Re: [Haskell-cafe] smallest double eps
last $ takeWhile (\x - 1 + x /= 1) (iterate (/2) 1) 2.220446049250313e-16 This works because of the way IEEE floating-point numbers are represented, so it's good for the majority of machines, but it is technically a hack, in that it depends on a representation of floating-point numbers in some form akin to the IEEE (1.mantissa bits)*2^(exponent). It would be nice for those of us interested in doing numerical work in Haskell if we could have machine epsilon available from the language, in case we're running on a machine where that assumption doesn't hold. I don't know where this would belong -- the Numeric library seems mainly concerned with reading and showing numbers, and System is more about interacting with the OS than making statements about the properties of the underlying hardware. --Grady Lemoine ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] smallest double eps
Hang on, hang on, now I'm getting confused. First you asked for the smallest (positive) x such that 1+x /= x which is around x=4.5e15. Then Joachim wondered if you wanted 1+x /= 1 which is around x=2.2e-16. But not you claim to be looking for the smallest positive number that a Double can represent. Which is a totally different beast. The smallest possible Double depends on if you want to accept denormalized numbers or not. If you don't, then it's about x=4.5e-308. Now what is the number you are looking for? -- Lennart On Sep 29, 2006, at 22:02 , Tamas K Papp wrote: On Fri, Sep 29, 2006 at 06:53:35PM -0700, Chad Scherrer wrote: Tamas, You might want to read Joachim's post more carefully - he's trying to help you, and I think he makes a good point. Chad, If his point is that there is no smallest positive number, then I think I understand it, thanks. I should have said that I was looking for the smallest positive number Double can represent, but thought that was clear from the context. If this is not his point, I'd really appreciate an explanation. Thanks, Tamas ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] smallest double eps
On Sat, Sep 30, 2006 at 04:19:50AM -0400, Lennart Augustsson wrote: Hang on, hang on, now I'm getting confused. First you asked for the smallest (positive) x such that 1+x /= x which is around x=4.5e15. Then Joachim wondered if you wanted 1+x /= 1 which is around x=2.2e-16. Oops, sorry, there was a typo in my original post. I was looking for the latter. Thanks for the help, Tamas ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] smallest double eps
Hang on, hang on, now I'm getting confused. First you asked for the smallest (positive) x such that 1+x /= x which is around x=4.5e15. Then Joachim wondered if you wanted 1+x /= 1 which is around x=2.2e-16. But not you claim to be looking for the smallest positive number that a Double can represent. Which is a totally different beast. The smallest possible Double depends on if you want to accept denormalized numbers or not. If you don't, then it's about x=4.5e-308. Now what is the number you are looking for? -- Lennart This is the point I was confused about also. Joachim seemed to be correcting what might have been a misstatement of the problem. ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] smallest double eps
Lennart Augustsson wrote: Hang on, hang on, now I'm getting confused. First you asked for the smallest (positive) x such that 1+x /= x which is around x=4.5e15. 1 + 0 /= 0 0 is smaller than 4.5e15 So I don't understand this at all... Regards, Brian. -- Logic empowers us and Love gives us purpose. Yet still phantoms restless for eras long past, congealed in the present in unthought forms, strive mightily unseen to destroy us. http://www.metamilk.com ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] smallest double eps
On 30 Sep 2006, at 17:19, Brian Hulley wrote: Lennart Augustsson wrote: Hang on, hang on, now I'm getting confused. First you asked for the smallest (positive) x such that 1+x /= x which is around x=4.5e15. 1 + 0 /= 0 0 is smaller than 4.5e15 So I don't understand this at all... But then 0 isn't positive. Bob ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] smallest double eps
