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Accuracy and the power function
02-16-2014, 02:43 PM (This post was last modified: 02-16-2014 05:31 PM by Dieter.)
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Accuracy and the power function
Recently I posted a HP35s program that determines the Bernoulli numbers \(B_n\). The formula includes powers of \(2\large\pi\) with exponents up to 372. Since \(\large\pi\) carries just 12 digits, the results with large exponents are somewhat inaccurate even though the 12-digit-value agrees with the true value rounded to 13 digits, so that the relative error in \(\large\pi\) is less than 6,6 E-14.

While this deviation can be compensated quite easily, there is another error that can reach magnitudes I did not expect. In the mentioned program it here and there may cause error peaks while most other results are within 2 ULP.

Let's assume \(2\large\pi\) = 6,28318530718, i.e. the best possible 12-digit value. Now evaluate some powers of this:

\(6,28318530718^{-76}  =  2,179 3651 6466 35... · 10^{-61}\)
\(6,28318530718^{ 200}  =  4,324 8761 1407 46... · 10^{ 159}\)
\(6,28318530718^{ 372}  =  8,373 5772 1132 00... · 10^{ 296}\)

Please try this with your 12-digit HP and see what you get.

Dieter
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Accuracy and the power function - Dieter - 02-16-2014 02:43 PM



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