select x from t where x >  low_x and x <  high_x  (open  , open  )
  select x from t where x >= low_x and x <= high_x  (closed, closed)
  select x from t where x >  low_x and x <= high_x  (open,   closed)
  select x from t where x >= low_x and x <  high_x  (closed, open)

  template:
  select x from t where x #low low_x and x #high high_x
  #low  in (">", ">=")
  #high in ("<", "<=")

  min_x = min (x) over all rows
  max_x = max (x) over all rows
  B = (max_x - min_x) / num_distinct (Band width)

  left band      : min_x     < x < min_x + B
  central region : min_x + B < x < max_x - B
  right band     : max_x - B < x < max_x

• For ranges completely contained in one of the two bands

       min_x     <= low_x < high_x <= min_x + B
    or max_x - B <= low_x < high_x <= max_x
  

  the cardinality can depend on both low_x and high_x, only low_x, only high_x, or none (= constant), depending on the value of #low and #high, in a somewhat counterintuitive way. We have found the relevant formulae.
• In all the other cases, a slightly modified (modifications underlined) version of Jonathan Lewis' "standard formula" applies:

    cardinality = num_rows * (high_x_effective - low_x_effective) / (max_x - min_x)
                + num_rows * correction_for_or_equal_operators
                - num_rows * correction_for_special_case
  
    if #low = ">=" and low_x inside the left band then 
      low_x_effective = min_x + B
    else
      low_x_effective = low_x
    end
  
    if #high = "<=" and high_x inside the right band
      high_x_effective = max_x - B
    else
      high_x_effective = high_x
    end
  
    correction_for_or_equal_operators = 0
      + 1/num_distinct if #low  = ">="
      + 1/num_distinct if #high = "<="
  
    correction_for_special_case = 0 
      + 1/num_distinct if low_x  = min_x and #low  = ">"    
      + 1/num_distinct if high_x = max_x and #high = "<"  
  

  This fitness test verifies the formula against the CBO, using its pl/sql implementation.