H
Harlan Grove
It's mysteries like this that make Excel so much more (and so much less) than
just a dry (reliable) mathematical tool.
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It's mysteries like this that make Excel so much more (and so much less) than
just a dry (reliable) mathematical tool.
M
Martin Brown
Jerry W. Lewis said:I got it by experimentation, and then saw that Arvi had also determined
the same limit. I find it interesting that MS makes no attempt to
explain such an unusual limit.
The mantissa of a long floating point number has 54 bits significance
and 2^27 is the value with exactly half that number of bits.
If you choose a particularly stupid and naive way of computing x MOD y
then things can go wrong when the denominator *and* quotient exceed the
limit 2^27. However, they only tested the quotient value x/y < 2^27.
The dodgy way to do it in floating point is frac(x/y)*y
But no one in their right mind would ever implement mod this way. And
anyway most high level languages have a correct mod library function.
An additional unusual limit that applies, is that MOD returns #NUM!
regardless of the quotient if the first argument exceeds
2.68873542664192E14 = 2^20+2^19+2^18+2^17+2^15+2^12+2^8+2^6
which is within the range of exact DP representation of whole numbers
by more than an order of magnitude.
???? I don't understand. 2.688E14 is a shade under 2^48
And I don't see any such odd limit in XL2k. It appears to work more or
less OK here for larger values up to around 2^54 of the numerator x and
denominator y. (provided that x/y < 2^27)
I'd believe it does go wrong for some specific large values though. It's
always hard to predict the behaviour of flawed algorithms
experimentally.
Since they have documented it I guess we can expect a fix in about 2014.Jerry
(based on the latency time for the recent fixes to the statistics bugs)
Regards,
I
Ian Smith
J.E. McGimpsey said:Given their big "mea culpa" on statistical functions,
http://support.microsoft.com/?kbid=828888
I'm hopeful.
Sadly, I'm less hopeful. It is undoubtedly a step in the right
direction but not everything in
http://support.microsoft.com/?kbid=828888 is correct, nor are all the
improvements discussed in it always improvements!
For example, it contains the following paragraph.
"In summary, users of earlier versions of Excel should not be troubled
in practice by numeric deficiencies because input values that cause
#NUM! results are extreme. Additionally, you can be assured that if
you receive a result other than an error message, the result is
accurate. Round off error is not a problem here."
Unfortunately this is not correct. A counter-example is
=NEGBINOMDIST(512,512,0.5) which gives 0 when the answer should be
approx 0.0125. Note the parameters are not "big" nor is the answer
"small" by anyone's standards (i.e. there can be no "extreme case"
excuses).
Additionally in the section "Continuous Distribution Inverse
Functions" the suggestion is that the only real problem was the lack
of a good binary search process. In fact, in EXCEL 2000, neither
gammadist nor betadist works even for moderately large parameters,
which is a pretty big hindrance for function inversion!
=GAMMADIST(7000,7000,1,TRUE) gives #NUM!,
=BETADIST(0.5,30000,30000,0,1) also gives #NUM!
I do not wish to unduly criticise the new code added for
BINOMDIST,HYPGEOMDIST & POISSON. In most cases it is an improvement,
albeit it will be very slow when large parameter values are supplied
to the functions. However, for cases such as =POISSON(126,230,TRUE)
the relative error will go from 3e-14 to 0.5. Worse still, cases such
as =POISSON(125,230,TRUE) and =POISSON(125,230,FALSE) will both return
the value 0 which is completely inaccurate. Admittedly, the values are
small in these cases, =POISSON(126,230,TRUE) should deliver approx
5e-14. However, unbelievable though it may seem to Microsoft, there
are wierdos about (self included) who want accurate calculations of
probabilities smaller than 5e-14, who want accurate calculations of
probability functions involving parameters bigger than 30000 and who
want these calculations done lots of times which means they have to be
reasonably efficient!
Ian Smith
J
James Silverton
Ian Smith said:"J.E. McGimpsey" <[email protected]> wrote in message
Sadly, I'm less hopeful. It is undoubtedly a step in the right
direction but not everything in
http://support.microsoft.com/?kbid=828888 is correct, nor are all the
improvements discussed in it always improvements!
For example, it contains the following paragraph.
"In summary, users of earlier versions of Excel should not be troubled
in practice by numeric deficiencies because input values that cause
#NUM! results are extreme. Additionally, you can be assured that if
you receive a result other than an error message, the result is
accurate. Round off error is not a problem here."
