Derivation

Split input into 2 regimes
if x < -449067.18520243093 or 161516.1252269543 < x
1. Initial program 59.6
  \[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
2. Taylor expanded around inf 0.3
  \[\leadsto \color{blue}{-\left(3 \cdot \frac{1}{{x}^{3}} + \left(3 \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right)\right)}\]
3. 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 -449067.18520243093 < x < 161516.1252269543
1. Initial program 0.1
  \[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
2. Using strategy rm
3. Applied frac-sub0.1
  \[\leadsto \color{blue}{\frac{x \cdot \left(x - 1\right) - \left(x + 1\right) \cdot \left(x + 1\right)}{\left(x + 1\right) \cdot \left(x - 1\right)}}\]
4. Taylor expanded around 0 0.0
  \[\leadsto \frac{\color{blue}{-\left(1 + 3 \cdot x\right)}}{\left(x + 1\right) \cdot \left(x - 1\right)}\]
5. Simplified0.0
  \[\leadsto \frac{\color{blue}{-(x \cdot 3 + 1)_*}}{\left(x + 1\right) \cdot \left(x - 1\right)}\]
Recombined 2 regimes into one program.
Final simplification0.0
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -449067.18520243093 \lor \neg \left(x \le 161516.1252269543\right):\\ \;\;\;\;(\left(\frac{3}{x} + 1\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{-(x \cdot 3 + 1)_*}{\left(x + 1\right) \cdot \left(x - 1\right)}\\ \end{array}\]

Runtime

herbie shell --seed 2018215 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))