GeezerVison is showing mathworld.wolfram.com — Digit-Extraction Algorithm -- from Wolfram MathWorld

Digit-Extraction Algorithm

A digit-extraction algorithm allows digits of a given number to be calculated without requiring the computation of earlier digits. The BBP formula permits individual hexadecimal digits or binary digits of pi to be calculated in this way (Bailey et al. 1997).

Plouffe (2022) proposes a formula for decimal digits of by defining

pi_n=((2(-1)^(n+1)(2n)!)/(2^(2n)B_(2n)(1-2^(-2n))(1-3^(-2n))(1-5^(-2n))(1-7^(-2n))))^(1/(2n)),

(1)

where is a positive integer and is a Bernoulli number. A candidate for the th digit to the right of the decimal point of for is

$d_n=int(10frac(10^(n-1)pi_(n-1))),$

(2)

where is the integer part and $frac(x)$ is the fractional part. To certify the digit, an error bound must exclude crossing a decimal truncation boundary. A small absolute error alone does not guarantee this. Similar approximations can be obtained using

(3)

and

(4)

where is an Euler number (Plouffe 2022). Plouffe uses the first approximation for octal (base 8) digits and the second for decimal digits, subject to the same truncation-boundary requirement. Related limits and formulas can also be obtained for (Plouffe 2022).

Explore with Wolfram|Alpha

More things to try:

References

Bailey, D. H.; Borwein, P. B.; and Plouffe, S. "On the Rapid Computation of Various Polylogarithmic Constants." Math. Comput. 66, 903-913, 1997. https://doi.org/10.1090/S0025-5718-97-00856-9.Plouffe, S. "A Formula for the