sbcl在将有理数制转换为十进制记数法时尽可能地保持精度

Question

我在这里有一个简单的lisp程序，它计算在单位间隔上随机选择的两个点之间的平均长度的近似值。如果我运行这个程序，我会得到一个有理数16666671666667/50000000000000，但是当我(天真地)尝试将有理数格式化为20位时，一些精度就被丢弃了0.33333343000000000000。我认为，在幕后，SBCL在格式化有理数之前将其转换为浮点数，但我真的不确定如何判断。我只是在使用表达式(format t "~20$~%" (scale-all-line-contributions 10000000 1)。有没有办法将有理数转换为小数记数法，同时保持尽可能高的精度？我知道格式系统是强大和可扩展的，但我很难找到专门与有理数相关的文档。

为了完整起见，下面是代码，因为它不是很长。

(defun number-of-pairs (n i)
  "get the number of pairs with distance x
  for non-zero distances we have to consider two cases"
  (cond
    ((= i 0) n)
    ((> i 0) (* 2 (- n i)))
    ((> i n) 0)))

(defun line-contribution (n i power)
  "get the number of segments of length i in a line of n segments and their weight combined"
  (let
    ((number-of-pairs (number-of-pairs n i))
     (weight-of-pair (expt i power)))
    (* number-of-pairs weight-of-pair)))

(defun all-line-contributions (n power)
  "get the line contributions for reach [0 .. n]"
  (loop for i from 1 upto (- n 1) summing (line-contribution n i power)))

(defun normalized-all-line-contributions (n power)
  "normalize line contributions by number of pairs"
  (let ((pair-count (expt n 2)))
    (/ (all-line-contributions n power) pair-count)))

(defun scale-all-line-contributions (n power)
  "scale the line contributions by the distance n
  this will guarantee convergence"
  (/ (normalized-all-line-contributions n power) (expt n power)))

(print (scale-all-line-contributions 10000000 1))
(format t "~20$~%" (scale-all-line-contributions 10000000 1))

编辑:修复代码中的逻辑错误。新的有理数，浮点数对为33333333333333/100000000000000 0.33333334000000000000