Unfortunately this is not correct. A counter-example is
=NEGBINOMDIST(512,512,0.5) which gives 0 when the answer should be
approx 0.0125. Note the parameters are not "big" nor is the answer
"small" by anyone's standards (i.e. there can be no "extreme case"
excuses).
You are certainly correct! NEGBINOMDIST does not give an error message in
Excel XP (2002, I suppose) either and returns zero. The formula seems to
fail for numbers larger than 511. I suppose that is because of the immense
numerical values of the combinations involved but I wonder why Microsoft
does not use their own suggestion?
NEGBINOMDIST(number_s, number_f, probability_s) =
BINOMDIST(number_s, number_f + number_s - 1, probability_s, false) *
probability_s.
This actually gives an apparently correct answer in Excel 2002.
H
Harlan Grove
You are certainly correct! NEGBINOMDIST does not give an error message in
Excel XP (2002, I suppose) either and returns zero. The formula seems to
fail for numbers larger than 511. I suppose that is because of the immense
numerical values of the combinations involved but I wonder why Microsoft
does not use their own suggestion?
Immense numerical values! If one uses a bone-headed algorithm for calculating (n
choose k) or p^a * (1-p)^b, then the calculations can blow up. However, outside
Redmond the following approach might be used.
NEGBINOMDIST(512,512,0.5)
is defined as
COMBIN(1023,511) * 0.5^512 * (1 - 0.5)^512
which can be rewritten as
EXP(LN(COMBIN(1023,511)) + LN(0.5)*512 + LN(0.5)*512)
which in this case simplifies to
EXP(GAMMALN(1024) - 2*GAMMALN(512) + 1024*LN(0.5))
which reduces to
EXP(705.397794316793 - 709.782712893384)
which evaluates to
0.0124639029464894
The zero result proves that Microsoft's Excel programmers don't know when to use
logarithms to avoid overflow/underflow (it's the 0.5^1024 that Excel can't
handle by direct calculation, not the COMBIN value). Pathetic.
J
James Silverton
Harlan Grove said:...
Immense numerical values! If one uses a bone-headed algorithm for calculating (n
choose k) or p^a * (1-p)^b, then the calculations can blow up. However, outside
Redmond the following approach might be used.
NEGBINOMDIST(512,512,0.5)
is defined as
COMBIN(1023,511) * 0.5^512 * (1 - 0.5)^512
which can be rewritten as
EXP(LN(COMBIN(1023,511)) + LN(0.5)*512 + LN(0.5)*512)
which in this case simplifies to
EXP(GAMMALN(1024) - 2*GAMMALN(512) + 1024*LN(0.5))
which reduces to
EXP(705.397794316793 - 709.782712893384)
which evaluates to
0.0124639029464894
The zero result proves that Microsoft's Excel programmers don't know when to use
logarithms to avoid overflow/underflow (it's the 0.5^1024 that Excel can't
handle by direct calculation, not the COMBIN value). Pathetic.
Thanks again! Bone-headed indeed but your formula looks good. I suppose
Microsoft might consider employing a few mathematicians in the Excel section
(g).
Jim.
D
Dana DeLouis
Just to share some thoughts. I may be wrong, but it doesn't look to me that
the following two terms can be combined into - 2*GAMMALN(512) .
I "think" the f term needs to be f+1 in the GammaLn. Here are my thoughts,
but I may be wrong.
Sub Demo()
Dim f, s, p, ans
f = 512
s = 512
p = 1 / 2
With WorksheetFunction
ans = f * Log(1 - p) + s * Log(p) - .GammaLn(s) - .GammaLn(f + 1) +
..GammaLn(f + s)
ans = Exp(ans)
End With
Debug.Print FormatNumber(ans, 16)
End Sub
returns: 0.0124639029469358
Unfortunately, the program Mathematica shows this to be accurate to only 11
sig. digits. :>(
the following two terms can be combined into - 2*GAMMALN(512) .
I "think" the f term needs to be f+1 in the GammaLn. Here are my thoughts,
but I may be wrong.
Sub Demo()
Dim f, s, p, ans
f = 512
s = 512
p = 1 / 2
With WorksheetFunction
ans = f * Log(1 - p) + s * Log(p) - .GammaLn(s) - .GammaLn(f + 1) +
..GammaLn(f + s)
ans = Exp(ans)
End With
Debug.Print FormatNumber(ans, 16)
End Sub
returns: 0.0124639029469358
Unfortunately, the program Mathematica shows this to be accurate to only 11
sig. digits. :>(
H
Harlan Grove
...Just to share some thoughts. I may be wrong, but it doesn't look to me that
the following two terms can be combined into - 2*GAMMALN(512) .
