Error

Try it out

Derivation

1. Split input into 2 regimes

2. if x < -12727.401168194334 or 12123.563172164775 < x

   1. Initial program 59.3

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]

   2. Taylor expanded around -inf 0.3

      \[\leadsto \color{blue}{-\left(3 \cdot \frac{1}{{x}^{3}} + \left(\frac{1}{{x}^{2}} + 3 \cdot \frac{1}{x}\right)\right)}\]

   3. Simplified0.0

      \[\leadsto \color{blue}{\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) + \frac{\frac{-3}{x}}{x \cdot x}}\]

   if -12727.401168194334 < x < 12123.563172164775

   1. Initial program 0.1

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
   2. Using strategy rm

   3. Applied flip--0.1

      \[\leadsto \color{blue}{\frac{\frac{x}{x + 1} \cdot \frac{x}{x + 1} - \frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}}\]
   4. Using strategy rm

   5. Applied flip3--0.1

      \[\leadsto \frac{\color{blue}{\frac{{\left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right)}^{3} - {\left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right)}^{3}}{\left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right) \cdot \left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right) + \left(\left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right) \cdot \left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right) + \left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right) \cdot \left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right)\right)}}}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}\]

   6. Applied associate-/l/0.1

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right)}^{3} - {\left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right)}^{3}}{\left(\frac{x}{x + 1} + \frac{x + 1}{x - 1}\right) \cdot \left(\left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right) \cdot \left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right) + \left(\left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right) \cdot \left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right) + \left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right) \cdot \left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right)\right)\right)}}\]
   7. Using strategy rm

   8. Applied flip--0.1

      \[\leadsto \frac{\color{blue}{\frac{{\left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right)}^{3} \cdot {\left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right)}^{3} - {\left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right)}^{3} \cdot {\left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right)}^{3}}{{\left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right)}^{3} + {\left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right)}^{3}}}}{\left(\frac{x}{x + 1} + \frac{x + 1}{x - 1}\right) \cdot \left(\left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right) \cdot \left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right) + \left(\left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right) \cdot \left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right) + \left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right) \cdot \left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right)\right)\right)}\]
   9. Using strategy rm

   10. Applied associate-*r/0.1

       \[\leadsto \frac{\frac{{\left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right)}^{3} \cdot {\left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right)}^{3} - {\left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right)}^{3} \cdot {\color{blue}{\left(\frac{\frac{x + 1}{x - 1} \cdot \left(x + 1\right)}{x - 1}\right)}}^{3}}{{\left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right)}^{3} + {\left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right)}^{3}}}{\left(\frac{x}{x + 1} + \frac{x + 1}{x - 1}\right) \cdot \left(\left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right) \cdot \left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right) + \left(\left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right) \cdot \left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right) + \left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right) \cdot \left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right)\right)\right)}\]

   11. Applied cube-div0.1

       \[\leadsto \frac{\frac{{\left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right)}^{3} \cdot {\left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right)}^{3} - {\left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right)}^{3} \cdot \color{blue}{\frac{{\left(\frac{x + 1}{x - 1} \cdot \left(x + 1\right)\right)}^{3}}{{\left(x - 1\right)}^{3}}}}{{\left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right)}^{3} + {\left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right)}^{3}}}{\left(\frac{x}{x + 1} + \frac{x + 1}{x - 1}\right) \cdot \left(\left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right) \cdot \left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right) + \left(\left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right) \cdot \left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right) + \left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right) \cdot \left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right)\right)\right)}\]

   12. Applied frac-times0.1

       \[\leadsto \frac{\frac{{\left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right)}^{3} \cdot {\left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right)}^{3} - {\color{blue}{\left(\frac{\left(x + 1\right) \cdot \left(x + 1\right)}{\left(x - 1\right) \cdot \left(x - 1\right)}\right)}}^{3} \cdot \frac{{\left(\frac{x + 1}{x - 1} \cdot \left(x + 1\right)\right)}^{3}}{{\left(x - 1\right)}^{3}}}{{\left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right)}^{3} + {\left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right)}^{3}}}{\left(\frac{x}{x + 1} + \frac{x + 1}{x - 1}\right) \cdot \left(\left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right) \cdot \left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right) + \left(\left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right) \cdot \left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right) + \left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right) \cdot \left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right)\right)\right)}\]