Thomas Davie wrote: On 30 Sep 2006, at 17:19, Brian Hulley wrote: Lennart Augustsson wrote: Hang on, hang on, now I'm getting confused. First you asked for the smallest (positive) x such that 1+x /= x which is around x=4.5e15. 1 + 0 /= 0 0 is smaller than 4.5e15 So I don't understand this at all... But then 0 isn't positive. Why not? In any case every positive number nust satisfy the above inequation so what about 0.1, which is certainly smaller than 4500? Regards, Brian. -- Logic empowers us and Love gives us purpose. Yet still phantoms restless for eras long past, congealed in the present in unthought forms, strive mightily unseen to destroy us. http://www.metamilk.com ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] smallest double eps
Hang on, hang on, now I'm getting confused. First you asked for the smallest (positive) x such that 1+x /= x which is around x=4.5e15. 1 + 0 /= 0 0 is smaller than 4.5e15 So I don't understand this at all... But then 0 isn't positive. Why not? In any case every positive number nust satisfy the above inequation so what about 0.1, which is certainly smaller than 4500? In math, every positive number must satisfy the above inequation, that is true. But as Chad said, the smallest number in Haskell (at least according to my GHC, it could be different with different processors, right?) that satisfies the equation is 2.2e-16. 1 + 2.2e-16 /= 1 True 1 + 2.2e-17 /= 1 False This is because the Double type only holds so much precision. After getting small enough, the type just can't hold any more precision, and the value is essentially 0. last $ takeWhile (\x - 1 + x /= 1) (iterate (/2) 1) 2.220446049250313e-16 ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] smallest double eps
Bryan Burgers wrote: Hang on, hang on, now I'm getting confused. First you asked for the smallest (positive) x such that 1+x /= x which is around x=4.5e15. 1 + 0 /= 0 0 is smaller than 4.5e15 So I don't understand this at all... But then 0 isn't positive. Why not? In any case every positive number nust satisfy the above inequation so what about 0.1, which is certainly smaller than 4500? People are confusing equality and inequality - the nontrivial thing here is to find the smallest positive x that satisfies the equation 1 + x == x. In math, every positive number must satisfy the above inequation, that is true. But as Chad said, the smallest number in Haskell (at least according to my GHC, it could be different with different processors, right?) that satisfies the equation is 2.2e-16. And you've changed the subject - the stuff above was talking about x + 1 /= x, you're demonstrating solutions to a different problem, finding the smallest x such that 1 + x == 1. That's the number often called epsilon. 1 + 2.2e-16 /= 1 True 1 + 2.2e-17 /= 1 False Let's stop confusing ourselves about this. Brandon ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] smallest double eps
Forwarded Message From: Victor Bandur [EMAIL PROTECTED] Reply-To: [EMAIL PROTECTED] To: Brandon Moore [EMAIL PROTECTED] Subject: Re: [Haskell-cafe] smallest double eps Date: Sat, 30 Sep 2006 20:17:05 -0400 Hi all, I'm new to this mailing list, so my response may be a little out of place, but I think either what's being asked is what is the smallest x such that 1 + x /= 1 (machine epsilon,) or the largest such that 1+x /= x. The bounds seem to be confused. Victor On Sat, 2006-30-09 at 16:10 -0700, Brandon Moore wrote: Bryan Burgers wrote: Hang on, hang on, now I'm getting confused. First you asked for the smallest (positive) x such that 1+x /= x which is around x=4.5e15. 1 + 0 /= 0 0 is smaller than 4.5e15 So I don't understand this at all... But then 0 isn't positive. Why not? In any case every positive number nust satisfy the above inequation so what about 0.1, which is certainly smaller than 4500? People are confusing equality and inequality - the nontrivial thing here is to find the smallest positive x that satisfies the equation 1 + x == x. In math, every positive number must satisfy the above inequation, that is true. But as Chad said, the smallest number in Haskell (at least according to my GHC, it could be different with different processors, right?) that satisfies the equation is 2.2e-16. And you've changed the subject - the stuff above was talking about x + 1 /= x, you're demonstrating solutions to a different problem, finding the smallest x such that 1 + x == 1. That's the number often called epsilon. 1 + 2.2e-16 /= 1 True 1 + 2.2e-17 /= 1 False Let's stop confusing ourselves about this. Brandon ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
[Haskell-cafe] smallest double eps
