[Rd] significant digits (PR#9682)

Pmc1

at Jun 3, 2008 at 1:30 am

To reply to my own message, that function wasn't quite right. I think
this one works better:

signif.string <- function(signum,sigdigs){
test <- abs(signum)
left <- nchar(trunc(test))
right <- nchar(test)-left-1
if (test<1) {left<-left-1}
if (right<0) {right<-0}
if (sigdigs<left) {out<-as.character(signif(signum,digits=sigdigs))}
else if (sigdigs==left & trunc(signum) %% 10 == 0)
{out<-paste(round(signum),".",sep="")}
else if (sigdigs<=left+right) {out<-format(signum,digits=sigdigs)}
else {out<-sprintf(paste("%.",sigdigs-left,"f",sep=""),signum)}
return(noquote(out))
}

But it should still have error checking and vector capability, yadda
yadda. Also, I forgot what year it was, so sorry, Scott, for spamming
you with something you're hopefully not still stuck on.

Pat Carr

reply | permalink

Duncan Murdoch

at Jun 3, 2008 at 10:36 am

⇧

pmc1 at cornell.edu wrote:
I came to report this same bug and found it already in the trash, but
I slightly disagree with that assessment. If it's not a bug, then
perhaps it's a feature request. Comments at the end.

On Mon, May 14, 2007, Duncan Murdoch wrote:

On 13/05/2007 8:46 PM, scott.wilkinson at csiro.au wrote:

In the example below round() does not report to the specified number of
digits when the last digit to be reported is zero: Compare behaviour for
0.897575 and 0.946251. Ditto for signif(). The number of sigfigs is
ambiguous unless the reader knows this behaviour. Is this a bug or
intended behaviour? Is there a work-around?

It's not a bug. It has nothing to do with round(), it is the way R
prints numbers by default. If you ask to print 0.90, you'll get

[1] 0.9

because 0.9 and 0.90 are the same number. If you want trailing zeros to
print, you need to specify a format to do that, e.g.

noquote(format(0.9, nsmall=2))

[1] 0.90

The noquote stops the "" from printing. You could also use sprintf() or
formatC() for more C-like format specifications.

All of those options require you to specify the number of digits after
the decimal, don't they? Unless sprintf would default to the number of
decimal places passed to it, but it doesn't:

That specification doesn't make sense. There is no "number of decimal
places passed to it". What sprintf() sees below is identical to what it
would see if you called

sprintf("%f", 0.9)

because signif(0.90, digits=2) == 0.9. Those two objects are identical.

sprintf("%f",signif(0.90, digits=2))

[1] "0.900000";

it defaults to 6. Although %g behaves differently,

sprintf("%g",signif(0.90, digits=2))

[1] "0.9",

this clearly still isn't the desired behavior.
Maybe not what you desired, but certainly reasonable behaviour.
To continue that vein, the same issue with rounding versus printing
occurs with vectors:

sapply(c(1:6),function(a){signif(c(18.423,0.90),digits=a)})

[,1] [,2] [,3] [,4] [,5] [,6]
[1,] 20.0 18.0 18.4 18.42 18.423 18.423
[2,] 0.9 0.9 0.9 0.90 0.900 0.900

Trying to get that and more complicated tables to print the correct
number of significant digits gets pretty hairy with sprintf(). I could
be wrong, but I would view the primary purpose of rounding to
significant digits the printed output of the number. That there
doesn't seem to be a way to do this without later having to specify
the number of decimal places would seem to render signif() as it's
written not particularly useful.

There are two solutions I can think of off the top of my head. The
first is to create a new data type of a fixed length real number but
the easier way would be to have a function that returns a string
something like this:

signif.string <- function(signum,sigdigs){
left <- nchar(trunc(signum))
right <- nchar(signum-trunc(signum))-2
if (abs(signum)<1 | signum<0) {left<-left-1}
if (right<0) {right<-0}
if (sigdigs<left) {return(as.character(signif(signum,digits=sigdigs)))}
else if (sigdigs==left) {return(paste(round(signum),".",sep=""))}
else if (sigdigs<=left+right) {return(format(signum,digits=sigdigs))}
else {return(sprintf(paste("%.",sigdigs-left,"f",sep=""),signum))}
}

This is just a skeleton that I think suits my needs for the moment and
might also cover the original poster's. One for production would need
to handle scientific notation and would probably want to fix the 5
round behavior on windows.
As far as I know, rounding is fine in Windows:
round(1:10 + 0.5)

[1] 2 2 4 4 6 6 8 8 10 10

looks okay to me. If you want biased rounding instead (getting 2:11 as
output), simply use trunc(x + 0.5) instead of round(x).

Duncan Murdoch

Pat Carr

version.string R version 2.7.0 (2008-04-22)

______________________________________________
R-devel at r-project.org mailing list
https://stat.ethz.ch/mailman/listinfo/r-devel

reply | permalink

Patrick Carr

at Jun 3, 2008 at 3:43 pm

⇧

On 6/3/08, Duncan Murdoch wrote:
because signif(0.90, digits=2) == 0.9. Those two objects are identical.

My text above that is poorly worded. They're identical internally,
yes. But in terms of the number of significant digits, 0.9 and 0.90
are different. And that matters when the number is printed, say as an
annotation on a graph. Passing it through sprintf() or format() later
requires you to specify the number of digits after the decimal, which
is different than the number of significant digits, and requires case
testing for numbers of different orders of magnitude.

