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Derivation

1. Initial program 32.9

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

3. Applied add-log-exp32.3

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

4. Taylor expanded around -inf 32.3

   \[\leadsto \log \color{blue}{\left(2 \cdot \frac{1}{{x}^{5}} + \left(2 \cdot \frac{1}{{x}^{3}} + 1\right)\right)}\]

5. Simplified32.3

   \[\leadsto \log \color{blue}{\left(\left(\frac{\frac{2}{x}}{x \cdot x} + \frac{2}{{x}^{5}}\right) + 1\right)}\]
6. Using strategy rm

7. Applied add-exp-log32.3

   \[\leadsto \color{blue}{e^{\log \left(\log \left(\left(\frac{\frac{2}{x}}{x \cdot x} + \frac{2}{{x}^{5}}\right) + 1\right)\right)}}\]
8. Using strategy rm

9. Applied pow132.3

   \[\leadsto e^{\log \color{blue}{\left({\left(\log \left(\left(\frac{\frac{2}{x}}{x \cdot x} + \frac{2}{{x}^{5}}\right) + 1\right)\right)}^{1}\right)}}\]

10. Applied log-pow32.3

    \[\leadsto e^{\color{blue}{1 \cdot \log \left(\log \left(\left(\frac{\frac{2}{x}}{x \cdot x} + \frac{2}{{x}^{5}}\right) + 1\right)\right)}}\]

11. Applied exp-prod32.3

    \[\leadsto \color{blue}{{\left(e^{1}\right)}^{\left(\log \left(\log \left(\left(\frac{\frac{2}{x}}{x \cdot x} + \frac{2}{{x}^{5}}\right) + 1\right)\right)\right)}}\]

12. Simplified32.3

    \[\leadsto {\color{blue}{e}}^{\left(\log \left(\log \left(\left(\frac{\frac{2}{x}}{x \cdot x} + \frac{2}{{x}^{5}}\right) + 1\right)\right)\right)}\]

13. Final simplification32.3

    \[\leadsto {e}^{\left(\log \left(\log \left(1 + \left(\frac{\frac{2}{x}}{x \cdot x} + \frac{2}{{x}^{5}}\right)\right)\right)\right)}\]

Runtime

Time bar (total: 54.8s)Debug logProfile

herbie shell --seed 2018255 +o rules:numerics
(FPCore (x)
  :name "3frac (problem 3.3.3)"

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

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