Derivation

Split input into 2 regimes
if x < -11709.23517546889 or 17646.51774600535 < x
1. Initial program 59.3
  \[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
2. Using strategy rm
3. Applied add-log-exp59.3
  \[\leadsto \color{blue}{\log \left(e^{\frac{x}{x + 1} - \frac{x + 1}{x - 1}}\right)}\]
4. Taylor expanded around -inf 0.3
  \[\leadsto \color{blue}{-\left(3 \cdot \frac{1}{{x}^{3}} + \left(\frac{1}{{x}^{2}} + 3 \cdot \frac{1}{x}\right)\right)}\]
5. Simplified0.0
  \[\leadsto \color{blue}{(\left(\frac{-1}{x \cdot x}\right) \cdot \left(\frac{3}{x}\right) + \left(\frac{-1}{x \cdot x} - \frac{3}{x}\right))_*}\]
if -11709.23517546889 < x < 17646.51774600535
1. Initial program 0.1
  \[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
2. Using strategy rm
3. Applied add-log-exp0.1
  \[\leadsto \color{blue}{\log \left(e^{\frac{x}{x + 1} - \frac{x + 1}{x - 1}}\right)}\]
Recombined 2 regimes into one program.
Final simplification0.1
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -11709.23517546889 \lor \neg \left(x \le 17646.51774600535\right):\\ \;\;\;\;(\left(\frac{-1}{x \cdot x}\right) \cdot \left(\frac{3}{x}\right) + \left(\frac{-1}{x \cdot x} - \frac{3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{x}{1 + x} - \frac{1 + x}{x - 1}}\right)\\ \end{array}\]

Runtime

herbie shell --seed 2018255 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))