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Associativity in floating point arithmetic failing by two values


Floating point binary arithmetic questionproperty of sum in floating point arithmeticAccurate floating-point linear interpolationFloating Point Calculationapproximation using floating point arithmeticTwo expressions using three-digit floating point arithmetic with rounding?calculating $sin x$ in floating point arithmetic16-bit Floating point range of different valuesCalculate the total number of floating-point numbers in a certain rangeUnderstanding Floating point arithmetic













5












$begingroup$


Assume all numbers and operations below are in floating-point arithmetic with finite precision, bounded exponent, and rounding to the nearest integer.




Are there $x,y$ positive such that $$beginalign(x+y)-x&>y\(x+y)-s(x)&>yendalign$$
where $s(x)$ denotes the successor of $x$?





This question appeared while designing a test for a software.



It is easy to write a program that searches for such an example, but it is unfeasible to test all possibilities and show that the example doesn't exist. So far my code hasn't got any example.



Example: In case seeing an example of $(x+y)-x>y$ helps somehow, take
$$
beginalign
x&=1.1234567891234568\
y&=1e-5text ( denoting 10^-5)
endalign
$$

Then $(x+y)-x=1.0000000000065512e-05 > y$. There are many examples of the first inequality.




Link to scicomp.stackexchange's copy of this post in case a solution appears there first. There is already a solution there.










share|cite|improve this question











$endgroup$











  • $begingroup$
    I don't see this question as being about mathematics.
    $endgroup$
    – 5xum
    Oct 3 '17 at 12:10






  • 3




    $begingroup$
    @5xum There is a field of mathematics called Numerical Analysis that studies floating point arithmetic. The question explores the magnitude of the failure of asociativity in floating point arithmetic.
    $endgroup$
    – EEE
    Oct 3 '17 at 12:14






  • 2




    $begingroup$
    @5xum And not only mathematics, it belongs to the far narrower scope that this website handles. Observe how the tag (floating-point) exists here.
    $endgroup$
    – EEE
    Oct 3 '17 at 12:28











  • $begingroup$
    Try also scicomp.stackexchange.com
    $endgroup$
    – lhf
    Oct 3 '17 at 14:04










  • $begingroup$
    I guess you want to work with rounding-to-nearest mode. For round-up mode there are examples for both.
    $endgroup$
    – gammatester
    Oct 3 '17 at 14:46















5












$begingroup$


Assume all numbers and operations below are in floating-point arithmetic with finite precision, bounded exponent, and rounding to the nearest integer.




Are there $x,y$ positive such that $$beginalign(x+y)-x&>y\(x+y)-s(x)&>yendalign$$
where $s(x)$ denotes the successor of $x$?





This question appeared while designing a test for a software.



It is easy to write a program that searches for such an example, but it is unfeasible to test all possibilities and show that the example doesn't exist. So far my code hasn't got any example.



Example: In case seeing an example of $(x+y)-x>y$ helps somehow, take
$$
beginalign
x&=1.1234567891234568\
y&=1e-5text ( denoting 10^-5)
endalign
$$

Then $(x+y)-x=1.0000000000065512e-05 > y$. There are many examples of the first inequality.




Link to scicomp.stackexchange's copy of this post in case a solution appears there first. There is already a solution there.










share|cite|improve this question











$endgroup$











  • $begingroup$
    I don't see this question as being about mathematics.
    $endgroup$
    – 5xum
    Oct 3 '17 at 12:10






  • 3




    $begingroup$
    @5xum There is a field of mathematics called Numerical Analysis that studies floating point arithmetic. The question explores the magnitude of the failure of asociativity in floating point arithmetic.
    $endgroup$
    – EEE
    Oct 3 '17 at 12:14






  • 2




    $begingroup$
    @5xum And not only mathematics, it belongs to the far narrower scope that this website handles. Observe how the tag (floating-point) exists here.
    $endgroup$
    – EEE
    Oct 3 '17 at 12:28











  • $begingroup$
    Try also scicomp.stackexchange.com
    $endgroup$
    – lhf
    Oct 3 '17 at 14:04










  • $begingroup$
    I guess you want to work with rounding-to-nearest mode. For round-up mode there are examples for both.
    $endgroup$
    – gammatester
    Oct 3 '17 at 14:46













5












5








5


2



$begingroup$


Assume all numbers and operations below are in floating-point arithmetic with finite precision, bounded exponent, and rounding to the nearest integer.




Are there $x,y$ positive such that $$beginalign(x+y)-x&>y\(x+y)-s(x)&>yendalign$$
where $s(x)$ denotes the successor of $x$?





This question appeared while designing a test for a software.



It is easy to write a program that searches for such an example, but it is unfeasible to test all possibilities and show that the example doesn't exist. So far my code hasn't got any example.



Example: In case seeing an example of $(x+y)-x>y$ helps somehow, take
$$
beginalign
x&=1.1234567891234568\
y&=1e-5text ( denoting 10^-5)
endalign
$$

Then $(x+y)-x=1.0000000000065512e-05 > y$. There are many examples of the first inequality.




