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Target

Derivation

1. Initial program 9.7

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

3. Applied div-inv9.7

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

5. Applied un-div-inv9.7

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

6. Applied frac-sub26.2

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

7. Applied frac-add25.3

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

8. Taylor expanded around 0 0.2

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

9. Taylor expanded around -inf 0.2

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

10. Final simplification0.2

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

Runtime

Time bar (total: 46.4s)Debug logProfile

herbie shell --seed 2018285 
(FPCore (x)
  :name "3frac (problem 3.3.3)"

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

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