The original complainant (and I) expected this behavior from signif(),
not merely rounding. As I said before, I wrote my own workaround so
this is somewhat academic, but I don't think we're alone.

As far as I know, rounding is fine in Windows:

round(1:10 + 0.5)

[1] 2 2 4 4 6 6 8 8 10 10
It might not be the rounding, then. (windows xp sp3)
signif(12345,digits=4) [1] 12340
signif(0.12345,digits=4)

[1] 0.1235

OS X (10.5.2/intel) does not have that problem. But (on both windows and OS X):

signif(12345.12345,digits)

[1] 12345.12

Pat Carr

reply | permalink

Duncan Murdoch

at Jun 3, 2008 at 6:48 pm

⇧

On 6/3/2008 11:43 AM, Patrick Carr wrote:

On 6/3/08, Duncan Murdoch wrote:

because signif(0.90, digits=2) == 0.9. Those two objects are identical.

My text above that is poorly worded. They're identical internally,
yes. But in terms of the number of significant digits, 0.9 and 0.90
are different. And that matters when the number is printed, say as an
annotation on a graph. Passing it through sprintf() or format() later
requires you to specify the number of digits after the decimal, which
is different than the number of significant digits, and requires case
testing for numbers of different orders of magnitude.

The original complainant (and I) expected this behavior from signif(),
not merely rounding. As I said before, I wrote my own workaround so
this is somewhat academic, but I don't think we're alone.

As far as I know, rounding is fine in Windows:

round(1:10 + 0.5)

[1] 2 2 4 4 6 6 8 8 10 10
It might not be the rounding, then. (windows xp sp3)
signif(12345,digits=4) [1] 12340
signif(0.12345,digits=4)

[1] 0.1235

It's easy to make mistakes in this, but a little outside-of-R
experimentation suggests those are the right answers. The number 12345
is exactly representable, so it is exactly half-way between 12340 and
12350, so 12340 is the right answer by the unbiased round-to-even rule.
The number 0.12345 is not exactly representable, but (I think) it is
represented by something slightly closer to 0.1235 than to 0.1234. So
it looks as though Windows gets it right.

OS X (10.5.2/intel) does not have that problem.

Which would seem to imply OS X gets it wrong. Both are supposed to be
using the 64 bit floating point standard, so they should both give the
same answer: but the actual arithmetic is being done by run-time
libraries that are outside our control to a large extent, and it looks
as though the one on the Mac is less accurate than the one on Windows.

But (on both windows and OS X):

signif(12345.12345,digits)

[1] 12345.12

This is a different problem. The number is correctly computed as
12345.12345 (or at least a representable number quite close to that),
and then the default display rounds it some more. Set
options(digits) to see it in its full glory.

Duncan Murdoch

reply | permalink

Patrick Carr

at Jun 3, 2008 at 8:07 pm

⇧

On 6/3/08, Duncan Murdoch wrote:
It's easy to make mistakes in this, but a little outside-of-R
experimentation suggests those are the right answers. The number 12345 is
exactly representable, so it is exactly half-way between 12340 and 12350, so
12340 is the right answer by the unbiased round-to-even rule. The number
0.12345 is not exactly representable, but (I think) it is represented by
something slightly closer to 0.1235 than to 0.1234. So it looks as though
Windows gets it right.

Well, right within the limitations of binary floating-point
arithmetic. Not right right.

In the grander scheme, this is a nicety which is largely
inconsequential--if I need a real measure precision (precise
precision?) I'll use a +/- notation of a propagated error and/or edit
the typography of the numbers by hand immediately before the final
output. But again, final printed output of the number is basically the
useful use I see for a function that returns significant digits. And
for that purpose I think it should be right right, and actually output
the number of significant digits requested.

signif(12345.12345,digits)

[1] 12345.12

This is a different problem. The number is correctly computed as
12345.12345 (or at least a representable number quite close to that), and
then the default display rounds it some more. Set options(digits) to see
it in its full glory.

Aha, my mistake; I missed that setting.

Pat Carr

reply | permalink

Simon Urbanek

at Jun 3, 2008 at 8:36 pm

⇧

On Jun 3, 2008, at 2:48 PM, Duncan Murdoch wrote:

On 6/3/2008 11:43 AM, Patrick Carr wrote:

On 6/3/08, Duncan Murdoch wrote:

because signif(0.90, digits=2) == 0.9. Those two objects are
identical.

My text above that is poorly worded. They're identical internally,
yes. But in terms of the number of significant digits, 0.9 and 0.90
are different. And that matters when the number is printed, say as an
annotation on a graph. Passing it through sprintf() or format() later
requires you to specify the number of digits after the decimal, which
is different than the number of significant digits, and requires case
testing for numbers of different orders of magnitude.
The original complainant (and I) expected this behavior from
signif(),
not merely rounding. As I said before, I wrote my own workaround so
this is somewhat academic, but I don't think we're alone.

As far as I know, rounding is fine in Windows:

round(1:10 + 0.5)

[1] 2 2 4 4 6 6 8 8 10 10
It might not be the rounding, then. (windows xp sp3)
signif(12345,digits=4) [1] 12340
signif(0.12345,digits=4)

[1] 0.1235

It's easy to make mistakes in this, but a little outside-of-R
experimentation suggests those are the right answers. The number
12345 is exactly representable, so it is exactly half-way between
12340 and 12350, so 12340 is the right answer by the unbiased round-
to-even rule. The number 0.12345 is not exactly representable, but
(I think) it is represented by something slightly closer to 0.1235
than to 0.1234. So it looks as though Windows gets it right.

