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Derivation

1. Split input into 2 regimes

2. if x < -6.95166373829725e+48 or 126507.6676441063 < x

   1. Initial program 59.8

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
   2. Using strategy rm

   3. Applied add-cbrt-cube59.8

      \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{x}{x + 1} \cdot \frac{x}{x + 1}\right) \cdot \frac{x}{x + 1}}} - \frac{x + 1}{x - 1}\]

   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

      \[\leadsto \color{blue}{\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) + \frac{\frac{-3}{x}}{x \cdot x}}\]

   if -6.95166373829725e+48 < x < 126507.6676441063

   1. Initial program 3.3

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
   2. Using strategy rm

   3. Applied frac-sub3.3

      \[\leadsto \color{blue}{\frac{x \cdot \left(x - 1\right) - \left(x + 1\right) \cdot \left(x + 1\right)}{\left(x + 1\right) \cdot \left(x - 1\right)}}\]

   4. Taylor expanded around -inf 0.0

      \[\leadsto \frac{\color{blue}{-\left(3 \cdot x + 1\right)}}{\left(x + 1\right) \cdot \left(x - 1\right)}\]

   5. Simplified0.0

      \[\leadsto \frac{\color{blue}{-1 - 3 \cdot x}}{\left(x + 1\right) \cdot \left(x - 1\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -6.95166373829725 \cdot 10^{+48} \lor \neg \left(x \le 126507.6676441063\right):\\ \;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) + \frac{\frac{-3}{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 - x \cdot 3}{\left(x - 1\right) \cdot \left(1 + x\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2018362 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))

Details

Time bar (total: 42.9s)Debug log