You're right. Mental typo on my part. I used LN(COMBIN(1023,511)) in my actual
calculations, but miswrote this as GAMMALN(1024)-2*GAMMALN(512). It should have
been GAMMALN(1024)-GAMMALN(512)-GAMMALN(513). However it remains the case that
NEGBINOMDIST(512,512,0.5) doesn't need to choke.
Unfortunately, the program Mathematica shows this to be accurate to only 11
sig. digits. :>(
FWIW, Mathematica and Maple differ in the results of their respective logarithm
of complete gamma functions at the 14th digit, so *prove* that Mathematica is
more accurate than Maple. Good luck finding standard mathematical tables for
this function beyond 10 significant digits.
Excel's GAMMALN function only agrees with Mathematica's and Maple's equivalents
to single precision (7 decimal digits), so there's some considerable loss of
precision using Excel's GAMMALN for something like this. Probably better to use
SUM(LN(ROW(512:1023)))-SUM(LN(ROW(2:512)))
assuming that Excel's LN is more precise (which it should be if it's a straight
pass through to the FPU's IEEE logarithm operator, but Microsoft has likely
"improved" upon IEEE).
D
Dana DeLouis
Hi Harlan. I think it's a good catch on your discussion of Excel failing on
the 0.5^1024. It appears that value is apr. 5.563 E-309, which is just a
"hair" outside of Excel's limits.(2.229E-308). Maybe Excel rounds to zero,
and returned a zero final answer. Who knows? I agree with you that Excel
should not choke on this. :>)
Just for discussion, the "other two programs" differing at the 14 digit does
not sound right to me, but of course, I can't tell. I don't have Maple. By
chance, were you testing with the inexact number of 0.5 so that it operated
at machine precision? I was testing at the more exact value of 1/2, thereby
using full arbitrary precision. From small testing in the past, I find that
Excel's GammaLn function to be a little more accurate than 7 digits, with a
usual range of about 9-11 digits.
Then there are the following two examples:
=GAMMALN(1)
=GAMMALN(2)
-4.1716E-11
-8.57678E-11
The Gamma of both 1 & 2 should be 1. And of course, the Log of 1 is zero.
I believe that Excel's GammaLn function should return an exact zero for both
examples. Excel is of course using an approximating function.
Here are just three different ways to calculate this in Mathematica. All
three methods agree out to 40 digits. I don't know why both programs would
give different answers. I'm not positive, but I would think they would
agree here. The Binomial example below is pretty exact. It's a 307 digit
integer divided by a 309 digit integer. (LogGamma here is similar to
Excel's GammaLn). Note that I used p of 1/2, and not 0.5.
data = {f -> 512, s -> 512, p -> 1/2};
(1)
N[PDF[NegativeBinomialDistribution[512, 1/2], 512], 40]
0.012463902946489771856117316100129344083043651786931274117`40.
(2)
N[Binomial[f + s - 1, s - 1]*p^s*(1 - p)^f /. data, 40]
0.012463902946489771856117316100129344083043651786931274117`40.
(3)
N[Exp[f*Log[1 - p] + s*Log[p] - LogGamma[1 + f] - LogGamma + LogGamma[f +
s] /. data], 40]
0.012463902946489771856117316100129344083043651786931274117`40.
Well anyway, interesting discussion. :>)
the 0.5^1024. It appears that value is apr. 5.563 E-309, which is just a
"hair" outside of Excel's limits.(2.229E-308). Maybe Excel rounds to zero,
and returned a zero final answer. Who knows? I agree with you that Excel
should not choke on this. :>)
Just for discussion, the "other two programs" differing at the 14 digit does
not sound right to me, but of course, I can't tell. I don't have Maple. By
chance, were you testing with the inexact number of 0.5 so that it operated
at machine precision? I was testing at the more exact value of 1/2, thereby
using full arbitrary precision. From small testing in the past, I find that
Excel's GammaLn function to be a little more accurate than 7 digits, with a
usual range of about 9-11 digits.
Then there are the following two examples:
=GAMMALN(1)
=GAMMALN(2)
-4.1716E-11
-8.57678E-11
The Gamma of both 1 & 2 should be 1. And of course, the Log of 1 is zero.
I believe that Excel's GammaLn function should return an exact zero for both
examples. Excel is of course using an approximating function.
Here are just three different ways to calculate this in Mathematica. All
three methods agree out to 40 digits. I don't know why both programs would
give different answers. I'm not positive, but I would think they would
agree here. The Binomial example below is pretty exact. It's a 307 digit
integer divided by a 309 digit integer. (LogGamma here is similar to
Excel's GammaLn). Note that I used p of 1/2, and not 0.5.
data = {f -> 512, s -> 512, p -> 1/2};
(1)
N[PDF[NegativeBinomialDistribution[512, 1/2], 512], 40]
0.012463902946489771856117316100129344083043651786931274117`40.