OS X (10.5.2/intel) does not have that problem.

Which would seem to imply OS X gets it wrong.

This has nothing to do with OS X, you get that same answer on pretty
much all other platforms (Intel/Linux, MIPS/IRIX, Sparc/Sun, ...).
Windows is the only one delivering the incorrect result here.

Both are supposed to be using the 64 bit floating point standard,
so they should both give the same answer:

Should, yes, but Windows doesn't. In fact 10000.0 is exactly
representable and so is 1234.5 which is the correct result that all
except Windows get. I don't have a Windows box handy, so I can't tell
why - but if you go through fprec this is what you get on the
platforms I tested (log10 may vary slightly but that's irrelevant here):

x = 0.123450000000000004174439
l10 = -0.908508905732048899217546, e10 = 4
pow10 = 10000.000000000000000000000000
x*pow10 = 1234.500000000000000000000000

Cheers,
Simon

but the actual arithmetic is being done by run-time libraries that
are outside our control to a large extent, and it looks as though
the one on the Mac is less accurate than the one on Windows.

But (on both windows and OS X):

signif(12345.12345,digits)

[1] 12345.12

This is a different problem. The number is correctly computed as
12345.12345 (or at least a representable number quite close to
that), and then the default display rounds it some more. Set
options(digits) to see it in its full glory.

Duncan Murdoch

______________________________________________
R-devel at r-project.org mailing list
https://stat.ethz.ch/mailman/listinfo/r-devel

reply | permalink

Duncan Murdoch

at Jun 3, 2008 at 9:12 pm

⇧

On 6/3/2008 4:36 PM, Simon Urbanek wrote:

On Jun 3, 2008, at 2:48 PM, Duncan Murdoch wrote:

On 6/3/2008 11:43 AM, Patrick Carr wrote:

On 6/3/08, Duncan Murdoch wrote:

because signif(0.90, digits=2) == 0.9. Those two objects are
identical.

My text above that is poorly worded. They're identical internally,
yes. But in terms of the number of significant digits, 0.9 and 0.90
are different. And that matters when the number is printed, say as an
annotation on a graph. Passing it through sprintf() or format() later
requires you to specify the number of digits after the decimal, which
is different than the number of significant digits, and requires case
testing for numbers of different orders of magnitude.
The original complainant (and I) expected this behavior from
signif(),
not merely rounding. As I said before, I wrote my own workaround so
this is somewhat academic, but I don't think we're alone.

As far as I know, rounding is fine in Windows:

round(1:10 + 0.5)

[1] 2 2 4 4 6 6 8 8 10 10
It might not be the rounding, then. (windows xp sp3)
signif(12345,digits=4) [1] 12340
signif(0.12345,digits=4)

[1] 0.1235

It's easy to make mistakes in this, but a little outside-of-R
experimentation suggests those are the right answers. The number
12345 is exactly representable, so it is exactly half-way between
12340 and 12350, so 12340 is the right answer by the unbiased round-
to-even rule. The number 0.12345 is not exactly representable, but
(I think) it is represented by something slightly closer to 0.1235
than to 0.1234. So it looks as though Windows gets it right.

OS X (10.5.2/intel) does not have that problem.

Which would seem to imply OS X gets it wrong.

This has nothing to do with OS X, you get that same answer on pretty
much all other platforms (Intel/Linux, MIPS/IRIX, Sparc/Sun, ...).
Windows is the only one delivering the incorrect result here.

Both are supposed to be using the 64 bit floating point standard,
so they should both give the same answer:

Should, yes, but Windows doesn't. In fact 10000.0 is exactly
representable and so is 1234.5 which is the correct result that all
except Windows get.

I think you skipped a step. The correct answer is either 0.1234 or
0.1235, not something 10000 times bigger. The first important question
is whether 0.12345 is exactly representable, and the answer is no. The
second question is whether it is represented by a number bigger or
smaller than the real number 0.12345. If it is bigger, the answer
should be 0.1235, and if it is smaller, the answer is 0.1234. My
experiments suggest it is bigger. Yours don't look relevant. It
certainly isn't exactly equal to 1234.5/10000, because that number is
not representable. It's equal to x/2^y, for some x and y, and it's a
pain to figure out exactly what they are.

However, I am pretty sure R is representing it (at least on Windows) as
the binary expansion

0.000111111001101001101011010100001011000011110010011111

while the true binary expansion (using exact rational arithmetic) starts
out

0.00011111100110100110101101010000101100001111001001111011101...

If you line those up, you'll see that the first number is bigger than
the second. (Ugly code to derive these is down below.)

Clearly the top representation is the correct one to that number of
binary digits, so I think Windows got it right, and all those other
systems didn't. This is probably because R on Windows is using extended
precision (64 bit mantissas) for intermediate results, and those other
systems stick with 53 bit mantissas.

Duncan Murdoch

# Convert number to binary expansion; add the decimal point manually

x <- 0.12345
while (x != 0) {
cat(trunc(x))
x <- x - trunc(x)
x <- x * 2
}

# Do the same thing in exact rational arithmetic

num <- 12345
denom <- 100000
for (i in 1:60) {
cat(ifelse(num > 100000, "1", "0"))
num <- num %% 100000
num <- 2*num
}

# Manually cut and paste the results to get these:

"0.000111111001101001101011010100001011000011110010011111"
"0.00011111100110100110101101010000101100001111001001111011101"

reply | permalink

Simon Urbanek

at Jun 3, 2008 at 9:33 pm

⇧

On Jun 3, 2008, at 5:12 PM, Duncan Murdoch wrote:

On 6/3/2008 4:36 PM, Simon Urbanek wrote:

On Jun 3, 2008, at 2:48 PM, Duncan Murdoch wrote:

On 6/3/2008 11:43 AM, Patrick Carr wrote:

On 6/3/08, Duncan Murdoch wrote:

because signif(0.90, digits=2) == 0.9. Those two objects are
identical.

