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Some preliminary comments
I thank the authors of this proposal for their work.
I'm not entirely sure that I understand the the thrust of this
proposal, or fully comprehend what this proposal encompasses, but I
would like to go through it carefully and propose certain questions
> Should the R5RS procedures for generic arithmetic (e.g. +)
remain in R6RS? Here are five possible answers, phrased in terms of
the + procedure:
> 1. + is not defined in R6RS.
> 2. + is defined to be a synonym for the ex+, so its domain
is restricted to exact arguments, and always returns an exact result.
> 3. + is defined as the union of the ex+ and in+ procedures,
so all of its arguments are required to have the same exactness, and
the exactness of its result is the same as the exactness of its
> 4. + is defined as in R5RS, but with the increased
portability provided by requiring the full numeric tower. This
alternative is described in the section R5RS-style Generic Arithmetic.
> 5. + is defined to return an exact result in all cases, even
if one or more of its arguments is inexact. This alternative is
described in the section Generic Exact Arithmetic.
> Will Clinger prefers the 4th possibility, Mike Sperber the 5th.
I have real difficulties with 5, which I hope to expand on at a later
time. Part of the problem is that 5 assumes that inexact->exact
gives reasonable results on inexact arguments, and I can't see how
this can be true if, for example, inexacts are implemented as the
> The real?, rational?, and integer? predicates must return false
for complex numbers with an imaginary part of inexact zero, as non-
realness is now contagious. This causes possibly unexpected behavior:
`(zero? 0+0.0i)' returns true despite `(integer? 0+0.0i)' returning
false. Possibly, new predicates realistic?, rationalistic?, and
integral? should be added to say that a number can be coerced to the
specified type (and back) without loss. (See the Design Rationale.)
I don't like the word contagious; I cannot think of a rigorous
definition in many contexts (including this one).
> Most Scheme implementations represent an inexact complex number
as a pair of two inexact reals, representing the real and imaginary
parts of the number, respectively. Should R6RS mandate the presence
of such a representation (while allowing additional alternative
representations), thus allowing it to more meaningfully discuss
semantic issues such as branch cuts?
> Should `(floor +inf.0)' return +inf.0 or signal an error
instead? (Similarly for ceiling, flfloor, infloor, etc.)
It should *be* an error, whether it should *signal* an error is
> The bitwise operations operate on exact integers only. Should
they live in the section on exact arithmetic?
I don't see this as an important issue?
> Should they carry ex prefixes?
> Or should they be extended to work on inexact integers as well?
Just exact integers.
> +nan.0 represents the result of (/ 0.0 0.0), and may represent
other NaNs as well. (This SRFI does not require read/write invariance
OK, but writing a NaN should give whatever info is in the
significand. There seems to be no way to get that information using
Scheme procedures described in this document.
Basically, I believe that the programming languages "Scheme", which
we can specify, should "play well with others", where "others"
include hardware and OS runtime libraries, which are specified either
by hardware companies, or by other standards bodies. The hardware
and runtime libraries of different systems generate different NaNs,
and if we're going to go to this amount of trouble to add what I hope
will be useful flonum arithmetic to Scheme, some way should be
provided in Scheme to distinguish NaNs that are different (in their
precision, range, or mantissa, i.e., according to the IEEE standard)
and to determine whether two are the same.
> Implementations that use binary floating point representations
of real numbers should represent x|p using a p-bit significand if
practical, or by a greater precision if a p-bit significand is not
practical, or by the largest available precision if p or more bits of
significand is not practical within the implementation.
> Note: The precision of a significand should not be confused
with the number of bits used to represent the significand. In the
IEEE floating point standards, for example, the significand's most
significant bit is implicit in single and double precision but is
explicit in extended precision. Whether that bit is implicit or
explicit does not affect the mathematical precision. In
implementations that use binary floating point, the default precision
can be calculated by calling the following procedure:
> (define (precision)
> (do ((n 0 (+ n 1))
> (x 1.0 (/ x 2.0)))
> ((= 1.0 (+ 1.0 x)) n)))
> Note: When the underlying floating-point representation is
IEEE double precision, the |p suffix should not be omitted for all
cases: Denormalized numbers have diminished precision, and therefore
should carry a |p suffix with the actual width of the signficand.