(2)
N[Binomial[f + s - 1, s - 1]*p^s*(1 - p)^f /. data, 40]
0.012463902946489771856117316100129344083043651786931274117`40.
(3)
N[Exp[f*Log[1 - p] + s*Log[p] - LogGamma[1 + f] - LogGamma
s] /. data], 40]
0.012463902946489771856117316100129344083043651786931274117`40.
Well anyway, interesting discussion. :>)
H
Harlan Grove
...
...
I was one of a few regulars in this ng who log of complete gamma functions from
several packages last Spring. The results may be found in a zipped XLS file at
ftp://members.aol.com/hrlngrv/gammaln8.zip
Appologies if this is a joke that I'm too ironly-impaired to recognize.
???!!!
Kindly show me any Intel-based PC and/or any software for such a machine that
uses IEEE binary floating point that treats 0.5 any differently than 1/2.
There's no difference whatsoever. Both have the identical internal binary
representation. Here's a small C program I just used to test this. Compiled with
the LCC compiler that comes with GNU's Small Eiffel language package for
Windows.
/* begin foo.c */
#include <stdio.h>
int main() {
double d;
unsigned char *pc;
pc = (unsigned char *) &d;
d = 0.5;
printf("%g\n%02hhX %02hhX %02hhX %02hhX %02hhX %02hhX %02hhX %02hhX\n\n", \
d, pc[0], pc[1], pc[2], pc[3], pc[4], pc[5], pc[6], pc[7]);
d = 1.0 / 2.0;
printf("%g\n%02hhX %02hhX %02hhX %02hhX %02hhX %02hhX %02hhX %02hhX\n\n", \
d, pc[0], pc[1], pc[2], pc[3], pc[4], pc[5], pc[6], pc[7]);
return 0;
}
/* end foo.c */
And it's little endian output.
00 00 00 00 00 00 E0 3F
0.5
00 00 00 00 00 00 E0 3F
Or perhaps you're aware of some subtle Excel functionality that renders 1/2
correctly but 0.5 as something else.
Half the time 8 or 9 bits, the other half the time 7 bits. See the comparison
worksheet in the Zip file the url for which appears above. Since single
precision gives more than exactly 7 decimal digits of precision, it's not
surprising that it does better than 7 decimal digits some of the time.
Since the complete gamma function is defined using a nonanalytic definite
integral, all methods of calculating it or its logarithm must be approximate.
Excel's approximation function is a rather poor one. What's new?! Why should
Excel's GAMMALN function be any higher quality than it's MOD function? Or its
NEGBINOMDIST function? Or its POISSON function? And so on . . .
FWLIW, the workbook in the linked file includes a VBA translation of a Perl
function based on FORTRAN code from http://www.netlib.org/specfun/gamma . Very
good accuracy. Pity no one at Microsoft seems to know about netlib. Instead, we
all get to enjoy - firsthand - the numerical teething pains of the Excel
developers. How long must they go on?
...
Just for discussion, the "other two programs" differing at the 14 digit does
not sound right to me, but of course, I can't tell. I don't have Maple.
I was one of a few regulars in this ng who log of complete gamma functions from
several packages last Spring. The results may be found in a zipped XLS file at
ftp://members.aol.com/hrlngrv/gammaln8.zip
By chance, were you testing with the inexact number of 0.5 so that it operated
at machine precision? I was testing at the more exact value of 1/2, thereby
using full arbitrary precision. . . .
Appologies if this is a joke that I'm too ironly-impaired to recognize.
???!!!
Kindly show me any Intel-based PC and/or any software for such a machine that
uses IEEE binary floating point that treats 0.5 any differently than 1/2.
There's no difference whatsoever. Both have the identical internal binary
representation. Here's a small C program I just used to test this. Compiled with
the LCC compiler that comes with GNU's Small Eiffel language package for
Windows.
/* begin foo.c */
#include <stdio.h>
int main() {
double d;
unsigned char *pc;
pc = (unsigned char *) &d;
d = 0.5;
printf("%g\n%02hhX %02hhX %02hhX %02hhX %02hhX %02hhX %02hhX %02hhX\n\n", \
d, pc[0], pc[1], pc[2], pc[3], pc[4], pc[5], pc[6], pc[7]);
d = 1.0 / 2.0;
printf("%g\n%02hhX %02hhX %02hhX %02hhX %02hhX %02hhX %02hhX %02hhX\n\n", \
d, pc[0], pc[1], pc[2], pc[3], pc[4], pc[5], pc[6], pc[7]);
return 0;
}
/* end foo.c */
And it's little endian output.
