float num with many zeros: 1.1795176680648731e+39 == 1179517668064873100000000000000008388608.0

manyzero.rb

def euclidsolve(cx, cy, c, small)
  return euclidsolve(cy, cx, c, small).reverse if cx > cy
  # cx <= cy
  # x * cx + y * cy = c + small
  # p [cx.to_f, cy.to_f, c.to_f, small.to_f]
  if small >= cy
    # cx <= cy, c < cy
    # cx*x + cy*y == c + small
    # x=0; cy*y == c + small
    return [0, 1]
  end
  x, y = euclidsolve(cx, cy % cx, c % cx, small)
  # x * cx + y * (cy%cx) == c%cx + small
  #   (x+dx) * cx + (y+dy) * (cy%cx + (cy/cx)*cx) == c%cx + (c/cx)*cx + small
  #   dx * cx + dy  + (y+dy) * (cy/cx)*cx == (c/cx)*cx
  [x + (c/cx) - y * (cy/cx), y]
end

x,y=euclidsolve(cx=2**622, cy=10**187, c=10**(187-16)/2, small=2**(621-53))
(53..1024).each do |bit|
  exp10 = (-400..400).bsearch{
    2**bit<10**it
  }-1
  x,y=euclidsolve(cx=2**bit, cy=10**exp10, c=10**(exp10-16)/2, small=2**(bit-1-53))
  if x.abs.bit_length <= 53
    f=(x*2**bit)
    match = f.to_s[/^\d{14}+0{5,}+[1-9]|^\d{14}9{5,}+[0-8]/]
    p [match.size, match, f.to_f] if match && match.size > 16
  end
rescue
end