My text above that is poorly worded. They're identical internally,
yes. But in terms of the number of significant digits, 0.9 and 0.90
are different. And that matters when the number is printed, say
as an
annotation on a graph. Passing it through sprintf() or format()
later
requires you to specify the number of digits after the decimal,
which
is different than the number of significant digits, and requires
case
testing for numbers of different orders of magnitude.
The original complainant (and I) expected this behavior from
signif(),
not merely rounding. As I said before, I wrote my own workaround so
this is somewhat academic, but I don't think we're alone.

As far as I know, rounding is fine in Windows:

round(1:10 + 0.5)

[1] 2 2 4 4 6 6 8 8 10 10
It might not be the rounding, then. (windows xp sp3)
signif(12345,digits=4) [1] 12340
signif(0.12345,digits=4)

[1] 0.1235

It's easy to make mistakes in this, but a little outside-of-R
experimentation suggests those are the right answers. The number
12345 is exactly representable, so it is exactly half-way between
12340 and 12350, so 12340 is the right answer by the unbiased
round- to-even rule. The number 0.12345 is not exactly
representable, but (I think) it is represented by something
slightly closer to 0.1235 than to 0.1234. So it looks as though
Windows gets it right.

OS X (10.5.2/intel) does not have that problem.

Which would seem to imply OS X gets it wrong.

This has nothing to do with OS X, you get that same answer on
pretty much all other platforms (Intel/Linux, MIPS/IRIX, Sparc/
Sun, ...). Windows is the only one delivering the incorrect result
here.

Both are supposed to be using the 64 bit floating point standard,
so they should both give the same answer:

Should, yes, but Windows doesn't. In fact 10000.0 is exactly
representable and so is 1234.5 which is the correct result that
all except Windows get.

I think you skipped a step.

I didn't - I was just pointing out that what you are trying to show is
irrelevant. We are dealing with FP arithmetics here, so although your
reasoning is valid algebraically, it's not in FP world. You missed the
fact that FP operations are used to actually get the result (*10000.0,
round and divide again) and thus those operation will influence it as
well.

The correct answer is either 0.1234 or 0.1235, not something 10000
times bigger. The first important question is whether 0.12345 is
exactly representable, and the answer is no. The second question is
whether it is represented by a number bigger or smaller than the
real number 0.12345. If it is bigger, the answer should be 0.1235,
and if it is smaller, the answer is 0.1234.

No. That was what I was trying to point out. You can see clearly from
my post that 0.12345 is not exactly representable and that the
representation is slightly bigger. This is, however, irrelevant,
because the next step is to multiply that number by 10000 (see fprec
source) and this is where your reasoning breaks down - the result is
exact representation of 1234.5, because the imprecision gets lost in
the operation on all platforms but Windows. The result is that Windows
is inconsistent with others, whether that is a bug or feature I don't
care. All I really wanted to say is that this has nothing to do with
OS X - if anything then it's a Windows issue.

My experiments suggest it is bigger.

I was not claiming otherwise.

Yours don't look relevant.

Vice versa as it turns out.

Cheers,
Simon

It certainly isn't exactly equal to 1234.5/10000, because that
number is not representable. It's equal to x/2^y, for some x and y,
and it's a pain to figure out exactly what they are.

However, I am pretty sure R is representing it (at least on Windows)
as the binary expansion

0.000111111001101001101011010100001011000011110010011111

while the true binary expansion (using exact rational arithmetic)
starts out

0.00011111100110100110101101010000101100001111001001111011101...

If you line those up, you'll see that the first number is bigger
than the second. (Ugly code to derive these is down below.)

Clearly the top representation is the correct one to that number of
binary digits, so I think Windows got it right, and all those other
systems didn't. This is probably because R on Windows is using
extended precision (64 bit mantissas) for intermediate results, and
those other systems stick with 53 bit mantissas.
However, this means that Windows doesn't conform
Duncan Murdoch

# Convert number to binary expansion; add the decimal point manually

x <- 0.12345
while (x != 0) {
cat(trunc(x))
x <- x - trunc(x)
x <- x * 2
}

# Do the same thing in exact rational arithmetic

num <- 12345
denom <- 100000
for (i in 1:60) {
cat(ifelse(num > 100000, "1", "0"))
num <- num %% 100000
num <- 2*num
}

# Manually cut and paste the results to get these:

"0.000111111001101001101011010100001011000011110010011111"
"0.00011111100110100110101101010000101100001111001001111011101"

reply | permalink

Karl Ove Hufthammer

at Jun 4, 2008 at 11:17 am

⇧

Duncan Murdoch:

The number 0.12345 is not exactly representable, but (I think) it is
represented by something slightly closer to 0.1235 than to 0.1234.

I like using formatC for checking such things. On my (Linux) system, I get:

$ formatC(.12345,digitsP)
[1] "0.12345000000000000417443857259058859199285507202148"

So it looks as though Windows gets it right.

