Error

Target

Derivation

1. Initial program 9.8

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

3. Applied associate-+l-9.8

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

5. Applied frac-sub26.2

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

6. Applied frac-sub25.3

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

7. Simplified8.8

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

8. Taylor expanded around 0 0.3

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

10. Applied associate-/r*0.1

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

11. Final simplification0.1

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

Reproduce

herbie shell --seed 2019053 
(FPCore (x)
  :name "3frac (problem 3.3.3)"

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

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