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Derivation

1. Initial program 0.0

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

3. Applied flip--0.0

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

4. Applied associate-/r/0.0

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

6. Applied associate-*l/0.0

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

7. Simplified0.0

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

9. Applied add-cbrt-cube0.0

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

10. Final simplification0.0

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

Reproduce

herbie shell --seed 2019007 
(FPCore (x)
  :name "Asymptote B"
  (+ (/ 1 (- x 1)) (/ x (+ x 1))))

Details

Time bar (total: 28.4s)Debug log