Link to scicomp.stackexchange's copy of this post in case a solution appears there first. There is already a solution there.










share|cite|improve this question











$endgroup$




Assume all numbers and operations below are in floating-point arithmetic with finite precision, bounded exponent, and rounding to the nearest integer.




Are there $x,y$ positive such that $$beginalign(x+y)-x&>y\(x+y)-s(x)&>yendalign$$
where $s(x)$ denotes the successor of $x$?





This question appeared while designing a test for a software.



It is easy to write a program that searches for such an example, but it is unfeasible to test all possibilities and show that the example doesn't exist. So far my code hasn't got any example.



Example: In case seeing an example of $(x+y)-x>y$ helps somehow, take
$$
beginalign
x&=1.1234567891234568\
y&=1e-5text ( denoting 10^-5)
endalign
$$

Then $(x+y)-x=1.0000000000065512e-05 > y$. There are many examples of the first inequality.




Link to scicomp.stackexchange's copy of this post in case a solution appears there first. There is already a solution there.







floating-point






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Mar 14 at 8:25









Rodrigo de Azevedo

13.2k41960




13.2k41960










asked Oct 3 '17 at 12:03









EEEEEE

915




915











  • $begingroup$
    I don't see this question as being about mathematics.
    $endgroup$
    – 5xum
    Oct 3 '17 at 12:10






  • 3




    $begingroup$
    @5xum There is a field of mathematics called Numerical Analysis that studies floating point arithmetic. The question explores the magnitude of the failure of asociativity in floating point arithmetic.
    $endgroup$
    – EEE
    Oct 3 '17 at 12:14






  • 2




    $begingroup$
    @5xum And not only mathematics, it belongs to the far narrower scope that this website handles. Observe how the tag (floating-point) exists here.
    $endgroup$
    – EEE
    Oct 3 '17 at 12:28











  • $begingroup$
    Try also scicomp.stackexchange.com
    $endgroup$
    – lhf
    Oct 3 '17 at 14:04










  • $begingroup$
    I guess you want to work with rounding-to-nearest mode. For round-up mode there are examples for both.
    $endgroup$
    – gammatester
    Oct 3 '17 at 14:46
















  • $begingroup$
    I don't see this question as being about mathematics.
    $endgroup$
    – 5xum
    Oct 3 '17 at 12:10






  • 3




    $begingroup$
    @5xum There is a field of mathematics called Numerical Analysis that studies floating point arithmetic. The question explores the magnitude of the failure of asociativity in floating point arithmetic.
    $endgroup$
    – EEE
    Oct 3 '17 at 12:14






  • 2




    $begingroup$
    @5xum And not only mathematics, it belongs to the far narrower scope that this website handles. Observe how the tag (floating-point) exists here.
    $endgroup$
    – EEE
    Oct 3 '17 at 12:28











  • $begingroup$
    Try also scicomp.stackexchange.com
    $endgroup$
    – lhf
    Oct 3 '17 at 14:04










  • $begingroup$
    I guess you want to work with rounding-to-nearest mode. For round-up mode there are examples for both.
    $endgroup$
    – gammatester
    Oct 3 '17 at 14:46















$begingroup$
I don't see this question as being about mathematics.
$endgroup$
– 5xum
Oct 3 '17 at 12:10




$begingroup$
I don't see this question as being about mathematics.
$endgroup$
– 5xum
Oct 3 '17 at 12:10




3




3




$begingroup$
@5xum There is a field of mathematics called Numerical Analysis that studies floating point arithmetic. The question explores the magnitude of the failure of asociativity in floating point arithmetic.
$endgroup$
– EEE
Oct 3 '17 at 12:14




$begingroup$
@5xum There is a field of mathematics called Numerical Analysis that studies floating point arithmetic. The question explores the magnitude of the failure of asociativity in floating point arithmetic.
$endgroup$
– EEE
Oct 3 '17 at 12:14




2




2




$begingroup$
@5xum And not only mathematics, it belongs to the far narrower scope that this website handles. Observe how the tag (floating-point) exists here.
$endgroup$
– EEE
Oct 3 '17 at 12:28





$begingroup$
@5xum And not only mathematics, it belongs to the far narrower scope that this website handles. Observe how the tag (floating-point) exists here.
$endgroup$
– EEE
Oct 3 '17 at 12:28













$begingroup$
Try also scicomp.stackexchange.com
$endgroup$
– lhf
Oct 3 '17 at 14:04




$begingroup$
Try also scicomp.stackexchange.com
$endgroup$
– lhf
Oct 3 '17 at 14:04












$begingroup$
I guess you want to work with rounding-to-nearest mode. For round-up mode there are examples for both.
$endgroup$
– gammatester
Oct 3 '17 at 14:46




$begingroup$
I guess you want to work with rounding-to-nearest mode. For round-up mode there are examples for both.
$endgroup$
– gammatester
Oct 3 '17 at 14:46










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