\[\frac{-\left(f + n\right)}{f - n}\]
Test:
subtraction fraction
Bits:
128 bits
Bits error versus f
Bits error versus n
Time: 4.5 s
Input Error: 0.1
Output Error: 0.1
Log: ⚲
Profile: 🕒
\(\frac{1}{{\left(\frac{f}{-\left(f + n\right)}\right)}^2 - {\left(\frac{n}{-\left(f + n\right)}\right)}^2} \cdot \left(\frac{f}{-\left(f + n\right)} + \frac{n}{-\left(f + n\right)}\right)\)
  1. Started with
    \[\frac{-\left(f + n\right)}{f - n}\]
    0.1
  2. Using strategy rm
    0.1
  3. Applied clear-num to get
    \[\color{red}{\frac{-\left(f + n\right)}{f - n}} \leadsto \color{blue}{\frac{1}{\frac{f - n}{-\left(f + n\right)}}}\]
    0.1
  4. Using strategy rm
    0.1
  5. Applied div-sub to get
    \[\frac{1}{\color{red}{\frac{f - n}{-\left(f + n\right)}}} \leadsto \frac{1}{\color{blue}{\frac{f}{-\left(f + n\right)} - \frac{n}{-\left(f + n\right)}}}\]
    0.1
  6. Using strategy rm
    0.1
  7. Applied flip-- to get
    \[\frac{1}{\color{red}{\frac{f}{-\left(f + n\right)} - \frac{n}{-\left(f + n\right)}}} \leadsto \frac{1}{\color{blue}{\frac{{\left(\frac{f}{-\left(f + n\right)}\right)}^2 - {\left(\frac{n}{-\left(f + n\right)}\right)}^2}{\frac{f}{-\left(f + n\right)} + \frac{n}{-\left(f + n\right)}}}}\]
    0.1
  8. Applied associate-/r/ to get
    \[\color{red}{\frac{1}{\frac{{\left(\frac{f}{-\left(f + n\right)}\right)}^2 - {\left(\frac{n}{-\left(f + n\right)}\right)}^2}{\frac{f}{-\left(f + n\right)} + \frac{n}{-\left(f + n\right)}}}} \leadsto \color{blue}{\frac{1}{{\left(\frac{f}{-\left(f + n\right)}\right)}^2 - {\left(\frac{n}{-\left(f + n\right)}\right)}^2} \cdot \left(\frac{f}{-\left(f + n\right)} + \frac{n}{-\left(f + n\right)}\right)}\]
    0.1

Original test:


(lambda ((f default) (n default))
  #:name "subtraction fraction"
  (/ (- (+ f n)) (- f n)))