Ln(x) using repeated square root extraction

03062022, 12:50 AM
Post: #11




RE: Ln(x) using repeated square root extraction
(03052022 08:39 PM)Thomas Klemm Wrote: For \(x = 2\) we end up with the following sequence of iterated square roots: It is more work, but we could improve accuracy by "pulling out" 1. CAS> f(x) := x/(sqrt(1+x)+1) // == sqrt(1+x)1 CAS> 1. // x1 = 21 = 1 CAS> f(Ans) 0.414213562373 0.189207115003 9.05077326653e−2 4.42737824274e−2 2.18971486541e−2 1.08892860517e−2 The recursive formula, log1p(x) = 2*log1p(x/(sqrt(1+x)+1)), is similar to atan formula (05312021 09:51 PM)Albert Chan Wrote: \(\displaystyle\arctan(x) = 2\arctan\left( {x \over \sqrt{1+x^2}+1} \right)\) 

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