Derivation

Split input into 2 regimes
if x < -6734.049548989393 or 9535.54553065378 < 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
  \[\leadsto \color{blue}{\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) + \frac{\frac{-3}{x}}{x \cdot x}}\]
4. Using strategy rm
5. Applied associate-+l-0.0
  \[\leadsto \color{blue}{\frac{-3}{x} - \left(\frac{1}{x \cdot x} - \frac{\frac{-3}{x}}{x \cdot x}\right)}\]
if -6734.049548989393 < x < 9535.54553065378
1. Initial program 0.1
  \[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
2. Using strategy rm
3. Applied add-cube-cbrt0.1
  \[\leadsto \frac{x}{x + 1} - \frac{x + 1}{\color{blue}{\left(\sqrt[3]{x - 1} \cdot \sqrt[3]{x - 1}\right) \cdot \sqrt[3]{x - 1}}}\]
4. Applied associate-/r*0.1
  \[\leadsto \frac{x}{x + 1} - \color{blue}{\frac{\frac{x + 1}{\sqrt[3]{x - 1} \cdot \sqrt[3]{x - 1}}}{\sqrt[3]{x - 1}}}\]
Recombined 2 regimes into one program.
Final simplification0.1
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -6734.049548989393 \lor \neg \left(x \le 9535.54553065378\right):\\ \;\;\;\;\frac{-3}{x} - \left(\frac{1}{x \cdot x} - \frac{\frac{-3}{x}}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x} - \frac{\frac{1 + x}{\sqrt[3]{x - 1} \cdot \sqrt[3]{x - 1}}}{\sqrt[3]{x - 1}}\\ \end{array}\]

Runtime

herbie shell --seed 2018285 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))

In
Out