Derivation

Split input into 2 regimes
if x < -12867.766118299465 or 15285.700431816435 < x
1. Initial program 59.2
  \[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
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 -12867.766118299465 < x < 15285.700431816435
1. Initial program 0.1
  \[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
2. Using strategy rm
3. Applied flip-+0.1
  \[\leadsto \frac{x}{\color{blue}{\frac{x \cdot x - 1 \cdot 1}{x - 1}}} - \frac{x + 1}{x - 1}\]
4. Applied associate-/r/0.1
  \[\leadsto \color{blue}{\frac{x}{x \cdot x - 1 \cdot 1} \cdot \left(x - 1\right)} - \frac{x + 1}{x - 1}\]
Recombined 2 regimes into one program.
Final simplification0.1
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -12867.766118299465 \lor \neg \left(x \le 15285.700431816435\right):\\ \;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) + \frac{\frac{-3}{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x \cdot x - 1} \cdot \left(x - 1\right) - \frac{1 + x}{x - 1}\\ \end{array}\]

Runtime

herbie shell --seed 2018349 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))

In
Out