A memorable approximation of pi

One of the best-known approximations of π is the fraction 22/7. Another is 355/113 which is easy to memorize by remembering the first three odd numbers, 1,3, and 5. Next, double the digits: 11 33 55, then group as 113 355, and finally slide the 113 under the 355.

Upon examining how close the result of 355/113 comes to the real value of π, it made sense to find another memorable fraction that approximates that difference which could then be summed. Repeating the process once more yields a formula accurate to the first 17 digits of π. (Perhaps we'll name it the "Dunn Approximation of Pi" after its discoverer.)

which, when computed carefully, yields: 3.1415926535897932 762597293465706

(which is greater than π by a mere 37.8 quintillionth, or more precisely, 3.7797085963291064091099708947939e-17)
(which is enough accuracy to compute the diameter of the earth with an error of only 1 nanometer. For reference, the thickness of a sheet of paper is about 100,000 nanometers!)

Most importantly, for memorization: the second term can be thought of as 33 78 99 (then slide the first three digits under the last three, just like we did with the first term). Next, think of the final term as 10 777 (this time, invert the slide, moving the last two 77s under the 107). What remains to be remembered are the signs and the adjusting multipliers of "7" (notice there are five of them in the equation) and "12" (which is, of course, five higher than 7).

In JavaScript, the formula looks like: var pi=355/113-1e-7*899/337+1e-12*107/77;
(yields 3.141592653589793 56009 )

In Excel or LibreOffice Calc, paste into a cell: =355/113-1e-7*899/337+1e-12*107/77
(3.14159265358979 00000)

In Java:
double t1=(double)355f/(double)113f;
double t2=(double)1e-7*(double)899f/(double)337f;
double t3=(double)1e-12*(double)107f/(double)77f;
double pi=t1-t2+t3;
(3.14159265358979 05)

In numcalc.com with 192-bit precision:
(yields 3.1415926535897932 76259729346570553471632750013044648164525)

Rational approximations were explored and computed using a modified version of this tool written in Perl.

See also this visualization of π approximations.

For reference, here are additional digits of π: 3.1415926535897932 3846264338327950288419716939937510 58209749445923078164062862089986280 348253421170679821480865132823066470 9384460955058223172535940 812848111745028410

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