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Derivation

1. Initial program 29.2

   \[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
2. Using strategy rm

3. Applied add-cube-cbrt29.7

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

4. Applied *-un-lft-identity29.7

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

5. Applied times-frac29.7

   \[\leadsto \frac{x}{x + 1} - \color{blue}{\frac{1}{\sqrt[3]{x - 1} \cdot \sqrt[3]{x - 1}} \cdot \frac{x + 1}{\sqrt[3]{x - 1}}}\]
6. Using strategy rm

7. Applied add-cube-cbrt29.7

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

8. Applied cbrt-prod29.7

   \[\leadsto \frac{x}{x + 1} - \frac{1}{\sqrt[3]{x - 1} \cdot \sqrt[3]{x - 1}} \cdot \frac{x + 1}{\color{blue}{\sqrt[3]{\sqrt[3]{x - 1} \cdot \sqrt[3]{x - 1}} \cdot \sqrt[3]{\sqrt[3]{x - 1}}}}\]

9. Applied add-cube-cbrt29.6

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

10. Applied times-frac29.6

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

Runtime

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herbie shell --seed 2019053 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))