0.5
00 00 00 00 00 00 E0 3F
0.5
00 00 00 00 00 00 E0 3F
Or perhaps you're aware of some subtle Excel functionality that renders 1/2
correctly but 0.5 as something else.
From small testing in the past, I find that Excel's GammaLn function to be a
little more accurate than 7 digits, with a usual range of about 9-11 digits.
Half the time 8 or 9 bits, the other half the time 7 bits. See the comparison
worksheet in the Zip file the url for which appears above. Since single
precision gives more than exactly 7 decimal digits of precision, it's not
surprising that it does better than 7 decimal digits some of the time.
...Then there are the following two examples:
=GAMMALN(1)
=GAMMALN(2)
-4.1716E-11
-8.57678E-11
The Gamma of both 1 & 2 should be 1. And of course, the Log of 1 is zero.
I believe that Excel's GammaLn function should return an exact zero for both
examples. Excel is of course using an approximating function.
Since the complete gamma function is defined using a nonanalytic definite
integral, all methods of calculating it or its logarithm must be approximate.
Excel's approximation function is a rather poor one. What's new?! Why should
Excel's GAMMALN function be any higher quality than it's MOD function? Or its
NEGBINOMDIST function? Or its POISSON function? And so on . . .
FWLIW, the workbook in the linked file includes a VBA translation of a Perl
function based on FORTRAN code from http://www.netlib.org/specfun/gamma . Very
good accuracy. Pity no one at Microsoft seems to know about netlib. Instead, we
all get to enjoy - firsthand - the numerical teething pains of the Excel
developers. How long must they go on?
D
Dana DeLouis
Thanks Harlan. I understand now. I appreciate the feedback. Thanks for
link also. That's interesting code. I'll probably be studying it for
awhile.
I recognized the const D1 of 0.57721... as the EulerGamma constant.
That part of 0.5 was only a guess as to why the two programs were giving
different answers. It wasn't meant for Excel. Sorry. Excel can't do
arbitrary precision. I see that you were testing both programs at machine
precision. (similar to double precision).
What I was thinking at the time was the following. As you know, Excel can't
do this, but the other programs treat the following two list as completely
different. The elements are different, and the algorithms used on them are
different.
exact = {f -> 512, s -> 512, p -> 1/2};
approx = {f -> 512., s -> 512., p -> 0.5};
For example...
Precision[1/2]
Infinity
Precision[0.5]
MachinePrecision
Here are the same equations, but with the different numbers from above,
InputForm[N[Binomial[f + s - 1, s - 1]*p^s*(1 - p)^f /. exact]]
0.012463902946489773
InputForm[N[Binomial[f + s - 1, s - 1]*p^s*(1 - p)^f /. approx]]
0.012463902946493622
As you can see, the last 5 digits are different. That's what I was trying
to point out as a "possible" explanation for the differences you observed.
However, I now see how it was tested, so it doesn't apply. Thanks. :>)
--
Dana DeLouis
Using Windows XP & Office XP
= = = = = = = = = = = = = = = = =
<snip>
link also. That's interesting code. I'll probably be studying it for
awhile.
I recognized the const D1 of 0.57721... as the EulerGamma constant.
That part of 0.5 was only a guess as to why the two programs were giving
different answers. It wasn't meant for Excel. Sorry. Excel can't do
arbitrary precision. I see that you were testing both programs at machine
precision. (similar to double precision).
What I was thinking at the time was the following. As you know, Excel can't
do this, but the other programs treat the following two list as completely
different. The elements are different, and the algorithms used on them are
different.
exact = {f -> 512, s -> 512, p -> 1/2};
approx = {f -> 512., s -> 512., p -> 0.5};
For example...
Precision[1/2]
Infinity
Precision[0.5]
MachinePrecision
Here are the same equations, but with the different numbers from above,
InputForm[N[Binomial[f + s - 1, s - 1]*p^s*(1 - p)^f /. exact]]
0.012463902946489773
InputForm[N[Binomial[f + s - 1, s - 1]*p^s*(1 - p)^f /. approx]]
0.012463902946493622
As you can see, the last 5 digits are different. That's what I was trying
to point out as a "possible" explanation for the differences you observed.
However, I now see how it was tested, so it doesn't apply. Thanks. :>)
--
Dana DeLouis
Using Windows XP & Office XP
= = = = = = = = = = = = = = = = =
<snip>
J
Jerry W. Lewis
Harlan said:...