--
Karl Ove Hufthammer

reply | permalink

Prof Brian Ripley

at Jun 4, 2008 at 12:18 pm

⇧

That (and simpler, sprintf()) merely tell you about your OS's sprintf
function. That is not required to accurately give a decimal
representation of more than DECIMAL_DIG digits, and certainly not 50.
The help for formatC does warn you about this.

On my F8 system DECIMAL_DIG is mentioned in math.h but not defined there.
(It is defined in the compiler, AFAICS -- SunStudio has it as 21, and gcc
computes __DECIMAL_DIG__ also as 21.) The C99 standard says that it
should be at least 10 and that 17 is appropriate for IEC60559 arithmetic.

On Wed, 4 Jun 2008, Karl Ove Hufthammer wrote:

Duncan Murdoch:

The number 0.12345 is not exactly representable, but (I think) it is
represented by something slightly closer to 0.1235 than to 0.1234.

I like using formatC for checking such things. On my (Linux) system, I get:

$ formatC(.12345,digitsP)
[1] "0.12345000000000000417443857259058859199285507202148"

So it looks as though Windows gets it right.

--
Karl Ove Hufthammer

______________________________________________
R-devel at r-project.org mailing list
https://stat.ethz.ch/mailman/listinfo/r-devel

--
Brian D. Ripley, ripley at stats.ox.ac.uk
Professor of Applied Statistics, http://www.stats.ox.ac.uk/~ripley/
University of Oxford, Tel: +44 1865 272861 (self)
1 South Parks Road, +44 1865 272866 (PA)
Oxford OX1 3TG, UK Fax: +44 1865 272595

reply | permalink

Simon Urbanek

at Jun 3, 2008 at 8:40 pm

⇧

On Jun 3, 2008, at 2:48 PM, Duncan Murdoch wrote:

On 6/3/2008 11:43 AM, Patrick Carr wrote:

On 6/3/08, Duncan Murdoch wrote:

because signif(0.90, digits=2) == 0.9. Those two objects are
identical.

My text above that is poorly worded. They're identical internally,
yes. But in terms of the number of significant digits, 0.9 and 0.90
are different. And that matters when the number is printed, say as an
annotation on a graph. Passing it through sprintf() or format() later
requires you to specify the number of digits after the decimal, which
is different than the number of significant digits, and requires case
testing for numbers of different orders of magnitude.
The original complainant (and I) expected this behavior from
signif(),
not merely rounding. As I said before, I wrote my own workaround so
this is somewhat academic, but I don't think we're alone.

As far as I know, rounding is fine in Windows:

round(1:10 + 0.5)

[1] 2 2 4 4 6 6 8 8 10 10
It might not be the rounding, then. (windows xp sp3)
signif(12345,digits=4) [1] 12340
signif(0.12345,digits=4)

[1] 0.1235

It's easy to make mistakes in this, but a little outside-of-R
experimentation suggests those are the right answers. The number
12345 is exactly representable, so it is exactly half-way between
12340 and 12350, so 12340 is the right answer by the unbiased round-
to-even rule. The number 0.12345 is not exactly representable, but
(I think) it is represented by something slightly closer to 0.1235
than to 0.1234. So it looks as though Windows gets it right.

OS X (10.5.2/intel) does not have that problem.

Which would seem to imply OS X gets it wrong.

This has nothing to do with OS X, you get that same answer on pretty
much all other platforms (Intel/Linux, MIPS/IRIX, Sparc/Sun, ...).
Windows is the only one delivering the incorrect result here.

Both are supposed to be using the 64 bit floating point standard,
so they should both give the same answer:

Should, yes, but Windows doesn't. In fact 10000.0 is exactly
representable and so is 1234.5 which is the correct result that all
except Windows get. I don't have a Windows box handy, so I can't tell
why - but if you go through fprec this is what you get on the
platforms I tested (log10 may vary slightly but that's irrelevant here):

x = 0.123450000000000004174439
l10 = -0.908508905732048899217546, e10 = 4
pow10 = 10000.000000000000000000000000
x*pow10 = 1234.500000000000000000000000

Cheers,
Simon

but the actual arithmetic is being done by run-time libraries that
are outside our control to a large extent, and it looks as though
the one on the Mac is less accurate than the one on Windows.

But (on both windows and OS X):

signif(12345.12345,digits)

[1] 12345.12

This is a different problem. The number is correctly computed as
12345.12345 (or at least a representable number quite close to
that), and then the default display rounds it some more. Set
options(digits) to see it in its full glory.

Duncan Murdoch

______________________________________________
R-devel at r-project.org mailing list
https://stat.ethz.ch/mailman/listinfo/r-devel

reply | permalink

Murdoch

at Jun 3, 2008 at 9:15 pm

⇧

On 6/3/2008 4:36 PM, Simon Urbanek wrote:

On Jun 3, 2008, at 2:48 PM, Duncan Murdoch wrote:

On 6/3/2008 11:43 AM, Patrick Carr wrote:

On 6/3/08, Duncan Murdoch wrote:

because signif(0.90, digits=2) == 0.9. Those two objects are
identical.

My text above that is poorly worded. They're identical internally,
yes. But in terms of the number of significant digits, 0.9 and 0.90
are different. And that matters when the number is printed, say as an
annotation on a graph. Passing it through sprintf() or format() later
requires you to specify the number of digits after the decimal, which
is different than the number of significant digits, and requires case
testing for numbers of different orders of magnitude.
The original complainant (and I) expected this behavior from
signif(),
not merely rounding. As I said before, I wrote my own workaround so
this is somewhat academic, but I don't think we're alone.

