Error

Try it out

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 add-sqr-sqrt25.2

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

5. Applied sqrt-div24.4

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

6. Applied sqrt-div26.3

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

7. Applied frac-times24.4

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

8. Applied frac-sub29.4

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

9. Applied frac-add28.5

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

10. Simplified28.4

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

11. Simplified25.2

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

12. Taylor expanded around inf 0.3

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

14. Applied associate-/r*0.1

    \[\leadsto \color{blue}{\frac{\frac{2}{x + -1}}{x + x \cdot x}}\]

15. Final simplification0.1

    \[\leadsto \frac{\frac{2}{x + -1}}{x + x \cdot x}\]

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018349 
(FPCore (x)
  :name "3frac (problem 3.3.3)"

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

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