..
You're right. Mental typo on my part. I used LN(COMBIN(1023,511)) in my actual
calculations, but miswrote this as GAMMALN(1024)-2*GAMMALN(512). It should have
been GAMMALN(1024)-GAMMALN(512)-GAMMALN(513). However it remains the case that
NEGBINOMDIST(512,512,0.5) doesn't need to choke.
FWIW, Mathematica and Maple differ in the results of their respective logarithm
of complete gamma functions at the 14th digit, so *prove* that Mathematica is
more accurate than Maple. Good luck finding standard mathematical tables for
this function beyond 10 significant digits.
Probably a rounding issue (Maple is sometimes optimistic about the
number of correct figures ... less experience with Mathematica). If I
request enough more digits in each package, then I can get arbitrarily
precise agreement between them.
Jerry
J
Jerry W. Lewis
Here is another one for the mystery books.
http://www.bygsoftware.com/issues/modbug.html
completely misses the point about what is happening, but does give an
interesting example.
=MOD(12.3,1.23)
returns 8.88178419700125E-16 (both Windows and Mac), which is an
extremly curious result, even considering binary approximations to the
inputs. If B(x) is the (IEEE double precision) binary approximation to
x, then
B(12.3)/B(1.23) = 10 + 2/x
where x=2769713770832855. B(10+2/x)=10, so I would have expected MOD to
return 0, instead of what it does return. The sign of MOD(12.3,1.23)
and 2/x are the same, which is promising, but 2/x = 7.22096276178964E-16
which is smaller than what MOD returned. Now 10+2/x in binary is
1.0100000000000000000000000000000000000000000000000000011010000001...B3
vs
1.010000000000000000000000000000000000000000000000000010B3
as the binary representation to 10+8.88178419700125E-16 = 10+2^-50.
Since all previous MOD results (that I have seen questioned) were
consistent with binary math, my best guess is that the worksheet MOD is
doing custom arithmetic that evaluates the quotient to 55 bits (vs. 53
bits for IEEE double precision).
Unfortunately that still does not lead me to a guess about the basis for
the two unexplained limits discussed in this (ancient) thread.
Jerry
[ the limit discussed in http://support.microsoft.com/kb/119083 ]
http://www.bygsoftware.com/issues/modbug.html
completely misses the point about what is happening, but does give an
interesting example.
=MOD(12.3,1.23)
returns 8.88178419700125E-16 (both Windows and Mac), which is an
extremly curious result, even considering binary approximations to the
inputs. If B(x) is the (IEEE double precision) binary approximation to
x, then
B(12.3)/B(1.23) = 10 + 2/x
where x=2769713770832855. B(10+2/x)=10, so I would have expected MOD to
return 0, instead of what it does return. The sign of MOD(12.3,1.23)
and 2/x are the same, which is promising, but 2/x = 7.22096276178964E-16
which is smaller than what MOD returned. Now 10+2/x in binary is
1.0100000000000000000000000000000000000000000000000000011010000001...B3
vs
1.010000000000000000000000000000000000000000000000000010B3
as the binary representation to 10+8.88178419700125E-16 = 10+2^-50.
Since all previous MOD results (that I have seen questioned) were
consistent with binary math, my best guess is that the worksheet MOD is
doing custom arithmetic that evaluates the quotient to 55 bits (vs. 53
bits for IEEE double precision).
Unfortunately that still does not lead me to a guess about the basis for
the two unexplained limits discussed in this (ancient) thread.
Jerry
Harlan said:
[ the limit discussed in http://support.microsoft.com/kb/119083 ]
G
Guest
-----------------------Jerry said:Here is another one for the mystery books.
http://www.bygsoftware.com/issues/modbug.html
completely misses the point about what is happening, but does give an
interesting example.
=MOD(12.3,1.23)
returns 8.88178419700125E-16 (both Windows and Mac), which is an
extremly curious result, even considering binary approximations to the
inputs. If B(x) is the (IEEE double precision) binary approximation to
x, then
B(12.3)/B(1.23) = 10 + 2/x
where x=2769713770832855. B(10+2/x)=10, so I would have expected MOD to
return 0, instead of what it does return. The sign of MOD(12.3,1.23)
and 2/x are the same, which is promising, but 2/x = 7.22096276178964E-16
which is smaller than what MOD returned. Now 10+2/x in binary is
1.0100000000000000000000000000000000000000000000000000011010000001...B3
vs
1.010000000000000000000000000000000000000000000000000010B3
as the binary representation to 10+8.88178419700125E-16 = 10+2^-50.