As far as I know, rounding is fine in Windows:

round(1:10 + 0.5)

[1] 2 2 4 4 6 6 8 8 10 10
It might not be the rounding, then. (windows xp sp3)
signif(12345,digits=4) [1] 12340
signif(0.12345,digits=4)

[1] 0.1235

It's easy to make mistakes in this, but a little outside-of-R
experimentation suggests those are the right answers. The number
12345 is exactly representable, so it is exactly half-way between
12340 and 12350, so 12340 is the right answer by the unbiased round-
to-even rule. The number 0.12345 is not exactly representable, but
(I think) it is represented by something slightly closer to 0.1235
than to 0.1234. So it looks as though Windows gets it right.

OS X (10.5.2/intel) does not have that problem.

Which would seem to imply OS X gets it wrong.

This has nothing to do with OS X, you get that same answer on pretty
much all other platforms (Intel/Linux, MIPS/IRIX, Sparc/Sun, ...).
Windows is the only one delivering the incorrect result here.

Both are supposed to be using the 64 bit floating point standard,
so they should both give the same answer:

Should, yes, but Windows doesn't. In fact 10000.0 is exactly
representable and so is 1234.5 which is the correct result that all
except Windows get.

I think you skipped a step. The correct answer is either 0.1234 or
0.1235, not something 10000 times bigger. The first important question
is whether 0.12345 is exactly representable, and the answer is no. The
second question is whether it is represented by a number bigger or
smaller than the real number 0.12345. If it is bigger, the answer
should be 0.1235, and if it is smaller, the answer is 0.1234. My
experiments suggest it is bigger. Yours don't look relevant. It
certainly isn't exactly equal to 1234.5/10000, because that number is
not representable. It's equal to x/2^y, for some x and y, and it's a
pain to figure out exactly what they are.

However, I am pretty sure R is representing it (at least on Windows) as
the binary expansion

0.000111111001101001101011010100001011000011110010011111

while the true binary expansion (using exact rational arithmetic) starts
out

0.00011111100110100110101101010000101100001111001001111011101...

If you line those up, you'll see that the first number is bigger than
the second. (Ugly code to derive these is down below.)

Clearly the top representation is the correct one to that number of
binary digits, so I think Windows got it right, and all those other
systems didn't. This is probably because R on Windows is using extended
precision (64 bit mantissas) for intermediate results, and those other
systems stick with 53 bit mantissas.

Duncan Murdoch

# Convert number to binary expansion; add the decimal point manually

x <- 0.12345
while (x != 0) {
cat(trunc(x))
x <- x - trunc(x)
x <- x * 2
}

# Do the same thing in exact rational arithmetic

num <- 12345
denom <- 100000
for (i in 1:60) {
cat(ifelse(num > 100000, "1", "0"))
num <- num %% 100000
num <- 2*num
}

# Manually cut and paste the results to get these:

"0.000111111001101001101011010100001011000011110010011111"
"0.00011111100110100110101101010000101100001111001001111011101"

reply | permalink

Simon Urbanek

at Jun 3, 2008 at 9:35 pm

⇧

On Jun 3, 2008, at 5:12 PM, Duncan Murdoch wrote:

On 6/3/2008 4:36 PM, Simon Urbanek wrote:

On Jun 3, 2008, at 2:48 PM, Duncan Murdoch wrote:

On 6/3/2008 11:43 AM, Patrick Carr wrote:

On 6/3/08, Duncan Murdoch wrote:

because signif(0.90, digits=2) == 0.9. Those two objects are
identical.

My text above that is poorly worded. They're identical internally,
yes. But in terms of the number of significant digits, 0.9 and 0.90
are different. And that matters when the number is printed, say
as an
annotation on a graph. Passing it through sprintf() or format()
later
requires you to specify the number of digits after the decimal,
which
is different than the number of significant digits, and requires
case
testing for numbers of different orders of magnitude.
The original complainant (and I) expected this behavior from
signif(),
not merely rounding. As I said before, I wrote my own workaround so
this is somewhat academic, but I don't think we're alone.

As far as I know, rounding is fine in Windows:

round(1:10 + 0.5)

[1] 2 2 4 4 6 6 8 8 10 10
It might not be the rounding, then. (windows xp sp3)
signif(12345,digits=4) [1] 12340
signif(0.12345,digits=4)

[1] 0.1235

It's easy to make mistakes in this, but a little outside-of-R
experimentation suggests those are the right answers. The number
12345 is exactly representable, so it is exactly half-way between
12340 and 12350, so 12340 is the right answer by the unbiased
round- to-even rule. The number 0.12345 is not exactly
representable, but (I think) it is represented by something
slightly closer to 0.1235 than to 0.1234. So it looks as though
Windows gets it right.

OS X (10.5.2/intel) does not have that problem.

Which would seem to imply OS X gets it wrong.

This has nothing to do with OS X, you get that same answer on
pretty much all other platforms (Intel/Linux, MIPS/IRIX, Sparc/
Sun, ...). Windows is the only one delivering the incorrect result
here.

Both are supposed to be using the 64 bit floating point standard,
so they should both give the same answer:

Should, yes, but Windows doesn't. In fact 10000.0 is exactly
representable and so is 1234.5 which is the correct result that
all except Windows get.

I think you skipped a step.

