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There is a formula for bisection method to estimate number of iterations that are needed to achieve a given precision (desired significant figures) in the interval $[a,b]$ $$ n\ge \frac{\log{(b-a)}-\log{\epsilon}}{\log2} $$ where $\epsilon$ is the given precision (for example $2^{-52}$. This is the machine epsilon of IEEE 754 double precision floating point format). This is one of advantages of bisection method. My question is that is there a similar formula for Newton's method?

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The Wackypedia article you link to has a section which gives an error estimate. If you can compute the $M$ specified, you can use it with $b-a$ and $\log\log\epsilon$ to compute the number of steps.

As a practical matter, it is often not possible to compute $M$, and sometimes the computed value is too pessimistic. Since Newton converges quadratically to the answer, the step converges linearly to the error. That is, the step size is approximately the error size. The step size is usually used as part of a stopping criterion.

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