It seems ironic to me that this program itself assumes generic
It is not clear to me how, under this proposal, a Scheme will
implement floating-point arithmetic with more than one precision.
This is an important part of some numerical algorithms (iterative
refinement in solving linear systems, for example).
The first paragraph seems to make the |p notation meaningless; it
seems to specify nothing about the relationship between p and the
precision of the resulting flonum.
> Implementations are not required to use floating-point
representations, but every implementation is required to designate a
subset of its inexact reals as flonums, and to convert certain
external representations into flonums.
What does this mean? In particular, can this subset be empty? Is
that what "Implementations are not required to use floating-point
> If a <decimal 10> does not contain a non-empty <mantissa
width> and does not contain one of the exponent markers s, f, d, or
l, but does contain a decimal point or the exponent marker e, then it
is an external representation for a flonum. Furthermore inf.0, +inf.
0, -inf.0, nan.0, +nan.0, and -nan.0 are external representations for
flonums. Some or all of the other external representations for
inexact reals may also represent flonums, but that is not required by
> If a <decimal 10> contains a non-empty <mantissa width> or one
of the exponent markers s, f, d, or l, then it represents an inexact
number, but does not necessarily represent a flonum.
Wow; what does this mean? The difference between flonums and
inexacts is confusing to me. (Sorry for "surfer dude" mode here.)
> Safe and Unsafe Mode
> This SRFI allows a Scheme implementation to run code in one of
two global modes: "safe" and "unsafe" mode. These affect the fixnum
and the flonum operations. In safe mode, these operations must check
that their arguments are actually fixnums or flonums respectively, or
perform possible additional checking as required by the
specifications of the operations. In unsafe mode, these operations
must provide no such checking. This distinction allows an
implementation to generate efficient numerical code at the cost of
avoiding run-time checking. The R6RS should require a Scheme
implementation to provide the safe mode.
I presume a Scheme "implementation" may include a compiler. I can't
imagine that the requirement that "In unsafe mode, these operations
must provide no such checking" means that a compiler cannot use
dataflow and control flow information to signal an error in code,
rather than compiling it to nonsense and deliberately crashing a
> Equivalence of Objects
> The R6RS specification of eqv? for numbers should be changed as
> The eqv? procedure returns #t if:
> * obj1 and obj2 are both exact numbers, and are numerically
equal (see =, section see section 6.2 Numbers).
> * obj1 and obj2 are both inexact numbers, are numerically
equal (see =, section see section 6.2 Numbers), and yield the same
results (in the sense of eqv?) when passed as arguments to any other
procedure that can be defined as a finite composition of Scheme's
standard arithmetic procedures.
> The eqv? procedure returns #f if:
> * one of obj1 and obj2 is an exact number but the other is
an inexact number.
> * obj1 and obj2 are numbers for which the = procedure
> * obj1 and obj2 yield different results (in the sense of
eqv?) when passed as arguments to any other procedure that can be
defined as a finite composition of Scheme's standard arithmetic
First, I presume that ", or" should be added to the end of the first
item of the first list and to the ends of the first and second items
of the second list.
OK, I just went through the whole document and checked for every
instance of = and could not find out whether
(let ((x (/ 0. 0.)))
(= x x))
returns #t or #f. It is not useful for it to return #t if IEEE
arithmetic is the underlying inexact representation and "=" means
"IEEE arithmetic =". So a definition of eqv? in terms of = seems
incomplete. (Or does = mean fl= here?)
The definition of eqv? is recursive for inexact numbers, and I don't
see where the recursion is based.
If we're going to go to all this trouble to insert flonums (and I
presume the main motivation is to have IEEE flonums) into Scheme, we
should add the inquiry functions of the IEEE standard (which I can't
look up at the moment, because it's not online) that return various
properties of the part of an IEEE flonum.
> Numerical Type Predicates
> Note: The behavior of these type predicates on inexact numbers
is unreliable, because any inaccuracy may affect the result.
Couldy you explain this statement? Or give an example?
> Integer Division
Why div+mod for reals and quotient+remainder for exact integers?
I'm losing it a bit; are the procedures in this section defined for
inexact numbers? If so, then the requirement for div+mod that
x1 = nd * x2 + xm.
may not be realizable due to precision issues; perhaps a proof is
The requirement that the sign of the second argument acts as a
selector to decide what function div+mod computes seems to give extra
meaning to (negate y) that should not be there.
