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Derivation

1. Initial program 0.0

   \[\frac{-\left(f + n\right)}{f - n}\]

2. Initial simplification0.0

   \[\leadsto -\frac{n + f}{f - n}\]
3. Using strategy rm

4. Applied *-un-lft-identity0.0

   \[\leadsto -\frac{\color{blue}{1 \cdot \left(n + f\right)}}{f - n}\]

5. Applied associate-/l*0.0

   \[\leadsto -\color{blue}{\frac{1}{\frac{f - n}{n + f}}}\]
6. Using strategy rm

7. Applied add-cbrt-cube0.0

   \[\leadsto -\color{blue}{\sqrt[3]{\left(\frac{1}{\frac{f - n}{n + f}} \cdot \frac{1}{\frac{f - n}{n + f}}\right) \cdot \frac{1}{\frac{f - n}{n + f}}}}\]
8. Using strategy rm

9. Applied add-cube-cbrt0.0

   \[\leadsto -\sqrt[3]{\left(\color{blue}{\left(\left(\sqrt[3]{\frac{1}{\frac{f - n}{n + f}}} \cdot \sqrt[3]{\frac{1}{\frac{f - n}{n + f}}}\right) \cdot \sqrt[3]{\frac{1}{\frac{f - n}{n + f}}}\right)} \cdot \frac{1}{\frac{f - n}{n + f}}\right) \cdot \frac{1}{\frac{f - n}{n + f}}}\]

10. Applied associate-*l*0.0

    \[\leadsto -\sqrt[3]{\color{blue}{\left(\left(\sqrt[3]{\frac{1}{\frac{f - n}{n + f}}} \cdot \sqrt[3]{\frac{1}{\frac{f - n}{n + f}}}\right) \cdot \left(\sqrt[3]{\frac{1}{\frac{f - n}{n + f}}} \cdot \frac{1}{\frac{f - n}{n + f}}\right)\right)} \cdot \frac{1}{\frac{f - n}{n + f}}}\]

11. Simplified0.0

    \[\leadsto -\sqrt[3]{\left(\color{blue}{\left(\sqrt[3]{\frac{f + n}{f - n}} \cdot \sqrt[3]{\frac{f + n}{f - n}}\right)} \cdot \left(\sqrt[3]{\frac{1}{\frac{f - n}{n + f}}} \cdot \frac{1}{\frac{f - n}{n + f}}\right)\right) \cdot \frac{1}{\frac{f - n}{n + f}}}\]
12. Using strategy rm

13. Applied add-cube-cbrt0.1

    \[\leadsto -\sqrt[3]{\left(\left(\sqrt[3]{\frac{f + n}{f - n}} \cdot \sqrt[3]{\frac{f + n}{f - n}}\right) \cdot \left(\sqrt[3]{\frac{1}{\frac{f - n}{n + f}}} \cdot \frac{1}{\frac{f - n}{n + f}}\right)\right) \cdot \frac{1}{\color{blue}{\left(\sqrt[3]{\frac{f - n}{n + f}} \cdot \sqrt[3]{\frac{f - n}{n + f}}\right) \cdot \sqrt[3]{\frac{f - n}{n + f}}}}}\]

14. Final simplification0.1

    \[\leadsto -\sqrt[3]{\frac{1}{\left(\sqrt[3]{\frac{f - n}{n + f}} \cdot \sqrt[3]{\frac{f - n}{n + f}}\right) \cdot \sqrt[3]{\frac{f - n}{n + f}}} \cdot \left(\left(\frac{1}{\frac{f - n}{n + f}} \cdot \sqrt[3]{\frac{1}{\frac{f - n}{n + f}}}\right) \cdot \left(\sqrt[3]{\frac{n + f}{f - n}} \cdot \sqrt[3]{\frac{n + f}{f - n}}\right)\right)}\]

Runtime

Time bar (total: 19.1s)Debug logProfile

herbie shell --seed 2018339 
(FPCore (f n)
  :name "subtraction fraction"
  (/ (- (+ f n)) (- f n)))