-Field)
(define (E_T n1 n2 cos-i cos-t s-polarization?)
  (if s-polarization?
      (/ (* 2 n1 cos-i)
         (+ (* n1 cos-i) (* n2 cos-t)))
      (/ (* 2 n1 cos-i)
         (+ (* n1 cos-t) (* n2 cos-i)))))
;;; The reflected voltage (E-Field)
(define (E_R n1 n2 cos-i cos-t s-polarization?)
  (if s-polarization?
      (/ (- (* n1 cos-i) (* n2 cos-t))
         (+ (* n1 cos-i) (* n2 cos-t)))
      (/ (- (* n2 cos-i) (* n1 cos-t))
         (+ (* n1 cos-t) (* n2 cos-i)))))

;;; Returns 2x2 matrix:
;;;
;;;            (   1      r      )
;;;    1       (           n-1,n )
;;; -------- * (                 )
;;;  t         ( r          1    )
;;;   n-1,n    (  n-1,n          )
(define (layer-interface n1 n2 th1 s-polarization?)
  (define cos-i (cos th1))
  (define cos-t (cos (Snell-law n1 n2 th1)))
  (let ((transmit (E_T n1 n2 cos-i cos-t s-polarization?))
        (reflect (E_R n1 n2 cos-i cos-t s-polarization?)))
    (let ((r/t (/ reflect transmit))
          (tinv (/ transmit)))
      (list (list tinv r/t) (list r/t tinv)))))

;;; Returns 2x2 matrix coding phase difference between reflected and
;;; transmitted paths.
;;;
;;;  (  -i*d_n    0    )
;;;  ( e               )
;;;  (           i*d_n )
;;;  (    0     e      )
(define (layer-phase h_j n_j th_j w)
  (define phase (exp (/ (* +2i pi h_j n_j (cos th_j)) w)))
  (list (list (/ phase) 0) (list 0 phase)))

;; Given the angle of impinging light,
;; returns the angle of the transmitted light
(define (Snell-law n1 n2 th-i)
  (asin (* (/ n1 n2) (sin th-i))))

How is this poorly designed?
Should it not use division?
Should it not use transcendental functions?

 | I believe that it indicates that
 | one or more of the uniform-numeric-vector SRFI's is
 | in conflict with Scheme's basic design principles
 | (the preference for exact calculation where possible)
 | and in need of redesign.

The adoption of SRFI-70 would fix that problem.

There seems to be growing agitation to remove exactness as a concept
from Scheme.  A proposal along these lines is "Cleaning up the Tower:
Numbers in Scheme"
http://www-pu.informatik.uni-tuebingen.de/users/sperber/papers/numerical-tower.pdf

SRFI-70 may be the best chance for retaining it.