   13. Applied cube-div0.1

       \[\leadsto \frac{\frac{{\left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right)}^{3} \cdot {\left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right)}^{3} - \color{blue}{\frac{{\left(\left(x + 1\right) \cdot \left(x + 1\right)\right)}^{3}}{{\left(\left(x - 1\right) \cdot \left(x - 1\right)\right)}^{3}}} \cdot \frac{{\left(\frac{x + 1}{x - 1} \cdot \left(x + 1\right)\right)}^{3}}{{\left(x - 1\right)}^{3}}}{{\left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right)}^{3} + {\left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right)}^{3}}}{\left(\frac{x}{x + 1} + \frac{x + 1}{x - 1}\right) \cdot \left(\left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right) \cdot \left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right) + \left(\left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right) \cdot \left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right) + \left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right) \cdot \left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right)\right)\right)}\]

   14. Applied frac-times0.1

       \[\leadsto \frac{\frac{{\left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right)}^{3} \cdot {\left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right)}^{3} - \color{blue}{\frac{{\left(\left(x + 1\right) \cdot \left(x + 1\right)\right)}^{3} \cdot {\left(\frac{x + 1}{x - 1} \cdot \left(x + 1\right)\right)}^{3}}{{\left(\left(x - 1\right) \cdot \left(x - 1\right)\right)}^{3} \cdot {\left(x - 1\right)}^{3}}}}{{\left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right)}^{3} + {\left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right)}^{3}}}{\left(\frac{x}{x + 1} + \frac{x + 1}{x - 1}\right) \cdot \left(\left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right) \cdot \left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right) + \left(\left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right) \cdot \left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right) + \left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right) \cdot \left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right)\right)\right)}\]

   15. Applied associate-*r/0.1

       \[\leadsto \frac{\frac{{\left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right)}^{3} \cdot {\color{blue}{\left(\frac{\frac{x}{x + 1} \cdot x}{x + 1}\right)}}^{3} - \frac{{\left(\left(x + 1\right) \cdot \left(x + 1\right)\right)}^{3} \cdot {\left(\frac{x + 1}{x - 1} \cdot \left(x + 1\right)\right)}^{3}}{{\left(\left(x - 1\right) \cdot \left(x - 1\right)\right)}^{3} \cdot {\left(x - 1\right)}^{3}}}{{\left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right)}^{3} + {\left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right)}^{3}}}{\left(\frac{x}{x + 1} + \frac{x + 1}{x - 1}\right) \cdot \left(\left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right) \cdot \left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right) + \left(\left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right) \cdot \left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right) + \left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right) \cdot \left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right)\right)\right)}\]

   16. Applied cube-div0.1

       \[\leadsto \frac{\frac{{\left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right)}^{3} \cdot \color{blue}{\frac{{\left(\frac{x}{x + 1} \cdot x\right)}^{3}}{{\left(x + 1\right)}^{3}}} - \frac{{\left(\left(x + 1\right) \cdot \left(x + 1\right)\right)}^{3} \cdot {\left(\frac{x + 1}{x - 1} \cdot \left(x + 1\right)\right)}^{3}}{{\left(\left(x - 1\right) \cdot \left(x - 1\right)\right)}^{3} \cdot {\left(x - 1\right)}^{3}}}{{\left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right)}^{3} + {\left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right)}^{3}}}{\left(\frac{x}{x + 1} + \frac{x + 1}{x - 1}\right) \cdot \left(\left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right) \cdot \left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right) + \left(\left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right) \cdot \left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right) + \left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right) \cdot \left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right)\right)\right)}\]

   17. Applied associate-*r/0.1

       \[\leadsto \frac{\frac{\color{blue}{\frac{{\left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right)}^{3} \cdot {\left(\frac{x}{x + 1} \cdot x\right)}^{3}}{{\left(x + 1\right)}^{3}}} - \frac{{\left(\left(x + 1\right) \cdot \left(x + 1\right)\right)}^{3} \cdot {\left(\frac{x + 1}{x - 1} \cdot \left(x + 1\right)\right)}^{3}}{{\left(\left(x - 1\right) \cdot \left(x - 1\right)\right)}^{3} \cdot {\left(x - 1\right)}^{3}}}{{\left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right)}^{3} + {\left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right)}^{3}}}{\left(\frac{x}{x + 1} + \frac{x + 1}{x - 1}\right) \cdot \left(\left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right) \cdot \left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right) + \left(\left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right) \cdot \left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right) + \left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right) \cdot \left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right)\right)\right)}\]