I didn't - I was just pointing out that what you are trying to show is
irrelevant. We are dealing with FP arithmetics here, so although your
reasoning is valid algebraically, it's not in FP world. You missed the
fact that FP operations are used to actually get the result (*10000.0,
round and divide again) and thus those operation will influence it as
well.

The correct answer is either 0.1234 or 0.1235, not something 10000
times bigger. The first important question is whether 0.12345 is
exactly representable, and the answer is no. The second question is
whether it is represented by a number bigger or smaller than the
real number 0.12345. If it is bigger, the answer should be 0.1235,
and if it is smaller, the answer is 0.1234.

No. That was what I was trying to point out. You can see clearly from
my post that 0.12345 is not exactly representable and that the
representation is slightly bigger. This is, however, irrelevant,
because the next step is to multiply that number by 10000 (see fprec
source) and this is where your reasoning breaks down - the result is
exact representation of 1234.5, because the imprecision gets lost in
the operation on all platforms but Windows. The result is that Windows
is inconsistent with others, whether that is a bug or feature I don't
care. All I really wanted to say is that this has nothing to do with
OS X - if anything then it's a Windows issue.

My experiments suggest it is bigger.

I was not claiming otherwise.

Yours don't look relevant.

Vice versa as it turns out.

Cheers,
Simon

It certainly isn't exactly equal to 1234.5/10000, because that
number is not representable. It's equal to x/2^y, for some x and y,
and it's a pain to figure out exactly what they are.

However, I am pretty sure R is representing it (at least on Windows)
as the binary expansion

0.000111111001101001101011010100001011000011110010011111

while the true binary expansion (using exact rational arithmetic)
starts out

0.00011111100110100110101101010000101100001111001001111011101...

If you line those up, you'll see that the first number is bigger
than the second. (Ugly code to derive these is down below.)

Clearly the top representation is the correct one to that number of
binary digits, so I think Windows got it right, and all those other
systems didn't. This is probably because R on Windows is using
extended precision (64 bit mantissas) for intermediate results, and
those other systems stick with 53 bit mantissas.
However, this means that Windows doesn't conform
Duncan Murdoch

# Convert number to binary expansion; add the decimal point manually

x <- 0.12345
while (x != 0) {
cat(trunc(x))
x <- x - trunc(x)
x <- x * 2
}

# Do the same thing in exact rational arithmetic

num <- 12345
denom <- 100000
for (i in 1:60) {
cat(ifelse(num > 100000, "1", "0"))
num <- num %% 100000
num <- 2*num
}

# Manually cut and paste the results to get these:

"0.000111111001101001101011010100001011000011110010011111"
"0.00011111100110100110101101010000101100001111001001111011101"

reply | permalink

Duncan Murdoch

at Jun 3, 2008 at 11:06 pm

⇧

simon.urbanek at r-project.org wrote:
On Jun 3, 2008, at 5:12 PM, Duncan Murdoch wrote:

On 6/3/2008 4:36 PM, Simon Urbanek wrote:

On Jun 3, 2008, at 2:48 PM, Duncan Murdoch wrote:

On 6/3/2008 11:43 AM, Patrick Carr wrote:

On 6/3/08, Duncan Murdoch wrote:

because signif(0.90, digits=2) == 0.9. Those two objects are
identical.

My text above that is poorly worded. They're identical internally,
yes. But in terms of the number of significant digits, 0.9 and 0.90
are different. And that matters when the number is printed, say
as an
annotation on a graph. Passing it through sprintf() or format()
later
requires you to specify the number of digits after the decimal,
which
is different than the number of significant digits, and requires
case
testing for numbers of different orders of magnitude.
The original complainant (and I) expected this behavior from
signif(),
not merely rounding. As I said before, I wrote my own workaround so
this is somewhat academic, but I don't think we're alone.

As far as I know, rounding is fine in Windows:

round(1:10 + 0.5)

[1] 2 2 4 4 6 6 8 8 10 10

It might not be the rounding, then. (windows xp sp3)
signif(12345,digits=4) [1] 12340
signif(0.12345,digits=4)

[1] 0.1235

It's easy to make mistakes in this, but a little outside-of-R
experimentation suggests those are the right answers. The number
12345 is exactly representable, so it is exactly half-way between
12340 and 12350, so 12340 is the right answer by the unbiased
round- to-even rule. The number 0.12345 is not exactly
representable, but (I think) it is represented by something
slightly closer to 0.1235 than to 0.1234. So it looks as though
Windows gets it right.

OS X (10.5.2/intel) does not have that problem.

Which would seem to imply OS X gets it wrong.

This has nothing to do with OS X, you get that same answer on
pretty much all other platforms (Intel/Linux, MIPS/IRIX, Sparc/
Sun, ...). Windows is the only one delivering the incorrect result
here.

Both are supposed to be using the 64 bit floating point standard,
so they should both give the same answer:

Should, yes, but Windows doesn't. In fact 10000.0 is exactly
representable and so is 1234.5 which is the correct result that
all except Windows get.

I think you skipped a step.

I didn't - I was just pointing out that what you are trying to show is
irrelevant. We are dealing with FP arithmetics here, so although your
reasoning is valid algebraically, it's not in FP world. You missed the
fact that FP operations are used to actually get the result (*10000.0,
round and divide again) and thus those operation will influence it as
well.

