$1/k_2$. Therefore, obtaining definite negative answer for constant known with machine precision of $\sigma$ in terms of Criterion 1 require testing of $\sigma^{-1}$ codes. For double precision machine epsilon it is above $10^{15}$. You need either hundreds of CPU cores, or a lot of patience (days of search). Positive identification can be much faster, of course, like for calculators 3, 4 in Figures \[SmokingGun\], \[Convergence2\].

[Statistical properties of the $\exp-\log$ numbers \[Criterion2\] ]{}

Observation from Fig. \[Convergence3\] and results of previous section lead to another view of constant identification problem. It can be described as a random process. Consider following numerical *Monte Carlo* experiment. We generate pseudo-random numbers $\xi>0$ from exponential distribution with some scale $\lambda$ and Probability Distribution Function (PDF): $$\label{Pexp}
P(\xi) = \frac{e^{-\xi/\lambda}}{\lambda}.$$ Next, we compare $\xi$ with our target number $z>0$, and generate sequence of progressively better approximations. Surprisingly, observed behavior is similar to Fig. \[Convergence2\], but with larger fluctuations. Let’s further assume $z$ is known with numerical precision $\sigma < z$. In other words, true number is in the interval $[z-\sigma,z+\sigma]$. Assuming we can easily obtain probability of random hit into vicinity $z$: $$P(z,\sigma, \lambda) = \int_{z-\sigma}^{z+\sigma} P(\xi) \; d \xi$$ what gives average number of 