We describe new fast algorithms intended for the evaluation of *π*(*x*) inspired by the harmonic and geometric mean integrals that can be used on any pocket calculator. In particular, the formula *h*(*x*) based on the harmonic mean is within ≈ 15 of the actual value for 3 ≤ *x* ≤ 10000*.* The approximation verifies the inequality *h*(*x*) ≤ Li(*x*) and, therefore, is better than Li(*x*) for small *x.* We show that *h*(*x*) and their extensions are more accurate than the other famous approximations, such as the Locker–Ernst or Legendre approximations also for large *x.* In addition, we derive another function *g*(*x*) based on the geometric mean integral that employs *h*(*x*) as an input and allows one to significantly improve the quality of this method. We show that *g*(*x*) is within ≈ 25 of the actual value for *x* ≤ 50*,* 000 (to compare, Li(*x*) lies within ≈ 40 for the same range) and asymptotically \(g\left(x\right)\sim \frac{x}{1\mathrm{n} x}\mathrm{exp}\left(\frac{1}{1\mathrm{n} x-1}\right).\)

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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, No. 9, pp. 1264–1273, September, 2022. Ukrainian DOI: https://doi.org/10.37863/umzh.v74i9.6193.

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Teruel, G.R.P. New Fast Methods To Compute The Number Of Primes Smaller Than A Given Value.
*Ukr Math J* **74**, 1441–1451 (2023). https://doi.org/10.1007/s11253-023-02145-2

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DOI: https://doi.org/10.1007/s11253-023-02145-2