But your multiplication isn't accurate: 10000 is 16 times 625.
Multiplying by 16 will shift the binary representation by 4 places, but
multiplying by an odd number should leave the last 1 bit in place. If
you go through the following series of operations, you get a different
answer than multiplying by 10000:

x <- 0.12345
x <- 16*x
y <- x
x <- 624*x
x + y - 1234.5

[1] 2.273737e-13

versus

x <- 0.12345
x <- 10000*x
x - 1234.5

[1] 0

I get identical results for both of these calculations on a Mac and on
Windows. Looks strange, but I think the explanation is that multiplying
by 625 (or 10000) needs one more bit than multiplying by 624 needs, so
we lose a bit at that point. Windows does the multiplication by 10000
accurately in its 64 bit registers, and then loses the last bit on
storing the result into x in 53 bits. But in the signif() calculation,
everything stays in 64 bit precision, so Windows gets the right result
(0.1235), while those systems that round intermediate results get it
wrong (0.1234).

It's all FP calculations, but in Windows we've programmed the run-time
to keep extra bits around for intermediate results and get more accurate
results in cases where things stay in registers, whereas the other
systems throw away the bits and get more consistent results regardless
of the execution path.

Duncan Murdoch

The correct answer is either 0.1234 or 0.1235, not something 10000
times bigger. The first important question is whether 0.12345 is
exactly representable, and the answer is no. The second question is
whether it is represented by a number bigger or smaller than the
real number 0.12345. If it is bigger, the answer should be 0.1235,
and if it is smaller, the answer is 0.1234.

No. That was what I was trying to point out. You can see clearly from
my post that 0.12345 is not exactly representable and that the
representation is slightly bigger. This is, however, irrelevant,
because the next step is to multiply that number by 10000 (see fprec
source) and this is where your reasoning breaks down - the result is
exact representation of 1234.5, because the imprecision gets lost in
the operation on all platforms but Windows. The result is that Windows
is inconsistent with others, whether that is a bug or feature I don't
care. All I really wanted to say is that this has nothing to do with
OS X - if anything then it's a Windows issue.

My experiments suggest it is bigger.

I was not claiming otherwise.

Yours don't look relevant.

Vice versa as it turns out.

Cheers,
Simon

It certainly isn't exactly equal to 1234.5/10000, because that
number is not representable. It's equal to x/2^y, for some x and y,
and it's a pain to figure out exactly what they are.

However, I am pretty sure R is representing it (at least on Windows)
as the binary expansion

0.000111111001101001101011010100001011000011110010011111

while the true binary expansion (using exact rational arithmetic)
starts out

0.00011111100110100110101101010000101100001111001001111011101...

If you line those up, you'll see that the first number is bigger
than the second. (Ugly code to derive these is down below.)

Clearly the top representation is the correct one to that number of
binary digits, so I think Windows got it right, and all those other
systems didn't. This is probably because R on Windows is using
extended precision (64 bit mantissas) for intermediate results, and
those other systems stick with 53 bit mantissas.

However, this means that Windows doesn't conform

Duncan Murdoch

# Convert number to binary expansion; add the decimal point manually

x <- 0.12345
while (x != 0) {
cat(trunc(x))
x <- x - trunc(x)
x <- x * 2
}

# Do the same thing in exact rational arithmetic

num <- 12345
denom <- 100000
for (i in 1:60) {
cat(ifelse(num > 100000, "1", "0"))
num <- num %% 100000
num <- 2*num
}

# Manually cut and paste the results to get these:

"0.000111111001101001101011010100001011000011110010011111"
"0.00011111100110100110101101010000101100001111001001111011101"

______________________________________________
R-devel at r-project.org mailing list
https://stat.ethz.ch/mailman/listinfo/r-devel

reply | permalink

Scott Wilkinson

at Jun 13, 2008 at 4:10 am

⇧

Thanks Patrick, your function is a neat work-around to include trailing
zeroes when specifying significant digits.
It worked better with this modification to the "format" arguments:
else if (sigdigs<=left+right)
{out<-format(signum,nsmall=sigdigs-left)}

Agree would be nice to include this in signif()
Regards, Scott.

-----Original Message-----
From: patcarr at gmail.com [mailto:patcarr at gmail.com] On Behalf
Of Patrick Carr
Sent: Tuesday, 3 June 2008 11:28 AM
To: r-bugs at r-project.org
Cc: Wilkinson, Scott (CLW, Townsville)
Subject: Re: significant digits (PR#9682)

To reply to my own message, that function wasn't quite right. I think
this one works better:

signif.string <- function(signum,sigdigs){
test <- abs(signum)
left <- nchar(trunc(test))
right <- nchar(test)-left-1
if (test<1) {left<-left-1}
if (right<0) {right<-0}
if (sigdigs<left) {out<-as.character(signif(signum,digits=sigdigs))}
else if (sigdigs==left & trunc(signum) %% 10 == 0)
{out<-paste(round(signum),".",sep="")}
else if (sigdigs<=left+right) {out<-format(signum,digits=sigdigs)}
else {out<-sprintf(paste("%.",sigdigs-left,"f",sep=""),signum)}
return(noquote(out))
}

But it should still have error checking and vector capability, yadda
yadda. Also, I forgot what year it was, so sorry, Scott, for spamming
you with something you're hopefully not still stuck on.

Pat Carr

reply | permalink

group	r-devel @
categories	r
posted	Jun 2, '08 at 11:40p
active	Jun 13, '08 at 4:10a
posts	16
users	6
website	r-project.org
irc	#r

responses	expanded	Hotkey: s
font	variable	Hotkey: f
user style	avatars	Hotkey: a

[Rd] significant digits (PR#9682)

Search Discussions

Related Discussions

6 users in discussion