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Target

Derivation

1. Initial program 9.5

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

3. Applied *-un-lft-identity9.5

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

4. Applied *-un-lft-identity9.5

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

5. Applied distribute-lft-out--9.5

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

7. Applied frac-sub26.0

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

8. Applied associate-*r/26.0

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

9. Applied frac-add25.2

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

10. Simplified25.3

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

11. Taylor expanded around 0 0.3

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

12. Taylor expanded around inf 0.2

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

13. Final simplification0.2

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

Reproduce

herbie shell --seed 2018362 
(FPCore (x)
  :name "3frac (problem 3.3.3)"

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

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

Details

Time bar (total: 1.3m)Debug log