I said,
Perhaps equivalent to [Bouwkamp's accelerated] circumscribing formula, simply expand the log of the quadrisected product,
The formulas turn out identical. In a desperate bid for originality, I trisected the rhs of the rearrangement inf inf 1 /===\ /===\ n + - | | x 2 x | | 2 x | | cos(-) = - sin(-) | | ----- sin(-----) | | n x 2 | | x 1 n = 3 n = 1 n + - 2 and got the same damn thing! But an obvious acceleration is to temporarily shed the first few terms of the product and start the series with a k-gon instead of a triangle: inf /===\ | | x log( | | cos(-)) = | | i i = k inf ==== i 4 i 2 i 2 i \ (- 1) (2 - 2 ) bern(2 i) hurwitz_zeta(2 i, k) x > -------------------------------------------------------- / 2 i (2 i)! ==== i = 0 E.g., starting with a pentagon, i 2 i 2 i - 1 (- 1) %pi 2 bern(2 i) 1 hurwitz_zeta(2 i, 5) = - -------------------------------- - ---- (2 i)! 2 i 4 1 1 - ---- - ---- - 1 2 i 2 i 3 2 - 2 i ~ 5 , giving a term ratio of 4/25 for x = pi. Finally, exponentiate, and restore the missing triangle and square factors (1/sqrt(8)). This (Hurwitz) zeta factor is the source of the severe subtractive cancellation mentioned last time, and grows worse for larger k *AND LARGER i*. I.e., for n digits of final accuracy, you need for the ith series term a precision of n + log_10(4/25) i ~ n - .796 i digits, so the zeta term requires n+log_10(4) i ~ n + .602 i digits of intermediate precision. If you are too lazy to adjust the precision of individual factors of individual terms, you need the worst case value of n/(1-log(2)/log(5)) ~ 1.76 n digits overall. --rwg Confidential to the Moderator: The elementwise product of two singular 2x2 matrices is singular, but the elementwise product of the three singular 3x3s (c27) map(transpose, [[1,1,1;1,1,1;0,0,1],[x,1,1;1,1,1;0,1,1],[y,1,1;0,1,1;0,1,1]]) [ 1 1 0 ] [ x 1 0 ] [ y 0 0 ] [ ] [ ] [ ] (d27) [[ 1 1 0 ], [ 1 1 1 ], [ 1 1 1 ]] [ ] [ ] [ ] [ 1 1 1 ] [ 1 1 1 ] [ 1 1 1 ] has determinant xy (c28) map(det,endcons(apply("*",%),%)); (d28) [0, 0, 0, x y] so the product of any number can be nonsingular.