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Derivation

1. Split input into 2 regimes

2. if x < -9416.721535543154 or 12753.559648949982 < x

   1. Initial program 59.1

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

   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 -9416.721535543154 < x < 12753.559648949982

   1. Initial program 0.1

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

   3. Applied add-log-exp0.1

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

   5. Applied add-sqr-sqrt0.1

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

   6. Applied log-prod0.1

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

   8. Applied sub-neg0.1

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

   9. Applied exp-sum0.1

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

   10. Applied sqrt-prod0.1

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

   11. Applied log-prod0.1

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

4. Final simplification0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -9416.721535543154 \lor \neg \left(x \le 12753.559648949982\right):\\ \;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) + \frac{\frac{-3}{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{e^{\frac{x}{1 + x} - \frac{1 + x}{x - 1}}}\right) + \left(\log \left(\sqrt{e^{\frac{-\left(1 + x\right)}{x - 1}}}\right) + \log \left(\sqrt{e^{\frac{x}{1 + x}}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019022 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))

Details

Time bar (total: 18.9s)Debug log