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Derivation

1. Initial program 9.6

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
2. Using strategy rm

3. Applied add-cube-cbrt25.6

   \[\leadsto \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \color{blue}{\left(\sqrt[3]{\frac{1}{x - 1}} \cdot \sqrt[3]{\frac{1}{x - 1}}\right) \cdot \sqrt[3]{\frac{1}{x - 1}}}\]
4. Using strategy rm

5. Applied cbrt-div25.1

   \[\leadsto \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \left(\sqrt[3]{\frac{1}{x - 1}} \cdot \sqrt[3]{\frac{1}{x - 1}}\right) \cdot \color{blue}{\frac{\sqrt[3]{1}}{\sqrt[3]{x - 1}}}\]

6. Applied cbrt-div24.6

   \[\leadsto \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \left(\sqrt[3]{\frac{1}{x - 1}} \cdot \color{blue}{\frac{\sqrt[3]{1}}{\sqrt[3]{x - 1}}}\right) \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{x - 1}}\]

7. Applied cbrt-div25.2

   \[\leadsto \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \left(\color{blue}{\frac{\sqrt[3]{1}}{\sqrt[3]{x - 1}}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{x - 1}}\right) \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{x - 1}}\]

8. Applied frac-times25.1

   \[\leadsto \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{x - 1} \cdot \sqrt[3]{x - 1}}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{x - 1}}\]

9. Applied frac-times24.5

   \[\leadsto \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \color{blue}{\frac{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}{\left(\sqrt[3]{x - 1} \cdot \sqrt[3]{x - 1}\right) \cdot \sqrt[3]{x - 1}}}\]

10. Applied frac-sub28.7

    \[\leadsto \color{blue}{\frac{1 \cdot x - \left(x + 1\right) \cdot 2}{\left(x + 1\right) \cdot x}} + \frac{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}{\left(\sqrt[3]{x - 1} \cdot \sqrt[3]{x - 1}\right) \cdot \sqrt[3]{x - 1}}\]

11. Applied frac-add25.6

    \[\leadsto \color{blue}{\frac{\left(1 \cdot x - \left(x + 1\right) \cdot 2\right) \cdot \left(\left(\sqrt[3]{x - 1} \cdot \sqrt[3]{x - 1}\right) \cdot \sqrt[3]{x - 1}\right) + \left(\left(x + 1\right) \cdot x\right) \cdot \left(\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}\right)}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(\left(\sqrt[3]{x - 1} \cdot \sqrt[3]{x - 1}\right) \cdot \sqrt[3]{x - 1}\right)}}\]

12. Applied simplify25.3

    \[\leadsto \frac{\color{blue}{x \cdot \left(\left(x + 1\right) + \left(x - 1\right)\right) - \left(x - 1\right) \cdot \left(2 + x \cdot 2\right)}}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(\left(\sqrt[3]{x - 1} \cdot \sqrt[3]{x - 1}\right) \cdot \sqrt[3]{x - 1}\right)}\]

13. Applied simplify25.3

    \[\leadsto \frac{x \cdot \left(\left(x + 1\right) + \left(x - 1\right)\right) - \left(x - 1\right) \cdot \left(2 + x \cdot 2\right)}{\color{blue}{\left(x \cdot x + x\right) \cdot \left(x - 1\right)}}\]

14. Taylor expanded around 0 0.3

    \[\leadsto \frac{\color{blue}{2}}{\left(x \cdot x + x\right) \cdot \left(x - 1\right)}\]

15. Taylor expanded around 0 0.3

    \[\leadsto \frac{2}{\color{blue}{{x}^{3} - x}}\]

Runtime

Time bar (total: 3.9m)Debug logProfile

herbie shell --seed '#(1072967564 1937075727 894099792 790700740 1036514779 1027793188)' 
(FPCore (x)
  :name "3frac (problem 3.3.3)"

  :herbie-target
  (/ 2 (* x (- (* x x) 1)))

  (+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1))))