Hi, Is there a built-in constant in Haskell (or, if it is compiler-specific, in ghc) that gives the smallest positive floating point number x such that 1+x /= x? Some languages refer to that as double.eps or similar. I need it for numeric algorithms. Thanks, Tamas ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] smallest double eps
Hi, Am Freitag, den 29.09.2006, 19:30 -0400 schrieb Tamas K Papp: the smallest positive floating point number x such that 1+x /= x? That would be the smallest positive number, woudn't it? Do you mean the smalles postive number x with 1+x /= 1? Greetings, Joachim -- Joachim Breitner e-Mail: [EMAIL PROTECTED] Homepage: http://www.joachim-breitner.de ICQ#: 74513189 ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] smallest double eps
On Sat, Sep 30, 2006 at 12:20:16AM +, Joachim Breitner wrote: Hi, Am Freitag, den 29.09.2006, 19:30 -0400 schrieb Tamas K Papp: the smallest positive floating point number x such that 1+x /= x? That would be the smallest positive number, woudn't it? Do you mean the smalles postive number x with 1+x /= 1? Hi Joachim, Specifically, I would be happy with the smallest Double that makes the statement true. I need it as a convergence bound for an iterative algorithm. Anyhow, thanks for the clarification, but I would be happier with an answer. Tamas ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] smallest double eps
Haskell doesn't provide such a value, but you could easily compute it from from the values given in the RealFloat class. It tells you the base, number of digits in mantissa, etc. As for using such an eps in a convergence test I'd be very careful. How do you know that your iteration doesn't make the value bounce back and forth with more than eps? -- Lennart On Sep 29, 2006, at 20:26 , Tamas K Papp wrote: On Sat, Sep 30, 2006 at 12:20:16AM +, Joachim Breitner wrote: Hi, Am Freitag, den 29.09.2006, 19:30 -0400 schrieb Tamas K Papp: the smallest positive floating point number x such that 1+x /= x? That would be the smallest positive number, woudn't it? Do you mean the smalles postive number x with 1+x /= 1? Hi Joachim, Specifically, I would be happy with the smallest Double that makes the statement true. I need it as a convergence bound for an iterative algorithm. Anyhow, thanks for the clarification, but I would be happier with an answer. Tamas ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] smallest double eps
On Fri, Sep 29, 2006 at 09:26:27PM -0400, Lennart Augustsson wrote: As for using such an eps in a convergence test I'd be very careful. How do you know that your iteration doesn't make the value bounce back and forth with more than eps? Hi Lennart, Thanks for the answer, I will try it. I am not using eps, but rather a value derived from eps by analyzing the algorithm (eg eps^0.25). Your answer will help me calculate that directly. Thanks, Tamas ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] smallest double eps
Tamas, You might want to read Joachim's post more carefully - he's trying to help you, and I think he makes a good point. -Chad Am Freitag, den 29.09.2006, 19:30 -0400 schrieb Tamas K Papp: the smallest positive floating point number x such that 1+x /= x? That would be the smallest positive number, woudn't it? Do you mean the smalles postive number x with 1+x /= 1? Hi Joachim, Specifically, I would be happy with the smallest Double that makes the statement true. I need it as a convergence bound for an iterative algorithm. Anyhow, thanks for the clarification, but I would be happier with an answer. Tamas -- ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe End of Haskell-Cafe Digest, Vol 37, Issue 92 -- Chad Scherrer Time flies like an arrow; fruit flies like a banana -- Groucho Marx ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] smallest double eps
On Fri, Sep 29, 2006 at 06:53:35PM -0700, Chad Scherrer wrote: Tamas, You might want to read Joachim's post more carefully - he's trying to help you, and I think he makes a good point. Chad, If his point is that there is no smallest positive number, then I think I understand it, thanks. I should have said that I was looking for the smallest positive number Double can represent, but thought that was clear from the context. If this is not his point, I'd really appreciate an explanation. Thanks, Tamas ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
