The dedoublifier, part one

oh right -- mediants! I've never heard of Stern-Brocot but I know what it is.

Why not just use continued fractions directly? https://en.wikipedia.org/wiki/Continued_fraction#Calculating...

e.g.

p = 3.141592653589793

  = 3 + 1 / 7.062513305931052

  = 3 + 1 / (7 + 1 / 15.996594406684103)

  = 3 + 1 / (7 + 1 / (15 + 1 / 1.0034172310150002))

  = 3 + 1 / (7 + 1 / (15 + 1 / (1 + 1/292.63459087501246)))

so the continued fraction coefficients are 3;7;15;1;292 (see http://mathworld.wolfram.com/PiContinuedFraction.html or https://en.wikipedia.org/wiki/Continued_fraction#Continued_f...) and can be converted to rationals at each step using the semiconvergent recursion formula https://en.wikipedia.org/wiki/Continued_fraction#Semiconverg... to get approximants 3/1, 22/7,

(3 + 15 * 22) / (1 + 15 * 7) = 333/106,

(22 + 1 * 333) / (7 + 1 * 106) = 355/113,

(333 + 292 * 355) / (106 + 292 * 113) = 103993/33102,

etc.