> Every implementation must define its fixnum range as a closed
interval [lo, hi] such that lo and hi are (mathematical) integers
with lo <= 0 < 1 <= hi. Every mathematical integer within an
implementation's fixnum range must correspond to an exact integer
that is representable within the implementation. The fixnum
operations of an implementation will perform arithmetic modulo hi-lo+1.
In mathematics, "arithmetic modulo hi-lo+1" means that results go
from 0 to hi-lo; I can't imagine that this is what is wanted here.
As a practical matter, signed integer arithmetic on all modern
processors wraps, so that
hardware-signed-+ (2^31-1) 1 => - 2^31,
yet the C standard says that such behavior is undefined. The C
standard does require that *unsigned* arithmetic wraps
hardware-unsigned-+ (2^32-1) 1 => 0
which is probably the reason why hardware signed arithemtic wraps the
way it does (so the processor can use a single + operation for both
signed and unsigned arithmetic).
> <definition of many operations that naturally can take multiple
arguments to take only two arguments>
Let's let the interpreter or compiler worry about doing the folding
of these operations; I don't see any reason to require the programmer
to do it by hand.
> library procedure: fxabs fx
> This procedure returns `(fx- 0 fx)' if fx is negative, fx
So, do you want
(fxabs least-fixnum) ==> least-fixnum
which is what this specifies?
> procedure: fxdiv+mod fx1 fx2
> library procedure: fxdiv fx1 fx2
> library procedure: fxmod fx1 fx2
> library procedure: fxquotient fx1 fx2
> library procedure: fxmodulo+remainder fx1 fx2
> library procedure: fxmodulo fx1 fx2
> library procedure: fxremainder fx1 fx2
> These procedures implement number-theoretic integer division
modulo hi-lo+1. See the section on Integer Division.
This seems to be a laundry list of ad-hoc, incompletely specified
procedures. I really don't know what you mean by these.
> procedure: fxbitwise-not fx
> Returns the fixnum which is the one's-complement of its
argument. congruent mod hi-lo+1.
I believe it should be "ones-complement" and that the "congruent mod
hi-lo+1" is unnecessary.
> procedure: fxarithmetic-shift fx1 fx2
> This procedure conceptually shifts the two's complement
representation of fx1 fx2 bits left when fx2 > 0, and -fx2 bits right
when fx2 < 0, extending the sign. (It returns fx1 when fx2 = 0.) It
returns the result of that shift congruent mod hi-lo+1.
Do you mean "(eq? fx1 (fxarithmetic-shift fx1 0))"?
> procedure: fl= fl1 fl2
> procedure: fl< fl1 fl2
> procedure: fl<= fl1 fl2
> procedure: fl> fl1 fl2
> procedure: fl>= fl1 fl2
> These procedures return #t if their arguments are
(respectively): equal, monotonically increasing, monotonically
decreasing, monotonically nondecreasing, or monotonically
nonincreasing, #f otherwise. These predicates are required to be
> (fl= +inf.0 +inf.0) ==> #t
> (fl= -inf.0 +inf.0) ==> #f
> (fl= -inf.0 -inf.0) ==> #t
> (fl= 0.0 -0.0) ==> unspecified
> (fl< 0.0 -0.0) ==> unspecified
> (fl= +nan.0 fl) ==> unspecified
> (fl= +nan.0 fl) ==> unspecified
> (fl< +nan.0 fl) ==> unspecified
> The following behavior is strongly recommended but not required
> (fl= 0.0 -0.0) ==> #t
> (fl< -0.0 0.0) ==> #f
> (fl= +nan.0 +nan.0) ==> #f
> procedure: flinteger? fl
> procedure: flzero? fl
> procedure: flpositive? fl
> procedure: flnegative? fl
> procedure: flodd? ifl
> procedure: fleven? ifl
> procedure: flnan? fl
> These numerical predicates test a flonum for a particular
property, returning #t or #f. Flinteger? tests it if the number is an
integer, flzero? tests if it is fl= to zero, flpositive? tests if it
is greater than zero, flnegative? tests if it is less than zero,
flodd? tests if it is odd, fleven? tests if it is even, flnan? tests
if it is a NaN.
