Derivation

Split input into 2 regimes
if x < -12356.370375557468 or 12767.72813954212 < x
1. Initial program 59.3
  \[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
2. Initial simplification59.3
  \[\leadsto \frac{x}{1 + x} - \frac{1 + x}{x - 1}\]
3. Taylor expanded around -inf 0.3
  \[\leadsto \color{blue}{-\left(3 \cdot \frac{1}{{x}^{3}} + \left(\frac{1}{{x}^{2}} + 3 \cdot \frac{1}{x}\right)\right)}\]
4. Simplified0.0
  \[\leadsto \color{blue}{(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_*}\]
if -12356.370375557468 < x < 12767.72813954212
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 *-un-lft-identity0.1
  \[\leadsto \frac{x}{1 + x} - \frac{\color{blue}{1 \cdot \left(1 + x\right)}}{x - 1}\]
5. Applied associate-/l*0.1
  \[\leadsto \frac{x}{1 + x} - \color{blue}{\frac{1}{\frac{x - 1}{1 + x}}}\]
Recombined 2 regimes into one program.
Final simplification0.0
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -12356.370375557468 \lor \neg \left(x \le 12767.72813954212\right):\\ \;\;\;\;(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x} - \frac{1}{\frac{x - 1}{1 + x}}\\ \end{array}\]

Runtime

herbie shell --seed 2018227 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))