Problem Statement

The Almkvist-Giullera formula for calculating   1/π2   is based on the Calabi-Yau differential equations of order 4 and 5,   which were originally used to describe certain manifolds in string theory.

The formula is:

This formula can be used to calculate the constant   π-2,   and thus to calculate   π. Note that, because the product of all terms but the power of 1000 can be calculated as an integer, the terms in the series can be separated into a large integer term: multiplied by a negative integer power of 10:

Let's start with the solution:

  [ $ "bigrat.qky" loadfile ] now!

  [ 1 swap times [ i^ 1+ * ] ] is !       ( n --> n   )

  [ dup dup 2 ** 532 *
    over 126 * + 9 +
    swap 6 * ! * 32 *
    swap ! 6 ** 3 * / ]        is intterm ( n --> n   )

  [ dup intterm 
    10 rot 6 * 3 + ** 
    reduce ]                   is vterm   ( n --> n/d )

  10 times [ i^ intterm echo cr ] cr
  
  0 n->v 
  53 times [ i^ vterm v+ ]
  1/v 70 vsqrt drop 
  70 point$ echo$ cr