   18. Applied frac-sub0.1

       \[\leadsto \frac{\frac{\color{blue}{\frac{\left({\left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right)}^{3} \cdot {\left(\frac{x}{x + 1} \cdot x\right)}^{3}\right) \cdot \left({\left(\left(x - 1\right) \cdot \left(x - 1\right)\right)}^{3} \cdot {\left(x - 1\right)}^{3}\right) - {\left(x + 1\right)}^{3} \cdot \left({\left(\left(x + 1\right) \cdot \left(x + 1\right)\right)}^{3} \cdot {\left(\frac{x + 1}{x - 1} \cdot \left(x + 1\right)\right)}^{3}\right)}{{\left(x + 1\right)}^{3} \cdot \left({\left(\left(x - 1\right) \cdot \left(x - 1\right)\right)}^{3} \cdot {\left(x - 1\right)}^{3}\right)}}}{{\left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right)}^{3} + {\left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right)}^{3}}}{\left(\frac{x}{x + 1} + \frac{x + 1}{x - 1}\right) \cdot \left(\left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right) \cdot \left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right) + \left(\left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right) \cdot \left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right) + \left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right) \cdot \left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right)\right)\right)}\]

   19. Simplified0.1

       \[\leadsto \frac{\frac{\frac{\color{blue}{\left({\left({\left(x + -1\right)}^{3}\right)}^{3} \cdot {\left(\frac{x \cdot x}{1 + x}\right)}^{3}\right) \cdot {\left(\frac{x}{1 + x} \cdot \frac{x}{1 + x}\right)}^{3} - \left({\left(\frac{1 + x}{x + -1}\right)}^{3} \cdot \left({\left(1 + x\right)}^{3} \cdot {\left(1 + x\right)}^{3}\right)\right) \cdot \left({\left(1 + x\right)}^{3} \cdot {\left(1 + x\right)}^{3}\right)}}{{\left(x + 1\right)}^{3} \cdot \left({\left(\left(x - 1\right) \cdot \left(x - 1\right)\right)}^{3} \cdot {\left(x - 1\right)}^{3}\right)}}{{\left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right)}^{3} + {\left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right)}^{3}}}{\left(\frac{x}{x + 1} + \frac{x + 1}{x - 1}\right) \cdot \left(\left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right) \cdot \left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right) + \left(\left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right) \cdot \left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right) + \left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right) \cdot \left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right)\right)\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -12727.401168194334 \lor \neg \left(x \le 12123.563172164775\right):\\ \;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) + \frac{\frac{-3}{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\left({\left(\frac{x \cdot x}{1 + x}\right)}^{3} \cdot {\left({\left(-1 + x\right)}^{3}\right)}^{3}\right) \cdot {\left(\frac{x}{1 + x} \cdot \frac{x}{1 + x}\right)}^{3} - \left(\left({\left(1 + x\right)}^{3} \cdot {\left(1 + x\right)}^{3}\right) \cdot {\left(\frac{1 + x}{-1 + x}\right)}^{3}\right) \cdot \left({\left(1 + x\right)}^{3} \cdot {\left(1 + x\right)}^{3}\right)}{\left({\left(x - 1\right)}^{3} \cdot {\left(\left(x - 1\right) \cdot \left(x - 1\right)\right)}^{3}\right) \cdot {\left(1 + x\right)}^{3}}}{{\left(\frac{1 + x}{x - 1} \cdot \frac{1 + x}{x - 1}\right)}^{3} + {\left(\frac{x}{1 + x} \cdot \frac{x}{1 + x}\right)}^{3}}}{\left(\frac{1 + x}{x - 1} + \frac{x}{1 + x}\right) \cdot \left(\left(\frac{x}{1 + x} \cdot \frac{x}{1 + x}\right) \cdot \left(\frac{x}{1 + x} \cdot \frac{x}{1 + x}\right) + \left(\left(\frac{1 + x}{x - 1} \cdot \frac{1 + x}{x - 1}\right) \cdot \left(\frac{x}{1 + x} \cdot \frac{x}{1 + x}\right) + \left(\frac{1 + x}{x - 1} \cdot \frac{1 + x}{x - 1}\right) \cdot \left(\frac{1 + x}{x - 1} \cdot \frac{1 + x}{x - 1}\right)\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019007 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))

Details

Time bar (total: 4.4m)Debug log