Since all previous MOD results (that I have seen questioned) were
consistent with binary math, my best guess is that the worksheet MOD is
doing custom arithmetic that evaluates the quotient to 55 bits (vs. 53
bits for IEEE double precision).
Unfortunately that still does not lead me to a guess about the basis for
the two unexplained limits discussed in this (ancient) thread.
Jerry
Harlan said:
[ the limit discussed in http://support.microsoft.com/kb/119083 ]
It's mysteries like this that make Excel so much more (and so much
less) than
just a dry (reliable) mathematical tool.
Perhaps I'm missing something, but 8.88178419700125E-16 looks extremely close to
0.000000000000000 to me.
If that's the biggest error one can find in Excel, I'd be content.
Bill
J
JE McGimpsey
It is close, but it's not quite zero, which means that, as in all other
floating point math, one can't blindly write things like:
=IF(MOD(A1,B1)=0,"Good","Bad")
Instead, it's good practice to use something like:
=IF(MOD(A1,B1)<1E-10,"Good","Bad")
where the comparison value is some number "close enough" to zero.
And if B1 can be negative:
=IF(ABS(MOD(A1,B1))<1E-10,"Good","Bad")
is appropriate.
Unfortunately, there are lots of applications out there where the
developer was naive about floating point math, and you can get some
incorrect results from very simple errors.
"Bill Martin -- (Remove NOSPAM from address)"
floating point math, one can't blindly write things like:
=IF(MOD(A1,B1)=0,"Good","Bad")
Instead, it's good practice to use something like:
=IF(MOD(A1,B1)<1E-10,"Good","Bad")
where the comparison value is some number "close enough" to zero.
And if B1 can be negative:
=IF(ABS(MOD(A1,B1))<1E-10,"Good","Bad")
is appropriate.
Unfortunately, there are lots of applications out there where the
developer was naive about floating point math, and you can get some
incorrect results from very simple errors.
"Bill Martin -- (Remove NOSPAM from address)"
H
Harlan Grove
Bill Martin -- (Remove NOSPAM from address) wrote...
....
That's just standard floating point rounding error. Far more obnoxious
is the call that began this thread,
MOD(12345678000,64)
This returns #NUM! in Excel, but 48 in Gnumeric, OpenOffice Calc, 123
and pretty much every other piece of non-Microsoft software I have that
can calculate modulus. Heck, even the Calculator applet that comes with
Windows returns 48. So at one time there was at least one programmer
somewhere in Microsoft who avoided the temptation to screw this up.
....
Perhaps I'm missing something, but 8.88178419700125E-16 looks extremely close to
0.000000000000000 to me.
If that's the biggest error one can find in Excel, I'd be content.
That's just standard floating point rounding error. Far more obnoxious
is the call that began this thread,
MOD(12345678000,64)
This returns #NUM! in Excel, but 48 in Gnumeric, OpenOffice Calc, 123
and pretty much every other piece of non-Microsoft software I have that
can calculate modulus. Heck, even the Calculator applet that comes with
Windows returns 48. So at one time there was at least one programmer
somewhere in Microsoft who avoided the temptation to screw this up.
G
Guest
Harlan said:Bill Martin -- (Remove NOSPAM from address) wrote...
...
That's just standard floating point rounding error. Far more obnoxious
is the call that began this thread,
MOD(12345678000,64)
This returns #NUM! in Excel, but 48 in Gnumeric, OpenOffice Calc, 123
and pretty much every other piece of non-Microsoft software I have that
can calculate modulus. Heck, even the Calculator applet that comes with
Windows returns 48. So at one time there was at least one programmer
somewhere in Microsoft who avoided the temptation to screw this up.
Now that's a more substantial error!
Bill
J
Jerry W. Lewis
Where did I call it an error? Indeed, I showed that this result is
consistent with the binary representations of the numbers involved. The
"mystery" is that getting this particular result requires more than IEEE
double preicison (which is presumably the basis of all Excel
calculations) but less than the 10-byte floating point precision
available internally in the processor.
Jerry
consistent with the binary representations of the numbers involved. The
"mystery" is that getting this particular result requires more than IEEE
double preicison (which is presumably the basis of all Excel
calculations) but less than the 10-byte floating point precision
available internally in the processor.
Jerry
-----------------------Jerry said:Here is another one for the mystery books.
http://www.bygsoftware.com/issues/modbug.html
completely misses the point about what is happening, but does give an
interesting example.
