Derivation

Split input into 2 regimes
if x < -1.0935487875710134e+18 or 276770.22637257783 < x
1. Initial program 60.0
  \[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
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
  \[\leadsto \color{blue}{\frac{-3}{x} - \frac{1 + \frac{3}{x}}{{x}^{2}}}\]
if -1.0935487875710134e+18 < x < 276770.22637257783
1. Initial program 0.9
  \[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
2. Using strategy rm
3. Applied frac-sub0.8
  \[\leadsto \color{blue}{\frac{x \cdot \left(x - 1\right) - \left(x + 1\right) \cdot \left(x + 1\right)}{\left(x + 1\right) \cdot \left(x - 1\right)}}\]
4. Taylor expanded around 0 0.0
  \[\leadsto \frac{\color{blue}{-\left(3 \cdot x + 1\right)}}{\left(x + 1\right) \cdot \left(x - 1\right)}\]
Recombined 2 regimes into one program.
Final simplification0.0
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0935487875710134 \cdot 10^{+18} \lor \neg \left(x \le 276770.22637257783\right):\\ \;\;\;\;\frac{-3}{x} - \frac{\frac{3}{x} + 1}{{x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\left(1 + 3 \cdot x\right)}{\left(x + 1\right) \cdot \left(x - 1\right)}\\ \end{array}\]

Runtime

herbie shell --seed 2018221 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))

In
Out