Error

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Derivation

1. Initial program 0.0

   \[\frac{-\left(f + n\right)}{f - n}\]
2. Using strategy rm

3. Applied add-cbrt-cube41.6

   \[\leadsto \frac{-\left(f + n\right)}{\color{blue}{\sqrt[3]{\left(\left(f - n\right) \cdot \left(f - n\right)\right) \cdot \left(f - n\right)}}}\]

4. Applied add-cbrt-cube41.8

   \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(-\left(f + n\right)\right) \cdot \left(-\left(f + n\right)\right)\right) \cdot \left(-\left(f + n\right)\right)}}}{\sqrt[3]{\left(\left(f - n\right) \cdot \left(f - n\right)\right) \cdot \left(f - n\right)}}\]

5. Applied cbrt-undiv41.8

   \[\leadsto \color{blue}{\sqrt[3]{\frac{\left(\left(-\left(f + n\right)\right) \cdot \left(-\left(f + n\right)\right)\right) \cdot \left(-\left(f + n\right)\right)}{\left(\left(f - n\right) \cdot \left(f - n\right)\right) \cdot \left(f - n\right)}}}\]

6. Simplified0.0

   \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{\left(-n\right) - f}{f - n}}{\frac{f - n}{n + f} \cdot \frac{f - n}{n + f}}}}\]
7. Using strategy rm

8. Applied add-cube-cbrt0.0

   \[\leadsto \sqrt[3]{\frac{\frac{\left(-n\right) - f}{f - n}}{\color{blue}{\left(\left(\sqrt[3]{\frac{f - n}{n + f}} \cdot \sqrt[3]{\frac{f - n}{n + f}}\right) \cdot \sqrt[3]{\frac{f - n}{n + f}}\right)} \cdot \frac{f - n}{n + f}}}\]

9. Applied associate-*l*0.0

   \[\leadsto \sqrt[3]{\frac{\frac{\left(-n\right) - f}{f - n}}{\color{blue}{\left(\sqrt[3]{\frac{f - n}{n + f}} \cdot \sqrt[3]{\frac{f - n}{n + f}}\right) \cdot \left(\sqrt[3]{\frac{f - n}{n + f}} \cdot \frac{f - n}{n + f}\right)}}}\]

10. Final simplification0.0

    \[\leadsto \sqrt[3]{\frac{\frac{\left(-n\right) - f}{f - n}}{\left(\sqrt[3]{\frac{f - n}{n + f}} \cdot \sqrt[3]{\frac{f - n}{n + f}}\right) \cdot \left(\frac{f - n}{n + f} \cdot \sqrt[3]{\frac{f - n}{n + f}}\right)}}\]

Reproduce

herbie shell --seed 2019026 
(FPCore (f n)
  :name "subtraction fraction"
  (/ (- (+ f n)) (- f n)))