Accuracy and the power function

02162014, 02:43 PM
(This post was last modified: 02162014 05:31 PM by Dieter.)
Post: #1




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 12digitvalue agrees with the true value rounded to 13 digits, so that the relative error in \(\large\pi\) is less than 6,6 E14.
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 12digit 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 12digit HP and see what you get. Dieter 

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