> (flnegative? -0.0) ==> #f
> Note: `(flnegative? -0.0)' must return #f, else it would lose
the correspondence with (fl< -0.0 0.0), which is #f according to the
> procedure: flmax fl1 fl2
> procedure: flmin fl1 fl2
> These procedures return the maximum or minimum of their arguments.
> procedure: fl+ fl1 fl2
procedure: fl- fl1 fl2
> procedure: fl* fl1 fl2
procedure: fl/ fl1 fl2
> These procedures return the flonum sum, difference, product, or
quotient of their flonum arguments. In general, they should return
the flonum that best approximates the mathematical sum, difference,
or quotient of their arguments. (For implementations that represent
flonums as IEEE binary floating point numbers, the meaning of "best"
is reasonably well-defined by the IEEE standards.)
> For undefined quotients, fl/ behaves as specified by the IEEE
> (fl/ 1.0 0.0) ==> +inf.0
> (fl/ -1.0 0.0) ==> -inf.0
> (fl/ 0.0 0.0) ==> +nan.0
I realize that "A foolish consistency is the hobgoblin of little
minds", but why in heaven's name do you justify some requirement by
saying "we must follow IEEE arithmetic" and at the same time totally
disregard other IEEE arithmetic requirments? Either implement the
IEEE spec for floating point arithmetic in Scheme flonum arithmetic
or forget it.
> library procedure: flnumerator fl
> library procedure: fldenominator fl
> These procedures return the numerator or denominator of their
argument as a flonum; the result is computed as if the argument was
represented as a fraction in lowest terms. The denominator is always
positive. The denominator of 0 is defined to be 1.
> (flnumerator +inf.0) ==> +inf.0
> (flnumerator -inf.0) ==> -inf.0
> (fldenominator +inf.0) ==> 1.0
> (fldenominator -inf.0) ==> 1.0
> (flnumerator 0.75) ==> 3.0 ; example
> (fldenominator 0.75) ==> 4.0 ; example
> The following behavior is strongly recommended but not required
> (flnumerator -0.0) ==> -0.0
If +inf.0 and +nan.o are not rational, I don't think it is reasonable
for them to be in the domain of flnumerator and fldenominator.
> procedure: flsqrt fl
> Returns the principal square root of z. For a negative
argument, the result may be +nan.0, or may be some meaningless
flonum. With flexp and fllog defined as above, the value of `(flsqrt
fl)' is defined as `(flexp (fl/ (fllog fl) 2.0))'.
Aren't the last two statements contradictory?
As a general question---why define the flonum versions of the
elementary functions to possibly return complex arguments rather than
to have restricted ranges or to return NaNs for arguments that would
not give a real result? If the goal of adding flonum arithmetic is
speed, why define these functions in such a way that the built-in
hardware or OS runtime library routines cannot be used directly?
> procedure: flexpt fl1 fl2
> Returns fl1 raised to the power fl2. For nonzero fl1
> fl1^fl2 = e^(z2 log z1)
> 0^fl is 1 if z = 0, and 0 if fl is positive. Otherwise, the
result may be +nan.0, or may be some meaningless flonum.
I take it z2 should be fl2 and z1 should be fl1, and z should be f1.
Does the "0" here mean "0.0"? Does the "1" here refer to "1.0"?
(Similar questions arise for insqrt.)
Enough for now.
[Egner et al. 2004] Sebastian Egner, Richard Kelsey, Michael
Sperber. Cleaning up the tower: Numbers in Scheme. In Proceedings of
the Fifth ACM SIGPLAN Workshop on Scheme and Functional Programming,
pages 109--120, September 22, 2004, Snowbird, Utah. Technical report
TR600, Computer Science Department, Indiana University
is cited as justification for several parts of this proposal.
Unfortunately, I believe this paper to be seriously flawed.
I'm bringing this up for the following reason: I don't believe that
this paper can be used, by itself, as justification for any aspect of
this proposal. It may be used as an *argument* for certain aspects
of the proposal, but as there has been no general discussion of the
merits of the paper, either in this forum or in others, I don't
believe that it can be used as "settled decisions". Thus, if the
authors of this SRFI wish to use arguments from that paper as
justification for certain aspects of *this* proposal, they should
repeat those arguments here for them to be discussed and debated as