Re: [Haskell-cafe] rounding errors with real numbers.

On Fri, Mar 03, 2006 at 01:29:06PM +0100, Henning Thielemann wrote:

To: Matthias Fischmann cc: haskell-cafe@haskell.org From: Henning Thielemann Date: Fri, 3 Mar 2006 13:29:06 +0100 (MET) Subject: Re: [Haskell-cafe] rounding errors with real numbers.

On Thu, 2 Mar 2006, Matthias Fischmann wrote:

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cancellation happens for instance here: 1 + 1e-50 - 1 == 0

the function again (in the wasteful original form, for clarity). where do you think cancellation may take place? isn't what you call canellation a generic rounding error?

Cancellation is a special kind of rounding error. Rounding errors appear everywhere, in (*), sin, exp and so on, but cancellations only arise on differences. They are especially bad, because as the example above shows, even if all numbers are given in double precision in the computation a+b-a, no digit of the result is correct, that is 100% rounding error! The danger of cancellation is everywhere where you subtract numbers of similar magnitude.

1 + epsilon - 1 == epsilon, which is zero except for a very small rounding error somewhere deep in the e-minus-somethings. how is the error getting worse than that, for which numbers? m.