'numeric' is a class but not a type -- so I think the FAQ is wrongly
worded but the concept is well defined (despite the presence of
is.numeric!) But it does not say that all such numbers can be represented
exactly, and only some can.
On Sat, 17 May 2008, Berwin A Turlach wrote:
G'day Erik,
On Fri, 16 May 2008 10:45:43 -0500
Erik Iverson <[EMAIL PROTECTED]> wrote:
[...]
The help page for '%%' addresses this a bit, but then caveats it with
'up to rounding error', which is really my question. Is there ever
'rounding error' with 2.0 %% 1 as opposed to 2 %% 1?
I am not in the position to give an authoritative answer, but I think
there should be no problem with rounding error in the situation that
you describe. At least I hope there is no problem, otherwise I would
consider this a serious issue. :)
However, my question is related to R FAQ 7.31, "Why doesn't R
think these numbers are equal?" The first sentence of that FAQ
reads, "The only numbers that can be represented exactly in R's
numeric type are integers and fractions whose denominator is a
power of 2."
Again, I did not write this FAQ answer and cannot give an authoritative
answer, but the word "integer" in that answer does not IMHO refer to
variables in R that are of integer type; in particular since the answer
discusses what kind of numbers "can be represented exactly in R's
numeric type". (Perhaps this should actually be plural since there are
several numeric types?)
My interpretation is that 2.0 and 2 are both *text constants* that
represent the integer 2, and that number is representable in a floating
point (and in an integer).
The paper by Goldberg, referenced in FAQ 7.31, contains a discussion on
whether it is possible (it is) to convert a floating point number
from binary representation to decimal representation and then back;
ending up with the same binary representation. This kind of questions
are important if you use commands like write.table() or write.csv()
which write out floating points in decimal representation, readable to
normal humans. When you read the data back in, you want to end up with
the exact same binary representation of the numbers. Goldberg is
indeed an interesting paper to read.
Possible to write out and read in *on the same computer*. R doesn't aspire
to that, as it assumes text files are for humans or transfer to unknown
other programs, and does provides binary save formats. (Human-friendly
numbers will be written out faithfully, but if there is a choice between a
short representation and a one with sequence of 9s, the short one will be
chosen.)
And the comments I made above are based on my understanding of
Goldberg, 2 and 2.0 are both decimal representation of the integer 2,
and this number has an exact representation (in integer type variables
and in floating point type variables). Hence, both these decimal
representation should lead to a binary representation that correspond
to that number exactly.
They should, at least for small integer values. In R this relies on
strtod for which C99 and POSIX say
If the subject sequence has the decimal form and at most DECIMAL_DIG
(defined in <float.h>) significant digits, the result should be
correctly rounded [...].
So this means that x.0 for x up to 10^15 or so should be represented
exactly and be different from x.y for y =1...9.
Note too that non-integer values > 10^16 or so will be represented as
integers.
Lots of 'should' here -- compiler and runtime writers do make mistakes and
may not have been working to the C99 standard (e.g. the Windows runtime
predates it).
Thus, I would expect
R> x <- 2.0
R> x %% 1 == 0
always to work and to return TRUE. It is things like:
R> x <- sqrt(2.0)^2
R> x %% 1 == 0
that FAQ 7.31 is about and what, IMHO, the comment in the help page
of %% warns about; if the variable x contains a value that was created
by some finite precision floating point calculations. But the
conversion from a textual representation of an integer to a binary
representation should not create problems.
Cheers,
Berwin
=========================== Full address =============================
Berwin A Turlach Tel.: +65 6515 4416 (secr)
Dept of Statistics and Applied Probability +65 6515 6650 (self)
Faculty of Science FAX : +65 6872 3919
National University of Singapore
6 Science Drive 2, Blk S16, Level 7 e-mail: [EMAIL PROTECTED]
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