=MOD(12.3,1.23)
returns 8.88178419700125E-16 (both Windows and Mac), which is an
extremly curious result, even considering binary approximations to the
inputs. If B(x) is the (IEEE double precision) binary approximation
to x, then
B(12.3)/B(1.23) = 10 + 2/x
where x=2769713770832855. B(10+2/x)=10, so I would have expected MOD
to return 0, instead of what it does return. The sign of
MOD(12.3,1.23) and 2/x are the same, which is promising, but 2/x =
7.22096276178964E-16 which is smaller than what MOD returned. Now
10+2/x in binary is
1.0100000000000000000000000000000000000000000000000000011010000001...B3
vs
1.010000000000000000000000000000000000000000000000000010B3
as the binary representation to 10+8.88178419700125E-16 = 10+2^-50.
Since all previous MOD results (that I have seen questioned) were
consistent with binary math, my best guess is that the worksheet MOD
is doing custom arithmetic that evaluates the quotient to 55 bits (vs.
53 bits for IEEE double precision).
Unfortunately that still does not lead me to a guess about the basis
for the two unexplained limits discussed in this (ancient) thread.
Jerry
Harlan said:...
I got it
[ the limit discussed in http://support.microsoft.com/kb/119083 ]
by experimentation, and then saw that Arvi had also determined the
same limit. I find it interesting that MS makes no attempt to
explain such an unusual limit.
An additional unusual limit that applies, is that MOD returns #NUM!
regardless of the quotient if the first argument exceeds
2.68873542664192E14 = 2^20+2^19+2^18+2^17+2^15+2^12+2^8+2^6
which is within the range of exact DP representation of whole
numbers by more than an order of magnitude.
It's mysteries like this that make Excel so much more (and so much
less) than
just a dry (reliable) mathematical tool.
Perhaps I'm missing something, but 8.88178419700125E-16 looks extremely
close to 0.000000000000000 to me.
If that's the biggest error one can find in Excel, I'd be content.
Bill
G
Guest
Correction: MOD uses at least 1-bit more than IEEE double precision. There
is no upper limit on the precision imposed by this example, because
MOD(B(12.3),B(1.23)) = (2/x)*B(1.23) not simply 2/x. In the Wintel world,
the most obvious source for more than double precision is the 10-byte
internal registers in the processor. Does anyone know if extended precision
is available in hardware on the Mac (since Excel on the Mac gives the same
answer)?
WAG alert:
I am not aware of commercial MS languages offering access to the processor's
extended precision (at least not in recent memory), so it is possible that
this cross-platform consistency is due to some non-standard software extended
precision. If so, then this extra precision on the mantissa and the
unexplained limits for MOD may all be related to fitting this hypothetical
custom FP precision into a convenient word size. It would be interesting to
see other examples that further define the size of the mantissa that MOD must
be using.
Jerry
is no upper limit on the precision imposed by this example, because
MOD(B(12.3),B(1.23)) = (2/x)*B(1.23) not simply 2/x. In the Wintel world,
the most obvious source for more than double precision is the 10-byte
internal registers in the processor. Does anyone know if extended precision
is available in hardware on the Mac (since Excel on the Mac gives the same
answer)?
WAG alert:
I am not aware of commercial MS languages offering access to the processor's
extended precision (at least not in recent memory), so it is possible that
this cross-platform consistency is due to some non-standard software extended
precision. If so, then this extra precision on the mantissa and the
unexplained limits for MOD may all be related to fitting this hypothetical
custom FP precision into a convenient word size. It would be interesting to
see other examples that further define the size of the mantissa that MOD must
be using.
Jerry
H
Harlan Grove
....Jerry W. Lewis said:WAG alert:
I am not aware of commercial MS languages offering access to the
processor's extended precision (at least not in recent memory), so
it is possible that this cross-platform consistency is due to some
non-standard software extended precision. If so, then this extra
precision on the mantissa and the unexplained limits for MOD may
all be related to fitting this hypothetical custom FP precision
into a convenient word size. It would be interesting to see other
examples that further define the size of the mantissa that MOD must
be using.
Warning - some cynicism to follow.
Microsoft's original commercial language was cassette BASIC. It morphed into
BASICA when Microsoft started selling operating systems. IIRC, BASICA had
only one floating point type, and it wasn't IEEE. A quick Google search
leads me to believe it was 4-byte/32-bit. Excel's MOD function dies at 2^27.
The cynic in me is tempted to leap to the conclusion that Microsoft used
it's BASIC/BASICA code in the original Excel for Mac 512Ks in the mid-1980s
and hasn't revised the code since. Surely that can't be?
End cynicism (and sarcasm).
Excel's MOD is clearly *NOT* IEEE-compliant. Why would they target a
non-IEEE virtual FPU?