Error

Derivation

1. Split input into 2 regimes

2. if x < -18457965852152.69 or 894676987.6521559 < x

   1. Initial program 60.1

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

   3. Applied add-log-exp60.1

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

   5. Applied div-inv60.1

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

   6. Applied exp-prod63.0

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

   7. Applied log-pow63.0

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

   8. Taylor expanded around inf 0.3

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

   9. Simplified0.0

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

   if -18457965852152.69 < x < 894676987.6521559

   1. Initial program 0.6

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

   3. Applied add-log-exp0.6

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

   5. Applied div-inv0.6

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

   6. Applied exp-prod1.7

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

   7. Applied log-pow1.7

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

   9. Applied associate-*l/1.7

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

   10. Applied frac-sub1.7

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

   11. Simplified0.0

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

   13. Applied flip-+0.0

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

   14. Applied associate-*l/0.0

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

   15. Applied associate-/r/0.0

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

4. Final simplification0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -18457965852152.69 \lor \neg \left(x \le 894676987.6521559\right):\\ \;\;\;\;(\left(\frac{-1}{x \cdot x}\right) \cdot \left(\frac{3}{x}\right) + \left(\frac{-1}{x \cdot x} - \frac{3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{(x \cdot \left(x - \left(2 + x\right)\right) + \left(-1 - x\right))_*}{\left(x \cdot x - 1\right) \cdot \left(x - 1\right)} \cdot \left(x - 1\right)\\ \end{array}\]

Runtime

Time bar (total: 2.3m)Debug logProfile

herbie shell --seed 2018274 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))