Error

Target

Derivation

1. Initial program 9.4

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

3. Applied associate-+l-9.4

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

5. Applied frac-sub25.0

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

6. Applied frac-sub24.1

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

7. Simplified24.6

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

8. Simplified24.6

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

9. Taylor expanded around -inf 0.3

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

11. Applied associate-/r*0.1

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

12. Final simplification0.1

    \[\leadsto \frac{\frac{2}{(x \cdot x + x)_*}}{x + -1}\]

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed 2018252 +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))))