Derivation

Split input into 2 regimes
if x < -11360.011079950777 or 10385.499811469535 < x
1. Initial program 59.4
  \[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
2. Initial simplification59.4
  \[\leadsto \frac{x}{1 + x} - \frac{1 + x}{x - 1}\]
3. Taylor expanded around inf 0.4
  \[\leadsto \color{blue}{-\left(3 \cdot \frac{1}{{x}^{3}} + \left(\frac{1}{{x}^{2}} + 3 \cdot \frac{1}{x}\right)\right)}\]
4. Simplified0.0
  \[\leadsto \color{blue}{\frac{-3}{x} - \frac{1 + \frac{3}{x}}{x \cdot x}}\]
if -11360.011079950777 < x < 10385.499811469535
1. Initial program 0.1
  \[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
2. Initial simplification0.1
  \[\leadsto \frac{x}{1 + x} - \frac{1 + x}{x - 1}\]
3. Using strategy rm
4. Applied div-inv0.1
  \[\leadsto \color{blue}{x \cdot \frac{1}{1 + x}} - \frac{1 + x}{x - 1}\]
Recombined 2 regimes into one program.
Final simplification0.1
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -11360.011079950777 \lor \neg \left(x \le 10385.499811469535\right):\\ \;\;\;\;\frac{-3}{x} - \frac{\frac{3}{x} + 1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + 1} \cdot x - \frac{x + 1}{x - 1}\\ \end{array}\]

Runtime

herbie shell --seed 2018225 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))

In
Out