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Derivation

1. Split input into 2 regimes

2. if x < -9521.382568401006 or 11061.746166762965 < 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

      \[\leadsto \color{blue}{\frac{-3}{x} - \frac{1 + \frac{3}{x}}{x \cdot x}}\]

   if -9521.382568401006 < x < 11061.746166762965

   1. Initial program 0.1

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

   3. Applied flip3--0.1

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

   5. Applied add-cbrt-cube0.1

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

   7. Applied cube-div0.1

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

   8. Applied cube-div0.1

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

   9. Applied cube-div0.1

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

   10. Applied associate-*r/0.1

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

   11. Applied frac-times0.1

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

   12. Applied cbrt-div0.1

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

   13. Applied frac-sub0.1

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

   14. Simplified0.1

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

4. Final simplification0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -9521.382568401006 \lor \neg \left(x \le 11061.746166762965\right):\\ \;\;\;\;\frac{-3}{x} - \frac{\frac{3}{x} + 1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(x - 1\right)}^{3} \cdot \sqrt[3]{\frac{{\left({x}^{3}\right)}^{3}}{{\left(x + 1\right)}^{3}}} - \sqrt[3]{{\left(x + 1\right)}^{3} \cdot {\left(x + 1\right)}^{3}} \cdot {\left(x + 1\right)}^{3}}{{\left(x - 1\right)}^{3} \cdot \sqrt[3]{{\left(x + 1\right)}^{3} \cdot {\left(x + 1\right)}^{3}}}}{\left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1} + \frac{x}{x + 1} \cdot \frac{x + 1}{x - 1}\right) + \frac{x}{x + 1} \cdot \frac{x}{x + 1}}\\ \end{array}\]

Runtime

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herbie shell --seed 2018220 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))