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Derivation

1. Initial program 14.5

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

3. Applied frac-sub13.9

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

4. Simplified0.4

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

6. Applied associate-/r*0.1

   \[\leadsto \color{blue}{\frac{\frac{-\left(1 + 1\right)}{x + 1}}{x - 1}}\]
7. Using strategy rm

8. Applied add-sqr-sqrt0.8

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

9. Applied distribute-rgt-neg-in0.8

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

10. Applied associate-/l*0.7

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

12. Applied add-cube-cbrt0.7

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

13. Applied *-un-lft-identity0.7

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

14. Applied times-frac0.7

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

15. Applied add-sqr-sqrt0.7

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

16. Applied sqrt-prod0.1

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

17. Applied times-frac0.1

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

18. Simplified0.1

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

19. Final simplification0.1

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

Runtime

Time bar (total: 8.4m)Debug logProfile

herbie shell --seed 2018242 
(FPCore (x)
  :name "Asymptote A"
  (- (/ 1 (+ x 1)) (/ 1 (- x 1))))